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Springer isomorphisms and the variety of elementary subalgebras

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Springer isomorphisms and the variety of elementary subalgebras

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Content Springer Isomorphisms and the Variety of Elementary Subalgebras Jared Warner Dissertation Advisor Eric Friedlander Dissertation Committee Eric Friedlander (Chair) Robert Guralnick Susan Montgomery Edmond Jonckheere A dissertation written in partial fulllment of the requirements for obtaining a Doc- tor of Philosophy degree in Mathematics from the University of Southern California on May 15th, 2015. In memory of Damon Freeman Contents Introduction 4 Acknowledgements 7 Chapter 1. Cohomology and support varieties of ane group schemes 9 1.1. Ane group schemes 9 1.2. Representations and cohomology 22 1.3. Support varieties 28 Chapter 2. Springer isomorphisms 35 2.1. Algebraic groups 35 2.2. Existence of Springer isomorphisms 40 2.3. In search of a canonical Springer isomorphism 41 Chapter 3. The variety of elementary subalgebras 44 3.1. Motivation - nite groups 45 3.2. The variety and category of elementary subalgebras 49 3.3. Rational points and orbits of the variety of elementary subalgebras 54 3.4. Irreducibility of the variety of elementary subalgebras 58 3.5. Computations for the general linear group 59 3.6. Computations for the upper-triangular unipotent group 63 Chapter 4. The commuting variety of one-parameter subgroups 70 4.1. Motivation and Preliminaries 70 4.2. Innitesimal one-parameter subgroups 72 4.3. Commuting nilpotent varieties 74 4.4. The commuting variety of one-parameter innitesimal subgroups 78 4.5. Pointwise and local operators on modules 80 Appendix A. Magma code 83 A.1. DrawQuillen 83 A.2. QuillenComplex 85 A.3. Detecting nilpotents in the variety of innitesimal subgroups 87 Appendix B. Visualizations of Quillen's category 89 Bibliography 98 3 Introduction IfG is a nite group scheme over an algebraically closed eldk of characteristic p, then the cohomology k-algebra of G with coecients in k is nitely generated. This theorem, due to Eric Friedlander and Andrei Suslin in 1997([15]), provided the necessary foundation to study geometric invariants associated to representations of nite group schemes. These geometric invariants, called \support varieties," were already well studied in the context of nite groups because the nite generation of cohomology in that setting had been known since 1959 ([53]). The content of this document is suitably placed in the historical framework of support varieties and their generalizations to the setting of nite group schemes. Hence, before describing our work, we begin with a brief historical review. Results on the nite generation of the cohomology algebra of a nite group dates back to 1959 when Golod showed that the cohomology of a nite p-group is nitely generated ([23]). Within the next two years, Venkov and Evens would establish the nite generation of cohomology for all nite groups ([53], [16]). Over a decade later, in the early 1980's, Carlson ([12]), Avrunin ([2]), and Alperin-Evens ([1]) employed nite generation to study geometric invariants asso- ciated to representations of a nite groups. If is a nite group with group algebra overk denoted byk, then to a nite-dimensionalk-moduleM one may associate a closed subset of the spectrum of the cohomology of . Specically, if we view k as a trivial k-module via the augmentation : k! k, then the graded abelian group Ext k (M;M) has the structure of a module over the even degree cohomology k-algebra Ext k (k;k) =: H (;k) (If p = 2, then H (;k) is the full cohomology algebra). The annihilator of this module denes a closed subset in the prime ideal spectrum of H (;k), which is called the support variety of M. As an example of the relationship between modules and their support varieties, Avrunin showed that the dimension of the support variety for M is equal to the complexity of M, which is a homological numerical invariant measured in terms of the growth rate of a minimal projective resolution. In 1981, Jon Carlson developed a new way of assigning geometric invariants to modules for an elementary abelian p-group, E. Carlson's \rank variety," was dened solely in terms of the representation theoretic properties of a kE-module M, and he conjectured that the rank variety of M and the support variety of M were isomorphic ([12]). Carlson's conjecture was shown to be true by Avrunin- Scott ([3]) thus showing that in this particular case, the support variety could be suciently dened without any cohomological considerations. Returning to nite group schemes, in 1986 Friedlander and Parshall established the nite generation of the cohomology of an innitesimal group scheme of height 1 ([20]), but it wasn't until 11 years later (and 38 years after the analogous result for nite groups!) that the nite generation of H (G;k) was established for an 4 INTRODUCTION 5 arbitrary nite group scheme (Friedlander-Suslin's method was to rst consider the case of G innitesimal of height r). With nite generation in hand, one could dene the cohomological support variety associated to a G-module as in the case of nite groups by considering the closed subset of Spec(H (G;k)) dened by the annihilator of the H (G;k)-module Ext kG (M;M). Here kG is the k-linear dual of the coordinate algebra of G, appro- priately termed the \group algebra" ofG, andM is a nite-dimensionalkG-module. The theories of innitesimal one-parameter subgroups of Suslin-Friedlander-Bendel ([51], [52]) and -points of Friedlander-Pevtsova ([18], [21]) provided a represen- tation theoretic description of the cohomological support variety in the spirit of Carlson's rank variety. There are many advantages to this new view of support varieties. For instance, if G is an innitesimal group scheme of height 1, then a special case of Theorem 5.2 of [52] identies the cohomological support variety of the trivial module k with the restricted nullcone of the Lie algebra of G (see Def- inition 2.1.3). Furthermore, Theorem 6.7 of [52] shows that, modulo nilpotents, a cohomology classes of even degree in H (G;k) is determined by the innitesimal one-parameter subgroups of G. In [13], Carlson-Friedlander-Pevtsova return to the study of innitesimal group schemes of height 1 by dening and studying E(r;g), the variety of elementary subalgebras of rank r of a restricted Lie algebra g for a xed positive integer r. In the caser = 1,E(r;g) is isomorphic to the projectivized restricted nullcone, and is thus (via Theorem 5.2 of [52]) a generalization of cohomological support varieties for innitesimal group schemes of height one. The main content of this document is concerned with the study of the geometric properties of E(r;g) for g the Lie algebra of a connected, reductive algebraic group. Our main tool in this study will be the canonical Springer isomorphism between the nullcone of g and the unipotent variety of G when p is large. Having appropriately placed our theory in its relevant place in the literature, we will now outline the structure of this document and highlight our main results. Chapter 1 develops some of the basic theory of ane group schemes, including representations and cohomology. We develop the relationships between an ane group schemeG and the many algebras associated to it (coordinate algebra, group algebra, distribution algebra, and restricted Lie algebra). We give plenty of exam- ples of ane group schemes, most of which play a role in the rest of the document. We develop the theory of cohomology using n-fold extensions and the Baer sum as opposed to derived functors. For our purposes, we feel it is easier to dene the module structure of Ext kG (M;M) from this point of view. In Section 1.3, we essentially develop in more detail the mathematics of the historical background just discussed. Figure 6 serves to summarize the relationship between the cohomological support variety and the many representation theoretic descriptions for various group schemes G. Chapter 2 begins with a review of algebraic groups, stating some of the theory of the structure of reductive groups. This builds the necessary background in which we explain the development of the canonical Springer isomorphism for large enough p (Example 2.3.1 shows what may go wrong in small characteristic). Our condition (?) on the prime p is precisely what is needed to establish Theorem 2.3.2, which appears piecemeal in the literature for simple groups, but is established here for connected, reductive groups. 6 INTRODUCTION As mentioned before, Chapter 3 contains most of our results. We begin the chapter by reviewing Quillen's stratication of the cohomology of a nite group, and applying his theory to study the cohomology of the nite groups GL 3 (F p ) and GL 4 (F p ) (Example 3.1.3 and Figure 9). In Section 3.2, we use the canonical Springer isomorphism to describe the F q -rational points of E(r;g) in terms of F q - linear elementary abelian p-subgroups of G(F q ) (see Denition 3.2.9). Here G is any connected, reductive group dened over F p , and g := Lie(G). Our treatment ofF q -linear subgroups leads to Corollary 3.2.15, which computes the maximal rank of an elementary subalgebra of g in terms of thep-rank ofG(F q ). Since thep-ranks of G(F q ) are known for simple G, this allows us to report the maximal ranks of elementary subalgebras for the simple Lie algebras in Table 1. In Section 3.3, we use the results from Section 3.2 together with Lang's theo- rem to prove Theorem 3.3.3, which establishes a bijection between the G-orbits of E(r;g) dened overF q and theG-conjugacy classes ofF q -linear elementary abelian p-subgroups of rank r in G(F q ). This allows us to prove Proposition 3.3.4, which resolves a conjecture of Eric Friedlander thatE(r;g) is a nite union ofG-orbits un- der certain hypotheses onr and the simple components of the derived group (G;G). Example 3.3.6 illustrates Theorem 3.3.3 and further solidies the connection be- tween orbits and conjugacy classes. In Section 3.4 we establish the irreducibility of E(r;gl n ) for certain pairs (r;n) (Theorem 3.4.1), and we make some computations in the nal two sections of Chapter 3 for GL n and the solvable group of unipo- tent upper triangular matrices, U n . The moral to these computations is that the geometric properties (dimension, irreducibile components, etc.) of E(r;g) can be determined in small cases, but a general result seems dicult. Finally, Chapter 4 presents ongoing work with Eric Friedlander which focuses on deningE(s;G r ), the commuting variety of one-parameter subgroups. E(s;G r ) simultaneously generalizes the variety of elementary subalgebras and the variety of innitesimal one-parameter subgroups (see Table 5). Section 4.2 reviews the scheme of innitesimal one-parameter subgroups, and establishes that in general this scheme is not reduced (Proposition 4.2.4). Section 4.3 and Section 4.4 are then concerned with motivating and dening E(s;G r ). The nal section of Chap- ter 4 constructs pointwise and local operators associated to modules of innitesimal group schemes (Proposition 4.5.3). These operators are the precursors to construct- ing vector bundles onE(s;G r ), following the constructions of [13] (see the opening discussion of Chapter 4). Appendix A.1 presents some of the code that was used in computing examples throughout the document. Appendix B presents visualizations of Quillen's category for dierent nite groups. Via the results of Chapter 3, these pictures have mathematical meaning described in detail in the opening discussion of Appendix B. Notation and Conventions. Throughout p will denote a prime number, k will denote an algebraically closed eld of characteristic p, d will be a positive integer, and q = p d will be a prime power. F q will denote the nite eld of q elements. All rings are associative and unital. By Sets, Grps, Alg, ASch, AGrpSch, and HopfAlg we mean the cate- gories of sets, groups, commutative k-algebras, ane schemes over k, ane group schemes over k, and commutative Hopf algebras over k, respectively. If C is ACKNOWLEDGEMENTS 7 a locally small category (cf. Remark 1.1.3) and A 2 Ob(C), then the func- tor Hom C (A;) : C ! Sets is dened on objects by B 7! Hom C (A;B) for all B 2 Ob(C). If B;B 0 2 Ob(C) and ' 2 Hom C (B;B 0 ), then Hom C (A;') : Hom C (A;B)! Hom C (A;B 0 ) is dened by f7!'f. We denote by id C the iden- tity functor onC. We say two categoriesC andD are equivalent (anti-equivalent) if there are covariant (contravariant) functors F :C!D and G :D!C such that the appropriate compositions of F and G are naturally isomorphic to the appropriate identity functors onC andD. If F;G :C!D, : F ! G is a nat- ural transformation, and A2 Ob(C), then we write A to denote the morphism F (A)! G(A). If A is clear from context, we at times will abuse notation and simply write :F (A)!G(A). IfA andB arek-algebras, then we employ the notationA B to signifyA k B. Similarly, the symbol Hom(A;B) represents thek-linear maps fromA toB viewed as vector spaces, whereas the symbol Hom k (A;B) represents the k-algebra maps from A to B. Sometimes, if we feel emphasis is needed, we write Hom k-alg (A;B) to denote the set of k-algebra maps fromA toB. IfA,B, andC are commutative k-algebras, andf :A!k andg :B!C arek-linear maps, then byf g we mean the composition f g :A B f g !k C!C where the last map sends a simple tensor c toc. IfR is a commutative ring, we denote by Specm(R) the maximal ideal spectrum ofR and Spec(R) the prime ideal spectrum of R. As a set, Specm(R) (Spec(R)) is the set of all maximal (prime) ideals of R. We place the Zarisiki topology on Spec(R) so that together with the structure sheafO R it has the structure of an ane scheme. Unless otherwise noted, all schemes are over k, that is, if X is a scheme we have a xed map of schemes X! Spec(k). In the case of an ane scheme dened by a ringR, this is equivalent to stating the R has the structure of a commutative k-algebra, that is, a map k!R. For an introduction to schemes from the topological point of view (i.e. as locally ringed spaces), see [25]. If S =S i is a graded ring, then we denote by Proj(S) the set of all prime homogeneous ideals not contained in the irrelevant ideal S + = i0 S i . This set is also given the structure of a scheme via the Zariski topology and the structure sheafO S . If m and n are positive integers, and R is a ring, we denote the set of mn matrices with entries in R by M m;n (R). If m = n, we write M n (R) in place of M n;n (R) for the ring of nn matrices with entries in R, where addition and multiplication are given by the familiar matrix operations. Ifm i;j 2R for 1im and 1jn, the equationM = (m i;j ) signies that the entry in the ith row and jth column of matrix M2M m;n (R) is m i;j . If M2M m;n (R), then M i;j denotes the element in R lying in the ith row and jth column of the matrix M. Acknowledgements It is often said that some things go without saying. I hope this is true of the many thanks I owe to my friends and family these last ve years. However, at times it is appropriate for \some things" to be said of some people. To say that Eric Friedlander's contributions to this document and to my aca- demic development are many would be a considerable understatement. If Eric hadn't been patient enough to explain, and re-explain, and re-explain again, I 8 INTRODUCTION would not have learned much of what is written here. I've grown accustomed to moments of epiphany these last ve years in which I nally realize what Eric has been trying to explain to me, only after the third time he's explained it, and only after a few weeks since we last spoke about it. These delayed epiphanies will surely continue, as there is plenty more Eric has tried to teach me that hasn't quite sunk in yet. His enthusiasm as an advisor and his expressed condence in me often served as much needed energy bars throughout this marathon. I'd also like to recognize the contributions of Paul Sobaje in both mathematics and morale. His physical presence bookended my time at USC, and so any claim I have to a good start or a good nish is certainly due in part to Paul's friendliness and encouragement. Besides sharing his contagious positivity, Paul provided a steady source of mathematical guidance and reference along the way. My parents are due a lot of credit as well. Growing up, my dad taught me the values of hard work and honesty, and I never doubted myself because my mom never did. In these last ve years, I rode my dad's old bike to and from USC, and my mom often provided welcome diversions from mathematics with sports conversations (my time as a graduate student began with the Lakers as repeat world champions to my mom's joy, and ended with repeat lottery appearances to her disgust). Actually, if my parents were a sports team, I'd be one of their biggest fans, and they'd certainly be repeat world champions themselves for the job they did in raising my sister and me. They're both incredible people and role models for me in more ways than one. Perhaps I can show my gratitude for all they've given me by assuring them that they need not read a single word on these pages (except those in the current paragraph). Finally, I acknowledge here that my wife Jessica, whose education and career have nothing to do with abstract mathematics, can, after years of practice and training, (somewhat) accurately give the mathematical denition of a group. On a good day she may even be able to tell you the dierence between an abelian group and a non-abelian one (giving examples of both!). She patiently humors my enthusiasm for mathematics on a regular basis, and she even took on my normal household chore of dish-washing during the nal stages of the writing of this docu- ment (knowing how much she hates washing dishes, I was certainly touched). Due to her support through diculties and her joy in small victories, I consider this document an accomplishment we achieved together. CHAPTER 1 Cohomology and support varieties of ane group schemes 1.1. Ane group schemes We begin by describing the main objects of study throughout the paper: ane group schemes. We'll take the view of ane schemes as representable functors, and show that the coordinate algebra of an ane group scheme is a Hopf algebra. The section is concluded with a number of examples that will be relevant throughout the entire document. Helpful references for this section are [29] and [54]. See [36] for a more thorough treatment of Hopf algebras. We start with the following familiar and motivating example. Example 1.1.1. Let n be a positive integer, let R be a commutative k-algebra, and let SL n (R) be the set of all nn matrices with entries in R with determinant equal to 1 = 1 R . If S is another commutative k-algebra, and ' :R!S is a map of k-algebras, then we obtain a map ' : SL n (R)! SL n (S) by applying ' to each entry of any matrix M2 SL n (R). Explicitly, we have ' (M) = ('(M i;j )). ' (M) has determinant equal to 1 because' is a map ofk-algebras. With these denitions of SL n (R) and' , it can be shown that we've given SL n the structure of a covariant functor from Alg to Sets. Now, let k[x i;j j 1 i;j n] be a polynomial ring in the n 2 indeterminates x i;j , let X2M n (k[x i;j ]) be the matrix dened by the equation X = (x i;j ), and let det2k[x i;j ] be the homogeneous polynomial of degree n given by the determinant of X. If we dene A :=k[x i;j ]=(det 1), then each matrix M2 SL n (R) denes a map f M :A!R given by the equation f M (x i;j ) =M i;j . This map is well-dened precisely because the determinant of M is 1. Conversely, suppose we are given a map of k-algebras f : A! R, and dene a matrix M f by M f = (f(x i;j )). Since f is a map of k-algebras, it can be shown that the determinant of M f is 1, so that M f 2 SL n (R). By denition, M f M =M for all M2 SL n (R) and f M f = f for all k-algebra maps f : A! R, so we have established the following bijection of sets: Elements of SL n (R) k-algebra maps A!R M f M M f f 9 10 1. COHOMOLOGY AND SUPPORT VARIETIES OF AFFINE GROUP SCHEMES The structure of this bijection is compatible with our view of SL n as a functor. We've already described above how a map of k-algebras ' :R!S induces a map of sets' : SL n (R)! SL n (S). We leave as an exercise to the reader the verication of the equality f '(M) = 'f M . Thus, as a functor, SL n is naturally isomorphic to the functor Hom k (A;). Example 1.1.1 illustrates the concept of a representable functor, which is a functor that is naturally isomorphic to a Hom functor. More precisely, ifC is a category, and F :C! Sets is a functor, then we say thatF is representable if there is an objectA2 Ob(C) such that F is naturally isomorphic to the Hom functor Hom C (A;). A representation ofF is a pair (A;) of an objectA and a natural isomorphism :F! Hom C (A;). Remark 1.1.2. If F is naturally isomorphic to Hom C (A;) and Hom C (A 0 ;) for distinct objects A;A 0 2 Ob(C), then A and A 0 are isomorphic as objects. We therefore can speak of \the" representing object of a representable functor, with the understanding that we're actually speaking of an isomorphism class of objects. We may also invoke the axiom of choice and simply choose, once and for all, a representation (A;) for each representable functor F . This allows us to x the bijection between the setsF (B) and Hom C (A;B) for all objectsB. In what follows, we will implicitly make use of such bijections when we view the elements of the set F (B) as morphisms A!B. Remark 1.1.3. Set theorists would object to the denition of a representabl func- tor we've just given. IfC is an arbitrary category, it is not always the case that Hom C (A;B)2 Ob(Sets) for any A;B 2 Ob(C). We should thus place further restrictions on the possible source categories of a representable functor. The nec- essary notion is that of a locally small category, which is a categoryC such that Hom C (A;B) is indeed a set for any A;B2 Ob(C). We will not bother justifying the following claim, which we need to dene ane schemes over k: Alg is locally small. Denition 1.1.4. An ane scheme over k is a representable functor with source Alg. IfX is an ane scheme overk, the representing commutative k-algebra ofX is denotedk[X], and is called the coordinate algebra ofX. IfX andY are two ane schemes over k, then a map of ane schemes over k is a natural transformation of functors, :X!Y . If A is a commutative k-algebra, then the set X(A) is called the set of A-points of X. Following Remark 1.1.2, we will consider the A-points of X to be k-algebra maps k[X]! A (i.e., we will consider the two sets X(A) and Hom k (k[X];A) to be equal). As discussed in the introduction, unless otherwise noted we assume all schemes are over k, so henceforth we employ the phrase \ane scheme" to signify \ane scheme overk." The procedure of assigning to an ane scheme its coordinate alge- bra has a reasonable inverse. IfA is a commutativek-algebra, then by the denition of a representable functor, Hom k (A;) is an ane scheme with representing ob- ject A. Similarly, the functors X and Hom k (k[X];) are naturally isomorphic. In fact, these procedures hint at the fact that ASch is anti-equivalent to Alg. The following lemma and it's corollary make this result precise. 1.1. AFFINE GROUP SCHEMES 11 Lemma 1.1.5. (Yoneda) Let X and Y be ane schemes. There is a bijection between the set of natural transformations :X!Y and the set ofk-algebra maps k[Y ]!k[X]. In other words, the sets Hom ASch (X;Y ) and Hom k (k[Y ];k[X]) are in bijection. Proof. Notice that the set X(k[X]) has the \special" element id : k[X]! k[X]: Given a natural transformation : X! Y , the symbol k[X] (id) represents an element of Y (k[X]) = Hom k (k[Y ];k[X]). Conversely, a map ' : k[Y ]! k[X] denes a natural transformation X ! Y as follows. Let A be a commutative k-algebra, and dene a map X(A) ! Y (A) by f 7! f ' for f 2 X(A) = Hom(k[X];A). The reader can check that these two processes are inverses of each other. Corollary 1.1.6. ASch is anti-equivalent to Alg. Proof. Dene a contravariant functorF : ASch! Alg which on an object is given by F (X) = k[X], and on a morphism : X ! Y is given by F () = k[X] (id k[X] ). F is full and faithful by Lemma 1.1.5, and essentially surjective because the k-algebra k[Hom k (A;)] is isomorphic to A (cf. Remark 1.1.2). It follows that F induces an anti-equivalence of categories. Alternatively, one may dene another contravariant functorG : Alg! ASch given on an object by G(A) = Hom k (A;) and on a morphism ' : A! B by G(') R (f) = f' for all commutative k-algebras R and all f : B ! R. The verication that FG (GF ) is naturally isomorphic to id Alg (id ASch ) is left as an exercise. Notice that the functor SL n discussed in Example 1.1.1 has more structure than we've discussed so far. Indeed, the R-points of SL n for any commutativek-algebra R have the structure of a group under matrix multiplication. Furthermore, the map ' : SL n (R)! SL n (S) associated to a map of k-algebras ' : R! S is in fact a map of groups. This shows that we may view SL n as a functor from Alg to Grps. This leads to the following denition of an ane group scheme. Denition 1.1.7. An ane group scheme G is a functor from Alg to Grps such that the composition ofG with the forgetful functor Grps! Sets is representable. The coordinate algebra of G composed with the forgetful functor is denoted k[G], and will simply be referred to as the coordinate algebra ofG. As with ane schemes, a map between two ane group schemes G and H is a natural transformation of functors :G!H, and the set G(A) for a commutative k-algebra A is called the A-points of G. Since the target category of an ane group scheme is Grps, the coordinate algebra of a group scheme inherits a richer structure. To explore this structure, we will use the following lemma. Lemma 1.1.8. (1) If G and H are functors from Alg to Sets, then we can dene a functor GH which to a commutative k-algebra R assigns the set G(R)H(R), and to a map ' :R!S assigns the map of sets G(')H('). If G and H are representable, with representing objects k[G] and k[H] respectively, thenGH is representable, with representing object given byk[G] k[H]. 12 1. COHOMOLOGY AND SUPPORT VARIETIES OF AFFINE GROUP SCHEMES (2) The ane group schemefeg which assigns to every k-algebra the trivial group is represented by k. The proof of (1) amounts to verifying that the sets Hom k (A;R) Hom k (B;R) and Hom k (A B;R) are in bijection (this relies on the commutativity of R). The proof of (2) follows from the fact that for any k-algebra R, there is a unique map of k-algebras k!R. Notice that for any ane group scheme G there is a unique map G!feg. Now letG be an ane group scheme. For any commutativek-algebraA, there are mapsm A :G(A)G(A)!G(A), A :G(A)!G(A), ande A :feg(A)!G(A) arising from the multiplication, inverse, and identity maps on the group G(A), respectively. These maps are compatible with the functorial structure of G, i.e., they dene natural transformations m :GG!G, :G!G, and e :feg!G. Furthermore, since these natural transformations are dened from the group laws of the A-points of G for various A, they satisfy the diagrams of Figure 1 derived from the axioms dening a group. The map G is dened as projection onto G. GGG GG GG G Associativity fegG GG G Left identity G GG feg G Left inverse m id G id G m m m e id G m G (; id G ) m e Figure 1. Diagrams satised by an ane group scheme Lemmas 1.1.5 and 1.1.8 together thus show there must be k-algebra maps : k[G]! k[G] k[G], S : k[G]! k[G], and " : k[G]! k corresponding to the maps of ane group schemes m, ; and e, respectively. We refer to , S, and " as the comultiplication, antipode, and augmentation of k[G], respectively. By Corollary 1.1.6 and Figure 1, the maps , S, and " must satisfy the diagrams of Figure 2. The map 1 k id k[G] is given by a7! 1 a for a2k[G]. For future use, we dene here the augmentation ideal of k[G] as the kernel of the augmentation, " :k[G]!k. A k-algebra A with maps , S, and " satisfying the diagrams of Figure 2 is called a Hopf algebra over k. Before moving on we dene some terms and discuss some properties associated to Hopf algebras. 1.1. AFFINE GROUP SCHEMES 13 k[G] k[G] k[G] k[G] k[G] k[G] k[G] k[G] Coassociativity k k[G] k[G] k[G] k[G] Augmentation k[G] k[G] k[G] k k[G] Antipode id k[G] id k[G] " id k[G] 1 k id k[G] S id k[G] " Figure 2. Diagrams satised by the coordinate algebra of an ane group scheme IfA is a Hopf algebra, and :A A!A A is the map which sends a simple tensor a b to b a, then we say that A is cocommutative if = . If A is nite-dimensional, then we can form the dual Hopf algebra A := Hom(A;k) as follows. For f;g2A and a;b2A dene: (1.1.1) (fg)(x) =f g (x) (f)(a b) =f(ab) The multiplicative unit of A is ", and the augmentation of A is the map " : A !k dened by " (f) =f(1). Notice that A is canonically isomorphic to (A ) via the map a7!ff7! f(a)g, and that A is cocommutative if and only if A is commutative. We omit the proof of the following theorem, which is nearly identical to the proof of Corollary 1.1.6. Theorem 1.1.9. AGrpSch is anti-equivalent to HopfAlg We conclude this section by dening certain specializations of ane group schemes, discussing the algebra of distributions associated to an ane group scheme, and giving a number of examples that will be relevant throughout this document. Denition 1.1.10. Let G be an ane group scheme. (1) G is algebraic if k[G] is nitely-generated. (2) G is nite if k[G] is nite-dimensional over k. (3) G is innitesimal if it is nite and k[G] is local. (4) G is reduced if k[G] contains no nilpotent elements. (5) G is abelian ifG(R) is abelian for all commutativek-algebrasR, or equiv- alently, if k[G] is cocommutative. 14 1. COHOMOLOGY AND SUPPORT VARIETIES OF AFFINE GROUP SCHEMES (6) The group algebra of G, denoted kG, is the k-linear dual of k[G]. (7) If G is innitesimal, and m k[G] is the unique maximal ideal, then m is nilpotent because k[G] is artinian. Thus, there is some minimal r such that x p r = 0 for all x2 m. This r is called the height of G. Notice that we may equivalentely dene G to be innitesimal if it is nite and if the augmentation ideal I is nilpotent. If k[G] is nite dimensional, then it is artinian, so that the Jacobson radical is nilpotent. If k[G] is local, then the Jacobson radical is I. Conversely, if I is nilpotent, then I is contained in the Jacobson radical, so that k[G] is local. Algebra of Distributions. To an ane group schemeG we've already associ- ated its coordinate ringk[G]. Here we introduce the (not necessarily commutative) algebra of distributions, Dist(G), which will play a central role in the representa- tion theory ofG. Along the way, we'll nd that a Lie algebra lies inside of Dist(G), which will be denoted Lie(G). In characteristic p, Lie(G) will come equipped with a generalized pth power map, giving it the structure of a restricted Lie algebra. More details can be found in [29]. Let G be an ane group scheme, and let I be the augmentation ideal. Let n be a nonnegative integer, and as a set, let Dist n (G) be all k-linear maps f : k[G]! k such that f(I n+1 ) = 0. Notice that scalar multiplication and pointwise addition give Dist n (G) the structure of a vector space over k. Within Dist n (G) is a subspace of codimension 1, denoted by Dist + n (G) and dened as those k-linear maps f :k[G]!k vanishing on I n+1 such that f(1) = 0. We then dene Dist(G) = [ n0 Dist n (G) and Dist + (G) = [ n0 Dist + n (G) We can dene a product on Dist(G) giving it the structure of a ltered associa- tive algebra. In other words, Dist(G) is ltered by Dist 0 (G) Dist 1 (G):::, and our product will satisfy Dist n (G) Dist m (G) Dist n+m (G). To do this, we will use the comultiplication on k[G]. Suppose that f2 Dist n (G) andg2 Dist m (G). Then dene (fg)(x) =f g (x). In other words, fg is the following composition: (1.1.2) fg :k[G] !k[G] k[G] f g !k We review the properties of Dist(G) with this product structure in the following proposition (without proof). The interested reader can work out the details. Proposition 1.1.11. Let G be an ane group scheme. (1) The product structure dened in Equation (1.1.2) turns Dist(G) into a ltered, associative algebra with multiplicative identity given by the aug- mentation, ". (2) Iff2 Dist n (G) andg2 Dist m (G), then [f;g] =fggf2 Dist n+m1 (G). In other words, the graded algebra associated to the ltered algebra Dist(G) is commutative. (3) Dist + (G) is a two-sided ideal in Dist(G). (4) If f2 Dist + 1 (G), then f p 2 Dist + 1 (G). (5) If H is another ane group scheme, and ' : G! H is a map of ane group schemes, then ' induces a map of ltered algebras d' : Dist(G)! Dist(H). (6) If G is innitesimal, then Dist(G) =kG :=k[G] . 1.1. AFFINE GROUP SCHEMES 15 (2) and (3) in Proposition 1.1.11 together show that Dist + 1 (G) is a Lie algebra, with bracket given by [f;g] =fggf. We will denote this Lie algebra by Lie(G). (4) shows that Lie(G) has the extra structure of a pth power map. Lie(G) is an example of a restricted Lie algebra, which we now dene abstractly. Restricted Lie algebras will play a central role in what follows. For further details, one may consult [28]. Denition 1.1.12. A restricted Lie algebra overk is a Lie algebra (g; [;]) over k together with a restriction map, () [p] : g! g satisfying the following for all 2k and for all x;y2 g: (1) (x) [p] = p x [p] (2) (x +y) [p] =x [p] +y [p] + P i = 1 p1 s i (x;y) whereis i (x;y) is the coecient of t i1 in the formal expression ad p1 tx+y (x) (3) [x [p] ;y] = ad p x (y) Here, ad x (y) = [x;y]. A map of restricted Lie algebrasf : g! h is a map of Lie algebras that respects the restriction map, that is,f(x [p] ) =f(x) [p] . If' :G!H is a map of ane group schemes, then the map d' discussed in (5) of Proposition 1.1.11 restricts to a map d' : Lie(G)! Lie(H), which is a map of restricted Lie algebras. We can equip any unital associative k-algebra A with the structure of a re- stricted Lie algebra by dening [a;b] := abba and a [p] := a p for all a;b2 A. That such a structure satises the axioms of 1.1.12 is not immediately clear, but a satisfying exercise. We writeL(A) if we view A as a restricted Lie algebra. There is an algebra, denotedu(g), together with an embeddingi : g,!L(u(g)) of restricted Lie algebras, which is in some sense the smallest unital associative algebra which admits such an embedding. This is made precise in the following universal property. If A is a unital, associative algebra, and f : g!L(A) is a map of Lie algebras, then there is a unique map of k-algebras g : u(g)! A such that f =gi. u(g) is called the restricted enveloping algebra, and is constructed as the quo- tient of the usual universal enveloping algebra of g by the ideal generated by all elements of the form x [p] x p . If g is nite dimensional with basis , then a basis for u(g) is the collection of all monomials in such that no single element in has degree greater than p 1. It follows that the k-dimension of u(g) is equal to the p dim k (g) . u(g) can be given the structure of a Hopf algebra by requiring that (x) = 1 x +x 1, S(x) =x, and (x) = 0 for all x2 g u(g). Since g generates u(g) as an algebra, these maps are uniquely extended to all of u(g). Examples. All functors dened in these examples have source Alg and target Grps. In some of the examples we report the structure of the ane group schemeG as a functor, and we discuss the objectsk[G],kG, Dist(G), and Lie(G). Verication of the details is left as an exercise. Example 1.1.13. (The additive group, G a ) Dene a functor G a as follows. If R is a commutative k-algebra, then G a (R) := (R; +), the additive group of R. If ' :R!S is a map of k-algebras, thenG a (') is simply ' (note that ' is in fact a map of the additive groups (R; +) and (S; +)). G a is represented by the k-algebra k[T ], and the Hopf algebra structure on k[T ] is given by (T ) = 1 T +T 1, S(T ) =T , and "(T ) = 0. Notice thatG a is algebraic. 16 1. COHOMOLOGY AND SUPPORT VARIETIES OF AFFINE GROUP SCHEMES Since "(T ) = 0, the augmentation ideal is (T ), so that for n 0, a k-linear map f :k[T ]!k is in Dist n (G a ) if and only if f vanishes on all polynomials in T with no term of degreed ford = 0;:::;n. If we dene n (T m ) = n;m , then the set f n g n0 is a k-basis of Dist(G a ). The multiplication in Dist(G a ) is given by n m = n +m n n+m Finally, notice that Lie(G a ) = Dist + 1 (G a ) = Dist 1 (G a ) is the one-dimensional space generated by 1 . Both the bracket and restriction map dened on Lie(G a ) are trivial. We will denote this one-dimensional trivial Lie algebra by g a . In general, an elementary algebra is any restricted Lie algebra with trivial bracket and trivial restriction. Such Lie algebras are at the heart of the study of this document. Example 1.1.14. (Constant group scheme) We will now show how AGrpSch is a generalization of the category of nite groups, that is, how we can identify the category of nite groups as a full subcategory of the category of ane group schemes (in particular of nite group schemes). Let be a nite group, and let k be the set of all functions (as sets) !k. Notice that under pointwise addition, multiplication, and scalar multiplication, k is a nite dimensional commutative k-algebra of dimensionjj. A basis for k consists of the functionse for all2 dened by the equatione () = ; . The following maps give k the structure of a commutative Hopf algebra: (e ) = X = e e S(e ) =e 1 "(e ) = ;id We then dene an ane group scheme G such that k[G ] = k , that is, to any commutative k-algebra R we set G (R) := Hom k (k ;R). Suppose that R has no nontrivial idempotents. Since each e is idempotent, any point f2G must map e to 0 or 1 for all 2 . Since P 2 e = 1 k , f can only send one e to 1, and the rest to 0. It follows that the elements of G (R) are in bijective correspondence with the elements of , and the reader may check using the Hopf algebra structure of k that the group structure dened on G (R) makes this bijection a group isomorphism. In other words,G (R) = for allR with no nontrivial idempotents. If R = R 1 R 2 splits as the direct product of two rings R 1 and R 2 with no nontrivial idempotents, then it can be shown that G (R) = . In general, G (R) is isomorphic to a product of 's indexed by the connected components of the topological space Spec(R). We brie y mention here the rather nice fact that the group algebra of G is isomorphic to the group algebra of , motivating the terminology introduced in Denition 1.1.10. In other words, thek-linear dual ofk[G ] equipped with the dual Hopf algebra structure, is isomorphic to k as a Hopf algebra. The Hopf algebra structure on k is dened as follows: (g) =g g, S(g) =g 1 , and "(g) = 1 for all g2 . Example 1.1.15. (Algebraic groups) We will take the view of [8] and [26] that an ane algebraic group is a reduced, algebraic ane group scheme. If G is an ane algebraic group, then we can recoverG from the data of itsk-points,G(k), as 1.1. AFFINE GROUP SCHEMES 17 follows. Suppose we choose a surjection ' :k[x 1 ;:::;x n ]k[G]. For any k-point k[G]!k, we can compose with ' to obtain a point (a 1 ;:::;a n )2k n where a i is the image ofx i under the composition. Thus, the k-points ofG have been realized as a Zariski-closed subset of k n . Eectively, our choice of surjection ' yields a particular embedding of the variety of k-points of G into some ane space. Now, with this particular embedding of G(k) in k n , let I k[x 1 ;:::;x n ] be the ideal generated by all functions which vanish on G(k), in other words, f2 I if and only if '(f) is in the kernel of all possible k-algebra maps k[G]!k. Since k is algebraically closed, the Hilbert Nullstellansatz shows that the intersection of all such kernels is the ideal of nilpotent elements of k[G], which is zero in our case sinceG is reduced. It follows thatf2I if and only if'(f) = 0, so thatI = ker('). We've thus recovered k[G] as the coordinate algebra k[x 1 ;:::;x n ]=I of the Zariski- closed setG(k)k n . Since an algebraic group G can be recovered by its k-points, we (along with the authors of [8] and [26]) often speak of an ane algebraic group G as a group (and not a functor), in essence identifying G with G(k). This also motivates the preferred terminology \ane algebraic group" as opposed to \ane algebraic group scheme." In the spirit of Corollary 1.1.6 and Theorem 1.1.9 we have an anti-equivalence of categories between ane algebraic groups and nitely generated, reduced Hopf algebras. We will revisit algebraic groups and their properties in Chapter 2. Example 1.1.16. (The general linear group, GL n ) For a particular example of an algebraic group, consider the functor GL n which on a commutative k-algebra R gives the group of invertiblenn matrices with entries inR. On a map' :R!S, we dene GL n (') : GL n (R)! GL n (S) as we have in Example 1.1.1. The coordinate algebra of GL n is k[x i;j ;tj 1 i;j n]=I where I is the ideal generated by the irreducible element dett 1. It follows that I is a radical ideal, so thatk[GL n ] is reduced and nitely generated, so that GL n is an algebraic group. The Hopf algebra structure onk[GL n ] is given by (x i;j ) = P n l=1 x i;l x l;j , "(x i;j ) = i;j , and S(x i;j ) = (1) i+j tX (j;i) where X (j;i) is the determinant of the (j;i) minor of the matrix of indeterminates X = (x l;m ). For any commutativek-algebraR, the group GL n (R) contains the subgroup of all upper-triangular matrices with 1's on the diagonal. We will denote this group by U n (R), and then dene a functor U n which maps R to U n (R). Notice that U n is representable, with k[U n ] being given by the quotient of k[GL n ] by the ideal generated by the elementsx i;j i;j forij. The Hopf algebra structure onk[U n ] is simply that inherited from k[GL n ]. To motivate Example 1.1.16, we mention the following result without proof which shows that in the setting of algebraic groups, the ane algebraic group GL n plays a similar role to that of the symmetric group in the setting of nite groups (recall that a nite group embeds in the symmetric groupS jj by the permutation representation induced by left multiplication of on itself). Theorem 1.1.17. LetG be an ane algebraic group. Then there is a surjective map of Hopf algebras k[GL n ] k[G] for some n. In other words, there is a map of ane algebraic groups G! GL n such that G(R)! GL n (R) is an embedding of groups for all commutative k-algebras R. 18 1. COHOMOLOGY AND SUPPORT VARIETIES OF AFFINE GROUP SCHEMES Remark 1.1.18. A map of ane group schemesH!G which induces a surjective map on Hopf algebras k[G]! k[H] is called a closed embedding. Theorem 1.1.17 states that any ane algebraic group admits a closed embedding into some GL n . We can dene a subgroup scheme H of an ane group scheme G to be an ane group scheme whose coordinate algebra is isomorphic to a quotient of k[G]. As in Theorem 1.1.17, this ensures that the induced mapH(R),!G(R) is an embedding of groups for all commutativek-algebrasR, which motivates the terminology \sub- group scheme." We've already given the example of the subgroup schemeU n GL n in Example 1.1.16 Before moving to the next example, we make a denition here for future pur- poses. Denition 1.1.19. An algebraic ane group scheme G is called unipotent if it is a closed subgroup scheme of U n for some n. In other words, we require that there is a surjection of Hopf algebras k[U n ]k[G] for some n. Example 1.1.20. (V a and GL V ) Here we dene a slight generalization of Exam- ple 1.1.16, which will be useful when dening representations of ane group schemes in Section 1.2. Let V be a xed vector space over k of dimension n. We dene the ane group scheme V a by V a (R) := (V R; +) for a commutative k-algebra R. Notice that a choice of basis for V yields an isomorphism V R =R n . It can be shown that V a is representable, and that k[V a ] =S(V ). Further, dene the ane group scheme GL V by GL V (R) = Aut R (V R), that is, to a commutativek-algebraR associate the group of allR-linear maps (viewing V R as an R-module). Notice that if we specify a basis of V , then there is an isomorphism GL V = GL n . Example 1.1.21. (Frobenius kernels) A nice treatment of Frobenius kernels can be found in [15]. We summarize the main points here. Let F : k! k be the Frobenius morphism of elds which maps any element x2 k to x p . Since k is algebraically closed, F is an isomorphism of elds. Let A be a commutative Hopf algebra, and dene the Frobenius twist of A to be vector space A (1) := A F k. In other words, A (1) is generated as a k vector space by elements of the forma 1 fora2A, where we have the following relations for any 2k and any a;b2A: a 1 +b 1 = (a +b) 1 and (a) 1 =a p For any a2 A dene a (1) := a 1. There is a natural isomorphism of vector spaces (A A) (1) = A (1) A (1) under which (a b) (1) corresponds to a (1) b (1) . It follows if m, , S, ", and u are the multiplication, comultiplication, antipode, augmentation, and unit in A (respectively), then A (1) inherits the structure of a commutative Hopf algebra from A in the following way: m (1) :A (1) A (1) ! (A A) (1) m id !A (1) (1) :A (1) id ! (A A) (1) !A (1) A (1) S (1) :A (1) S id !A (1) " (1) :A (1) " p id !k u (1) :k 1 id !A (1) 1.1. AFFINE GROUP SCHEMES 19 With this Hopf algebra structure on A (1) , we have a map of Hopf algebras dened by: (1.1.3) A (1) ! A a 7! a p Notice this map is well dened as (a +b) p =a p +b p and because the simple tensors (a) 1 and a p (which are equal in A (1) ) are both mapped to p a p . Now, suppose thatG is an ane group scheme. Then the commutative Hopf al- gebra k[G] (1) represents an ane group scheme we denote by G (1) (i.e., k[G (1) ] := k[G] (1) ). By Lemma 1.1.5, the Hopf algebra morphism k[G (1) ]! k[G] just de- scribed gives a morphism of ane group schemes G!G (1) which by abuse of no- tation we also denote by F . This morphism is called the Frobenius morphism. We now dene the Frobenius kernel to be the closed subgroup schemeG (1) G dened by G (1) (R) = kerfG(R)! G (1) (R)g. If we inductively dene G (n) = (G (n1) ) (1) , then we can consider ther-fold iterate of the Frobenius morphism for some positive integer r. We thus obtain a map F r :G!G (r) whose kernel is denoted G (r) and called the rth Frobenius kernel. It will be helpful to identify the representing algebra of G (r) as a quotient of k[G] in order to study the properties of Frobenius kernels. For this, we will need the notion of a cokernel in the category of Hopf algebras. To begin, a Hopf ideal of a Hopf algebra A is an ideal I such that (I) I A +A I, S(I) I and "(I) = 0. It can be shown that these are precisely the conditions an ideal requires so thatA=I inherits the structure of a Hopf algebra from that of A. It follows that the Hopf ideals are in bijection with kernels of surjective homomorphisms. Now, for an arbitrary map of Hopf algebras ' : A! B, it is not necessarily true that the cokernel inherits the structure of a Hopf algebra, that is, the image of ' is not always a Hopf ideal. However, it is true that '(I A )B is a Hopf ideal, where I A is the augmentation ideal of A. It follows that B=('(I A )B) is a Hopf algebra, which we'll call the Hopf cokernel of '. We state without proof (although one can be found in [54],x2.1) that k[G (1) ] is isomorphic to the Hopf cokernel of the map k[G (1) ]!k[G], and more generally, thatk[G (r) ] is isomorphic to the Hopf cokernel of the mapk[G (r) ]!k[G]. We also omit the proof of the fact that the Lie algebra of G and G (r) coincide. We end this example by claiming that k[G (r) ] is nite dimensional and local, and that the maximal ideal mk[G (r) ] has the property that any x2 m satises x p r = 0. It will follow that G (r) is an innitesimal group scheme of height r. To see this, we note that for an algebraically closed eld k of characteristic p, every element 2 k has a unique pth root. It follows that every element of A (1) can be written uniquely asa (1) =a 1 for somea2A. IfI (1) is the augmentation ideal ofA (1) , from the denition of the augmentation" (1) , we see thata 12I (1) if and only ifa p 2I. However,I is maximal, and therefore prime, so thata 12I (1) if and only if a2 I. It follows that the image of I (1) under the map in 1.1.3 is fa p ja2Ig. Recalling the notion of Hopf cokernel, we see that the Hopf cokernel of A (1) ! A = kI is nite dimensional whose augmentation ideal is nilpotent. Furthermore, if x is in the augmentation ideal of the Hopf cokernel of A (1) ! A, then we can see that x p = 0. If we replace A with k[G] in this discussion, we see that we've shown our claim for r = 1. The general case is extrapolated from what is done here. 20 1. COHOMOLOGY AND SUPPORT VARIETIES OF AFFINE GROUP SCHEMES Example 1.1.22. (Frobenius kernel ofG a ) In this example, we apply the machin- ery of Example 1.1.21 to the additive group G a . From Example 1.1.13 we know that k[G a ] is isomorphic to the k-algebra k[T ], so that k[G (1) a ] is isomorphic to the k-algebra k[T ] f k. Now k[T ] = F p [T ] Fp k, so that k[T ] (1) = k[T ] via the composition: k[T ] (1) :=k[T ] F k! (F p [T ] Fp k) F k!F p [T ] Fp (k F k)!F p [T ] k!k[T ] which maps the simple tensor ( P a i T i ) to the polynomial P a p i T i (in the above composition, the map identifyingk F k withk is given by 7! p ). The inverse of this isomorphism sends a polynomial P b i T i to the element P (T i b i ). These maps are in fact maps of Hopf algebras, so thatG (1) a =G a as ane group schemes. With this identication of k[T ] (1) with k[T ], the map given in Equation (1.1.3) sends a polynomial P b i T i to the polynomial P b i T pi . The ideal generated by the image of the augmentation ideal is thus (T p ), so that the Hopf cokernel of the map from 1.1.3 isk[T ]=(T p ). It follows thatG a(1) is innitesimal of height 1 (as expected from the discussion at the end of Example 1.1.21), and that G a(1) (R) :=fa2Rj a p = 0g. We leave the verication that G a(r) is represented by by k[T ]=(T p r ) to the reader. For future use, we report here the structure of the distribution algebra. Since G a(r) is innitesimal, Dist(G a(r) ) =kG a(r) :=k[G a(r) ] . One can show thatkG a(r) is isomorphic to the k-algebra k[u 0 ;:::;u r1 ]=(u p 0 ;:::;u p r1 ) where the element u i reads o the coecient of T p i . Remark 1.1.23. In Example 1.1.22, we saw that writing k[T ] asF p [T ] Fp k led to an isomorphism k[T ] (1) =k[T ] so thatG (1) a =G a as ane group schemes. This fact is true in general: if G is an ane group scheme dened over F p (i.e. there is an F p -algebra F p [G] and an isomorphism k[G] = F p [G] Fp k), then G (1) = G as ane group schemes. Thus in this case one may write Frobenius as F :G!G. Example 1.1.24. (Base change) Let G be an ane group scheme over k, and let A be a k-algebra. Notice that if R is a commutative A-algebra, then we can give R the structure of a commutative k-algebra via restricting along the map k ,! A. We denote by R k the k-algebra obtained in this way. It follows that we can dene a functorG A from the category of commutativeA-algebras to groups by G A (R) =G(R k ). In some sense, G A is the functor G restricted to A-algebras. Since maps of k-algebras k[G]! R k are in one-to-one correspondence with maps ofA-algebras k[G] A!R, G A is representable, with coordinate A-algebra given by k[G] A. Example 1.1.25. (Restricted Lie algebras) The restricted Lie algebras we en- countered in Denition 1.1.12 give us another example of ane group schemes. If g is a restricted Lie algebra, then the restricted enveloping algebra u(g) is a - nite dimensional, cocommutative Hopf algebra. It follows from Equation (1.1.1) that the k-linear dual u(g) is a nite dimensional, commutative Hopf algebra. I claim further that u(g) is a local algebra, with unique maximal ideal given by m :=ff :u(g)!kjf(1) = 0g. To see this, one can show that any f2 m satises f p = 0, and any f = 2 m has inverse given by f(1) 1 (fS). It follows that the ane group scheme dened by g(R) := Hom k (u(g) ;R) is innitesimal of height 1. 1.1. AFFINE GROUP SCHEMES 21 Conversely, if G is an innitesimal group scheme of height 1, then the aug- mentation ideal satises I p = 0 so that k[G] = Dist(G) =[ p1 n=0 Dist n (G). No- tice that since k[G] = Ik, we know that Dist 0 (G) = k. We state without proof that Lie(G) = Dist + 1 (G) generates Dist(G) as a k-algebra, and in fact that u(Lie(G)) = Dist(G). It follows that k[G] = u(Lie(G)) . Along with the previous paragraph, this discussion amounts to the fact that the category of restricted Lie algebras is equivalent to the category of height one innitesimal group schemes. Further details can be found inx9:6 in [29]. Example 1.1.26 (A classication of Restricted Lie algebras of dimension 1). Let g be a restricted Lie algebra of dimension 1. Then it must be abelian, so we only need to determine the restriction map. Choose a nonzero element x2 g. Since g is one dimensional, we must have x [p] = x for some 2 k. By scaling we can assume that either = 0 or = 1. In the case = 0, we get the trivial restricted Lie algebra g a which is the Lie algebra of the groupG a . If = 1, then for any 2 k we have (x) [p] = p x [p] = p x. It follows that g is isomorphic toL(k), where k is the unique one-dimensional k-algebra. This isomorphism sends x to 1. We will denote this Lie algebra by g m . It follows from our discussion here that u(g m ) =k[x]=(x p x). I claim that the height one group scheme corresponding to g m is G m;(1) , the rst Frobenius kernel of G m , where G m is the ane group scheme dened by the Hopf algebra k[t;t 1 ] with (t) = t t. One can show that k[G m;(1) ] = k[T ]=(T p 1) andkG m;(1) =k Z=pZ so to prove the claim, it suces to exhibit a Hopf algebra isomorphism f : k[x]=(x p x)! k Z=pZ : The appropriate map is given by x7! P p1 i=1 ie i : Since algebra maps preserve multiplicative identities, we must have 17! P p1 i=0 e i . Extending this map multiplicatively, we see that x n 7! P p1 i=1 i n e i because thee i are mutually orthogonal idempotents. Thatx p x maps to zero is a result of Fermat's little theorem. It can also be shown that f preserves the counit, comultiplication, and the antipode. If we letf1;x;x 2 ;:::;x p1 g andfe 0 ;e 1 ;e 2 ;:::;e p1 g be ordered bases for the k-algebras k[x]=(x p x) and k Z=pZ respectively, then the matrix of f, viewed as a linear transformation is given by: f = 0 B B B B B B B @ 1 0 0 0 ::: 0 1 1 1 1 ::: 1 1 2 4 8 ::: 2 p1 1 3 9 27 ::: 3 p1 . . . . . . . . . . . . ::: . . . 1 p 1 (p 1) 2 (p 1) 3 ::: (p 1) p1 1 C C C C C C C A where the entries are to be reduced modp. This matrix has a non-zero determinant because it has the same determinant as its p 1p 1 Vandermonde sub-matrix excluding the rst row and column. This sub-matrix has non-zero determinant because each row is generated by powers of the distinct numbers 1; 2; 3;:::;p 1. Thus, the matrix is invertible andf is an isomorphism. The inverse is given by the following matrix: 22 1. COHOMOLOGY AND SUPPORT VARIETIES OF AFFINE GROUP SCHEMES f 1 = 0 B B B B B B B B B @ 1 0 0 0 ::: 0 0 1 2 p2 3 p2 ::: (p 1) p2 0 1 2 p3 3 p3 ::: (p 1) p3 . . . . . . . . . . . . ::: . . . 0 1 4 9 ::: (p 1) 2 0 1 2 3 ::: (p 1) 1 1 1 1 ::: 1 1 C C C C C C C C C A We've written the matrices with entries that suggest the pattern involved, ne- glecting to reduce modp. Here I write out the matrices for the casep = 5, reducing the entries appropriately: f = 0 B B B B @ 1 0 0 0 0 1 1 1 1 1 1 2 4 3 1 1 3 4 2 1 1 4 1 4 1 1 C C C C A f 1 = 0 B B B B @ 1 0 0 0 0 0 4 2 3 1 0 4 1 1 4 0 4 3 2 1 4 4 4 4 4 1 C C C C A Notice that f 1 gives a system of p mutually orthogonal idempotents in the k- algebra k[x]=(x p x) as the images of the e i . To be explicit, we have: e 0 7! 1x p1 e n 7! p1 X i=1 n i1 x pi n = 1; 2;:::;p 1 1.2. Representations and cohomology Building towards the denition of a support variety in the next section, we now discuss the notion of a representation of an ane group scheme. Although our main point of view throughout the rest of the document will consider support varieties as dened through \-points" (see Denition 1.3.3), in this section we'll include a brief discussion of cohomology to give an equivalent denition of support varieties. Having introduced ane group schemes in Section 1.1, we now drop the descriptor \ane" and simply refer to a \group scheme" G with the assumption that G is ane. Denition 1.2.1. Let V be a xed vector space over k of dimension n. A rep- resentation of a group scheme G on V is a morphism GV a ! V a such that for any commutativek-algebraR, the associated mapG(R)V a (R)!V a (R) gives an action of the group G(R) on the R-module V a (R) =V R which is R-linear. Notice that a representation of G on V gives a map G(R)! Aut R (V R) for every commutative k-algebra R. In other words, a representation of G on V is equivalent to a morphism of group schemes G! GL V . If V comes with a xed basis, then a representation of G on V = k n is a morphism of group schemes G! GL n . The next proposition shows that the data of a representation ofG onV is equivalent to the data of a \k[G]comodule" (to be dened after the statement of the proposition). Proposition 1.2.2. Let V be a nite dimensional vector space over k, and let G be a group scheme. Then representations of G on V correspond to k-linear maps :V !V k[G] such that the diagrams in Figure 3 are satised. 1.2. REPRESENTATIONS AND COHOMOLOGY 23 Proof. We sketch the proof without including the details. If : G! GL V is a representation of G, then the element k[G] (id k[G] ) 2 Aut k[G] (V k[G]) is determined by its restriction to V k = V . This restriction is the corresponding map :V !V k[G]. Denition 1.2.3. LetA be a Hopf algebra. A comodule forA is a vector spaceV together with a map :V !V A which satises the diagrams in Figure 3 V V k[G] V k[G] V k[G] k[G] V V k[G] V k id V id V id V " v7!v 1 Figure 3. Diagrams satised by a k[G]-comodule Now if V is nite-dimensional (which will be a standard assumption we will maintain) and G is nite, then (V k[G]) = V k[G] = V kG, so we can dualize the diagrams in Figure 3 to obtain those in Figure 4 (we've replaced V by V ). In Figure 4, we've denoted by the transpose of the map . These diagrams dene a kG-module structure on V . This further motivates the notation kG and the terminology \group algebra" for the cocommutative Hopf algebra k[G] when G is nite. Recall that in the category of nite groups, the category of representa- tions is equivalent to the category of modules over the group algebra. Here, we see that for nite group schemes G, the category of nite-dimensional representations of G is equivalent to the category of nite-dimensional k[G]-comodules (by Propo- sition 1.2.2) which is equivalent to the category of nite-dimensional kG-modules (by dualizing). V V kG V kG V kG kG V V kG V k id V m id kG id V u v 17!v Figure 4. Diagrams satised by a kG-module We now move towards a denition of cohomology. A helpful reference for nite group cohomology is [10], and much of the basic denitions and theory developed 24 1. COHOMOLOGY AND SUPPORT VARIETIES OF AFFINE GROUP SCHEMES for nite groups applies in the case of nite group schemes. For the rest of Chap- ter 1, we'll assume that all group schemes are nite and all representations are nite-dimensional. In this setting, we've reduced the study of G representations to kG-modules. kG is a nice nite-dimensional, cocommutative Hopf algebra to which we can associate its algebra of extensions. This algebra is the cohomology algebra of G. As mentioned before, we will mostly prefer to think of a support variety as dened by -points (Denition 1.3.3), so we will not spend too much time here developing the theory of cohomology. We only do so to emphasize one of the main results of [18] that the denition of a support variety via -points agrees with that via cohomology. For a reference dealing with the cohomology of nite- dimensional cocommutative Hopf algebras, see [7]. We will work with an arbitrary nite-dimensional cocommutative Hopf algebra A, keeping in mind that our appli- cation will be to consider A =kG for a nite group scheme G. All A-modules are nitely generated, left modules. We'll develop the theory of cohomology by considering extensions of modules. A more standard approach in the literature is to use derived functors. The resulting algebras are naturally isomorphic, a nice result we won't state precisely as we will not discuss derived functors. Denition 1.2.4. Let M and N be A-modules. (1) An n-fold extension of M by N is an exact sequence of A-modules: N!M n1 !M n2 !:::!M 1 !M 0 !M We will writefM i g to denote this extension. (2) Two n-fold extensionsfM i g andfM 0 i g are equivalent if there are maps f i :M i !M 0 i such that the following diagram commutes: 0 // N // M n1 // ::: // M 0 // M // 0 0 // N // M 0 n1 // ::: // M 0 0 // M // 0 Notice that the relation just dened is not necessarily symmetric, but we may complete it to an equivalence relation. (3) The set of equivalence classes of n-fold extensions of M by N is denoted Ext n A (M;N). We now aim to give Ext n A (M;N) the structure of an abelian group. To do so, consider the following constructions. Let f : N ! N 0 be a module map. This induces a map f : Ext 1 kG (M;N)! Ext 1 kG (M;N 0 ) as follows: E : 0 // N f d // M 0 // M // 0 f E : 0 // N 0 // NM 0 im(f;d) // M // 0 Applying f to the three leftmost terms of an n-fold extension gives a map f : Ext n kG (M;N)! Ext n kG (M;N 0 ). 1.2. REPRESENTATIONS AND COHOMOLOGY 25 Similarly, let f : M 0 ! M be a module map. This induces a map f : Ext 1 kG (M;N)! Ext 1 kG (M 0 ;N) as follows: f E : 0 // N // M 0 M M 0 // M 0 f // 0 E : 0 // N // M 0 d // M // 0 Again, applyingf to the three rightmost terms of ann-fold extension gives a map f : Ext n kG (M;N)! Ext n kG (M 0 ;N). The abelian group operation on Ext n kG (M;N) is dene as follows. Consider the module maps: f :M!MM m7! (m;m) g :NN!N (n 1 ;n 2 )7!n 1 +n 2 LetE andE 0 be two n-fold extensions of M by N, withEE 0 the extension of MM by NN obtained by summingE andE 0 . Then dene the Baer sum, E +E 0 to be the extension f g (EE 0 ). See Figure 5 for a summary of the Baer sum of extensions. Proposition 1.2.5. The operation of Baer sum of extensions gives Ext n A (M;N) the structure of an abelian group. We can consider k as a trivial module for A via the augmentation. Notice that in this instance Ext 0 A (k;k) = (k; +). We now aim to dene a product on the abelian group Ext A (k;k) = n0 Ext n A (k;k) which will give it the structure of a graded ring. The product is easier to dene than the sum, as we can simply splice together anm-fold extension of k byk and ann-fold extension of k byk to obtain an m +n-fold extension of k by k. For instance, iffM i g is an m-fold extension of k by k andfM 0 j g is and n-fold extension of k by k, then the extension: 0 k M m1 M 0 M 0 n1 M 0 0 k 0 is an m +n-fold extension, where the map M 0 !M 0 n1 is the composition M 0 ! k ! M 0 n1 . This product is called the Yoneda composition, the Yoneda splice, or the Yoneda product. Notice we can generalize the Yoneda product to a map of groups Ext m A (M;M 0 )Ext n A (M 0 ;N)! Ext m+n A (M;N). This generalized denition gives Ext A (M;k) the structure of a module over the k-algebra Ext A (k;k). All of this is made precise in the statement of the following proposition. Proposition 1.2.6. (1) The operations of Baer sum and Yoneda composition give Ext A (k;k) the structure of a graded-commutative ring. That is, if a;b2 Ext A (k;k) are homogenous elements of degreesjaj andjbj respectively, then ab = (1) jaj+jbj ba (2) Ext 0 A (k;k) =k; in particular, Ext A (k;k) is a k-algebra. (3) Yoneda composition gives Ext A (M;k) the structure of a left, graded mod- ule over the k-algebra Ext A (k;k). Denition 1.2.7. Let G be a nite group scheme. The cohomology algebra of G, or simply the cohomology of G is the k-algebra Ext kG (k;k). We will denote this graded-commutative algebra by H (G;k). 26 1. COHOMOLOGY AND SUPPORT VARIETIES OF AFFINE GROUP SCHEMES 0 N E M 0 0 N E 0 M 0 E;E 0 2 Ext n kG (M;N) 0 NN EE 0 MM 0 EE 0 2 Ext n kG (MM;NN) g 0 N g (EE 0 ) MM 0 g (EE 0 )2 Ext n kG (MM;N) f 0 N f g (EE 0 ) M 0 E +E 0 :=f g (EE 0 )2 Ext n kG (M;N) Figure 5. The Baer sum of extensions We will need a few more facts about extensions before we are ready to dene a cohomological support variety. Given an element 2 Ext n A (k;k), we can form the n-fold extension of M k =M by M k =M by tensoring with M. This gives a map M : Ext n A (k;k)! Ext n A (M;M). Denote by M the image of under this map. Corollary 3.2.2 of [7] shows that for any 2 Ext m A (M;M) we have (1.2.1) M = (1) mn M 1.2. REPRESENTATIONS AND COHOMOLOGY 27 Thus if n is even, we see that the image of M is central. It follows that Ext A (M;M) is a graded module over the graded k-algebra n0 Ext 2n A (k;k). So far, our discussion concerning cohomology has been completely algebraic. To a nite group schemeG we've associated a graded-commutativek-algebraH (G;k). If we want to associate an algebraic variety to G (or more generally to a represen- tation of G), we must restrict ourselves to considering nitely-generated, commu- tative k-algebras. If p = 2, notice that the notions of graded-commutativity and commutativity coincide, so that H (G;k) is already a commutative k-algebra. For p6= 2, if we consider the subalgebra of H (G;k) generated by homogeneous terms of even degree, we nd this subalgebra to be commutative. This brief discussion is summarized in the following denition. Denition 1.2.8. Let G be a nite group scheme. Dene H (G;k) := ( H ev (G;k) = L i0 H 2i (G;k) p6= 2 H (G;k) p = 2 Is H (G;k) nitely generated? For G associated to a nite group as in Example 1.1.14, the nite generation of H (G;k) was established in [16]. For G innitesimal of height 1, that is, for G such that k[G] = u(g) for some restricted Lie algebra g (see Example 1.1.25), nite generation of H (G;k) is shown in [20]. For a general nite group schemeG, nite generation ofH (G;k) is proved in [15], in which Friedlander and Suslin remark that the proof of nite generation in the general case of a nite group schemeG \has proved surprisingly elusive." We state the main theorem of [15] here. Theorem 1.2.9. (Friedlander-Suslin, 1997) Let G be a nite group scheme over k and let M be a nite dimensional rational G-module. Then H (G;k) is a nitely generated k-algebra and H (G;M) is a nite H (G;k)-module. Remark 1.2.10. A rational G-module is exactly what we've termed a representa- tion in Denition 1.2.1. The following example of a group cohomology computation will be important in our discussion of Quillen's stratication in Chapter 3. Example 1.2.11. Let r be a nonnegative integer, and let E = (Z=pZ) r be an elementary abelian p-group of rank r. Then H (E;k) = ( k[x 1 ;:::;x r ] [y 1 ;:::;y r ] p6= 2 k[x 1 ;:::;x r ] p = 2 where [y 1 ;:::;y r ] is the exterior algebra onr generators ([y 1 ;:::;y r ] is the quo- tient of the free k-algebra F in the non-commuting variables y 1 ;:::;y r by the two-sided ideal generated by elements of the form a 2 for all a2F ). For p6= 2, the grading is dened by deg(x i ) = deg(y i )+1 = 2, and forp = 2, the grading is dened by deg(x i ) = 1. In particular, notice that if r = 1, we have H (Z=pZ;k) = k[x] for allp. Similarly, notice that the maximal ideal spectrum of H (E;k) is an ane space of dimension equal to the rank of E. This will be important later when we describe a stratication of the cohomology of a nite group via the ane pieces com- ing from elementary abelian p-subgroups (see Proposition 3.1.2 and the discussion following). 28 1. COHOMOLOGY AND SUPPORT VARIETIES OF AFFINE GROUP SCHEMES To a nite group scheme G, we've thus successfully associated a nitely gener- ated, commutative k-algebra, H (G;k). This step is precisely what opens up the theory to techniques from algebraic geometry, as the maximal ideals of H (G;k) form the points of an algebraic variety, which we'll denote byjGj. Furthermore, to a representation M, we will associate a subvarietyjGj M jGj which encodes certain properties of M. We pause here to emphasize that the main results in this document are not directly related to the study of support varieties themselves, but instead to spaces likejGj within which support varieties live. 1.3. Support varieties Since we've just now discussed cohomology, we'll rst give the cohomological denition of a support variety. Recall that if G is a nite group scheme, and M is a representation of G, then the abelian group Ext kG (M;M) is given the structure of a graded module over the commutative k-algebra H (G;k) by tensoring as in Equation (1.2.1). Denition 1.3.1. (1) LetG be a nite group scheme. Denote byjGj the maximal ideal spectrum of the nitely generated, commutative k-algebra H (G;k). (2) Let M be a nitely generated representation of a nite group scheme G. DenejGj M to be the closed subset ofjGj dened by the ideal I M := Ann H (G;k) (Ext kG (M;M)). That is, letjGj M denote those maximal ideals m inH (G;k) such thatI M m. We calljGj M the cohomological support variety of M. We immediately remark that since Ext kG (M;M) is a graded module, the ideal I M is homogeneous, and thus the varietyjGj M is conical. From a geometric point of view, if we choose a surjection k[x 1 ;:::;x n ] H (G;k), we obtain an isomor- phism k[x 1 ;:::;x n ]=I = H (G;k). Via the Nullstullenzats, this choice provides an embedding of Specm(H (G;k)) into the ane space A n . Specically, we iden- tify a maximal ideal m = (x 1 a 1 ;:::;x n a n ) k[x 1 ;:::;x n ]=I with the point (a 1 ;:::;a n )2A n . Then, a subvariety ofjGjA n is conical if it is a union of lines through the origin, that is, if (a 1 ;:::;a n )2jGj, then (a 1 ;:::;a n )2jGj for every 2k. We describe the meaning of \conical" completely algebraically, free of any choice of embedding. Further details of the following discussion can be found in [6]. Fix 2 k and dene a map m : H (G;k)! H (G;k) which on a homogeneous element a of degree r takes the value r a. The corresponding map m :jGj!jGj is dened by m (m) = m 1 (m). We say that a subvariety V ofjGj is conical if m (m)2V for all m2V and all 2k. If an ane variety V is conical, we can projectivize it in the following way. Geometrically, embedding V inside of some ane space A n , we have k[V ] = k[x 1 ;:::;x n ]=I, where I is a homogeneous ideal (because V is conical). Thus k[V ] is graded, so we can consider the projective variety Proj(k[V ]). As a set, Proj(k[V ]) consists of all nontrivial homogeneous ideals that are maximal amongst such ideals. At times we will use the notation Proj(V ) to denote Proj(k[V ]). Geometrically, the projectivization of V is the variety inP n1 consisting of those points (lines) which vanish on I. 1.3. SUPPORT VARIETIES 29 Remark 1.3.2. To justify the terminology \support variety," we mention the fol- lowing denition and fact. If M is a module over a commutative ring R, then the support ofM, denoted Supp R (M), is the subset of Spec(R) consisting of those prime ideals p such thatM p 6= 0. The maximal ideal support Suppm R (M) is dened similarly, considering only maximal ideals. It can be shown that if M is nitely generated, then Suppm R (M) = Z(Ann R (M)), that is, a maximal ideal m R contains the annihilator of M if and only M m 6= 0. It follows that our denition ofjGj M is exactly the maximal ideal support of the module Ext (M;M) over the ring H (G;k). Before mentioning what support varieties tell us about representations (or vice versa), we now introduce another view via p-points (later to be generalized to - points). Much of the following is a summary of the main results of [21]. Iff :R!S is a map of rings, and M is an S-module, then we denote by f (M) the R-module dened by rm := f(r)m for any r2 R, m2 M. Furthermore, we say that f is left at if it equips S with the structure of a left, at module over R. Denition 1.3.3. Let G be a nite group scheme, with group algebra kG. (1) A p-point of G is a left at map of k-algebras : kZ=pZ! kG which factors through the group algebrakC of some abelian unipotent subgroup scheme CG. (2) Twop-points and are equivalent if for all nite dimensionalG-modules M, (M) is free if and only if (M) is free. LetP (G) denote the set of all equivalence classes of p-points of G. (3) If M is a nite-dimensional G-module, then let P (G) M P (G) be the set of equivalence classes of p-points such that for any representative , (M) is not free. The following proposition, found in [21], states that the setsP (G) M satisfy the axioms of the closed sets of a topological space. Proposition 1.3.4. (Friedlander-Pevtsova, 2005) Let G be a nite group scheme. Then the class of subsets fP (G) M P (G) :M is a nite dimensional G-moduleg is closed under nite unions and arbitrary intersections. It follows thatP (G) is a topological space, whose topology is dened via the rep- resentation theory of G. We now establish the relationship between P (G) andjGj. Notice that by the contravariance of the functor H (;k), a p-point :kZ=pZ! kG denes a map of commutative k-algebras : H (G;k)! H (Z=pZ;k). By Example 1.2.11, we know that H (Z=pZ;k) = k[x], so that : H (G;k)! k[x]. The next lemma shows that all possible nontrivial maximal homogeneous ideals of H (G;k) have the form ker . Lemma 1.3.5. If A = i0 A i is a commutative, graded, connected k-algebra, then the maximal nontrivial prime homogeneous ideals of A are in one-to-one cor- respondence with nontrivial, surjective graded maps A! k[x], where k[x] is given a grading by dening the degree of x to be the smallest positive i such that A i 6= 0. Proof. By a nontrivial prime homogeneous ideal we mean a prime homoge- neous ideal properly contained in the trivial homogeneous ideal A + := i1 A i . 30 1. COHOMOLOGY AND SUPPORT VARIETIES OF AFFINE GROUP SCHEMES The lemma is concerned with such ideals that are contained in no other. We em- phasize here that such ideals are not maximal among the set of all ideals as they are properly contained in the trivial ideal. Notice that all homogeneous elements in k[x] have degree divisible by deg(x). Suppose that ' : A! k[x] is a nontrivial, surjective graded map. To ' we associate the homogeneous ideal ker'. We now show that ker' is maximal among nontrivial prime homogeneous ideals. First, since ' is nontrivial, it follows that ker'6=A + . Also, since A is connected (i.e., A 0 =k), we must have ker'(A + . Now, suppose that m is a homogeneous ideal properly containing ker'. We aim to show that m =A + . Notice that since m properly contains ker', it must contain some homogeneous elementm such that'(m) =x n . Now, leta be a homogeneous element in A + . If '(a) = 0, then a2 ker' m. If '(a)6= 0, then we must have '(a) = x d for some 06= 2 k and some positive integer d. We thus have '(( 1 a) n m d ) = 0. It follows that ( 1 a) n m d 2 ker' so that 1 a2 m (here we have used thathm; ker'i2 m and that m is prime). It follows that A + m, so indeed A + = m. Conversely, suppose that I is a maximal nontrivial prime homogeneous ideal, and consider the quotient A=I. I claim that A=I = k[x]. Since I is not equal to the trivial ideal, there is some homogeneous element of minimal degree that is not contained in I. Such an element must be unique up to scalar multiplication. The map ofk-algebras sendingx ink[x] to the coset of this element inA=I can be shown to be an isomorphism. We'd now like to dene a map G :P (G)! ProjjGj by G ([]) = ker . In order for this map to be well-dened, we must show that ker is not trivial for any p-point , and that if two p-points and are equivalent, then ker = ker . All of this was shown in [21], and we restate the results here. Lemma 1.3.6. (Friedlander-Pevtsova, 2005) (1) Let G be a nite group scheme and : kZ=p! kG be a p-point of G. Then :H (G;k)!H (Z=p;k) is nontrivial and surjective. (2) LetG be a nite group scheme. Then sending ap-point :kZ=p!kG to kerf :H (G;k)!H (Z=p;k)g determines a well dened map of sets G :P (G)! ProjjGj; 7! kerf g We've now setup the necessary machinery to present the main theorem of [21]. Theorem 1.3.7. (Friedlander-Pevtsova, 2005) Let G be a nite group scheme. Then G :P (G) ! ProjjGj is a homeomorphism satisfying the property that 1 G (ProjjGj M ) =P (G) M for every nitely generated G-module M. In summary, the theory of p-points of a nite group scheme gives a repre- sentation theoretic description of the topological spacejGj and the cohomological support varietiesjGj M . This result provides a generalization of representation the- oretic descriptions of support varieties associated to elementary abelian p-groups and innitesimal group schemes. For the sake of completeness, and to place the 1.3. SUPPORT VARIETIES 31 work of [21] in its proper context, we now brie y discuss rank varieties associ- ated to elementary abelian subgroups and innitesimal one-parameter subgroups of innitesimal group schemes. Rank varieties. Let E be an elementary abelian p-group of rank r, and let M be a nite dimensional representation of E. To the module M we already have a denition ofjG E j M via cohomology. Notice that by the Nullstullensatz and Example 1.2.11, the constant group scheme G E associated to the nite group E satisesjG E j =A r , so thatjG E j M is a subvariety ofA r . The following construction due to Jon Carlson (see [12]) can be seen to be a special case of the theory of p- points we've just described. Here we follow the discussion found in [3]. Choose generatorsg 1 ;:::;g r forE, and consider the elementst i =g i 1 inkE. Thent p i = 0 fori = 1;:::;r, and thet i generatekE as ak-algebra. Notice that the linear span of thet i is an ane space of dimensionr inside ofkE; let's denote this ane space by V . V will play the role ofjG E j in the following description of the rank variety (that is, the rank variety associate to a moduleM will be a subvariety of V ). Let 2 V be any point, and notice that p = 0 so that (1 +) p = 1. Let h1+i denote the subalgebra ofkE generated by 1+. This algebra is isomorphic to kZ=pZ, and is called a cyclic shifted subgroup ofE. It is technically not a subgroup of E, but there is an algebra automorphism of kE which maps the p-elements (1 +) i for i = 0;:::;p 1 to a cyclic subgroup of E. Now, given any nite dimensional E-module M, we dene V r M V to be the union of 0 with the set of all 2 V such that the module M restricted to the subalgebrah1 +i is not free. In [12], Carlson shows that V r M is a variety (called the rank variety ofM), and conjectures thatV r M =jG E j M . He makes some progress towards a proof of his theorem, which is fully established in [3], and is reproduced below. Theorem 1.3.8. (Carlson's Conjecture, Avrunin-Scott, 1982) Let E be an ele- mentary abelian p-group of rank r, and let M be a nite-dimensional module of E. Then Carlson's rank variety V r M is isomorphic to the usual cohomological support varietyjG E j M . If the reader thinks long enough, and traces through the denitions of p-points in the case that G =G E (and reads [21]), she will nd that the theory of p-points specializes to that of rank varieties. Innitesimal one-parameter subgroups. We'll discuss innitesimal one- parameter subgroups of innitesimal group schemes further in Chapter 4, but for now we give the denitions and state the main results, again with the motivation to see that P (G) is a generalization of the theory of innitesimal-one parameter subgroups. The denitions and results stated here appear in [51] and [52]. Let G be an innitesimal group of height r, and let G a(r) denote the rth Frobenius kernel ofG a . Recall that from Example 1.1.22 thatk[G a(r) ] =k[T ]=(T p r ) and kG a(r) =k[u 0 ;:::;u r1 ]=(u p 0 ;:::;u p r1 ) where u i reads o the coecient of T p i . Denition 1.3.9. Let G be an innitesimal group scheme of height r, and let M be a nite dimensional G-module. 32 1. COHOMOLOGY AND SUPPORT VARIETIES OF AFFINE GROUP SCHEMES (1) A(n) (innitesimal) one-parameter subgroup ofG is a morphism of group schemesG a(r) !G. (2) Dene a functor V r (G) which to a commutative k-algebra R associates the set of all morphisms of group schemes over R,G a(r);R !G A . (3) Let V r (G) M be the set of all one-parameter subgroups : G a(r) ! G such that the restriction of M to the subalgebra k[u r1 ]=(u r1 ) p along the maps k[u r1 ]=(u r1 ) p ,!kG a(r) !kG is not free. Based on our discussion of p-points and rank varieties, the following results of [51] and [52] are expected. Theorem 1.3.10. (Suslin-Friedlander-Bendel, 1997) Let G be an innitesimal group scheme of heightr. (1) The functor V r (G) is representable, and the k-points of V r (G) are the one-parameter subgroups of G. (2) Ifk[V r (G)] is thek-algebra representingV r (G), then there is a homeomor- phism G : Specm(k[V r (G)])!jGj. (3) If M is a nite dimensional G-module, then 1 G (jGj M ) =V r (G) M . In summary, for any nite group scheme G and nite dimensional G-module M, we've introduced the varietiesjGj andjGj M via cohomological considerations. We then took a representation theoretic approach via the theory of p-points to discuss the projective varieties P (G) and P (G) M . Due to [21], we found that our two considerations were in fact the same. We then showed how the theory of p- points was a generalization of two previous representation theoretic descriptions of cohomological support varieties: those for constant group schemes associated to elementary abelian p-groups and innitesimal group schemes. To motivate why a representation theorist might care about the geometric invariantjGj M associated to a representation, we present some properties of support varieties below (although we won't use these throughout the rest of the document). If M is a nite-dimensional representation of G, the complexity of M, denoted cx(M), is the smallest nonnegative integer c such that there exists a constant with dim k P n n c1 for alln, whereP !M is the minimal projective resolution of M. Proposition 1.3.11. Let G be a nite group scheme and let M and N be nite dimensional G-modules. The support variety P (G) M satises the following proper- ties: (1) If k is viewed as the trivial G-module, then P (G) k =P (G). (2) P (G) MN =P (G) M [P (G) N . (3) P (G) M N =P (G) M \P (G) N . (4) P (G) M =? is and only if M is projective. (5) dimP (G) M = cx(M) 1. -points. We end this section with a discussion of a generalization of the theory ofp-points. In [18], the theory ofp-points has been \improved" to the theory of -points. We don't review the details of -points, but we roughly describe the advantage to this new point of view. 1.3. SUPPORT VARIETIES 33 First of all, thus far in all of our considerations we've only considered maximal ideal spectra. In other words, all of the varieties we've discussed (jGj,jGj M ,P (G), P (G) M , V , V r M , V r (G), and V r (G) M ) are the maximal ideal spectra of certain nitely generated commutativek-algebras (in the case ofP (G) we've used the Proj construction). It follows that these varieties are actually just the closed points of their underlying schemes, which are given by the prime ideal spectra. The theory of -points recovers the prime ideals. Also, we've thus far only considered support varieties for nite-dimensionalG-modulesM. The theory of-points will allow for the study of arbitrary (possibly innite dimensional) modules. We give the following denition formally, and then discuss the theory of - points informally in the following paragraph. For further details, one may consult [18]. IfG is a nite group scheme overk, andK=k is an extension of elds, then we denote byKG the group algebra of the group schemeG K obtained by base change. Specically, KG =kG K. Denition 1.3.12. (Friedlander-Pevtsova, 2007) Let G be a nite group scheme over k, and let K=k be an extension of elds. A -point of G dened over K is a left at map of K-algebras K : K[t]=t p ! KG which factors through the group algebra KC K KG K = KG of some unipotent abelian subgroup scheme C K of G K . This denition is clearly reminiscent of Denition 1.3.3. In a similar manner to how we proceeded with p-points, we may place an equivalence relation on the set of -points to form the set (G). This equivalence relation is dened by the restriction of nite dimensional G-modules M along -points. To a nite dimen- sional moduleM we associate a subset (G) M (G), and it turns out that such sets are the closed subsets of a topology on (G) which make it homeomorphic to Proj(H (G;k)), where here we consider all homogeneous prime ideals properly contained in the irrelevant ideal. We thus may refer to (G) M as the \support scheme" of M. This homeomorphism also preserves the support as expected, that is, the image of (G) M under this homeomorphism is equal to the set of those homogeneous prime ideals properly contained in the irrelevant ideal which contain the annihilator of the H (G;k)-module Ext kG (M;M). One may further consider arbitrary (possibly innite dimensional) G-modules and dene their support (G) M . It turns out that every subset of (G) is re- alized as the support of some G-module, so if one used arbitrary G-modules to dene a topology on (G), one would obtain the discrete topology. Even when M is innite dimensional, it turns out that we have the following generalization of Proposition 1.3.11. Proposition 1.3.13. LetG be a nite group scheme and letM andN be arbitrary G-modules. The scheme (G) M satises the following properties: (1) If k is viewed as the trivial G-module, then (G) k = (G). (2) (G) MN = (G) M [ (G) N . (3) (G) M N = (G) M \ (G) N . (4) (G) M =? is and only if M is projective. Figure 6 summarizes our discussion of representation theoretic descriptions of cohomological support varieties for modules of nite group schemes. The arrows point in directions of increasing levels of particularity. 34 1. COHOMOLOGY AND SUPPORT VARIETIES OF AFFINE GROUP SCHEMES This gure sets the framework and motivation for the objects of study in Chap- ter 3 and Chapter 4. Notice that even in the section labeled "1-parameter sub- groups," we could further consider the case of height 1 innitesimal subgroups. As mentioned in Example 1.1.25, the category of height 1 innitesimal group schemes is equivalent to the category of restricted Lie algebras. In this case, it can be shown that V 1 (G) is isomorphic to the \restricted nullcone" of Lie(G), an object to be studied in Chapter 2. -points (G) M (G) = = Z(I M ) Proj(H (G;k)) - G any nite group scheme - Recovers prime ideal spectrum - Invariants for innite-dimensional modules p-points P (G) M P (G) = = ProjjGj M ProjjGj - G any nite group scheme - Recovers only maximal ideal spectrum - Modules must be nite-dimensional 1-parameter subgroups V r (G) M V r (G) = = jGj M jGj - G innitesimal of heightr Rank varieties V E (M) V E = = jG E j M jG E j - E elementary abelian p-group Figure 6. A graphical summary of representation theoretic de- scriptions of cohomological support varieties. Arrows point in de- creasing levels of generality. CHAPTER 2 Springer isomorphisms In this section we develop the existence of a Springer isomorphism, which will be our main tool in translating information between groups and Lie algebras in Chapter 3. 2.1. Algebraic groups We've already discussed algebraic groups brie y in Chapter 1. Recall that an algebraic group G is a group scheme whose coordinate algebra k[G] is nitely generated and reduced. We emphasize that we're only considering ane algebraic groups, so that elliptic curves and abelian varieties are beyond the scope of this document. Since k is algebraically closed, we've shown in Example 1.1.15 that an algebraic group G can be recovered by the data of its k-points, G(k). For this section, when we write G, we will mean G(k). Further, Theorem 1.1.17 states that any algebraic group embeds into the group scheme GL n for some n, so we can keep in mind matrices and matrix multiplication as we discuss the algebraic group G and its Lie algebra g (which in turn is a sub-Lie algebra of gl n ). Finally, we'll view k[G] as \functions on G" in the following manner: given any g2 G = G(k) := Hom k-alg (k[G];k) and any f 2 k[G], we (admittedly confusingly) dene f(g) :=g(f)2k. We now discuss some properties of algebraic groups, leading up to the denition and structure of a reductive algebraic group. None of the statements made here are justied, but one may consult [8], [26], and [34] for further details. First we recall a few facts from linear algebra. IfV is a nite dimensional vector space over a eld, then an endomorphism of V is called semisimple if its minimal polynomial has no repeated roots. An endomorphism is called unipotent if its only eigenvalue is 1 and nilpotent if its only eigenvalue is 0. If V is innite dimensional, then we say that an endomorphism is semisimple (unipotent, nilpotent) if V is the union of nite-dimensional subspaces stable under and the restriction of to all such subspaces is semisimple (unipotent, nilpotent). G acts on k[G] via right translation as follows. If g;h2G and f2k[G], then let ( g f)(h) = f(hg). The map : G! GL(k[G]) is a group homomorphism. Giveng2G there exist unique commuting elementsg s ;g u 2G such thatg =g s g u , gs is semisimple, and gu is unipotent. g s is called the semisimple part of g and g u is called the unipotent part of g. If g =g s , then g is semisimple, and if g =g u , then g is unipotent. The identity e2G is contained in a unique connected component of G, which we'll denote G . If G = G , then G is called connected. Let G m (k) = GL 1 (k) be the group of invertible elements of k under multiplication. We denote by X(G) the set of all homomorphisms of algebraic groups G!G m and we give X(G) the 35 36 2. SPRINGER ISOMORPHISMS structure of a group under pointwise multiplication. X(G) is called the group of characters of G. Any algebraic group G possesses a largest, connected, normal, solvable sub- group called the radical ofG, denotedR(G). The unipotent elements ofR(G) form a normal subgroup of G denoted R u (G). We say that G is semisimple if R(G) is trivial, and we say thatG is reductive ifR u (G) is trivial. We mention the following result which will be useful later. Theorem 2.1.1. Let G be a reductive algebraic group. (1) The derived subgroup (G;G) is semisimple. (2) G is the product of its center and derived subgroup, the intersection being nite: G =Z(G) (G;G),jZ(G)\ (G;G)j<1. An algebraic group is called almost simple if it has no nontrivial, proper, closed, connected, normal subgroups. If G 1 ;:::;G n are algebraic groups which are sub- groups of G, and the product map G 1 :::G n ! G is surjective with nite kernel, thenG is said to be an almost direct product of theG i . A surjective map of algebraic groupsf :H!G with nite kernel is called an isogeny. Here is a helpful result involving semisimple groups, the proof of which may be found inx27:5 of [26]. Theorem 2.1.2. Let G be semisimple, and let fG i ;i 2 Ig be the minimal closed connected normal subgroups of positive dimension. Then I is nite and G is an almost direct product of the G i . We denote by D n (k) the group of invertible n n diagonal matrices with entries in k. Any algebraic group isomorphic to D n (k) is called a torus. A torus T contained in an algebraic group G is called maximal if it is contained in no other torus. A maximal connected solvable subgroup of G is called a Borel subgroup. It turns out that all Borel subgroups of G are conjugate, and that the maximal tori of an algebraic group are exactly those maximal tori of its Borel subgroups. It is also true that all maximal tori in an algebraic group G are conjugate, so they all have the same dimension as an algebraic variety. IfT is a maximal torus ofG, then we dene the rank of G to be the dimension of T . A closed subgroup containing a Borel subgroup is called a parabolic subgroup. For a torus T (or more generally for any closed subgroup of a torus), we can dene the group of one-parameter subgroups ofT as the abelian group under point- wise multiplication of all morphismsG m !T . We denote this groupY (T ). Notice thatY (T ) is dual to the character groupX(T ), dened above. In fact, if2Y (T ) and 2 X(T ), then the composition is a map G m !G m . It can be shown that all such maps are given by x7! x n for some integer n, so that X(G m ) =Z. We dene a pairing X(T )Y (T )!Z byh;i = n where (x) = x n . More generally, if T is a torus of rank n, then X(T ) is free abelian of rank n. Further, we say that a torusTG (not necessarily maximal) is regular if it is contained in only nitely many Borel subgroups, and we say that a one-parameter subgroup of T is regular if the torus (G m )T is regular. The lower central series of a groupG is dened inductively by 0 (G) =G and i+1 (G) = (G; i (G)). We thus have a decreasing chain of subgroups: G = 0 (G) 1 (G) 2 (G)::: 2.1. ALGEBRAIC GROUPS 37 If there is some n such that n (G) = e, then G is called nilpotent. If n is the smallest such integer such that n (G) =e, then we dene the nilpotence class ofG to be equal to n. Denition 2.1.3. Let G(= G(k)) be an algebraic group, and let g be the Lie algebra of G. (1) An element g2G is unipotent if g p t =e for some t2N. Here e denotes the identity in the group G. If t can be taken to be 1, then g is called p-unipotent. (2) Denote byU(G) the set of all unipotent elements ofG, andU p (G)U(G) the subset of all p-unipotent elements. (3) An element x2 g is nilpotent if x [p] t = 0 for some t2 N. Here x [p] t denotes the tth iterate of the map () [p] : g! g. If t can be taken to be 1, then x is called p-nilpotent. (4) Denote byN (g) the set of all nilpotent elements of g, andN p (g)U(G) the subset of all p-nilpotent elements. N (g) is called the nullcone of g andN p (g) is called the restricted nullcone or p-nullcone. Remark 2.1.4. We note that ifG is reductive, then any unipotent element g2G is contained in (G;G). Similarly, any nilpotent element x2 g is contained in [g;g]. It follows thatU(G) = U((G;G)), where as noted in Theorem 2.1.1, (G;G) is semisimple. In Section 2.2 we will discuss U(G) andN (g) in further detail, and there we will assume thatG is semisimple. We lose no generality in this assumption as a result of this remark. ViewingG as a closed subgroup of GL n , we see that the condition thatg2U(G) is a polynomial condition on the entries of g. It follows thatU(G) is an ane algebraic variety, withU p (G) a subvariety. Similarly, since the restriction map in gl n is given by the pth-power of a matrix, the condition that x2 g is nilpotent is also given by polynomial equations on the entries ofx (viewed as a matrix). Hence N (g) is an ane variety withN p (g) a subvariety. We note here for later purposes that if we've embedded G ,! GL n (k), and if p n, then an element g2 G is p-unipotent if and only if it is unipotent (a similar statement holds for nilpotent elements in g). The moral is that for large enough characteristic, the notions of p-unipotence and p-nilpotence coincide with the notions of unipotence and nilpotence, respectively. U(G) andN (g) have a bit more structure than just that of an algebraic variety. To see this we now discuss the adjoint action. Denition 2.1.5. Let G be an algebraic group, with Lie algebra g. (1) For every g2 G, dene a group isomorphism Ad g : G! G which maps h!g 1 hg. The map Ad :G! Aut(G) is called the adjoint action of G on itself. (2) For every g2 G, consider the dierential d(Ad g ) : g! g. This gives a map G! Aut(g) which is called the adjoint action of G on g. If the context is clear, we will at times use the notation Ad g to denote both the adjoint of action of G on itself and on its Lie algebra. We note here that for G = GL n (k), the adjoint action of G on g = gl n (k) is given by conjugation. Notice that conjugation by g is a polynomial map on the entries of g and (det(g)) 1 . Furthermore, conjugation preserves the properties of 38 2. SPRINGER ISOMORPHISMS (p-)unipotence and (p-)nilpotence. It follows that the adjoint actions of G restrict to automorphisms of the varietiesU(G) andN (g). We thus have givenU(G) and N (g) the structures of G-varieties. Fix a maximal torus TG and letT act on g under the adjoint action. Since AdT Autg is diagonalizable, we can write g as a direct sum of eigenspaces. To do this, for every 2X(T ) dene the subspace g :=fx2 gj Ad t (x) =(t)x for all t2Tg Notice that if 12 X(T ) is the trivial character of G which maps all elements to 12G m (k) = k , then g 1 is equal to all x2 g such that Ad t (x) = x for all t2T . We can then write: g = g 1 a 2 g where = (G;T ) X(T ) is the set of those nontrivial characters for which g 6= 0. If we view inside of the real vector space X(T ) Z R (recall that X(T ) is free abelian of rank equal to the rank of G), then satises the axioms of an abstract root system (see [26], Appendix). If2Y (T ) is a one-parameter subgroup, it can be shown that is regular if and only ifh;i6= 0 for all 2 . Given a regular one-parameter subgroup , we make the following denitions of subsets of : + () :=f2 jh;i> 0g () :=f2 + ()j is not the sum of two elements of + ()g With these denitions, it can be shown that () is a basis of . In other words, () =f 1 ;:::; n g wheren = rank(G) and all elements of are uniquely express- ible as P c i i where the c i all have the same sign. + () are precisely those roots where the c i are positive, and are called the positive roots. For each i 2 (), there is a unique linear transformation i 2 Aut(X(T ) Z R) which xes a codi- mension 1 subspace, maps i to i , and maps into itself. We call the element i the simple re ection associated to the root i . The group generated inside of Aut(X(T ) Z R) by all of the simple re ections for i = 1;:::;n is called the Weyl group of G, denoted W =W (G;T;), and is independent of choice of T and . It can be shown that W is nite. A coxeter element of W is a product of the n simple re ections in any order. All coxeter elements of W are conjugate, and thus have the same order. We call this order the coxeter number of G. There is an inner product on X(T ) Z R relative to which W consists of or- thogonal transformations. We denote this inner product by (;). The set of all 2X(T ) Z R such that the expression 2(;) (;) is an integer for all 2 is called an abstract weight of . The abstract weights form a lattice X(T ) Z R which contains the lattice spanned by , denoted r . Notice that we have r X(T ) . In the following denition, we discuss two dierent notions of the fundamental group, which should be kept distinct in the reader's mind despite similar terminology. 2.1. ALGEBRAIC GROUPS 39 Denition 2.1.6. Let G be a reductive group, T a maximal torus, = (G;T ) the root system of G relative to T . Let be the lattice of all abstract weights, X(T ) be the character group of T , and r be the lattice spanned by the roots . (1) The fundamental group of , denoted (), is the quotient = r . (2) The fundamental group of G, denoted (G), is the quotient =X(T ). (3) G is simply connected if (G) =e. (4) G is of adjoint type if (G) =(), that is, if X(T ) = r . We brie y mention that if is a root system, then there exist semisimple algebraic groups G sc andG ad which have root system and are simply connected and of adjoint type, respectively. In fact, these groups are unique, due to the discussion that follows. The notion of the fundamental group of an algebraic group helps classify all almost-simple algebraic groups. If G is almost simple, then its root system is irreducible, that is, is not the disjoint union of two orthogonal subsets. Due to the rigid conditions dening a root system, the irreducible root systems have been completely classied. There are those in the four \classical families" of types A n , B n , C n , and D n . Furthermore, there are the ve \exceptional" root systems G 2 , F 4 , E 6 , E 7 , and E 8 . For a discussion of this classication, see Lecture 21 of [22]. However, an almost simple group is not completely determined by its irreducible root system. For example, the algebraic groups SL n and SL n =(Z(SL n )) =: PSL n are not isomorphic, but they have the same root system: A n1 . We therefore need another invariant to help classify the almost simple algebraic groups. This invariant is provided by the fundamental group of G. The next theorem makes this precise. Theorem 2.1.7. (1) If G and G 0 are almost simple algebraic groups with isomorphic root sys- tems and isomorphic fundamental groups, then G =G 0 . (2) Let be a root system, and let G be an almost-simple group with root system . Then there are isogenies G sc 1 !G 2 !G ad where G sc and G ad are the simply connected and adjoint forms for the root system , respectively. (3) The covering map 1 is separable if and only if the dierential d 1 : Lie(G sc )! Lie(G) is surjective if and only if p does not divide the order of (G). If is an irreducible root system with a chosen basisf 1 ;:::; n g, then there is a unique root 2 with the property that + i is not a root for alli = 1;:::;n. We call this root the highest root. If = P c i i , and p does not divide c i for all i, then p is said to be a good prime for . Otherwise, p is said to be a bad prime. The bad primes are easily classied in all cases. Irreducible root systems of type A have no bad primes. 2 is a bad prime for all types except A. 3 is a bad prime for the exceptional root systems, and 5 is a bad prime for E 8 . All other primes are good. If G is a reductive group, with root system , then we say p is good for G if it is good for all irreducible components of . Finally, if p is good and does not divide the order of the fundamental group of , then we say p is very good. Notice that in all cases except for groups with a component of type A, a prime is good if and only if it is very good. For groups with a component of type A, a prime is 40 2. SPRINGER ISOMORPHISMS very good if and only if it is good and the rank of any component of type A is not divisible by p. 2.2. Existence of Springer isomorphisms In preparation for the development of the history of the Springer isomorphism, in this section we'll discuss the structure of theG-varietiesN (g) andU(G). Helpful references are [14], [27], and [30]. We will assume that G is semisimple (see Remark 2.1.4). Let g := Lie(G). We begin with the unipotent variety,U(G). It can be shown that the unipotent variety is irreducible of dimension dim(G) rank(G) (seex4.2 of [27]). This will be of note later when we discussN (g). We would also like to show that inside of U(G) is a large, open (so necessarily dense) orbit. This orbit will consist of all those unipotent elements that are regular, a notion we now dene. Denition 2.2.1. Let G be a semisimple algebraic group. An element g2 G is called regular if the dimension of the centralizer of g is equal to the rank of G: dimC G (g) = rank(G). Let G reg denote the set of all regular elements of G. It can be shown that G reg is nonempty and open in G, and therefore dense if G is connected. It follows that the regular unipotent elements, denotedU(G) reg are open in the unipotent varietyU(G) asU(G) reg =U(G)\G reg . However, it is not entirely clear thatU(G) reg is nonempty. The existence of regular unipotent elements in G is a nontrivial result of Steinberg in [49]. It follows thatU(G) reg is nonempty, open and dense in the irreducible varietyU(G). In fact, all regular unipotent elements are conjugate, so thatU(G) reg is a single orbit in theG-variety U(G). Furthermore, it can be shown thatU(G) is a normal variety (seex4.14 and x4.24 of [27]). We now turn to the nilpotent varietyN (g) which we will henceforth refer to as the nullcone. To highlight their similarities, we develop our understanding of the nullcone in a manner re ecting the discussion just completed concerning the unipotent varietyU(G). First of all, x a Borel subgroup B of G and let U be the unipotent radical of B. If u is the Lie algebra ofU, then it can be shown that dim(N (g)) = 2 dim(u) = dim(G) rank(G). As before,N (g) is irreducible, and so we hope to locate a nonempty, open (and therefore dense) orbit of regular nilpotent elements. In light of Denition 2.2.1, the following denition is not surprising. Denition 2.2.2. If x2 g, then the centralizer of x in G, denoted C G (x), is the subgroup of all elements g2G such that Ad g (x) =x. An element x2 g is regular if dimC G (x) = rank(G). It should also not be surprising that regular nilpotent elements exist, that all such elements are conjugate, and that they form an open and therefore dense orbit of the nullcone. The normality of the nullcone is proved in [4] (see Remark 9.3.6(c)). This discussion has been directed in such a way as to notice the similarities between the G-varietiesU(G) andN (g). They both have nitely many G-orbits, they are both irreducible of the same dimension, and each has a unique open orbit consisting of regular elements. Furthermore, each is a normal variety. In this light, the following result might be expected. 2.3. IN SEARCH OF A CANONICAL SPRINGER ISOMORPHISM 41 Theorem 2.2.3. (Springer, 1969) Let G be a simple algebraic group, and as- sume that p is very good for G. Then there exists a G-equivariant isomorphism :N (g)!U(G). Remark 2.2.4. Springer's original proof appears in [48], but an overview is given in [9]. For a discussion of how bad primes complicate matters, consult [32]. Inx6.20 of [27], Humphreys notes that Springer's original statement was weaker than that provided here. Springer assumed G was simply connected, and only showed to be a homeomorphism. However, since both varieties are normal, it follows that is an isomorphism. The full generality of Theorem 2.2.3 is proven in [4]. Any G-equivariant isomorphism :N (g)!U(G) will be called a Springer isomorphism. Although Springer showed that G-equivariant isomorphisms N (g) ! U(G) exist, a note of Serre in ([35],x10) mentions that in general they are not unique. In fact, they are parametrized by a variety of dimension equal to rank(G). Example 2.2.5 ([46],x3). Let G = SL n . Then the Springer isomorphisms are parameterized by the variety a 1 6= 0 in A n1 by (a1;:::;an1) (X) = 1 +a 1 X + ::: +a n1 X n1 whereX n = 0. It follows that dierent Springer isomorphisms can behave very dierently. Here, since ph(SL n ) =n, we have the particularly nice choice of Springer isomorphism (X) = 1 +X + X 2 2! +::: + X p1 (p 1)! This is just the truncated exponential series, which we will denote exp. The note of Serre in [35] does prove that although there are many Springer isomorphisms, they all induce the same bijection on classes. That is, if and 0 are two distinct Springer isomorphisms, and if C is a conjugacy class of nilpotent ele- ments in g, then(C) = 0 (C). Thus, although one may not speak of the Springer isomorphism, one can speak of the Springer correspondence between unipotent con- jugacy classes in G and nilpotent conjugacy classes in g. In the next section we discuss how one might require further conditions of Springer isomorphisms in order to nd a canonical such map satisfying the new conditions. 2.3. In search of a canonical Springer isomorphism In this section, as always let k be an algebraically closed eld of characteristic p > 0, and let G be a connected, reductive algebraic group over k, with coxeter number h = h(G). Followingx2 in [50], we let = (G) denote the fundamental group of G 0 = (G;G). We will often require that p satises the following two conditions, which will be collectively referred to as condition (?): (?) (1)ph (2)p-jj We make three remarks about condition (?). First, (1) implies (2) in all cases except when p =h and G 0 has an adjoint component of type A. Second, (2) is equivalent to the separability of the universal cover G 0 sc ! G 0 ([50],x2.4). For example, the canonical map SL p ! PSL p is not separable in characteristicp, so we must exclude the case G = PSL p . Third, (?) implies that p is non-torsion for G (cf.x2 in [39]), which we require to use Theorem 2.2 of [39] in our proof of Theorem 2.3.2. 42 2. SPRINGER ISOMORPHISMS For our purposes in this document, it is not enough that we locate some Springer isomorphism between the nilpotent variety and the unipotent variety. Notice that if X;Y 2N (g) and X and Y commute, then it is also the case that X +Y 2N (g). Similarly, notice that if g;h2U(G), and g;h commute, then gh2U(G). We would like for our Springer isomorphism to be compatible with this extra structure, that is, we would like (X +Y ) =(X)(Y ) if [X;Y ] = 0. Notice the following example shows this is not true in general. Example 2.3.1. Letp = 2 and consider the restricted Lie algebra gl 4 . Let be the truncated exponential as dened in Example 2.2.5; in other words (X) = 1 +X for any matrix X2 gl 4 . The simplicity of our Springer isomorphism is a result of how small p is. Dene two matrices as follows: X = 0 B B @ 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 C C A and Y = 0 B B @ 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 C C A Then a computation shows that [X;Y ] =X 2 =Y 2 = (X +Y ) 2 = 0, but (X +Y ) = 0 B B @ 1 1 1 0 0 1 0 1 0 0 1 1 0 0 0 1 1 C C A 6= 0 B B @ 1 1 1 1 0 1 0 1 0 0 1 1 0 0 0 1 1 C C A =(X)(Y ) Examples such as Example 2.3.1 disappear when p is large enough. The trun- cated exponential series considered in Example 2.2.5 has the following convenient property for pn: (2.3.1) [X;Y ] = 0 =) exp(X +Y ) = exp(X) exp(Y ) We give a brief proof of (2.3.1). Serre records in ([44], (4.1.7)) that exp(X) exp(Y ) = exp(X +YW p (X;Y )) for two commuting elements X;Y and arbitrary p, where W p (X;Y ) = 1 p ((X +Y ) p X p Y p ) Forpn, we haveX p =Y p = (X +Y ) p = 0 so thatW p (X;Y ) = 0 and we recover (2.3.1). We now turn to an arbitrary connected, reductive group G to nd a canonical Springer isomorphism which satises (X +Y ) = (X)(Y ) for commuting ele- ments X;Y 2 Lie(G). Proposition 5:3 in [43] states for any parabolic subgroup P inG whose unipotent radicalU P has nilpotence class less than p, there is a unique P -equivariant isomorphism " P : Lie(U P ) = u P !U P satisfying the following con- ditions. (1) " P is an isomorphism of algebraic groups, where u P has the structure of an algebraic group via the Baker-Campbell-Hausdor formula (notice the condition on the nilpotence class of u P is required for this group law to make sense). (2) The dierential of " P is the identity on u P . 2.3. IN SEARCH OF A CANONICAL SPRINGER ISOMORPHISM 43 In Theorem 3 of [11], the authors uniquely extend this isomorphism on u P to all ofN (g) for G simple, with weaker conditions on p than we consider in this paper. Specically, they require that p is good, thatN (g) is normal, and that G sc !G is separable. Condition (1) of (?) implies thatp is good and thatN (g) is normal, and we've already noted that condition (2) is equivalent to the separability ofG sc !G. We now show that under our assumptions onp, the result of Theorem 3 in [11] may be extended to reductive groups, and the canonical isomorphism obtained sends sums to products much like the truncated exponential (cf (2.3.1)). Theorem 2.3.2. ForG a connected, reductive algebraic group, and forp satisfy- ing condition (?), there is a (necessarily) unique Springer isomorphism :N (g)! U(G) which restricts to the canonical isomorphism of [43] on all u P for P any parabolic subgroup of G. This Springer isomorphism has the following properties: (1) [X;Y ] = 0 if and only if ((X);(Y )) =e. (2) If G is dened over F p , then so is . (3) If [X;Y ] = 0, then (X +Y ) =(X)(Y ). Proof. First, we note thatU(G)G 0 = (G;G) andN (g) [g;g] Lie(G 0 ), so that an isomorphism of the nilpotent and unipotent varieties of G 0 is also one for G. Furthermore, since G is the product of Z(G) and G 0 , a G 0 -equivariant map N (g)!U(G) is also G-equivariant map. Hence, we may assume that G = G 0 is semisimple. If G is semisimple, there is an isogeny H!G where H is a product of simple groups. By our assumptions on p, H! G is separable, so that the induced map on Lie algebras is an isomorphism. Hence we may assume that G is a product of simple groups. For simple groups G (and hence products of simple groups) Theorem 3 of [11] states that there is a unique Springer isomorphism with properties 1 and 2 which restricts to the canonical isomorphism of [43] for all parabolic subgroups P whose unipotent radical has nilpotence class less than p. Sinceph, all parabolic subgroups satisfy this criteria. Finally, to see that has property 3, supposeX andY are commuting nilpotent elements. Theorem 2:2 in [39] states that there is some Borel subgroupBG with unipotent radical U such that X;Y 2 Lie(U): Since restricts to the canonical isomorphism on u B , it follows that (XY ) = (X)(Y ), where is the group operation dened by the Baker-Campbell-Hausdor formula, which for commuting elements satises XY =X +Y . Property 3 follows. In the next section, we will use the canonical Springer isomorphism to study the projective varietyE(r;g), as dened in [13]. We mention here work of Sobaje in [46] which proves the existence of the canonical Springer isomorphism for weaker conditions onp than we consider here. Specically, Sobaje shows that forG simple and for p very good for G, there is a unique G-equivariant isomorphism which satises (1) and (2) of 2.3.2 among other conditions. In what follows, we will make use of (3), so we require that p satisfy condition (?). CHAPTER 3 The variety of elementary subalgebras In [13], J. Carlson, E. Friedlander, and J. Pevtsova initiated the study ofE(r;g), the projective variety of rank r elementary subalgebras of a restricted lie algebra g. The authors demonstrate that the study of E(r;g) informs the representation theory and cohomology of g. This is all reminiscent of the case of a nite group G, where the elementary abelian p-subgroups play a signicant role in the story of the representation theory and cohomology ofG, as rst explored by Quillen in [41] and [42]. In this section, we further explore the structure of E(r;g) and its relationship with elementary abelian subgroups. Theorem 3.2.6 shows in the case that g is the Lie algebra of a connected, reductive group G dened overF p , the category ofF q - expressible subalgebras (Denitions 3.2.3 and 3.2.5) is isomorphic to a subcategory of Quillen's category of elementary abelian p-subgroups of G(F q ), where q = p d . Specically, we introduce the notion of an F q -linear subgroup (Denition 3.2.9), and we show in Corollary 3.2.13 that the F q -expressible subalgebras of rank r are in bijection with the F q -linear elementary abelian subgroups of rank rd in G(F q ). This bijection leads to Corollary 3.2.15, which allows us to compute the largest integer R =R(g) such thatE(R;g) is nonempty for a simple Lie algebra g. These values are presented in Table 1. The results and denitions inx3.2 rely on the canonical Springer isomorphism :N (g)!U(G), which has been shown to exist in Theorem 2.3.2, a result of work that appears in [43], [11], [47], and [35]. Together with Lang's theorem, Theorem 3.2.6 implies Theorem 3.3.3, which establishes a natural bijection between the G- orbits ofE(r;g) dened overF q and theG-conjugacy classes ofF q -linear elementary abelian subgroups of rank rd inG(F q ). Example 3.3.9, due to R. Guralnick, shows that E(r;g) may be an innite union of G-orbits (in fact this is usually the case). However, Proposition 3.3.4 demonstrates thatE(R(g);g) is a nite union of orbits for all connected, reductive G such that (G;G) is an almost-direct product of sim- ple groups of classical type (recall that R(g) is the rank of the largest elementary subalgebra of g). In work unpublished during the writing of this document, Julia Pevtsova and Jim Stark have shown, under weaker conditions on p than we con- sider here, that E(R(g);g) is a nite union of orbits for all connected, reductive groups. Our interest in describing the G-orbits is motivated by the results ofx6 in [13], where the authors construct algebraic vector bundles on G-orbits ofE(r;g) associated to a rational G-module M via the restriction of image, cokernel, and kernel sheaves. Through personal communication with the author, E. Friedlander asked for conditions implying that E(r;g) is irreducible. In the case that g = gl n , Theorem 3.4.1 presents certain ordered pairs (r;n) for which E(r;g) is irreducible. This theorem relies on previous results concerning the irreducibility ofC r (N (gl n )), the 44 3.1. MOTIVATION - FINITE GROUPS 45 variety of r-tuples of pair-wise commuting, nilpotent nn matrices (see [37] for a nice summary of these results). Finally, in Section 3.5 and Section 3.6, we compute a few examples for GL n and forU n . Some of the computations depend on Conjecture 3.5.1, which supposes the dimension of an orbit is related to the size of the corresponding G-conjugacy class. Equation (3.5.1) computes the dimension of E(r;gl n ) for all (r;n) such that C r (N (gl n )) is irreducible, and surprisingly this equation agrees with computations of dim(E(r;gl n )) even for ordered pairs whereC r (N (gl n )) is known to be reducible. Proposition 3.5.3 computes the dimension of the open orbit dened by a regular nilpotent element, as rst considered in Proposition 3:19 of [13]. For n 5, we bound the number of G-orbits in E(r;gl n ) dened over F q and compute their di- mensions. 3.1. Motivation - nite groups We begin with a brief discussion of Quillen's work with nite groups. We'll also include a few computations using Magma. A helpful reference for this section is [17] Let be a nite group, and let p continue to be a prime number. In [41] and [42], Quillen studied the maximal ideal spectrum of the cohomology of with coecients in the nite eldF p . He resolved a conjecture of Atiyah (independently conjecture by Swan) that the dimension of this topological space is equal to the p-rank of , i.e., the rank of the largest elementary abelian p-subgroup of . The following construction helped Quillen describe his result. Denition 3.1.1. Let be a nite group, and let p be a prime number. De- ne a categoryE() as follows. The objects ofE() are the elementary abelian p-subgroups of , and morphisms between subgroups are group homomorphisms obtained from conjugations composed with inclusions. We callE() the category of elementary abelian p-subgoups of . If we pause to consider this denition, we see that the categoryE() encodes the information of the elementary abelian p-subgroups of , their rank, how they t inside of one another, and how they conjugate to one another. We'll make two notes about Denition 3.1.1. First, notice that Hom E() (E;E 0 )6=? if and only if E is conjugate to a subgroups of E 0 . Second, notice that Hom E() (E;E) has the structure of a group under composition of morphisms, and is naturally isomorphic to the group N (E)=C (E). Now, if :E! is the natural inclusion of an elementary abelian p-subgroup E into , then induces a map of commutativek-algebras :H (;k)!H (E;k). If we dene X := SpecmH (;k) then by contravariance of the Specm functor, we have an induced map E :X E ! X . The following result shows why the structure ofE() helps us describe the space X . Recall that if g2N (E)=C (E) then g induces an action on H (E;k) and thus on X E . We'll denote this action by g . Proposition 3.1.2. (1) X = S E2Ob(E()) E (X E ) (2) E (X E ) E 0(X E 0) if and only if Hom E() (E;E 0 )6=?. 46 3. THE VARIETY OF ELEMENTARY SUBALGEBRAS (3) Let p;q2X E . Then E (p) = E (q) if and only if there is some g2 such that g (p) = q. The proposition allows us to view the spaceX as a nite union of the subspaces E (X E ). Recall from Example 1.2.11 thatX E is an ane space of dimension equal to the rank of E, so that (1) in the proposition shows us that X is a nite cover by ane pieces associated to the spaces X E for the elementary abelian p-groups E . (2) in the proposition describes how these ane pieces \glue" together, and (3) shows that E may not be an inclusion; there might be some self-identications of E (X E ) which are described by the groupN (E)=C (E). However, since the group N (E)=C (E) is nite, this quotienting will not aect the dimension of E (X E ) as a topological space. This can all be compactly described in the formula X = lim ! E2E() X E Notice that Atiyah's and Swan's conjecture is resolved by this formula. In fact, by (2) of Proposition 3.1.2, we see that X E is covered by the pieces coming only from maximal elementary abelian p-subgroups, and only one from each conjugacy class. We will now discuss a ner stratication of X . For a xed E, consider all subgroups F ,!E, and consider the complement of the union of all F (X F ): X + E =X E n [ FE F (X F ) It is shown inx9.1 of [17] that X + E is an open subset of X E , and that X + ;E := E (X + E ) is a locally closed ane subspace of X . Then we have the following decomposition of X , known as the Quillen stratication: X = [ E2I X + ;E whereI is a family of elementary abelianp-subgroups ofI, one from each conjugacy class. To illustrate Quillen's theorem in action, let's take a toy example. Example 3.1.3. Let be the nite group GL 3 (F p ). That is, is the group of 3 3 invertible matrices with entries in the eld of p elements. Since we want p to satisfy condition (?), we'll assume thatp6= 2. Then a computation in Magma shows that there are three conjugacy classes of elementary abelianp-groups of rank three, two conjugacy classes of rank one, and one conjugacy class of rank 0 (the trivial subgroup). A picture of these conjugacy classes, along with their \inclusions" can be found in Figure 7. A few notes are in order. First, the number of dots in each conjugacy class re ects the size of the conjugacy class (specically, a conjugacy class represented by n dots has roughly p n elements). Second, the inclusions between conjugacy classes is not the classical notion of inclusion of subgroups. A line is drawn from one conjugacy classC to anotherC 0 if there is some group inC which is a subgroup of C 0 . Notice that by denition of E(), if E;F 2 C and E 0 ;F 0 2 C 0 , then Hom E() (E;E 0 ) 6= ? if and only if Hom E() (F;F 0 ) 6= ?. Since the images of conjugate subgroups inX are equal, we only need to consider the conjugacy classes to piece together X . 3.1. MOTIVATION - FINITE GROUPS 47 r = 0 r = 1 r = 2 Figure 7. E(GL 3 (F p )) Using Quillen's theorem, and the fact that X E is an ane space of dimension equal to the rank of E, we can draw a \picture" of the topological space X . This picture is seen in Figure 8. Figure 8. SpecH (GL 3 (F p )) Notice that the three ane planes intersect in the ane line associated to the black conjugacy class, whereas only the yellow plane contains the ane line corresponding to the green conjugacy class. This is consistent with the inclusions of conjugacy classes seen in Figure 7. One should note that the picture in Figure 8 is only meant to illustrate Quillen's result concerning how the ane pieces coming from elementary abelianp-subgroups t together. As mentioned before, the ane pieces do not always inject intoX , so that the ane planes in Figure 8 should actually have self-identications coming from the groups N (E)=C (E). We also note here that the size of the conjugacy class re ected in the number of dots has no signicance in Quillen's theorem. We will later conjecture in that the sizes of the conjugacy class reveal information about the dimensions of the orbits in the variety of elementary subalgebras (see Denition 3.2.1 and the discussion following). E(GL 4 (F p )) for p> 3 is shown in Figure 9. The nodes each represent a conju- gacy class, and the number in the node represents the size of each conjugacy class; if the numbern appears in a node, then the size of the corresponding conjugacy class is roughly p n . If a node is colored, then the size of its conjugacy class is roughly 48 3. THE VARIETY OF ELEMENTARY SUBALGEBRAS 1 2 p n . Notice these conjugacy classes come in pairs, which suggests they actually merge into one conjugacy class in the group GL 4 (F q ) for some pth power q. 0 7 11 5 9 8 11 7 8 10 10 5 5 8 9 9 10 9 8 7 3 6 3 7 7 8 4 Figure 9. E(GL 4 (F p )) One can see from examining Figure 9 that the maximal ideal spectrum of the cohomology of GL 4 (F p ) is comprised of a four-dimensional ane piece along with seven 3-dimensional ane pieces (those nodes in the second row from the top that are not contained in the unique conjugacy class of rank 4). As before, the picture encodes how these pieces glue together. For instance, the two ane planes corresponding to the blue nodes glue together along an ane line corresponding to the node marked with a 7 in the second row from the bottom. We again note that the sizes of the conjugacy classes re ected by the numbers in the node are currently of no signicance (however, see Conjecture 3.5.1). For more pictures of Quillen's category of elementary abelian p-subgroups for various nite groups and primes p, see Appendix B. 3.2. THE VARIETY AND CATEGORY OF ELEMENTARY SUBALGEBRAS 49 3.2. The variety and category of elementary subalgebras In the previous section we explored the signicance of elementary abelian p- subgroups of a nite group in describing the spectrum of the cohomology algebra. Of particular importance was the way that the elementary abelianp-subgroups were included in each other, and how they conjugated to each other. In this section we'll introduce the notion of an elementary subalgebra, which will play the analogue of an elementary abelian subgroup in the setting of restricted Lie algebras. We will use the canonical Springer isomorphism given in Theorem 2.3.2 to study the projective variety E(r;g), as dened in [13]. The following discussion is relevant for an arbitrary restricted Lie algebra (g; [;]; () [p] ), but we are only concerned with the case g = Lie(G). Denition 3.2.1. [[13], Denition 1.2] An elementary subalgebra g is an abelian Lie subalgebra of g with trivial restriction, i.e., x [p] = 0 for all x2. Let E(r;g) be the set of elementary subalgebras of rank r in g. Considering ,! g as an inclusion of vector spaces, there is an embeddingE(r;g),! Grass(r;g). This is a closed embedding so thatE(r;g) has the structure of a projective subvariety of Grass(r;g) ([13], Proposition 1.3). If g is the Lie algebra of an algebraic group G, thenE(r;g) is aG-variety via the adjoint action of G on g. Specically, for any 2E(r;g) and any g2G, the image of under Ad g : g! g is elementary of rank r. We note for later purposes the following construction, which appears in Propo- sition 1.3 of [13] and its proof. LetC r (N (g)) denote the variety of r-tuples of pairwise-commuting, nilpotent, linearly independent elements of g. By taking thek- span of elements in anr-tuple, any (X 1 ;:::;X r )2C r (N (g)) denes an elementary subalgebra of rank r, so there is a map of algebraic varietiesC r (N (g)) E(r;g). The following denitions are motivated by ([11],x3). In all that follows we suppose that G has a xed F p -structure, i.e. G = G 0 Fp Spec k for some xed algebraic groupG 0 overF p . It follows that g := Lie(G) has anF p -structure coming from g 0 := Lie(G 0 ) given by g = g 0 Fp Spec k. For q = p d , by abuse of notation we writeG(F q ) (resp. g(F q )) to denote theF q -rational points ofG 0 (resp. g 0 ). For another point of view, we may consider G(F q ) to be the subgroup consisting of all k-points of G obtained from the base-change of an F q -point of G 0 (and similarly for the Lie algebra). Here, we view the vector space g as a scheme over k via the following standard construction. Given a nite dimensional vector space V over k, give V the structure of a linear scheme over k with coordinate algebra S (V # ). Then the k-points of V with this scheme structure are naturally identied with the elements of the vector space V . In particular, in the setting just discussed, g(F q ) = g 0 . Denition 3.2.2 ([11],x3, Denition 1). An element X2 g isF q -expressible if it can be written as X = P c i X i with c i 2k, X i 2 g(F q ), X [p] i = 0 = [X i ;X j ]: This denition can be extended to the notion of anF q -expressible subalgebra. Denition 3.2.3. We call an elementary subalgebra anF q -expressible subalgebra, if it has a basis of the formfX 1 ;:::;X r g g(F q ), i.e., =(F q ) Fq k. To speak of F q -rational points of E(r;g), we require a rationality condition on g. Notice that if g has anF p -structure given by g = g 0 Fp k, thenE(r;g) is dened over 50 3. THE VARIETY OF ELEMENTARY SUBALGEBRAS F p . This can be seen as follows. Fix an embedding g 0 ,! gl n (F p ). Then g ,! gl n is determined by linear equations with coecients in F p . The equations dening the nilpotent, commuting, and linearly independent conditions are all homogeneous polynomials with coecients inF p as well. We claim that theF q -rational points ofE(r;g) are precisely theF q -expressible subalgebras of g. To see this, notice that E(r;g)(F q ) = E(r;g)\ Grass(r;g)(F q ). The claim follows from the fact that the F q -rational points of the Grassmannian are those r-planes with a basis in g(F q ). We recall the category of elementary abelianp-subgroups of a nite group, rst considered by Quillen. For g2G, let c g :G!G be dened by c g (h) =ghg 1 Denition 3.2.4. Let be a nite group, and let p be a prime dividing the order of . DeneE() to be the category whose objects are the elementary abelian p-subgroups of , and whose morphisms are group homomorphisms E!E 0 which can be written as the composition of an inclusion followed by c g for someg2 . In particular we have a morphismE!E 0 if and only ifE is conjugate to a subgroup of E 0 . Motivated by Denition 3.2.4, we make similar denitions for restricted Lie algebras. Denition 3.2.5. (1) Let G be an algebraic group dened over k, and let g = Lie(G). Dene E(g) to be the category whose objects are the elementary abelian sub- algebras of g, and whose morphisms are inclusions followed by Ad g for g2G. (2) Inside ofE(g), letE Fq (g) be the subcategory whose objects consist of the F q -expressible subalgebras, and whose morphisms are inclusions composed with Ad g for g2G(F q ). The following theorem further emphasizes that elementary subalgebras of a re- stricted Lie algebra are the appropriate cohomological and representation theoretic analog to elementary abelian subgroups of a nite group. The existence of the canonical Springer isomorphism is used in the proofs of this section, so throughout we assume p satises condition (?). Theorem 3.2.6. Let G be a reductive, connected group. Then for q =p d , the categoryE Fq (g) is isomorphic to a subcategory ofE(G(F q )). For d = 1, Quillen's categoryE(G(F p )) is isomorphic toE Fp (g). Proof. Let be the canonical Springer isomorphism from Theorem 2.3.2, and dene a fully faithful functorF :E Fq (g)!E(G(F q )) as follows. F() =((F q )) F(,! 0 ) =((F q )),!( 0 (F q )) F(Ad g ) =c g where we note that if 0 , then ((F q ))( 0 (F q )). Since is dened overF p , we have((F q ))G(F q ). The fact that((F q )) is an elementary abelianp-group in G(F q ) follows from property 3 of Theorem 2.3.2. Notice if has dimension r, thenj(F q )j = q r = p rd , so that ((F q )) has rank rd. ThatF is fully faithful follows from the denition of the morphisms in the categoriesE Fq (g) andE(G(F q )). 3.2. THE VARIETY AND CATEGORY OF ELEMENTARY SUBALGEBRAS 51 In the cased = 1, it remains to show thatF is surjective on objects. Let E be any elementary abelian subgroup of rank r in G(F p ). I claim that V = 1 (E) g(F p ) is anr-dimensional subspace overF p . ThatV is closed under addition follows from properties 1 and 3 of Theorem 2.3.2. If 2F p , then 1 (g) = 1 (g ), so thatV is closed underF p -scalar multiplication. SinceV hasp r elements, it follows that it is an F p -subspace of dimension r. Then = V Fp k is an r-dimensional k-space such thatF() =E. Example 3.2.7. As in Example 3.1.3, we continue to take the group GL 3 as our illustrating example. In this case, Theorem 3.2.6 says that the way the elementary subalgebras of gl 3 dened overF p t together is exactly how the elementary abelian p-subgroups of GL 3 (F p ) t together. This is illustrated in Figure 10. = Figure 10. E(GL 3 (F p )) =E Fp (gl 3 ) Notice that in the proof of Theorem 3.2.6, the symbol g is meaningless for 2 F q nF p , which is whyF is only an isomorphism for d = 1. The following example shows whyF fails to be surjective for d> 1. Example 3.2.8. LetG = SL 3 , letd = 2, and let2F q nF p . In this case, we have (X) =I +X + 1 2 X 2 . Consider the elementary abelian subgroup of rank 2 dened as follows: E = * g = 0 @ 1 1 0 0 1 0 0 0 1 1 A ;h = 0 @ 1 0 1 0 1 0 0 0 1 1 A + G(F p )G(F q ) Then any subalgebra with EF() must contain the elements X = 1 (g) = (gI) and Y = 1 (h) = (hI). It follows that I +X;I +Y 2F(), but I +X;I +Y = 2E, so that there is no subalgebra withF() =E. To determine the image ofF for d > 1, it will be helpful to dene g := ( 1 (g)) for g2U(G) and 2 k. One can check using properties 1 and 3 of Theorem 2.3.2 that the following familiar formulas hold for all g;h;k2U(G) with gh =hg, and all ;2k: (3.2.1) g g =g + (g ) =g g h =h g (gkg 1 ) =gk g 1 Denition 3.2.9. Call an elementary abelian subgroupEGF q -linear ifg 2E for any g2E and any 2F q . 52 3. THE VARIETY OF ELEMENTARY SUBALGEBRAS Notice that any elementary abelian subgroup isF p -linear, asg is just given by the group operation in E whenever2F p . Also notice that E in Example 3.2.8 is notF q -linear as g =I +X = 2E. Lemma 3.2.10. IfEG is a niteF q -linear elementary abelianp-subgroup, then the rank of E is divisible by d. Proof. Choose any g 1 2E, and let E g1 =fg 1 j2F q g. Then, E g1 E by F q -linearity of E. Also, using (3.2.1) we see that E g1 is an F q -linear elementary abelian subgroup of rank d. Chooseg 2 2EnE g1 , and note that theF q -linearity of E g1 ensures thatE g2 \E g1 =e. ThusE g1 E g2 E isF q -linear of rank 2d. Since E is nite, this process stops, and so there is a generating set g 1 ;:::;g r such that E =E g1 :::E gr has rank rd. Remark 3.2.11. The motivation behind the terminology \F q -linear" can be seen in Lemma 3.2.10 and its proof. Given any niteF q -linear elementary abelian subgroup E, there is a decomposition E = E g1 :::E gr for an appropriate choice of generators g 1 ;:::;g r . The rank of E as an elementary abelian p-group is then rd. Viewing E gi as the \span" of g i , this decomposition is analogous to decomposing anr-dimensionalF q -vector space into the direct sum of one dimensional subspaces spanned by vectors in a basis. Proposition 3.2.12. The image ofF :E Fq (g)!E(G(F q )) is the full subcategory ofF q -linear elementary abelian subgroups inE(G(F q )). Proof. First, let beF q -expressible, and choose anyg2E =((F q )). Write g =(X) forX2(F q ) and let2F q . Theng =(X) =(X)2E so thatE isF q -linear. Conversely, suppose E G(F q ) is F q -linear. We proceed as in the proof of the d = 1 case in Theorem 3.2.6. The F q -linearity hypothesis on E is precisely what is needed to show that V = 1 (E) is closed underF q -scalar multiplication. Notice that for 2 F q and for 1 (g)2 V , we have 1 (g) = 1 (g )2 V because g 2E. It follows that theF q -expressible subalgebra =V Fq k satises F() =E, which completes the proof. Corollary 3.2.13. There is a bijection betweenF q -expressible elementary subalge- bras of g = Lie(G) of rank r and F q -linear elementary abelian subgroups of G(F q ) of rank rd. Proof. The maps 7! ((F q )) and E7! 1 (E) Fq k used in the proof of Proposition 3.2.12 are inverse to each other. Denition 3.2.14. Let R = R(g) denote the largest integer such that E(R;g) is nonempty. Corollary 3.2.15. Any elementary abelian subgroup E G(F q ) is contained in an F q -linear elementary abelian subgroup of G(F q ). In particular, any maximal elementary abelian subgroup is F q -linear. Also, the largest rank of an elementary abelian subgroup of G(F q ) is R(Lie(G))d. Proof. To any EG(F q ), consider V =h 1 (E)i; theF q -subspace of g(F q ) generated by 1 (E). Then (V ) =F(V Fq k) isF q -linear, and E(V ). For the last statement of the corollary, let be an elementary subalgebra of rank R =R(Lie(G)). Then Corollary 3.2.13 shows that((F q )) is elementary abelian of 3.2. THE VARIETY AND CATEGORY OF ELEMENTARY SUBALGEBRAS 53 rank Rd. If there exists EG(F q ) of larger rank, then E must lie in anF q -linear elementary abelian subgroup E 0 of rankR 0 d forR 0 >R. Then 1 (E 0 ) Fq k is an elementary subalgebra of rank R 0 , contradicting the maximality of R. Remark 3.2.16. In the proof of Corollary 3.2.15, one could also construct the groupfg jg2E; 2F q g as anF q -linear elementary abelian subgroup containing E. In fact, we have the equality (h 1 (E)i) =fg j g 2 E; 2 F q g. This motivates the notationhEi Fq for the groupfg j g2 E; 2 F q g, which is the smallestF q -linear subgroup containing E. Corollary 3.2.15 allows us to relate the maximal rank of an elementary abelian p-subgroup in G(F q ), known as the p-rank of G(F q ), with R(Lie(G)). The p-ranks of the nite simple groups of Lie type are known (cf. Table 3.3.1 in [24]). This leads to Table 1, which presents R(g) for the simple Lie algebras. Table 1. Dimension of largest elementary subalgebra of simple g Type R(g) A n j n+1 2 2 k B n , n = 2; 3 (2n 1) B n , n 4 1 + n 2 C n , n 3 n+1 2 D n , n 4 n 2 Type R(g) E 6 16 E 7 27 E 8 36 F 4 9 G 2 3 Example 3.2.17. Let G = SO 3 , p 3, and r = 1. Then Lie(SO 3 ) = so 3 is the collection of skew-symmetric 3 3 matrices. A skew-symmetric 3 3 nilpotent matrix has the form: 0 @ 0 x y x 0 z y z 0 1 A wherex 2 +y 2 +z 2 = 0. It follows thatE(1;so 3 ) is the irreducible projective variety in P 2 of all points [x : y : z] satisfying x 2 +y 2 +z 2 = 0. This equation has q 2 solutions overF q (exercise!), one of which is (0; 0; 0). This leaves us with a setS of q 2 1 non-trivial solutions, each of which spans a one dimensional F q -expressible subalgebra. Each F q -expressible subalgebra contains exactly q 1 elements of S, so that there are q + 1 dierent F q -expressible subalgebras. A quick computation in Magma shows it is also true that there are q + 1 subgroups of the form (Z=pZ) d in SO 3 (F q ), for many small values of q (q < 400) as we expect from Corollary 3.2.13. Notice here that d is the maximal rank of an elementary abelian subgroup in SO 3 (F q ), so that all such subgroups areF q -linear by Corollary 3.2.15. Remark 3.2.18. In this section we have chosen to use Quillen's category of ele- mentary abelian subgroups as a motivation for dening our category of elementary 54 3. THE VARIETY OF ELEMENTARY SUBALGEBRAS subalgebras. Our reasoning behind this is due to the importance of Quillen's cat- egory in the cohomology of the group G(F p ). It is the author's hope that the isomorphic categoryE Fp (g), or the larger categoryE(g) might hold a similar im- portance in the cohomology of g. Another approach for this section would be to motivate our denitions by the Quillen complex of elementary abelian subgroups, that is, the complex associated to the poset of elementary abelian subgroups ofG ordered by inclusion. With a similar denition of the complex of elementary subalgebras, we nd that Quillen's complex for G(F p ) is isomorphic to the subcomplex of F p -expressible subalgebras. Using the appropriate analogues of group-theoretic notions (for example, the Frattini subalgebra of g as studied in [33]), much of the machinery developed in Part 2 of [45] for groups can be developed for Lie algebras. In particular, it is true that the subcomplex of p-subalgebras is G-homotopy equivalent to the subcomplex of elementary subalgebras, and the proof follows that of the analogous statement for groups (see ([45], Theorem 4.2.4) for a proof using Quillen's Fiber Theorem). These connections further emphasize that in the study of restricted Lie algebra cohomology, elementary subalgebras of restricted Lie algebras are the appropriate analogue to elementary abelian subgroups of nite groups. 3.3. Rational points and orbits of the variety of elementary subalgebras In this section we use Lang's theorem to show that F q -rational points exist in the G-orbits of E(r;g) that are dened over F q . By the previous sections, these points correspond exactly to F q -linear elementary abelian p-groups of rank rd in G(F q ). This leads to Theorem 3.3.3, which gives a bijection between the G-orbits ofE(r;g) dened over F q and G-conjugacy classes of F q -linear elementary abelian p-subgroups of rankrd inG(F q ). We clarify the phrase "G-conjugacy classes ofF q - linear elementary abelian p-groups of rank rd in G(F q )." Two elementary abelian p-subgroups of rankr inG(F q ), sayE r andE 0 r , areG-conjugate if there isg2G(k) such that g conjugates E r to E 0 r . Notice this is not the same as the standard notion of conjugate subgroups in G(F q ), as there may be non-conjugate subgroups H;K G(F q ) which are conjugate when viewed as subgroups in G(F q e). Also notice that by equation (3.2.1), the conjugate of anF q -linear subgroup isF q -linear, so that the notion of a conjugacy class ofF q -linear subgroups is well-dened. E. Friedlander has asked for sucient conditions such that E(r;g) is a nite union of G-orbits. Theorem 3.3.3 shows that if G is connected and reductive, and if p satises condition (?) thenE(r;g) has nitely many G-orbits dened over F q . This of course does not resolve the question, but as Example 3.3.9 shows, any list of sucient conditions is sure to be fairly restrictive. E. Friedlander has conjectured that if R =R(g) is the largest integer such thatE(R;g) is non-empty, thenE(R;g) is a nite union ofG-orbits. Proposition 3.3.4 reduces this conjecture to showing that the number of conjugacy classes of elementary abelian subgroups of rankRd is bounded asd grows. This is known to be true for all simple groups of classical type, so we obtain an armative answer to the conjecture for a large class of groups, namely those G whose derived subgroup is an almost-direct product of simple groups of classical type. Theorem 3.3.3 also provides a method for bounding the number of G-orbits dened overF q . In section 3.5, we use Magma to bound the number of GL n -orbits dened overF p inE(r;gl n ) for n 5. 3.3. RATIONAL POINTS AND ORBITS OF THE VARIETY OF ELEMENTARY SUBALGEBRAS 55 As mentioned in the introductory paragraph to this section, Theorem 3.3.3 follows from the previous sections and the following theorem of Lang. Theorem 3.3.1 ([31], Theorem 2). Let G be an algebraic group dened over a nite eld F , and let V be a variety dened over F on which G acts morphically and transitively. Then V has an F -rational point. We should clarify that in Theorem 3.3.1, an action is transitive if there is a v2 V such that V = Gv. We do not require the map G=G v ! V to be an isomorphism of varieties. The proof of the following lemma is immediate from Theorem 3.3.1. Lemma 3.3.2. Letr be such thatE(r;g) is nonempty, and letO be anyG-orbit of E(r;g). IfO is dened overF q , then the set ofF q -rational points ofO is non-empty. Theorem 3.3.3. Let G be connected and reductive, and let p satisfy condition (?). The G-orbits of E(r;g) dened over F q are in bijection with the G-conjugacy classes ofF q -linear elementary abelian p-groups of rankrd inG(F q ). In particular, E(r;g) contains nitely many G-orbits dened over F q . Furthermore, the number of F q -rational points of a G-orbit dened over F q is equal to the size of its corre- sponding G-conjugacy class. Proof. LetO be any G-orbit of E(r;g) dened over F q . By Lemma 3.3.2, O(F q ) is non-empty. Furthermore, by Corollary 3.2.13,O(F q ) is in bijective cor- respondence with a collection of F q -linear elementary abelian p-subgroups of rank rd in G(F q ). By G-equivariance, these elementary abelian p-subgroups form a G-conjugacy class. Conversely, starting with a G-conjugacy class of F q -linear el- ementary abelian p-subgroups of rank rd, Corollary 3.2.13 gives us a G-orbit of E(r;g) whose F q -rational points correspond to elements of the given G-conjugacy class. Proposition 3.3.4. Fix r, and let N d be the number of conjugacy classes of ele- mentary abelian p-groups of rank rd in G(F q ). If N d is bounded, thenE(r; Lie(G)) is a nite union of G-orbits. In particular, N d is bounded for simple groups of classical type when r = R, so E(R(Lie(G)); Lie(G)) is a nite union of G-orbits whenever (G;G) is an almost-direct product of simple groups of classical type. Proof. Suppose to the contrary that E(r; Lie(G)) is an innite union of G- orbits. Then the number of G-orbits dened over F q approaches innity as d gets large. By Theorem 3.3.3, the number of G-orbits dened over F q is at most N d , which is bounded, providing a contradiction. If G is a simple group of classical type, and r = R(g), it is actually the case thatN d is a constant sequence (see [5]), so thatE(r;g) is a nite union ofG-orbits. Now, supposeG is a direct product of simple groups of classical typeG 1 :::G m . By Proposition 1:12 in [13], we have an isomorphism of projective G-varieties m Y i=1 E(R i ;g i )!E m X i=1 R i ; m M i=1 g i ! where g i = Lie(G i ) and R i = R(g i ). This isomorphism is given by sending the m-tuple ( 1 ;:::; m ) to 1 ::: m . NowG =G 1 :::G m acts componentwise, and the action of eachG i has nitely many orbits, so the same is true of the action of G. 56 3. THE VARIETY OF ELEMENTARY SUBALGEBRAS If G is an almost-direct product of simple groups of classical type, then the isogenyG 1 :::G m G is separable by condition (?), so we have g = Lie(G) = g 1 ::: g m , and the orbits of the action of G on E(r;g) coincide with those of the action of G 1 :::G m onE(r;g). Finally, if G is connected and reductive, then since G = Z(G) (G;G), the G-orbits ofE(r;g) coincide with the (G;G)-orbits. Remark 3.3.5. We also expect, but are unable to verify that N d is constant for the exceptional simple groups. If this were true, then E(R(Lie(G)); Lie(G)) is a nite union of G-orbits for all connected, reductive groups G. Example 3.3.6. Earlier, in Example 3.1.3, we considered the nite group GL 3 (F p ). Figure 7 shows how the conjugacy classes of elementary abelian p-subgroups t together in Quillen's category. Recalling that all elementary abelian p-subgroups ofG(F p ) areF p -linear, we know by Theorem 3.3.3 that the conjugacy classes are in one-to-one correspondence with the orbits overF p . This is illustrated in Figure 11. E(GL 3 (F p )) =E Fp (gl 3 ) r = 1 r = 2 E(1;gl 3 ) E(2;gl 3 ) Figure 11. Orbits ofE(r;gl 3 ) dened overF p for r = 1; 2 Example 3.3.7. For an example illustrating that our reductive hypothesis may be unnecessary, consider the non-reductive group U 3 , the unipotent radical of B 3 , the group of upper triangular 3 3 matrices. Example 1:7 in [13] shows that E(2;u 3 ) =P 1 . Explicitly, any elementary subalgebra has a basis of the form: (3.3.1) 8 < : 0 @ 0 0 1 0 0 0 0 0 0 1 A ; 0 @ 0 a 0 0 0 b 0 0 0 1 A 9 = ; and this basis is unique up to scalar multiple of the vector (a;b). A computation shows that each such subalgebra is xed under conjugation by U 3 , so that the G- varietyE(2;u 3 ) =P 1 has innitely manyG-orbits (each point is an orbit). However, only theF q -rational points ofP 1 , of which there are nitely many, are orbits dened overF q . 3.3. RATIONAL POINTS AND ORBITS OF THE VARIETY OF ELEMENTARY SUBALGEBRAS 57 Example 3.3.8. Let G = GL 3 , p 3, and r = 2. Any elementary subalgebra 2E(2;gl 3 ) can be put in upper-triangular form, so is conjugate to a subalgebra 0 with basis given by (3.3.1) for some [a :b]2P 1 . In general, the element [a :b] is not dened by , as conjugating 0 by 0 @ 0 0 0 1 0 0 0 1 1 A ; 6= 0 gives an upper triangular subalgebra corresponding to [a : b]. It follows that all subalgebras 2 E(2;gl 3 ) that have an element of rank 2 are conjugate. The dimension of this orbit is shown to be 4 in Example 3:20 of [13]. The only other subalgebras are those whose non-zero elements all have rank equal to 1. These subalgebras are conjugate to upper-triangular subalgebras corre- sponding to the points [1 : 0] and [0 : 1]. These two upper-triangular subalgebras are not conjugate (in short, the conditions for a matrix A to conjugate [1 : 0] to [0 : 1] require that det(A) = 0, a contradiction). The dimension of each of these two distinct orbits is 2 (Example 3.20, [13]). We have thus veried thatE(2;gl 3 ) is the union of three GL 3 -orbits, all of which are dened over F p . As expected from Theorem 3.3.3, any elementary abelian p-subgroup of rank 2d in GL 3 (F q ) is conju- gate to exactly one of the groupshI +E 12 +E 23 ;I +E 13 i,hI +E 23 ;I +E 13 i, andhI +E 12 ;I +E 13 i. HereE ij is the matrix whose only non-zero entry is a 1 in theith row andjth column. This example will be further developed in Proposition 3.5.7 below. Example 3.3.9. The following example (due to R. Guralnick) shows that even ifG is connected and reductive,E(r;g) may be an innite union of orbits. LetG = GL 2n and let be the elementary subalgebra of g = gl 2n of dimensionn 2 whose matrices only have nonzero entries in the upper-right nn block. For any rn 2 , we have Grass(r;)E(r;g) so that dim(E(r;g)) dim(Grass(r;)) = (n 2 r)r: If r and n 2 are such that (n 2 r)r > 4n 2 , then dim(E(r;g)) > dim(G), so that E(r;g) is not a nite union of G-orbits. Question 3.3.10. As with nilpotent orbits of g, we can place a partial ordering on the G-orbits of E(r;g) byO O 0 if and only ifO O 0 . For classical Lie algebras, the ordering on nilpotent orbits (r = 1) corresponds to the dominant ordering on Jordan type. Forr> 1, given twoG-conjugacy classesC andC 0 ofF q - linear elementary abelianp-groups of rankrd inG(F q ) with corresponding orbitsO andO 0 dened overF q , is there some group-theoretic condition on C and C 0 that determines whenOO 0 ? In other words, can we describe the partial ordering on orbits in the group setting? Notice that the existence of a unique maximal element in the partial order implies thatE(r;g) is irreducible. Describing the partial order in the group setting might allow us to nd further examples of groups G such that E(r; Lie(G)) is irreducible. Example 3.3.11. For the case G = GL n and r = 1, the answer to Question 3.3.10 is already known. For each unipotent g2 GL n (F q ), we know by the theory of Jordan forms that g is conjugate to a direct sum of Jordan blocks, all with eigenvalue 1. As in the nilpotent case, the orbits are ordered by Jordan type of the corresponding unipotent elements. This result is expected, since Springer has shown thatU(GL n ) andN (gl n ) are isomorphic as GL n -varieties. 58 3. THE VARIETY OF ELEMENTARY SUBALGEBRAS Remark 3.3.12. Example 3.3.11 suggest that the answer to Question 3.3.10 for classical Lie algebras may lie in the Jordan types of the elements of C and C 0 . That is, there may be a condition on the Jordan types of elements in C and C 0 that determines whenOO 0 . 3.4. Irreducibility of the variety of elementary subalgebras E. Friedlander has asked for sucient conditions such thatE(r;g) is irreducible. Here we use known results on the irreducibility of the commuting variety of nilpotent matrices to deduce irreducibility for E(r;g). In fact, the question of irreducibility of the commuting variety of nilpotent matrices has been resolved in all cases except for triples of nn matrices for n = 7;:::; 12 (see [38]). Work of A. Premet in [40] shows that E(2;gl n ) is irreducible for all n. This is observed in Example 1.6 of [13]. Premet shows that under less restrictive hy- potheses on g and p than we consider here, the variety of pairs of commuting nilpotent elements in g is equidimensional, having irreducible components which are in one-to-one correspondence with distinguished nilpotent orbits. For g = gl n , there is only one distinguished nilpotent orbit, namely, the regular orbit. It follows thatC 2 (N (gl n )) is irreducible. Since open sets of irreducible sets are themselves irreducible, and continuous images of irreducible sets are irreducible, the map of algebraic varietiesC 2 (N (gl n )) E(2;gl n ) discussed at the end ofx3.2 shows that E(2;gl n ) is irreducible. This same argument shows thatE(1;g) is irreducible for all g, as the restricted nullconeN (g) is irreducible. It is also known thatC r (N (gl n )) is irreducible for r = 3 and n 6, so by similar reasoning, it follows thatE(r;gl n ) is irreducible for the corresponding pairs (r;n). We should note that we have proven the implication C r (N (g)) irreducible =)E(r;g) irreducible however, the converse is not true. In Corollary 4 of ([37]) it is shown thatC r (N (gl n )) is reducible for allr 4 andn 4, but Theorem 2.9 in [13] shows thatE(n 2 ;gl 2n ) is irreducible for alln. We summarize the above discussion in the following theorem. Theorem 3.4.1. The variety E(r;gl n ) is irreducible for the following ordered pairs (r;n): (1;n) for anyn, (2;n) for anyn, (3;n) forn 6, and (n 2 ; 2n) for any n. Question 3.4.2. The reducibility of the variety ofr-tuples of pairwise-commuting matricesC r (gl n ) and the variety of r-tuples of pairwise-commuting nilpotent ma- tricesC r (N (gl n )) has been extensively studied. As we've already observed, since C r (N (gl n )) is open inC r (N (gl n )), the irreducibility of the latter implies that of the former. Are there counterexamples to the converse? Also, as in the case of C r (N (gl n )), isC r (N (gl n )) reducible for large enough r and n? Question 3.4.3. IfX2 gl n is a regular nilpotent element and is then 1-plane with basis given byfX;X 2 ;:::;X n1 g, is the orbit of X dense in E(n 1;gl n )? Proposition 3.19 and Example 3.20 of [13] show that it is open in general and dense in the case n = 3. We have shown above that the question also has an armative answer for the cases n = 4. If the orbit of X is indeed dense, thenE(n 1;gl n ) is irreducible for all n 1. 3.5. COMPUTATIONS FOR THE GENERAL LINEAR GROUP 59 3.5. Computations for the general linear group Since the G-orbits of E(r;g) dened over F q are in bijection with the G- conjugacy classes of F q -linear elementary abelian p-groups of rank rd in G(F q ), we can bound the number of suchG-orbits by computing in the nite groupG(F q ). In this section we make some computations for G = GL n , and d = 1, using the \ElementaryAbelianSubgroups" function in Magma. The values appearing in Tables 2 and 3 below, as well as the computation of Example 3.5.8, rely on Conjecture 3.5.1. Before stating the conjecture, we intro- duce some notation. Any reductive algebraic group G has a Chevalley Z-form, denoted G Z . Let G p = G Z Z Spec F p . Also, if ' : G! G 0 is a map of re- ductive groups dened over Z, then the dierential d' induces a map from the G p -orbits of E(r; Lie(G p )) to the G 0 p -orbits of E(r; Lie(G 0 p )). This follows because d' Ad g = Ad '(g) d'. By abuse of notation, we will also denote the induced map on orbits by d'. Conjecture 3.5.1. (1) For every reductive algebraic group G and for any pair of primes p;p 0 satisfying condition (?), there is a natural dimension-preserving bijection f p;p 0 between the G p -orbits ofE(r; Lie(G p )) dened overF p d and the G p 0- orbits of E(r; Lie(G p 0)) dened over F p 0d. By natural, we mean that for any primesp,p 0 , andp 00 satisfying condition (?) and any map' :G!G 0 of reductive groups dened over Z we have: (a) f p;p 00 =f p 0 ;p 00f p;p 0 and (b) d'f p;p 0 = f 0 p;p 0d', where f p;p 0 and f 0 p;p 0 are the bijections for G and G 0 respectively. (2) Fix a prime p satisfying condition (?) and a G p -orbit of E(r; Lie(G p )) of dimension e dened over F p d, denotedO p . Then for all primes p 0 satisfying condition (?) the counting function p 0 to #f p;p 0(O p )(F p 0) is a polynomial in p 0 of degree ed. The usefulness of assuming the conjecture in what follows is that it allows us to compute the dimension of an orbit by nding the degree of the polynomial which counts the orbit's rational points. These numbers appear in the nodes of Figure 9, and are equal to the number of dots in the groupings of Figure 11. Although not explicitly calculated for the large examples of Appendix B, there the sizes of the conjugacy classes (and therefore the dimension of the orbits) is related to the color of the node. Table 2 below records experimental results for upper bounds on the number of GL n -orbits of E(r;gl n ) dened over F p for varying r and n. To be specic, Table 2 records (roughly) the number of conjugacy classes of elementary abelian subgroups of rank r in GL n (F p ). In light of Conjecture 3.5.1, the primes used in the computation have been suppressed, although it is important to note that in all cases p n = h(GL n ). The numbers recorded are upper bounds because it is not clear to the author how to determine when two G(F p )-conjugate subgroups merge in some G(F q ) (in certain cases, it can be inferred from the sizes of the conjugacy classes which classes merge. It is the detecting of this merging that causes the numbers in Table 2 to be slightly less than the number of conjugacy classes of elementary abelian subgroups of rank r in GL n (F p )). By Theorem 3.4.1, all the varieties represented here are irreducible, except for those corresponding to 60 3. THE VARIETY OF ELEMENTARY SUBALGEBRAS (r;n) = (4; 5); (5; 5) and (5; 6). E(6;gl 5 ) is known to be reducible, which follows from the fact that for n > 1, E(n(n + 1);gl 2n+1 ) is the disjoint union of two connected components both isomorphic to Grass(n; 2n + 1) ([13], Theorem 2.10). It is not known to the author ifE(5;gl 5 ) orE(4;gl 5 ) are irreducible. Table 3 records the dimension of the largest orbit from Table 2. In all known cases, this dimension is equal to the dimension of E(r;g), suggesting the dimension of E(r;g) may be determined by its largest orbit dened overF p . Table 2. Upper bounds forjfG-orbits ofE(r;gl n ) dened overF p gj n r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 2 1 0 0 0 0 0 3 2 3 0 0 0 0 4 4 10 8 1 0 0 5 6 35 67 32 5 2 Table 3. Dimension of largest orbit ofE(r;gl n ) dened overF p n r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 2 1 0 0 0 0 0 3 5 4 0 0 0 0 4 11 11 9 4 0 0 5 19 20 19 16 11 6 Question 3.5.2. Can we expect a formula for the (r;n) entry of either table? Example 3.3.11 shows that the entry in the nth row of column r = 1 in Table 2 is exact, and is equal to p(n) 1, where p(n) is the number of partitions of the integern. We lose the trivial partition 1 + 1 +::: + 1 =n because this corresponds to the trivial subgroup. For r 2, we would need more exact data to determine if there is a closed formula for the number of orbits dened over F p . This closed formula may involve p(n). There is at least hope for an armative answer to Question 3.5.2 for Table 3, as evidenced by the following. Since E(1;gl n ) is the projectivized nullcone, the fact that the entries in column r = 1 are n 2 n 1 follows from the well-known formula dim(N (g)) =n 2 n. Furthermore, in Example 1:6 of [13] the authors use work of Premet in [40] to establish that the entries in column r = 2 are n 2 5, which agrees with our computation. Also in [13], the identications ofE(n 2 ;gl 2n ) with Grass(n; 2n) andE(n(n + 1);gl 2n+1 ) with Grass(n; 2n + 1) give the following formulas (which agree with Table 3): dim(E(n 2 ;gl 2n )) =n(2nn) =n 2 dim(E(n(n + 1);gl 2n+1 ) =n(2n + 1n) =n(n + 1) It is also known that ifC r (N (gl n )) is irreducible, then it has dimension (n + r 1)(n 1), as reviewed in Proposition 1 and Corollary 2 of [37]. This together 3.5. COMPUTATIONS FOR THE GENERAL LINEAR GROUP 61 with the mapC r (N (g)) E(r;g) discussed at the end ofx3.2, whose bers are GL r -torsors, show that (3.5.1) dim(E(r;gl n )) = (n +r 1)(n 1)r 2 for all ordered pairs (r;n) for whichC r (N (gl n )) is irreducible (all known such or- dered pairs are presented inx3.4). Notice that equation (3.5.1) subsumes the results for r = 1 and r = 2, and surprisingly it even agrees with Table 3 in entries (4; 5) and (5; 5) for whichC r (N (gl n )) is known to be reducible. In fact, the only entries of Table 3 that don't agree with equation (3.5.1) are (3; 3), (4; 4), and (6; 5). The following proposition is another piece of evidence that the entries of Table 3 may have a closed form. Proposition 3.5.3. LetO be the open GL n -orbit of E(n 1;gl n ) containing the subalgebra spanned by the powers of a regular nilpotent element X (cf. [13], Proposition 3:19). Then dim(O) = (n 1) 2 . Proof. LetG = GL n . We will show the dimension of the stabilizerG of has dimension 2n 1, from which we obtain dim(O) = dim(G) dim(G ) =n 2 (2n 1) = (n1) 2 . We may chooseX to be the Jordan block of sizen with eigenvalue 0, so thatG consists solely of upper-triangular matrices. In this case, I claim that G is isomorphic to the (2n1)-dimensional quasi-ane variety dened byx 1 6= 0,x 2 6= 0 inA 2n1 . The map dening the isomorphism is given by sending a matrixA = (a ij ) which normalizesO to the point (a 11 ;a 22 ;a 12 ;a 23 ;a 13 ;a 24 ;:::;a 1(n1) ;a 2n ;a 1n ). For injectivity, we must show that the entries in the top two rows of A along with the conditionA2G completely determineA. For surjectivity, we must show that any choice of entries in the top two rows of A dene a matrix A2 G as long as a 11 6= 0, a 22 6= 0 and a 21 = 0. What follows is rather tedious, but the basic idea is that the entries along a super-diagonal are determined uniquely by the rst two entries in the super- diagonal, and these rst two entries may be arbitrary (except in the case of the diagonal, in which case the entries must be non-zero). By a super-diagonal, we mean any collection of entries of the form a i;j where ji = k for some xed k = 1;:::;n 1. First, notice that A2G if and only if AXA 1 2. For j >i, the (i;j) entry of AXA 1 is (3.5.2) a ii a i+1;j a i+1;i+1 a jj a i;i+1 a i+2;j a i+2;i+2 a jj +::: a i;j2 a j1;j a j1;j1 a jj + a i;j1 a jj We've assumed i + j is odd, else we should switch the parity of all the signs. Notice that a ii 6= 0 for all i as A is upper-triangular and invertible. For xed k = 1;:::;n1, the condition thatAXA 1 2 requires that (3.5.2) is independent of choice of (i;j) such that ji = k. For ji = 1, (3.5.2) simplies to a ii =a jj . If this expression is to be independent of (i;j) such that ji = 1, once we choose a 11 6= 0 anda 22 6= 0, thena ii is determined fori = 3;:::;n. Fork = 2;:::;n 2, if (3.5.2) is independent of (i;j) such thatji =k, then once we choose arbitrarya 1;k and a 2;k+1 , then a i;k+i1 is determined for i = 3;:::;nk + 1. Finally, a 1n does not appear in (3.5.2) for any (i;j), so it may be chosen arbitrarily. This shows that A is uniquely determined by its top two rows (injectivity), and any choice of top two rows a 11 6= 0, a 22 6= 0 and a 21 = 0 denes a matrix A2G (surjectivity). 62 3. THE VARIETY OF ELEMENTARY SUBALGEBRAS Remark 3.5.4. A dierent approach to the computation in Proposition 3.5.3 was shown to the author by E. Friedlander and J. Pevtsova. They noticed that two regular nilpotent elements X and Y dene the same subalgebra if and only if Y =a 1 X +a 2 X 2 +::: +a n1 X n1 , with a 1 6= 0. These n 1 degrees of freedom together with the fact that the regular nilpotent orbit is n 2 n shows that the dimension ofO is n 2 n (n 1) = (n 1) 2 . If Question 3.4.3 has an armative answer, or if any subalgebra dened by a regular nilpotent element is in the orbit of largest dimension, then Proposition 3.5.3 computes the dimension of E(n 1;gl n ) to be (n 1) 2 , which is veried through n = 5 in Table 3, and agrees with equation (3.5.1). By considering a smaller subspace dened by a regular nilpotent element, the following proposition gives a lower bound on the dimension of E(r;gl n ) for the intermediate region 1r<n 1. Proposition 3.5.5. LetX be a regular nilpotent element in gl n and for 1r<n 1 consider the elementary subalgebra r = SpanfX;X 2 ;:::;X r g. Then dim(GL n r ) =n 2 n 1. In particular, dim(E(r;gl n )n 2 n 1 for 1r<n 1. Proof. I claim that the map sending a matrix A2 G r to the n-tuple of elements ink given by (a 11 ;a 22 ;a 12 ;a 13 ;:::;a 1n ) is an isomorphism onto the quasi- ane variety in A n+1 dened by the condition that the rst two coordinates are nonzero. From this claim we have dim(GL n r ) = n 2 (n + 1). The proof of the claim follows similar reasoning of the proof of Proposition 3.5.3, except that the superdiagonals corresponding to ji =k >r must now be 0. It follows that g2 G r if and only if g2 N G (SpanfXg). To see this, note if g2 N G ( r ) and gXg 1 = P c i X i with some c i 6= 0 for i > 1, then gX r g 1 will have nonzero entries in the superdiagonal corresponding to k =r + 1. Hence, for k> 1 a choice of a 1;k determines a i;k+i1 for i = 2;:::;nk + 1. For k = 1, we can still choose a 1;1 and a 2;2 arbitrarily. It follows that dimG r = (n 1) + 2 = n + 1, so that dim(GL n r ) =n 2 (n + 1). Notice we have shown that the corresponding bound on dim(E(r;gl n ) is sharp in the limiting case r = 1. However, Table 3 shows that we may have strict inequality for r = 2;:::;n 2. Motivated by Conjecture 3.5.1, we include two propositions computing the sizes of the dierent conjugacy classes found in Table 2. Proposition 3.5.6. There is one orbit inE(1;gl 2 ), and the number ofF p -rational points is p + 1. Proof. A Sylow p-subgroup of GL(2;p) is isomorphic to Z=pZ, so the Sylow theorems show there is a unique G-conjugacy class. One such group is represented by the matrices 1 a 0 1 a2F p whose stabilizer under conjugation is the group a b 0 c a;c2F p ; b2F p This stabilizer has order p(p 1) 2 , so that the size of the orbit is jGL(2;p)j p(p 1) 2 = (p 2 p)(p 2 1) p(p 1) 2 =p + 1 3.6. COMPUTATIONS FOR THE UPPER-TRIANGULAR UNIPOTENT GROUP 63 The result then follows from Theorem 3.3.3. Proposition 3.5.7. There are three G-orbits in E(2;gl 3 ), two with p 2 +p + 1 F p -rational points, and one with (p 2 +p + 1)(p + 1)(p 1)F p -rational points. Proof. We know from Example 3.3.8 that there are 3 G-orbits of E(2;gl 3 ). The orbit consisting of subalgebras with elements of rank 2 has representative E = * 0 @ 1 0 1 0 1 0 0 0 1 1 A ; 0 @ 1 1 0 0 1 1 0 0 1 1 A + in GL 3 (F p ). The normalizer of E is N GL3(Fp) (E) = 8 < : 0 @ a b c 0 d e 0 0 f 1 A af =d 2 9 = ; which has order p 3 (p 1) 2 . Orbit-stabilizer then gives the size of the orbit: jGL(3;p)j p 3 (p 1) 2 = (p 3 p 2 )(p 3 p)(p 3 1) p 3 (p 1) 2 = (p 2 +p + 1)(p + 1)(p 1) The result for the large orbit follows from Theorem 3.3.3. The proof for the sizes of the other two conjugacy classes is similar, and omitted. Notice that the dimensions computed in Example 3.3.8 and the degree of the polynomials in Proposition 3.5.7 provide evidence for the veracity of Conjecture 3.5.1. Example 3.5.8. If Conjecture 3.5.1 is true, then a computation with Magma shows that E(3;gl 4 ) contains 8 G-orbits dened over F p of dimensions 3, 3, 6, 7, 7, 7, 8, and 9 (compare with Figure 9). The two orbits of dimension 3 must be closed, and by irreducibility, all orbits of degree less than 9 lie in the boundary of the orbit of dimension 9. Further understanding of the partial order on the orbits is necessary to determine which intermediate orbits lie in the closure of others. For example, is the orbit of dimension 6 closed, or is one or more of the 3 dimensional orbits found in its closure? Can this question be answered by some group theoretic condition on the G-conjugacy classes of subgroups corresponding to the orbits of dimensions 3, 3, and 6, per Question 3.3.10? Table 4 records how manyF p -rational points lie in each orbit. 3.6. Computations for the upper-triangular unipotent group In this section we look at the Lie algebra u n which is the Lie algebra of the unipotent radical of the upper triangular matrices. We will compute E(r;u n ) for variousr andn. The moral to this section is that such computations are hard, not particularly pleasant, and do not give much promise for a general theory (at least not one easily accesible from the computations shown here). Example 3.6.1 (E(2;u 3 )). Let g = u 3 , the set of all strictly upper triangular 3 3 matrices. u 3 is the Lie algebra of the Lie group U 3 of unipotent upper triangular 3 3 matrices, so that U 3 acts on u 3 by conjugation. Consider E(2;u 3 ). Let A = (a ij ) and B = (b ij ) be two basis vectors for some 2E(2;u 3 ), so A and B have the form: 64 3. THE VARIETY OF ELEMENTARY SUBALGEBRAS Table 4. Orbits ofE(3;gl 4 ) dened overF p dim(O) #O(F p ) 3 (p 2 + 1)(p + 1) 3 (p 2 + 1)(p + 1) 6 (p 2 +p + 1)(p 2 + 1)(p + 1) 2 7 (p 2 +p + 1)(p 2 + 1)(p + 1)p(p 1) 7 (p 2 +p + 1)(p 2 + 1)(p + 1) 2 (p 1) 7 (p 2 +p + 1)(p 2 + 1)(p + 1) 2 (p 1) 8 (p 2 +p + 1)(p 2 + 1)(p + 1)p 2 (p 1) 9 (p 2 +p + 1)(p 2 + 1)(p + 1) 2 p(p 1) 2 A = 0 @ 0 a 12 a 13 0 0 a 23 0 0 0 1 A ; B = 0 @ 0 b 12 b 13 0 0 b 23 0 0 0 1 A Since A and B commute, we know that a 12 b 23 a 23 b 12 = 0. We will assume that p > 2 so that the requirement that A and B have trivial [p]-restriction is automatically satised, and leads to no new equations. We split into two cases: Case 1. Suppose a 12 = 0. Then it follows that either a 23 = 0 or b 12 = 0. If a 23 = 0, we know that a 13 6= 0 so we can normalize A to assume it has the form: A = 0 @ 0 0 1 0 0 0 0 0 0 1 A Then by subtracting b 13 A from B, we can assume B has the form: B = 0 @ 0 b 12 0 0 0 b 23 0 0 0 1 A Any choice of (b 12 ;b 23 )6= (0; 0) (up to scalar multiple) gives a unique elemen- tary subalgebra. In the case that b 12 = 0, a dimension argument shows that the space spanned by A and B must be the space spanned by A = 0 @ 0 0 1 0 0 0 0 0 0 1 A ; B = 0 @ 0 0 0 0 0 1 0 0 0 1 A Notice this space is already covered in the rst subcase if we choose b 12 = 0 and b 23 = 1. 3.6. COMPUTATIONS FOR THE UPPER-TRIANGULAR UNIPOTENT GROUP 65 Case 2. Suppose a 12 6= 0. Then as before, we can normalize both A and B and assume they are of the form: A = 0 @ 0 1 a 13 0 0 a 23 0 0 0 1 A ; B = 0 @ 0 0 b 13 0 0 b 23 0 0 0 1 A Our commutation relation now requires that b 23 = 0, so that after normalization we have the matrices: A = 0 @ 0 1 0 0 0 a 23 0 0 0 1 A ; B = 0 @ 0 0 1 0 0 0 0 0 0 1 A The subalgebra generated by these matrices is the same as that considered in the rst case under the choice b 12 = 1: Here A and B have switched roles. In summary, the elementary abelian subalgebras of dimension 2 inside of u 3 are given up to scalar multiple by a choice of nonzero element in A 2 , ie, as sets, E(2;u 3 ) =P 1 . To show thatE(2;u 3 ) embeds intoP 2 as the closed subvarietyP 1 , lets look at the Pl ucker coordinates. Let E 12 , E 13 , and E 23 be an ordered basis for u 3 , and write A = (0; 1; 0) and B = (b 12 ; 0;b 23 ) as vectors. Form the 3 2 matrix whose columns are A and B, and embed intoP 2 using the Pl ucker coordinates: 0 @ 0 b 12 1 0 0 b 23 1 A 7! [b 12 : 0 :b 23 ]2P 2 The image of the Pl ucker embedding is thus the copy of P 1 sitting in P 2 whose second homogeneous coordinate vanishes. Since u 3 is not commutative, it follows thatE(3;u 3 ) =?. The following propo- sition computesE(1;u 3 ). Proposition 3.6.2. Let pn. ThenE(1;u n ) = Grass(1;u n ) =P(u n ). Also: E (n)(n 1) 2 ;u n = 8 > < > : ? n 3 * 0 1 0 0 !+ n = 2 Proof. Since pn, all matrices in u n have trivial [p]-restriction. Also, since all one-dimensional subalgebras are abelian, it follows that any one-dimensional space in u n generates an elementary abelian subalgebra. Now, as for the second claim, notice that dim(u n ) = n(n 1)=2, and that u n is abelian if and only if n = 2. Proposition Proposition 3.6.2 helps us deal with the limiting cases of u n . We've now computed E(r;u 3 ) for all r, and E(r;u 4 ) for r = 1 and r = 6. Let us now consider u 4 for r = 2; 3; 4; 5. As before, we assume p 5 so that the [p]-restriction condition adds no new requirements to our subalgebras. In considering u 4 , it will be helpful to write down the general commutation relations. If A and B are strictly upper triangular, the condition that AB = BA leads to the following three equations: (3.6.1) a 12 b 23 =b 12 a 23 66 3. THE VARIETY OF ELEMENTARY SUBALGEBRAS (3.6.2) a 34 b 23 =b 34 a 23 (3.6.3) a 12 b 24 +a 13 b 34 =b 12 a 24 +b 13 a 34 Since A and B are arbitrary, all matrices we consider must pairwise satisfy the above three equations. Suppose that a 23 6= 0. Then Equation (3.6.1) and Equation (3.6.2) yield: a 12 b 23 a 23 =b 12 a 34 b 23 a 23 =b 34 Notice that ifa 12 = 0 thenb 12 = 0, and also ifa 12 6= 0 then (b 12 ;b 23 ) =(a 12 ;b 23 ). The same argument holds for a 34 , so that in all cases we have (b 12 ;b 13 ;b 23 ) = (a 12 ;a 13 ;a 23 ) for some 2 k. We have thus proven the following useful lemma for our calculations. Lemma 3.6.3. Let A i , i = 1;:::;r, be a collection of matrices in u 4 that pair- wise commute. Suppose there is some i with A i = A, and a 23 6= 0. Then each superdiagonal of the A i is a scalar multiple of the superdiagonal of A. In the case that there is some matrix whose (2; 3) entry is non-zero, the Lemma 3.6.3 allows us to normalize the rest of the matrices and assume they have no superdiagonal. Let's use this result to calculateE(r;u 4 ) for r = 3; 4; and 5. Example 3.6.4 (E(5;u 4 )). Let A, B, C, D, and E be linearly independent, pair- wise commuting matrices in u 4 . If all (2; 3) entries are zero, a dimension argument shows that the space spanned by these matrices must be the ve dimensional space: 8 > > < > > : 0 B B @ 0 a b c 0 0 0 d 0 0 0 e 0 0 0 0 1 C C A j (a;b;c;d;e)2A 5 9 > > = > > ; This space does not commute. It follows that there is some matrix, say A, with a 23 6= 0. Normalizing, it follows that B, C, D, and E have no superdiagonal. This contradicts linear independence. In conclusion, we haveE(5;u 4 ) =?: Example 3.6.5 (E(4;u 4 )). Let A, B, C, and D be four linearly independent, pairwise commuting matrices in u 4 . First assume there is some matrix, sayA, with a 23 6= 0. Normalizing we can assume that B = E 13 ; C = E 14 , and D = E 24 . Equation (3.6.3) applied to A and B, and A and D yields a 34 = 0 and a 12 = 0 respectively. Thus we may assumeA =E 23 so that we obtain the four dimensional elementary subalgebra =hE 13 ;E 14 ;E 23 ;E 24 i. Next assume that all (2; 3) entries are 0. If also all the (1; 2) entries are zero, we obtain the nonabelian four dimensional subalgebra k hE 13 ;E 14 ;E 24 ;E 34 i. Hence we may assume a 12 6= 0 and by normalizing, a 12 = 1 and b 12 =c 12 =d 12 = 0. If all of the (3; 4) entries are 0, then again we obtain a nonabelian subalgebra, so some (3; 4) entry is nonzero. If the only such one is a 34 , then B, C, and D generate the three dimensional subalgebra k hE 13 ;E 14 ;E 24 i; but this then does not commute with A. Hence, we may assume thatb 34 6= 0. Normalizing all four matrices and checking commutation relations will show that C and D must both be E 14 , violating linear independence. We conclude that there is only one elementary subalgebra of u 4 of dimension 4, so thatE(4;u 4 ) is just a point. 3.6. COMPUTATIONS FOR THE UPPER-TRIANGULAR UNIPOTENT GROUP 67 Example 3.6.6 (E(3;u 4 )). LetA,B, andC be three pairwise commuting, linearly independent matrices in u 4 . First assume all (1; 2) entries and all (3; 4) entries are 0. Then Equation (3.6.1), Equation (3.6.2), and Equation (3.6.3) are automatically satised, and the matrices A,B andC multiply like 22 matrices (only the upper right 22 block is nonzero). Thus the corresponding elementary subalgebras are Grass(3;gl 2 ) = Grass(1;gl 2 ) = P 3 ) (to identify Grass(3;gl 2 ) with Grass(1;gl 2 ) one must choose an isomorphism between gl 2 and its dual). One family of subalgebras in this copy ofP 3 is of the form k haE 13 +E 14 ;bE 13 + E 23 ;cE 13 +E 24 i, where (a;b;c)2A 3 . The Pl ucker embedding gives: 0 B B @ a b c 1 0 0 0 1 0 0 0 1 1 C C A 7! [c :b :a : 1]2P 3 We see that this corresponds to the ane open set in P 3 where x 3 doesn't vanish. There are three other families like this, and they form the other three standard ane open sets ofP 3 . Let E 14 be the matrix whose only nonzero entry is a 1 in the rst row and fourth column. A tedious calculation of checking cases and normalizing matrices shows that those elementary subalgebras not in this copy of P 3 are in one of the following families: F 1 = 8 > > < > > : k * 0 B B @ 0 a b 0 0 0 c 0 0 0 0 1 0 0 0 0 1 C C A ; 0 B B @ 0 0 a 0 0 0 0 1 0 0 0 0 0 0 0 0 1 C C A ;E 14 + (a;b;c)2A 3 9 > > = > > ; F 2 = 8 > > < > > : k * 0 B B @ 0 1 0 0 0 0 a b 0 0 0 c 0 0 0 0 1 C C A ; 0 B B @ 0 0 1 0 0 0 0 c 0 0 0 0 0 0 0 0 1 C C A ;E 14 + (a;b;c)2A 3 9 > > = > > ; F 3 = 8 > > < > > : k * 0 B B @ 0 0 a 0 0 0 0 b 0 0 0 1 0 0 0 0 1 C C A ; 0 B B @ 0 1 b 0 0 0 0 c 0 0 0 0 0 0 0 0 1 C C A ;E 14 + (a;b;c)2A 3 9 > > = > > ; One can check that each set of three matrices are nilpotent and pairwise com- muting. F 1 andF 2 intersect in A 3 A 2 (this intersection is F 1 fa = 0g = F 2 fc = 0g).F 3 is disjoint fromF 1 andF 2 . In conclusion, we have the very rough description: E(3;u 4 ) =P 3 G (F 1 [ F 2 ) G F 3 =P 3 G (A 3 G A 2 ) G A 3 Example 3.6.7 (E(2;u 4 ). Without giving the details of the computation, any elementary subalgebra of dimension 2 inside of u 4 is in one of the following disjoint 68 3. THE VARIETY OF ELEMENTARY SUBALGEBRAS families: Grass(2; 4) = 8 > > < > > : k * 0 B B @ 0 0 0 0 0 0 0 0 0 0 0 0 1 C C A ; 0 B B @ 0 0 0 0 0 0 0 0 0 0 0 0 1 C C A + 9 > > = > > ; F 1 = 8 > > < > > : k * 0 B B @ 0 1 a b 0 0 0 c 0 0 0 0 0 0 0 0 1 C C A ; 0 B B @ 0 0 d e 0 0 0 a 0 0 0 1 0 0 0 0 1 C C A + (a;b;c;d;e)2A 5 9 > > = > > ; F 2 = 8 > > < > > : k * 0 B B @ 0 1 a 0 0 0 b c 0 0 0 d 0 0 0 0 1 C C A ; 0 B B @ 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 C C A + (a;b;c;d)2A 4 9 > > = > > ; F 3 = 8 > > < > > : k * 0 B B @ 0 1 0 b 0 0 a c 0 0 0 d 0 0 0 0 1 C C A ; 0 B B @ 0 0 1 e 0 0 0 d 0 0 0 0 0 0 0 0 1 C C A + (a;b;c;d;e)2A 5 9 > > = > > ; F 4 = 8 > > < > > : k * 0 B B @ 0 0 a 0 0 0 b c 0 0 0 1 0 0 0 0 1 C C A ; 0 B B @ 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 C C A + (a;b;c)2A 3 9 > > = > > ; F 5 = 8 > > < > > : k * 0 B B @ 0 0 a b 0 0 c 0 0 0 0 1 0 0 0 0 1 C C A ; 0 B B @ 0 0 0 d 0 0 0 1 0 0 0 0 0 0 0 0 1 C C A + (a;b;c;d)2A 4 9 > > = > > ; Above, there is a subcollection of Grass(2; 4) such that its union withF 2 and F 4 isP 4 . Together these are represented by the family: 8 > > < > > : k * 0 B B @ 0 a b 0 0 0 c d 0 0 0 e 0 0 0 0 1 C C A ; 0 B B @ 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 C C A + [a :b :c :d :e]2P 4 9 > > = > > ; The last observation of the previous example leads us to the following general result. Proposition 3.6.8. P ( n 2 )2 ,!E(2;u n ). Proof. For every element ofP ( n 2 )2 there is a unique one-dimensional subspace with no (1;n) entry. Paired with the matrixE 1;n which spans the center of the Lie algebra u n , this subspace is an elementary subalgebra. Remark 3.6.9. We end this section by noting that an inclusion of restricted Lie algebras h,! g induces an inclusion of projective varieties E(r;h),!E(r;g). This follows because if is an elementary subalgebra of h of dimension r, then '() is an elementary subalgebra of g of dimensionr. So in some sense,E(r;) is a functor 3.6. COMPUTATIONS FOR THE UPPER-TRIANGULAR UNIPOTENT GROUP 69 from the category of restricted Lie algebras and injective Lie algebra maps to the category of projective varieties and injective maps of varieties. This observation may help further computations, by piecing together E(r;g) from the E(r;h) for certain h g. CHAPTER 4 The commuting variety of one-parameter subgroups 4.1. Motivation and Preliminaries If G is an innitesimal group scheme of height 1, then we've seen two vari- eties associated to G which house support varieties for G-modules. The rst is the variety of innitesimal one-pararmeter subgroups V 1 (G) and the second is the variety of elementary subalgebras E(r; Lie(G)). It is shown in [52] that V 1 (G) is homeomorphic to the restricted nullconeN p (Lie(G)), and it follows from denition that E(1; Lie(G)) is the projectivized restricted nullcone. We thus have a natural homeomorphism ProjV 1 (G) =E(1;g) In this section, we make use of this observation to develop a theory that unites those of innitesimal one-parameter subgroups and elementary subalgebras. We pause to mention a notational change. It is unfortunate that the letter r has been commonly used in the literature to describe both the height of an innitesimal group scheme, and the rank of an elementary subalgebra. In this section, we will need to vary each of these parameters independently, so we have to change notation. Since the study of innitesimal group schemes precedes that of elementary subalgebras, we will allow r to continue to describe the height of an innitesimal group scheme G. We will use s to denote the dimension of an elementary subalgebra in the case r = 1. In other words, we will use the notation E(s;g) to refer to the variety of s-dimensional elementary subalgebras of g. This variety has a natural inclusion into the Grassmannian of s-planes in g. Our goal is as follows. To any positive integers r;s, we aim to dene the commuting variety of one-parameter subgroups, denoted E(s;G r ), such that there are natural isomorphisms (4.1.1) E(1;G r ) = ProjV r (G) and E(s;G 1 ) =E(s;g) We've used the notation G r to describe an innitesimal group scheme of height r so that the height of the group scheme is encoded in the notation. One may think of G r as the rth Frobenius kernel of an arbitrary group scheme G. Table 5 summarizes howE(s;G r ) unies the theories of innitesimal one-parameter subgroups and elementary subalgebras. In preparation for the denition ofE(s;G r ), we dene exponential maps asso- ciated to r-tuples of pairwise commuting p-nilpotent matrices. We will also prove Lemma 4.1.2 which will play a role in Section 4.5 when we consider representations ofG. In Section 4.2, we review the theory of innitesimal one-parameter subgroups, and we observe nilpotents in the representing algebra of V r (GL n ). In Section 4.3, 70 4.1. MOTIVATION AND PRELIMINARIES 71 Table 5. E(s;G r ), a new support space s = 1 s> 1 r = 1 E(1;g) = ProjV 1 (G) E(s;g) r> 1 ProjV r (G) E(s;G r ) we prove some equivalences involving the properties of the exponential maps dis- cussed here and the matrices used to dene them. This all builds to Section 4.4 in which we dene the projective variety of commuting one-parameter innitesi- mal subgroups, E(s;G r ). Finally, in Section 4.5, we associate to a G-module M and a k-rational point in E(s;G r ) an action of the k-algebra k[t 1 ;:::;t s ]=(t p i ) in a \continuous" way. By continuous, we mean that for each m = 1;:::;s and for each k-rational point 2E(s;G r ), there is an open set U containing and a local operator m :k[U] M!k[U] M that specializes to the action of t m at (see Figure 12). This construction is motivated by similar constructions made in [13] forE(s;g). In that paper, the authors use these local operators and suciently nice modulesM with certain constant properties to construct vector bundles onE(s;g). Although we don't construct vector bundles here, it is our hope that the local op- erators we dene are the rst step towards obtaining vector bundles on E(s;G r ). The work presented in this chapter is ongoing and joint with Eric Friedlander. We brie y recall some notions from Chapter 1. Iff :G!H is a map of group schemes over k, then to each commutative k-algebra R we obtain a map of groups f R :G(R)!H(R). In what follows, if we deem there is no possibility for confusion, we suppress the notation f R in favor of the simpler notation f. If I G k[G] is the kernel of the counit " : k[G]! k, then we dene the algebra of distributions on G to be the set of all k-linear maps ' : k[G]! k such that there is some n with '(I n G ) = 0. The algebra of distributions is denoted Dist(G) and has the structure of an associative algebra whose product is dened using the comultiplication in k[G]. A map f : G! H induces a map f : Dist(G)! Dist(H) as follows. If '2 Dist(G), thenf (') ='f :k[H]!k. It can be checked that'f vanishes on I n H if ' vanishes on I n G so that f is indeed a map from Dist(G)! Dist(H). Dist(G) has a natural ltration given by dening Dist n (G) :=f' : k[G]! kj '(I n+1 G ) = 0g, and it can be shown that Dist n (G) Dist m (G) Dist n+m (G). We thus have a covariant functor Dist from the category of ane group schemes over k to the category of ltered associative algebras over k. If we dene Dist + n (G) to be those '2 Dist n (G) such that '(1) = 0, then it can be shown that for ' 2 Dist + n (G) and 2 Dist + m (G) we have ['; ] := ' ' 2 Dist + n+m1 (G). In particular, Dist + 1 is a Lie algebra, which is naturally isomorphic to the usual Lie algebra structure of G as dened by left- invariant derivations of k[G]. It is also the case that f restricted to Dist + 1 (G) is equal to df : Lie(G)! Lie(H) under the natural isomorphism just mentioned. We will therefore use f and df interchangeably. Example 4.1.1. In what follows, the additive group,G a , plays a central role. IfR is a commutative k-algebra, then we deneG a as a functor to beG a (R) = (R; +). The coordinate algebra of G a is isomorphic to k[T ], with (T ) = 1 T +T 1 and"(T ) = 0. It follows that I Ga = (T ), and tracing through the denitions shows 72 4. THE COMMUTING VARIETY OF ONE-PARAMETER SUBGROUPS that Dist(G a ) is the innite-dimensional vector space with basis given by n for n2 N, where n (T m ) = n;m . The product structure on Dist(G a ) is given by n m = n+m n n+m . Let GL n be the ane group scheme which assigns to a commutative k-algebra R the group ofnn invertible matrices with entries inR. IfA is ap-nilpotentnn matrix (i.e.,A p = 0), then we can dene a map of group schemes exp A :G a ! GL n given by (4.1.2) exp A (s) := p1 X i=0 (sA) i i! for any commutative k-algebra R and any s2R. More generally, letA = (A 0 ;:::;A r1 ) be anr-tuple of pairwise commuting,p- nilpotent matrices. Then we can dene a map of group schemes exp A :G a ! GL n given by (4.1.3) exp A (s) := r1 Y i=0 exp Ai (s p i ) for any commutative k-algebra R and any s2R. Lemma 4.1.2. Let G and H be two ane group schemes over k and let f;g : G! H be two maps of ane group schemes. If for all commutative k-algebra R the images of f R :G(R)!H(R) and g R :G(R)!H(R) commute in H(R), then the images of f and g commute in Dist(H). Furthermore, if for all commutative k-algebra R and all r2 R the elements f R (r) and g R (r) commute in H(R), then f (x) and g (x) commute in Dist(H) for any x2 Dist(G). Proof. The proof of both statements follows from applying the covariant func- tor Dist to the following diagrams (which commute by hypotheses): GG fg // HH m H ## H GG gf // HH m H ;; GG fg // HH m H ## G D ;; D ## H GG gf // HH m H ;; 4.2. Innitesimal one-parameter subgroups The commuting variety of innitesimal one-parameter subgroups,E(s;G r ), gen- eralizes the variety of innitesimal subgroups as dened and studied in [51] and [52]. Before we make our denition ofE(s;G r ), we brie y review the scheme of in- nitesimal one-parameter subgroups. We show in Proposition 4.2.4 that in general, this scheme is not reduced. Let G be an ane group scheme over an algebraically closed eld k of char- acteristic p. In [51], A. Suslin, E. Friedlander, and C. Bendel explored the ane 4.2. INFINITESIMAL ONE-PARAMETER SUBGROUPS 73 scheme V r (G) of innitesimal 1-parameter subgroups of height r. In particular, they computed the coordinate algebra ofV r (G) for groupsG with an embedding of exponential type intoGL n . In this section, we observe nilpotents in the coordinate algebra ofV r (G) for classicalG, ie, we show theV r (G) is not reduced. Throughout, k is an algebraically closed eld of characteristic p. The Scheme V r (G). Denition 4.2.1. LetG be an ane group scheme over k, and letr be a positive integer. Dene a functor V r (G) from commutative k-algebras to sets as follows: V r (G)(A) := Hom(G a(r);A ;G A ) for all commutativek-algebrasA, where the morphisms are in the category of ane group schemes over A. Recall thatG a(r) is ther th Frobenius kernel of the additive groupG a , thus its coordinate algebra is the Hopf algebrak[G a(r) ] =k[x]=(x p r ), wherex is a primitive element. Also, recall that for any ane group scheme G over k and any k-algebra A, we can form the ane group scheme G A over A whose coordinate algebra is k[G A ] =A k k[G]. In the case G =GL n , a map of ane group schemesG a(r);A !GL n;A gives a linear representation of G a(r);A on A n . This is equivalent to giving a map A n ! A n A[G a(r);A ] givingA n the structure of a comodule for A[G a(r);A ] =A[x]=(x p r ) ([54],x3.1,3.2). This observation led to the following computation. Proposition 4.2.2 ([51], Proposition 1.2). Consider the polynomial ring in rn 2 indeterminates R = k[x k ij j 1 i;j n; 1 k r]. Let A k 2 M n (R) be dened by (A k ) ij = x k ij : Then the functor V r (GL n ) is representable, with representing k- algebra given by R=I where I is generated by the rn 2 homogeneous polynomial equations of degree p given by A p k = 0 for k = 1;:::;r together with the r 2 n 2 homogeneous polynomial equations of degree 2 given by [A k ;A k 0] =A k A k 0A k 0A k = 0 for 1k<k 0 r. Here we take 1 2 = 0. In short, the proposition states thatV r (GL n ) is an ane scheme overk with coordinate ringR=I as described above. The signicance of this proposition is partly seen in [51], Theorem 1.5, in which the authors prove that the coordinate algebra of V r (G) is a nitely generated commutative k-algebra for any G, by choosing some embedding G,!GL n . Nilpotents. Here we demonstrate thatV r (GL n ) is not reduced. Equivalently, this amounts to showing that the coordinatek-algebraR=I contains nilpotents, i.e., that I is not a radical ideal. We will require the use of the following computation. Lemma 4.2.3. Let A = (a ij )2M n (k). Then Tr(A m ) = X 1i1;:::;imn a i1i2 a i2i3 :::a imi1 : In particular, Tr(A p ) = Tr(A) p . Proof. The rst claim follows from the formula (4.2.1) (A m ) ij = X 1i2;:::;imn a ii2 a i2i3 :::a imj 74 4. THE COMMUTING VARIETY OF ONE-PARAMETER SUBGROUPS which can be proven by induction. To show that Tr(A p ) = Tr(A) p , let us examine one particular choice of indices i 1 ;i 2 ;:::;i p and determine how often the terma i1i2 a i2i3 :::a ipi1 appears in the sum of Tr(A p ). Sincep is prime, we have only two possibilities. Ifi 1 =i 2 =::: =i p =i, then the term a p ii appears only once in the sum. If the indices are not all equal, then any cyclic permutation necessarily produces a dierent ordered set of indices (here we use the fact thatp is prime). Sincek is commutative, we see that the term a i1i2 a i2i3 :::a ipi1 has a coecient of p = 0 in the sum. Thus we have Tr(A p ) = n X i=1 a p ii = n X i=1 a ii ! p = Tr(A) p Proposition 4.2.4. Using the notation of Proposition 4.2.2, for all k = 1;:::;r, the element Tr(A k ) p 2I but Tr(A k )62I. Hence the ideal I is not radical, and the ane scheme V r (GL n ) is not reduced. Proof. By Lemma 4.2.3, Tr(A k ) p = Tr(A p k ) = 0 by the conditions on A k . Hence Tr(A k ) p 2 I. Now notice that Tr(A k ) is a homogeneous polynomial of degree 1. I is a homogeneous ideal generated by homogeneous elements of degrees 2 and p. Thus, any homogeneous element in I has degree greater than or equal to 2, so that Tr(A k )62I. Example 4.2.5. Let's consider the case r = 1. Then R is a polynomial ring over k in the n 2 indeterminates x ij , and (4.2.1) shows that I is generated by the n 2 homogeneous equations of degree p of the form f ij = X 1i2;:::;ipn x ii2 x i2i3 :::x ipj In other words, for r = 1, we only require that A p 1 = 0. However, note that this is equivalent to the condition that ch(A 1 ) = 0. For t = 0;:::;n, let g t be the homogeneous polynomial of degree nt in the x ij 's which is the coecient of t in det(I n A 1 ). Then if J = (g 0 ;:::;g n ), it follows that J and I dene the same variety in A n 2 , so that p J = p I. It can be shown that J is a radical ideal, and since I6=J (by considering degrees), it follows that I is not radical. 4.3. Commuting nilpotent varieties Here we dene a number of commuting nilpotent varieties which lead up to the denition of E(s;G r ). We work rst explicitly with the case G = GL n , and then extend our denitions to arbitrary innitesimal group schemes of heightr. Proposition 4.3.1. Suppose A and B are p-nilpotent matrices. The following are equivalent: (1) exp A (s) and exp B (t) commute in GL n (R) for all commutative k-algebras R and all s;t2R. (2) exp A (X) and exp B (Y ) commute in GL n (k[X;Y ]). (3) exp A (s) and exp B (s) commute in GL n (R) for all commutative k-algebras R and all s2R. (4) exp A (X) and exp B (X) commute in GL n (k[X]). (5) [A;B] = 0. 4.3. COMMUTING NILPOTENT VARIETIES 75 Proof. If (1) holds, then taking R = k[X;Y ], s = X, and t = Y , we obtain (2). Assuming (2), let R be any commutative k-algebra and let s2 R. Dene a map k[X;Y ]! R taking both X and Y to s. This gives a group homomorphism GL n (k[X;Y ])! GL n (R), and the images of exp A (X) and exp B (Y ) under this homomorphism are exp A (s) and exp B (s), respectively. (3) follows. If (3) holds, then taking R = k[X] and s = X, we obtain (4). Assuming (4), consider the following formula for the coecient of X 2 in exp A (X) exp B (X) (viewed as a polynomial in X with coecients in GL n (k)): (Coecient of X 2 in exp A (X) exp B (X)) = 8 < : AB p = 2 A 2 +B 2 2 +AB p6= 2 A similar formula arises for exp B (X) exp A (X), and equating the two shows that [A;B] = 0. Finally, assume (5) and let R be any commutative k-algebra. For any s;t2R we may view s;t2 GL n (R) via the identication s$sI n and t$tI n . Then the elements A; B; s; and t pairwise commute so that any two polynomials in these elements commute as well. (1) then follows from the denition of exp A (s) and exp B (t). Proposition 4.3.2. Let A = (A 0 ;:::;A r1 ) and B = (B 0 ;:::;B r1 ) be two r- tuples of p-nilpotent, pairwise commuting matrices. Then the following are equiva- lent: (1) exp A (s) and exp B (t) commute in GL n (R) for all commutative k-algebras R and all s;t2R. (2) exp A (X) and exp B (Y ) commute in GL n (k[X;Y ]). (3) [A i ;B j ] = 0 for all 0i;jr 1. Proof. If (1) holds, then taking R = k[X;Y ], s = X, and t = Y , we obtain (2). Now we assume (2) and x i;j. A computation shows that the coecient of X p i Y p j in exp A (X) exp B (Y ) is AiBj i!j! . Similarly, the coecient of X p i Y p j in exp B (T ) exp A (S) is BjAi i!j! . By hypothesis, these two coecients must be equal, which shows [A i ;B j ] = 0. Finally, assume (3) and let R be any commutative k-algebra. For any s;t2R we may view s;t2 GL n (R) via the identication s$ sI n and t$ tI n . Then the elements A i ; B j ; s; and t pairwise commute so that any two polynomials in these elements commute as well. (1) then follows from the denition of exp A (s) and exp B (t). In preparation for the proof of the next theorem, let i;j be the coecient of p i in the base p expansion of j, i.e., j = X i0 i;j p i Proposition 4.3.3. Let A = (A 0 ;:::;A r1 ) and B = (B 0 ;:::;B r1 ) be two r- tuples of p-nilpotent, pairwise commuting matrices. Then the following are equiva- lent: (1) exp A (s) and exp B (s) commute in GL n (R) for all commutative k-algebras R and all s2R. 76 4. THE COMMUTING VARIETY OF ONE-PARAMETER SUBGROUPS (2) exp A (X) and exp B (X) commute in GL n (k[X]). (3) The following formula holds for all 0mp r 1: (4.3.1) X i+j=m 0i;j A 0;i 0 :::A r1;i r1 B 0;j 0 :::B r1;j r1 0;i !::: r1;i ! 0;j !::: r1;j ! = X i+j=m 0i;j B 0;i 0 :::B r1;i r1 A 0;j 0 :::A r1;j r1 0;i !::: r1;i ! 0;j !::: r1;j ! Proof. We rst show that (1) is equivalent to (2). If (1) holds, then takingR = k[X] ands =X, we obtain (2). Now we assume (2), and letR be any commutative k-algebra. Any s2R denes a homomorphism k[X]!R which sends X!s. By functoriality, this homomorphism denes a map of groups GL n (k[X])! GL n (R), and the images of exp A (X) and exp B (X) under this homomorphism are exp A (s) and exp B (s), respectively. (1) follows. We nish the proof by showing that (2) is equivalent to (3). A computation shows that the coecient of X m in exp A (X) exp B (X) is given by the left hand side of Equation (4.3.1). Similarly, the coecient of X m in exp B (X) exp A (X) is the right hand side of Equation (4.3.1). Then exp A (X) and exp B (X) commute in GL n (k[X]) if and only if these two coecients are equal for all m. Remark 4.3.4. For a slight simplication, we need only consider terms in Equa- tion (4.3.1) such that i and j are strictly positive. The two terms on either side corresponding to (i;j) = (0;m) and (i;j) = (m; 0) will always cancel. Ignoring these two terms, we write out the left hand side explicitly for a few representative values of m in the following table. m Coecient of X m in exp A (X) exp B (X) 2 A 0 B 0 3 A 2 0 B 0 +A 0 B 2 0 2 p p1 X i=1 A i 0 B pi 0 i!(pi)! p + 1 A 1 B 0 +A 0 B 1 + p2 X i=1 A i+1 0 B pi 0 (i + 1)!(pi)! We note that the entry form = 2 shows that [A 0 ;B 0 ] = 0. It's not clear to the author if the equations described in (3) have some simpler representation. In fact, we're even unsure if these equations imply something as simple as [A 1 ;B 1 ] = 0. For the following proposition, by a bracket-word of length p, we mean any symbol with p 1 nested brackets. For example, [A[A[B[A;B]]]] is a bracket-word of length 5. Proposition 4.3.5. Let A = (A 0 ;:::;A r1 ) and B = (B 0 ;:::;B r1 ) be two r- tuples of p-nilpotent, pairwise commuting matrices. Then the following are equiva- lent: (1) A +B = (A 0 +B 0 ;:::;A r1 +B r1 ) is an r-tuple of p-nilpotent, pairwise commuting matrices for all ;2k. (2) [A i ;B j ] = [A j ;B i ] for all 0i;jr 1 and for each i = 0;:::;r 1, all bracket-words of length p in A i and B i vanish. 4.3. COMMUTING NILPOTENT VARIETIES 77 Proof. Fixi andj. Then [A i +B i ;A j +B j ] =([A i ;B j ]+[B i ;A j ]) = 0 for all ; 2 k if and only if [A i ;B j ] = [A j ;B i ]. Similarly, (A i +B i ) [p] = bracket-words of length p in A i and B i . If all bracket-words of length p vanish, it follows that (A i +B i ) [p] = 0. Conversely, if (A i +B i ) [p] = 0 for all;2k, x = 1 so that we have a polynomial expression in whose coecients are the bracket-words of lengthp inA i andB i . By assumption, this expression is identically 0, so it has innitely many roots. Therefore, each coecient must vanish, so that each bracket-word of length p vanishes. Proposition 4.3.6. Let A = (A 0 ;:::;A r1 ) and B = (B 0 ;:::;B r1 ) be two r- tuples of p-nilpotent, pairwise commuting matrices. If p6= 2, then the following are equivalent: (1) A and B satisfy the conditions of Proposition 4.3.2. (2) A andB satsify the conditions of Proposition 4.3.3 and Proposition 4.3.5. Proof. If [A i ;B j ] = 0 for all 0i;jr 1, Equation (4.3.1) is satised as we may slide all of the A's past the B's on the left hand side, and switch the roles ofi andj to obtain the right hand side. Also, if [A i ;B j ] = 0 for all 0i;jr 1, then [A i ;B j ] = [A j ;B i ] = 0. It follows that (1) implies (2). We use induction on maxfi;jg to prove the converse. If (2) is true, then Remark 4.3.4 shows that [A 0 ;B 0 ] = 0, so that [A i ;B j ] = 0 if maxfi;jg< 1. Now, assume that [A i ;B j ] = 0 for all (i;j) such that maxfi;jg<l. We must show that [A lk ;B k ] = 0 for all k = 0;:::;l. We show this by induction on k. Let k = 0 and consider Equation (4.3.1) for m =p l + 1, which is of the form A l A 0 +A l B 0 +A 0 B l +B 0 B l + (products of A i 's and B j 's with i;j <l) = B l B 0 +B l A 0 +B 0 A l +A 0 A l + (products of B i 's and A j 's with i;j <l) By the induction hypothesis on l, the rightmost terms of both sides are equal, so we're left with the equation A l B 0 +A 0 B l = B l A 0 +B 0 A l which rearranges to [A l ;B 0 ] = [B l ;A 0 ]. However, since we're assuming (2), we also know that [A l ;B 0 ] = [A 0 ;B l ]. Since p6= 2, these two equalities show that [A l ;B 0 ] = [B l ;A 0 ] = 0. Now for some 0<k<l, suppose that [A l ;B q ] = [B l ;A q ] = 0 for all q<k and consider Equation (4.3.1) for m =p l +p k , which is of the form A l A k +A l B k +A k B l +B k B l + (products of A i 's and B j 's where i;j <l or i +j <l +k) = B l B k +B l A k +B k A l +A k A l + (products of B i 's and A j 's where i;j <l or i +j <l +k) By the induction hypothesis on k, the rightmost terms of both sides are equal, so we're left with the equation A l B k +A k B l =B l A k +B k A l which rearranges to [A l ;B k ] = [B l ;A k ]. However, since we're assuming (2), we also know that [A l ;B k ] = [A k ;B l ]. Since p6= 2, these two equalities show that [A l ;B k ] = [B l ;A k ] = 0. The double induction is complete, and we've shown that (2) implies (1). Remark 4.3.7. Notice that thep-nilpotent condition is unnecessary for the proof. 78 4. THE COMMUTING VARIETY OF ONE-PARAMETER SUBGROUPS Remark 4.3.8. Here is another way to phrase Proposition 4.3.6. Consider the three varieties V 1 , V 2 , and V 3 of all pairs of r-tuples of p-nilpotent pairwise com- muting matrices dened by the conditions in Proposition 4.3.2, Proposition 4.3.3, and Proposition 4.3.5, respectively. Then Proposition 4.3.6 states thatV 1 =V 2 \V 3 . The following example illustrates that V 2 andV 3 are indeed dierent varieties. Example 4.3.9. Let n = 4, p = 2, and r = 2. Let E ij be the elementary matrix whose kl-th entry is ki lj . Let A 0 = E 12 , A 1 = E 43 , B 0 = E 14 , and B 1 = E 23 . We leave as a computation to the reader the verication that indeed exp A;R (t) and exp B;R (t) commute in GL n (R) for any commutative k-algebra R and any t2 R. However, one can check that [A 0 ;B 1 ] = [B 0 ;A 1 ]6= 0. 4.4. The commuting variety of one-parameter innitesimal subgroups In this section we nally dene E(s;G r ). We begin by making the denition for groups of exponential type. We rst recall the denition of a group with an embedding of exponential type. For further details, one may consult [19] and [51]. Denition 4.4.1. Let j : G ,! GL n be a map of ane group schemes. Then j is an embedding of exponential type if for every k-algebra A and every p-nilpotent element x2 g A the map exp dj(x) :G a (A)! GL n (A) factors through G(A)! GL n (A). It is shown in [51] that if j : G ,! GL n is an embedding of exponential type, then theA-points of the schemeV r (G) are given byr-tuples of pairwise commuting, nilpotent elements in Lie(G) A. This shows that for a group of exponential type (that is, a group with an embedding of exponential type), the innitesimal one- parameter subgroupsG a !G are completely determined by the Lie algebra Lie(G). For this reason, in the denition that follows we only consider the height 1 structure of G. Let g be a restricted Lie algebra over k, and let r be a positive integer. There is a natural action of S r on g r given by permuting the components. Denition 4.4.2. (1) For 2S r , dene the -bracket on g r to be the following composition: [;] : g r g r id ! g r g r [;] ! g r (2) A subalgebra g r is -elementary if [p] = 0 and [;] = 0. More generally, if is a subset of S r , then is -elementary if [p] = 0 and [;] = 0 for all 2 . If = S r , we say is r-elementary instead of S r -elementary. Remark 4.4.3. Notice that a subalgebra is r-elementary if and only if it is - elementary for = h(1; 2;:::;r)i. Furthermore, notice that a subalgebra is 1- elementary if and only if it is elementary. Denition 4.4.4. Let G = G (r) be an innitesimal group scheme of height r equipped with an embedding of exponential type, let g = Lie(G), and let s be a positive integer. As a set, dene E(s;G r ) to be all r-elementary subalgebras of dimension s. 4.4. THE COMMUTING VARIETY OF ONE-PARAMETER INFINITESIMAL SUBGROUPS 79 In order to prove that E(s;G r ) is a projective subvariety of Grass(s;g r ), we will need the following notion of ans-tuple of totally pairwise commuting elements. Denition 4.4.5. Let s be a positive integer. (1) Let (a 1 ;:::;a s ) be ans-tuple of elements of g r . We say that (a 1 ;:::;a s ) is totally pairwise commuting if [a i ;a j ] = 0 for all i;j = 1;:::;s and all 2S r . (2) LetTC s (N p (g r )) be the set of all totally pairwise commutings-tuples of p-nilpotent elements in g r . Similarly, letTC s (N p (g r )) be the set of all linearly independent, totally pairwise commuting s-tuples of p-nilpotent elements in g r . Notice that TC s (N p (g r )) is Zariski closed inside ofN p (g r ) s because both the bracket and restriction map are given by polynomial equations. It follows that TC s (N (g r )) is Zariski closed inside ofN p (g r ) s . The proof of the following Theorem thus follows that of Proposition 1:3 in [13]. Theorem 4.4.6. The image of the natural embeddingE(s;G r ),! Grass(s;g r ) is a closed subvariety. We now wish to extend our denition of E(s;G r ) to arbitrary innitesimal group schemes of height r. We begin with the observation that if j : G ,! GL n is an embedding of exponential type, and A is an r-tuple of pairwise commuting, p-nilpotent elements in Lie(G), then d exp A (u i ) = A i 2 Dist + 1 (G) = g. Recall that u i 2 kG a is the linear functional on k[T ]=(T p r ) which equals 1 on T p i and 0 elsewhere. Denition 4.4.7. Let G be an innitesimal group scheme of height r, and let 1 ;:::; s :G a !G be s innitesimal one-parameter subgroups of G. (1) We say that 1 and 2 commute if 1 (s) and 2 (t) commute inG(R) for all commutative k-algebras R and all s;t2R. (2) We say that 1 ;:::; s are linearly independent if the s elements ( i (u 0 );:::; i (u r1 ))2 Dist(G) r for i = 1;:::;s are linearly independent in Dist(G) r . (3) Dene an equivalence relation on the set of pairwise commuting, linearly independent s-tuples of innitesimal one-parameter subgroups by declar- ing ( 1 ;:::; s ) ( 0 1 ;:::; 0 s ) if and only if the s-plane in Dist(G) r spanned by the setf( i (u 0 );:::; ( i (u r1 ))g s i=1 is equal to the span of f( 0 i (u 0 );:::; 0 i (u r1 ))g s i=1 . Denition 4.4.8. For an arbitrary group scheme, let E(s;G r ) denote the set of equivalence classes of pairwise commuting, linearly independent s-tuples of inni- tesimal one-parameter subgroups. Due to the equivalences proved in Section 4.3, this denition agrees with Def- inition 4.4.4 for G,! GL n an embedding of exponential type. Notice also that by denition we have the desired natural isomorphisms of Equation (4.1.1). (4.4.1) E(1;G r ) = ProjV r (G) and E(s;G 1 ) =E(s;g) 80 4. THE COMMUTING VARIETY OF ONE-PARAMETER SUBGROUPS 4.5. Pointwise and local operators on modules Throughout this section G = GL n(r) , so Lie(G) = g = gl n . Let kG be the k-linear dual of k[G], and notice that kG = Dist(G) since G is innitesimal. Let M be a nitely generated left module for the k-algebra kG, and unless otherwise noted, tensor products are over k. To every k-rational point 2 E(s;G r ) and a choice of basis for , we will dene an action of k[t 1 ;:::;t s ]=(t p i ) onM. We do this in such a way that for an ane open set of U E(s;G r ) with a chosen section q : U! TC s (N p (g r )), there are s local operators i : k[U] k M! k[U] k M that specialize at a point to the action associated to t i for the basis of given by the section q. Let x l i;j be a basis for gl r n , with 1 i;j n and 0 l r 1. Let M := M n 2 r;s (k) be the set ofn 2 rs matrices with entries ink, and letM denote the set of n 2 rs matrices with linearly independent columns. For future notational purposes, we denote the ring of coordinate functions on M in the following unorthodox way. Let T l;m i;j be the coordinate function which reads o the entry in position (n 2 l + n(i 1) +j;m). Here we have 1ms. Then k[M] =k[T l;m i;j ]. We have a mapM Grass(s;g r ) given by sending a matrix to thek-span of its columns. For any f1;:::;n 2 rg of cardinality s, let U denote all points in Grass(s;g r ) such that the Pl ucker coordinate associated to is non-zero. There is a sectionq :U !M which sends a planeU to the unique matrix whose columns spanU and whose submatrix is the identity. q allows us to identifyU with an ane space of dimensions(n 2 rs), so we may writek[U ] =k[T l;m i;j ]=I whereI is generated by the polynomials (T l;m i;j F(n 2 l+n(i1)+j);m ) forn 2 l +n(i 1) +j2 , where F : !f1;:::;sg is the unique bijection which is increasing as a map from N ! N. Let V = U \E(s;G (r) ), and let Y l;m i;j be the image of T l;m i;j under the composition k[T l;m i;j ] k[U ] k[V ]. We note that the denition of Y l;m i;j = Y l;m; i;j depends on the choice of , but we omit this dependence in the (admittedly horrible) notation. Lemma 4.5.1. Let 2E(s;G (r) ), and choose a basis for denoted byfA l;m g, 0 lr 1, 1ms. For xed m, let A m denote the r-tuple (A 0;m ;:::;A r1;m ). The s elements dexp A m(u r1 ) for m = 1;:::;s are p-nilpotent and pairwise com- mute in kG. Proof. Since dexp A m is a map of k-algebras, and since u r1 is p-nilpotent in kG a , it follows that dexp A m(u r1 ) is p-nilpotent for m = 1;:::;s. By denition of E(s;G r ), thes-tuple of elements in g r (A 1 ;:::;A s ) is totally pairwise commuting so that by Proposition 4.3.2 the elements exp A m(u r1 ) pairwise commute in G. It follows from Lemma 4.1.2 that thes elementsdexp A m(u r1 ) also pairwise commute in kG. Using Lemma 4.5.1, we may dene a pointwise operator at eachk-rational point 2E(s;G r ) given a choice of basis. To review, we aim to give M the structure of a module over the k-algebra k[t 1 ;:::;t s ]=(t p i ). Denition 4.5.2. Let 2 E(s;G r ) be a k-rational point, with basis fA l;m g, where m = 1;:::;s and l = 0;:::;r 1. Fix some m, and let t m act on M via dexp A m(u r1 )2 kG. By Lemma 4.5.1, this denition gives M the structure 4.5. POINTWISE AND LOCAL OPERATORS ON MODULES 81 of a module over the k-algebra k[t 1 ;:::;t s ]=(t p i ). Notice that the module structure depends on the choice of basisfA l;m g. For now we ignore this choice and use the notation M when we consider M as a module over k[t 1 ;:::;t s ]=(t p i ) in this way. We'd like to show that the denition of the pointwise operators is \continuous" in the sense that over an ane open set U E(s;G r ) equipped with a section q : U! TC s (N p (g r )), the pointwise action of any 2 U associated to the basis dened by q() is the specialization of a local operator at . We now aim to dene this local operator over open ane sets dened by the nonvanishing of a Pl ucker coordinate. Fix m such that 1 m s. Dene a map k[V ]-linear map m; = m : k[V ] kG a ! k[V ] kG as follows. First, to dene m , it suces to dene m (1 f) for some f 2 kG a . Further, we may view k[V ] kG as a subset of Hom k[V] (k[V ] k[G];k[V ]) via the identication a f7!a id f. Now we dene ( m (1 f)). For l = 0;:::;r 1 let X l;m be the matrix of indeterminates in k[V ] whose i;j-th entry is Y l;m i;j . For T2k[T ]=(T p r ) =kG a(r) , the symbol exp X m(T ) := r1 Y l=0 exp X l;m(T p l ) represents an element of GL n(r) (k[V ] k[T ]=(T p r )) = Hom kalg (k[G];k[V ] k[T ]=(T p r )). Using this, we dene a map d exp X m :kG a ! Hom k (k[G];k[V ]) by d exp X m(f) = id f exp X m(T ) Finally, dene m (1 f) := id d exp X m(f) To any k-rational point 2 V , we can specialize m at via the following process. LetfA l;m g be the basis of which corresponds to the columns of the matrixq (), i.e., thei;jth entry ofA l;m is the entry in rown 2 l +n(i 1) +j and column m of the matrix q (). denes a map of schemes : Speck!V which gives a map of k-algebras :k[V ]!k. This in turn gives isomorphisms: k k[V] (k[V ] k kG a ) =kG a k k[V] (k[V ] k kG) =kG which can both be dened as (a b f)7!a (b)f. Then dene: m; :kG a =k k[V] (k[V ] k kG a )!k k[V] (k[V ] k kG) =kG by a b f7!a ( m (b f)). Notice the role of in the isomorphisms. For use in the proof of the following theorem, we note the identity d exp X m =d exp A m as maps from kG a to kG. Proposition 4.5.3. The map m; coincides with d exp A m. Proof. We've dened m (1 f) to be the map id d exp X m(f)2 k[V ] kG Hom k[V] (k[V ] k[G];k[V ]). Continue to view k[V ] kG as a subset of Hom k[V] (k[V ] k[G];k[V ]), and leta2k and'2 Hom k[V] (k[V ] k[G];k[V ]). The isomorphismk k[V] (k[V ] k kG) =kG is then given by (a ')7!a 'i where i :k[G]!k[V ] k[G] sends g to 1 g. Thus we have m; (f) = m (1 f)i = (id d exp X m(f))i = d exp X m(f) =d exp A m(f) 82 4. THE COMMUTING VARIETY OF ONE-PARAMETER SUBGROUPS E(s;G r ) V m :k[V ] M!k[V ] M m; =t m :M !M Specialize at Figure 12. Continuity property of the local operators m for m = 1;:::;s (Proposition 4.5.4) Now, to dene the local operator, notice that a simple tensora b2k[V ] kG denes a k[V ]-linear map k[V ] M! k[V ] M via 1 m7! a bm. We can extend linearly so that any element in k[V ] kG denes a k[V ]-linear map k[V ] M ! k[V ] M. Finally, we dene the local operator m to be the element m (1 u r1 )2 k[V ] kG. As above, for a k-rational point 2 V , we can specialize m to to obtain a map m; :M !M . Proposition 4.5.4. Fix 2V . LetfA l;m g be the basis of given by the columns of the matrix q (). Then the map m; :M !M is given by the action of t m as dened in Denition 4.5.2 using the basisfA l;m g. Proof. By Proposition 4.5.3, we have m; = m; (u r1 ) =d exp A m(u r1 ) = t m . Figure 12 summarizes the discussion of this section and the result of Proposi- tion 4.5.4. APPENDIX A Magma code A.1. DrawQuillen The following code was used to create the pictures seen in Appendix B. The code is implemented in Magma, and the output is a list of commands for the graphics program TikZ. The output from Magma was copied and pasted into TikZ and compiled to see the pictures of Appendix B. The code uses the information from Magma's computations to appropriately scale the sizes of the nodes, the spacing between the nodes, the thickness of the edges, and the rectangular frame. A slight adjustment is necessary if the input group is abelian. DrawQuillen:=procedure(G,p);//Input is a finite group G and a prime p //finding the p-rank of G n:=1; while Gcd(p^(n+1),#G) eq p^(n+1) do n:=n+1; end while; //computing the conjugacy classes of elementary abelian p subgroups A:=ElementaryAbelianSubgroups(G:OrderDividing:=p^n); //The maximum and minimum sizes of conjugacy classes //(for the purpose of coloring; adjustment needed if G is abelian) m:=Max({Log(p,A[i]`length): i in [1..#A]}); min:=Min({Log(p,A[i]`length): i in [1..#A]}); if min eq 0 then min:=Min({Log(p,A[i]`length): i in [2..#A]}); end if; //If G is abelian, replace the above section of code with the following m:=Max({Log(p,#A[i]`subgroup): i in [1..#A]}); min:=Min({Log(p,#A[i]`subgroup): i in [1..#A]}); if min eq 0 then min:=Min({Log(p,#A[i]`subgroup): i in [2..#A]}); end if; //Parsing the conjugacy classes by rank and assigning colors //If G is abelian, replace A[j]`length with #A[j]`subgroup below R:=[]; M:=[[]:i in [2..n+2]]; for i in [2..n+1] do T:={@@}; a:=1; for j in [1..#A] do if #A[j]`subgroup eq p^(i-1) then 83 84 A. MAGMA CODE T:=T join {@A[j]`subgroup@}; M[i][a]:=(Log(p,A[j]`length)-min)*300/(m-min)+400; a:=a+1; end if; end for; R[i-1]:=T; end for; S:=[]; for i in [1..#R] do if #R[i] ne 0 then S[i]:=R[i]; end if; end for; R:=S; //Appropriately sizing nodes, spacing between nodes, and edge thickness D:=Max({#R[i]:i in [1..#R]}); w:=10.66/(D-1); c:=Min(3.5*w,3); t:=c/2; h:=Max(w,3/2); H:=h*#R+1; //opening TikZ code and printing rectangular frame and nodes print "\\begin{figure}\n\\centering\n\\begin{tikzpicture}"; printf "\\draw[fill=blue!5!white] (-6.33,-1) rectangle (6.33,\%o);\n",H; printf "\\node[fill=white, circle, outer sep=0pt, inner sep=\%opt](0) at (0,0){};\n",c; v:=1; for i in [1..#R] do d:=-(#R[i]+1)*w/2; for j in [1..#R[i]] do printf "\\definecolor{new}{wave}{\%o}\\node[fill=new, circle, outer sep=0pt, inner sep=\%opt](\%o) at (\%o,\%o){};\n",M[i+1][j], c,v,d+j*w,h*i; v:=v+1; end for; end for; //printing paths to identity subgroup for i in [1..#R[1]] do printf "\\definecolor{top}{wave}{\%o}\\connect{0}{\%o}{bottom color=white, top color=top, minimum height=\%opt}\n",M[2][i],i,t; end for; //printing paths between all other conjugacy classes for i in [1..(#R-1)] do for l in [1..#R[i+1]] do B:=ElementaryAbelianSubgroups(R[i+1][l]:OrderEqual:=#R[i][1]); for k in [1..#R[i]] do for j in [1..#B] do if IsConjugate(G,R[i][k],B[j]`subgroup) then a:=&+[#R[m]:m in [1..i]]-#R[i]+k; b:=&+[#R[m]:m in [1..i]]+l; printf "\\definecolor{top}{wave}{\%o} \\definecolor{bottom}{wave}{\%o} \\connect{\%o}{\%o}{bottom color=bottom, top color=top,minimum height=\%opt}\n", A.2. QUILLENCOMPLEX 85 M[i+2][l],M[i+1][k],a,b,t; break j; end if; end for; end for; end for; end for; //ending TikZ code print "\\end{tikzpicture}\n\\caption{}\n\\end{figure}\n"; end procedure; The output can then be copied and pasted into a TikZ compiler with the following in the preamble. \usepackage[rgb]{xcolor} \usepackage{tikz} \usetikzlibrary{calc} \def\connect#1#2#3{ \path let \p1 = ($(#2)-(#1)$), \n1 = {veclen(\p1)}, \n2 = {atan2(\x1,\y1)} in (#1) -- (#2) node[#3, midway, sloped, shading angle=\n2+90, minimum width=\n1, inner sep=0pt, #3] {}; } A.2. QuillenComplex The following code computes the Quillen complex of a nite group G with respect to a prime p. In other words, the output is the simplicial complex associ- ated to the partially ordered set of elementary abelian p-subgroups of G. If G is connected, reductive, dened over F p and if p satises condition (?) from Chap- ter 2, then the Quillen complex for G(F p ) is isomorphic to the Quillen complex of elementary subalgebras (see Remark 3.2.18). QuillenComplex:=function(G,p); n:=1; //compute p-rank of G while Gcd(p^(n+1),#G) eq p^(n+1) do n:=n+1; end while; //compute elementary abelian p subgroups of G A:=ElementaryAbelianSubgroups(G:OrderDividing:=p^n); //parse elementary abelian p subgroups by rank R:=[]; U:={i:i in [1..#A]}; for i in [1..n] do T:={@@}; for j in U do if #A[j]`subgroup eq p^i then T:=T join Class(G,A[j]`subgroup); Exclude(~U,j); end if; 86 A. MAGMA CODE end for; R[i]:=T; end for; //Create graph with paths between ranks differing by 1 S:=&join R; Q:=Digraph<S|>; for i in [1..(#R-1)] do for k in [1..#R[i]] do for l in [1..#R[i+1]] do if R[i][k] subset R[i+1][l] then Q+:=[Vertices(Q)!R[i][k],Vertices(Q)!R[i+1][l]]; end if; end for; end for; end for; Q:=StandardGraph(Q); //Create paths for all inclusions using those paths constructed above while Q ne SatGraph(Q) do Q:=SatGraph(Q); end while; Q:=UnderlyingGraph(Q); //Return the complex associated to the graph return CliqueComplex(Q); end function; SatGraph:=function(G); E:=IndexedSetToSet(Edges(G)); R:=[]; for i in [1..Order(G)] do T:=[]; for j in E do if InitialVertex(j) eq VertexSet(G).i then Append(~T,j); Exclude(~E,j); end if; end for; R[i]:=T; end for; for i in [1..Order(G)] do for j in [1..#R[i]] do l:=Index(TerminalVertex(R[i][j])); for k in [1..#R[l]] do G+:=[InitialVertex(R[i][j]),TerminalVertex(R[l][k])]; end for; end for; end for; return G; end function; A.3. DETECTING NILPOTENTS IN THE VARIETY OF INFINITESIMAL SUBGROUPS 87 A.3. Detecting nilpotents in the variety of innitesimal subgroups The following three programs return the ideal dening the scheme V r (G) for GL n , SL n , and Sp 2n . They can be used to make conjectures concerning the nilpo- tent index of k[V r (G)]. No progress has been made in proving anything about the \size" of the nilpotents in k[V r (G)]. VrGLn:=function(r,n,p); R:=PolynomialRing(GF(p),r*n^2); X:=[Matrix(R,n,n,[R.i:i in [(j-1)*n^2+1..j*n^2]]):j in [1..r]]; Q:=[]; for i in [1..r-1] do for j in [i+1..r] do Q cat:=ElementToSequence(X[i]*X[j]-X[j]*X[i]); end for; end for; for i in [1..r] do Q cat:=ElementToSequence(X[i]^p); end for; I:=ideal<R|Q>; return I; end function; VrSLn:=function(r,n,p); R:=PolynomialRing(GF(p),r*n^2); X:=[Matrix(R,n,n,[R.i:i in [(j-1)*n^2+1..j*n^2]]):j in [1..r]]; Q:=[]; for i in [1..r-1] do for j in [i+1..r] do Q cat:=ElementToSequence(X[i]*X[j]-X[j]*X[i]); end for; end for; for i in [1..r] do Q cat:=ElementToSequence(X[i]^p); Append(~Q,Trace(X[i])); end for; I:=ideal<R|Q>; return I; end function; VrSPn:=function(r,m,p); n:=2*m; R:=PolynomialRing(GF(p),r*n^2); X:=[Matrix(R,n,n,[R.i:i in [(j-1)*n^2+1..j*n^2]]):j in [1..r]]; Q:=[]; for i in [1..r-1] do for j in [i+1..r] do Q cat:=ElementToSequence(X[i]*X[j]-X[j]*X[i]); end for; end for; A:=[Submatrix(X[i],1,1,m,m):i in [1..r]]; B:=[Submatrix(X[i],1,1+m,m,m):i in [1..r]]; C:=[Submatrix(X[i],1+m,1,m,m):i in [1..r]]; D:=[Submatrix(X[i],1+m,1+m,m,m):i in [1..r]]; for i in [1..r] do Q cat:=ElementToSequence(X[i]^p); Q cat:=ElementToSequence(A[i]+Transpose(D[i])); Q cat:=ElementToSequence(B[i]-Transpose(B[i])); 88 A. MAGMA CODE Q cat:=ElementToSequence(C[i]-Transpose(C[i])); end for; I:=ideal<R|Q>; return I; end function; APPENDIX B Visualizations of Quillen's category The pictures seen here have been drawn according to the discussion in Ap- pendix A.1. The input required to draw such a picture is a nite group and a prime p. The nodes in the pictures represent the conjugacy classes of elementary abelian p-groups. The identity subgroup is placed at the bottom, and the nodes are arranged vertically by rank. The color of a node is assigned based on the size of the conjugacy class and with respect to the wavelength of visible light. That is, a small conjugacy class corresponds to the color violet (which has the smallest wavelength), and a large conjugacy class will be colored red (which has the largest wavelength). A conjugacy class of an arbitrary size is colored based on its place within the spectrum red-orange-yellow-green-blue-violet. The exceptions to this are the pictures for GL 2 (F 128 ) and (Z=2Z) 5 . For GL 2 (F 128 ), all conjugacy classes (except for the maximal one) have the same size. For the abelian group (Z=2Z) 5 , all conjugacy classes have a unique element. In both of these cases, we've colored the nodes according to their rank. An edge is connected between a conjugacy class C of rank r and a conjugacy class C 0 of rank r + 1 if there are groups E2C and E 0 2C 0 such that EE 0 . Besides being nice to look at, these picture encode the following information about cohomology and elementary subalgebras. (1) The number of layers (which is equal to the p-rank of ) is the Krull dimension of the cohomology of . For example, the Krull dimension of the cohomology of Sp 6 (F 2 ) is 6. (2) The maximal ideal spectrum of the cohomology of is stratied by the elementary abelian p-subgroups as discussed in Proposition 3.1.2. For example the maximal ideal spectrum of the cohomology of S 12 for p = 2 is composed of four ane pieces of dimension 6 and two ane pieces of dimension 5 (notice the two nodes of rank 5 near the center that are not contained in any node of rank 6). The way these pieces glue is encoded in the edges (see Example 3.1.3, Figure 7, and Figure 8). (3) If is the group of F p points of a connected reductive group, and if p satises condition (?), then the nodes of rank r represent the orbits of E(r;g) dened over F p , and the color of the node corresponds to the dimension of the orbit (Section 3.5). The careful observer will note that for most of the pictures, each row of nodes proceeds left to right along the visible spectrum from violet to red. This is because Magma automatically arranges conjugacy classes in order of increasing size. This leaves Figures B.1 - B.6 and B.8 with a pleasant asymmetry. However, for Figures B.7, B.9, and B.10, we've shued the placement of the nodes in a way to recapture some of the symmetry. 89 90 B. VISUALIZATIONS OF QUILLEN'S CATEGORY Here we make a few observations about some of the pictures. This list of observations is certainly not exhaustive (there is a lot more information in the pictures than is worth explicitly writing down), but we include these observations as illustrating examples of the mathematical information contained in the pictures. For some of the pictures, one must zoom in to clearly see the nodes and connecting edges. The quality of the pictures is very high, so one may zoom in quite close without losing clarity due to pixelation. (1) Figure B.1 shows Quillen's category for GL 4 (F 5 ) and p = 5. We see that the Krull dimension of the cohomology is 4, and that the spectrum of the cohomology is composed of an ane piece of dimension 4 along with seven ane pieces of dimension 3 (those rank 3 conjugacy classes not contained in the class of rank 4). There are 12 orbits ofE(2;gl 4 ) dened overF 5 . (2) Figure B.2 shows Quillen's category for SL 4 (F 5 ) and p = 5. As with GL 4 (F 5 ), we see that the Krull dimension of the cohomology is 4, however here the spectrum of the cohomology is composed of an ane piece of dimension 4 along with ten ane pieces of dimension 3. Also, there are now 17 orbits ofE(2;gl 4 ) dened overF 5 . (3) Figure B.3 shows Quillen's category for SL 3 (F 9 ) and p = 3. We see that the Krull dimension of the cohomology is 4, and that the spectrum of the cohomology is composed of three ane pieces of dimension 4. Notice that these ane pieces only glue along subspaces of dimension 2, as no rank 3 subgroup is contained in two rank 4 subgroups. In order to determine the orbits of E(r;sl 3 ) dened over F 9 , we would have to know which nodes correspond to F 9 -linear subgroups. This computation is not dicult for Magma, but not implemented here. (4) Figure B.4 shows Quillen's category for GL 5 (F 2 ) and p = 2. Here the Krull dimension is 6, and the cohomology is a union of two ane pieces of dimension 6 and four ane pieces of dimension 4 (note that the rank 5 conjugacy classes are all contained in a rank 6 conjugacy class). (5) Figure B.5 shows Quillen's category for GL 2 (F 128 ) andp = 2. We see that the Krull dimension of the cohomology is 7, and that the spectrum of the cohomology is composed of one ane piece of dimension 7. Similar observations can be made about Figures B.6 - B.10. Or one can just appreciate their beauty. B. VISUALIZATIONS OF QUILLEN'S CATEGORY 91 Figure B.1. GL 4 (F 5 ), p=5 Figure B.2. SL 4 (F 5 ), p=5 92 B. VISUALIZATIONS OF QUILLEN'S CATEGORY Figure B.3. SL 3 (F 9 ), p=3 Figure B.4. GL 5 (F 2 ), p=2 B. VISUALIZATIONS OF QUILLEN'S CATEGORY 93 Figure B.5. GL 2 (F 128 ), p=2 94 B. VISUALIZATIONS OF QUILLEN'S CATEGORY Figure B.6. Sp 6 (F 2 ), p = 2 B. VISUALIZATIONS OF QUILLEN'S CATEGORY 95 Figure B.7. S 12 , p=2 96 B. VISUALIZATIONS OF QUILLEN'S CATEGORY Figure B.8. A 14 , p=2 B. VISUALIZATIONS OF QUILLEN'S CATEGORY 97 Figure B.9. The extraspecial group of order 128 with two ele- ments of order 4, p = 2 Figure B.10. (Z=2Z) 5 , p = 2 Bibliography [1] J. L. Alperin and L. Evens. Varieties and elementary abelian groups. Journal of Pure and Applied Algebra, 26(3):221{227, 1982. [2] George S. Avrunin. Annihilators of cohomology modules. Journal of Algebra, 69(1):150{154, 1981. [3] George S. Avrunin and Leonard L. Scott. Quillen stratication for modules. Inventiones Mathematicae, 66(2):277{286, 1982. [4] P. Bardsley and R. W. Richardson. Etale slices for algebraic transformation groups in char- acteristic p. 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Abstract (if available)

Abstract Over a field of large enough characteristic, we use the canonical Springer isomorphism between the unipotent variety of a connected, reductive group and the nilpotent variety of the group's Lie algebra to study the projective variety of elementary subalgebras, as defined by Jon Carlson, Eric Friedlander, and Julia Pevtsova. When our structures are defined over finite fields, we relate certain computable information about elementary abelian subgroups of Chevalley groups with rational points and orbits of the variety of elementary subalgebras. We also construct the commuting variety of one-parameter subgroups which simultaneously generalizes the variety of elementary subalgebras and the variety of one-parameter subgroups, as defined by Andrei Suslin, Eric Friedlander, and Christopher Bendel. We employ Magma to make many computations and draw certain visualizations of the theory.

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Creator Warner, Harry Jared (iv)

Core Title Springer isomorphisms and the variety of elementary subalgebras

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Mathematics

Publication Date 04/03/2015

Defense Date 03/13/2015

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag affine group schemes,algebraic groups,elementary Abelian subgroups,elementary subalgebras,OAI-PMH Harvest,representation theory,restricted Lie algebras,Springer isomorphism,support varieties

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Friedlander, Eric M. (committee chair), Guralnick, Robert M. (committee member), Jonckheere, Edmond A. (committee member), Montgomery, M. Susan (committee member)

Creator Email hjwarner@usc.edu,JaredWarner4@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c3-541250

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Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

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