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Blocks of finite group schemes

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Blocks of finite group schemes

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Content BLOCKS OF FINITE GROUP SCHEMES by Paul Sobaje A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulllment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (MATHEMATICS) August 2011 Copyright 2011 Paul Sobaje Table of Contents Abstract iii Introduction 1 Chapter 1: Background 4 1.1 Ane Group Schemes . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Blocks of an Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 Blocks of Group Algebras . . . . . . . . . . . . . . . . . . . . . . . 9 Chapter 2: Cohomological Support Varieties 14 2.1 Recollections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2 Blocks and Subvarieties of V G . . . . . . . . . . . . . . . . . . . . . 16 Chapter 3: p-Points 20 3.1 Denitions and Theorems . . . . . . . . . . . . . . . . . . . . . . . 20 3.2 The Factorization Requirement . . . . . . . . . . . . . . . . . . . . 22 3.3 Invariants of Blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Chapter 4: Hochschild Cohomology 30 4.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.2 Comparing The Varieties . . . . . . . . . . . . . . . . . . . . . . . . 33 Chapter 5: Reductive Algebraic Groups 38 5.1 Notation and Background Results . . . . . . . . . . . . . . . . . . . 38 5.2 Rank Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 5.3 Block Decompositions . . . . . . . . . . . . . . . . . . . . . . . . . 42 5.4 G 1 of Reductive Groups . . . . . . . . . . . . . . . . . . . . . . . . 43 5.5 Blocks of SL 2(r) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 Bibliography 47 ii Abstract We study the block theory of a nite group scheme G over an algebraically closed eld of positive characteristic. Our primary interest will be in studying blocks by means of invariants for modules. These invariants include cohomological support varieties as well as p-point support spaces, the latter developed by E. Friedlander and J. Pevtsova. iii Introduction Let G be a nite group scheme over an algebraically closed eld k of character- istic p > 0, and let k[G] denote the coordinate algebra (or representing algebra) of G. The algebra k[G] is a nite dimensional commutative Hopf algebra, and representations of G are equivalent to right co-modules for k[G], which in turn are equivalent to left modules of the \group algebra" kG := Hom k (k[G];k). As kG is nite dimensional, it can be decomposed uniquely as an algebra into the direct product of its indecomposable two-sided ideals, which we call its \blocks," and write as kG =B 0 B r . Any kG-module breaks up as the direct sum of modules which \lie in a block," thus the study of kG-mod reduces to the study of the categories B i -mod. In the case thatG is a nite group, the study of blocks has a rich theory which dates back to the work of R. Brauer. The key notion in this theory is that of a defect group of a block B, which can be dened as the smallest p-subgroup DG with the property that all modules lying in B are summands of modules induced from kD. The subgroupD is unique up to conjugacy inG, and determines much about the structure of a blockB as an algebra, as well as about its representation theory. In particular, the more complicated the defect group, in terms of its structure as a group, the more complicated the representation theory of the block. 1 Based on the eectiveness of defect groups, it would seem desirable to adapt this theory to the more general setting of nite group schemes, with unipotent sub- group schemes taking the place ofp-subgroups (which are the unipotent subgroup schemes for nite groups). However, key features of p-subgroups which do not generalize to arbitrary unipotent subgroups seem to stand in the way of making such a generalization. Beyond that, results from R. Farnsteiner and A. Skowronski [FS02] in the case of p-restricted Lie algebras suggest that a general defect theory may not even be the correct thing to look for. Fortunately there have been developments in recent years in the representation theory of nite group schemes that provide other ways in which to study blocks. In [FP05] and [FP07], E. Friedlander and J. Pevtsova developed the theory of p- points, which gives a \representation-theoretic" description of support spaces for modules of nite groups schemes. These papers not only extend familiar properties of support varieties for nite groups and innitesimal group schemes to arbitrary nite group schemes, but the methods used have additionally been exploited to create newer, and possibly more powerful, invariants for modules. It is therefore the aim of this thesis to make an investigation into the block theory of arbitrary nite group schemes, with most of our inquiries centered around the interaction between blocks and the afore mentioned invariants for modules. This thesis is organized into ve chapters which we now brie y describe, high- lighting some of our main results. The rst chapter contains the relevant back- ground information on blocks of algebras and nite group schemes. The chapter ends with a result, Proposition 1.3.5, which shows that the structure of B as a module under the left-adjoint action yields information about the p-divisibility of the dimensions of all B-modules. In Chapter 2, we introduce cohomological support varieties and look at the \support variety of a block", denoted V G (B), a 2 variety which has previously been considered by Farnsteiner and others. We then prove, in three seperate propositions, various properties and characteristics of these block varieties. In Chapter 3 we look at the theory ofp-points, recalling how block theory actually gures into the denition of a p-point. The main result of this chapter is Theorem 3.2.3, which proves that we need only require that a p-point factors through a unipotent subgroup scheme in order for the space of p-points to provide a representation-theoretic description of the cohomological variety. The Hochschild cohomology ring of a block B, denoted as HH (B), is the primary object of investigation in Chapter 4. We prove in Theorem 4.2.3 that if G is a nite group scheme whose principal block B 0 has only one simple module, then there is an isomorphism modulo nilpotent elements between HH (B 0 ) and H (G;k). In Proposition 4.2.7, we prove that for any nite group scheme G and any kG-block B, the Krull dimension of HH (B) is equal to the dimension of the block variety V G (B). In the nal chapter, we make some explicit computations of block varieties for Frobenius kernels of reductive algebraic groups. In particular, we compute in Theorem 5.5.2 the block varieties for blocks of various Frobenius kernels of SL 2 . 3 Chapter 1 Background The aim of this rst chapter is to give some background on the two mathematical concepts present in the title of this thesis: nite group schemes and blocks of an algebra. In the rst section we will dene and cite some basic results about nite group schemes, while the second section will be devoted to the notion of blocks of an arbitrary nite dimensional algebra. The third section combines the previous two, looking at special features of blocks of a Hopf algebra, and specically those which are \group algebras" of a nite group scheme. In reviewing this material, we will be closely following the presentation from a few main sources: nite group schemes from [Jan03] and [Wat79] and block theory from [Alp86]. As said in the introduction, we will always work under the assumption that k is an algebraically closed eld. 1.1 Ane Group Schemes Denition 1.1.1. An ane group scheme G over a eld k is a representable covariant functor from the category of commutative k-algebras to groups. We denote the representing algebra by k[G], and call it the coordinate algebra of G. If k[G] is nite dimensional, then G is called a nite (ane) group scheme. Remark 1.1.2. Strictly speaking, the functor Hom k-alg (k[G]; ) is naturally iso- morphic to the functor which is G followed by the forgetful functor from groups to sets. 4 It can be shown, as it is elegantly in [Wat79], that the commutative k-algebras representing group functors are precisely those which carry the additional structure of a Hopf algebra. We recall that a k-algebra A is a Hopf algebra if there are k- algebra morphisms :A!A A (co-multiplication) :A!k (co-unit) :A!A op (antipode or co-inverse) which satisfy the following commutative diagrams: A // A A 1 A A 1 // A A A A = && M M M M M M M M M M M M // A A 1 A A 1 // A A && M M M M M M M M M M M M // A A 1 A A 1 // k A (right) co-module of a Hopf algebra A is a k-vector space M, together with a linear map M :M!M A which satises a commutative diagram similar to the one for co-associativity listed above. We can now give the following denition of a representation of G: 5 Denition 1.1.3. A representation of an ane group schemeG is a k-vector space M which is a co-module for the Hopf algebra k[G]. If G is a nite group scheme, then its coordinate algebra k[G] is nite dimen- sional, and so the dual spacek[G] also has the structure of a Hopf algebra. We will denote k[G] by kG, and call it the group algebra of G. As the co-multipliation of kG comes from dualizing the multiplication of k[G], we see that kG is co- commutative. The category of (right) co-modules for k[G] is equivalent to the category of left-modules forkG (see [Jan03]), and so in this sense the study of rep- resentations of nite group schemes boils down to the study of modules of nite dimensional co-commutative Hopf algebras. Example 1.1.4. IfG is an abstract nite group, then its group algebrakG (in the usual sense of the denition) is a well-known example of a co-commutative Hopf algebra, and thus corresponds to a nite group scheme. Though perhaps confusing, we will also call the corresponding functor G. Note also that the term \group algebra" is unambiguous here, as both notions agree. The coordinate algebra k[G] = kG = k jGj ; that is, as an algebra k[G] is justjGj copies of k, and the co-algebra structure comes from dualizing the multiplication in kG. As a functor, for any local k-algebra A, G(A) =G (the nite group). Example 1.1.5. Suppose G is any ane group scheme such that the coordinate algebra k[G] is a nitely generated algebra. We can associate to it various nite group schemes in the following way. For any r > 0, the ideal I r = (fx p r jx2I g), where I is the augmentation ideal of k[G], is itself a Hopf ideal, hence k[G]=I r is also a commutative Hopf algebra, whose corresponding group functor we will denote by G r . These group algebras will be nite dimensional since they will be nitely generated by elements whose p r -th powers lie inside the image of k ,! 6 k[G]=I r . This also shows every element is either nilpotent or invertible, so k[G]=I r is local. The G r are called the Frobenius kernels of G, as they correspond to the kernels of various iterations of the Frobenius mapF :G!G (see [Jan03] for more details on this). As we are working under the assumption that k is algebraically closed, in this case the nite group schemes which correspond to nite groups are precisely the ones whose coordinate algebra is reduced, which we call constant group schemes. At the other extreme we have the Frobenius kernels, which are exam- ples of nite group schemes whose coordinate algebras are local, which we call innitesimal group schemes. According to a fundamental structure theorem, every nite group scheme is a semi-direct product of an innitesimal group scheme and a constant group scheme. 1.2 Blocks of an Algebra Let A be a nite dimensional k-algebra. By nite dimensionality, it is clear that there exists a decomposition of k-algebras A =A 1 A n , with each A i inde- composable. Such a decomposition is given by a setfe i g of central primitive orthogonal idempotents which sum to 1 A , and we have Ae i = A i . It is not too hard to show that such a decomposition is unique ([Alp86, 4.1]), in the sense that if A = B 1 B m is another such decomposition with corresponding idempo- tents f j , then m =n, and after reordering, e i =f i . These unique indecomposable summands are called the blocks of A. Since the decomposition ofA is given by central idempotents, an algebra decom- poses if and only if its center does. By the following well-known result (cf. [Wat79, 7 6.2]), nite dimensional commutative algebras decompose whenever they are not local. Lemma 1.2.1. Every nite dimensional commutative k-algebraA is a direct prod- uct of local algebras. Remark 1.2.2. When working over an algebraically closed eld, this result then says that every element in the center of a block can be written as an element in k plus a nilpotent element. The blocks of A break up the module category of A according to the action of the primitive central idempotents. Specically, we have for any A-moduleM that M =e 1 Me n M ([Alp86, 4.2]). If turns out for somei thatM =e i M, then M is said to lie in the block A i . Clearly any indecomposable module must lie in a block. It is also true that M lies in A i if and only if all submodules and quotients of M lie in A i . In particular, this is true if and only if all composition factors of M lie in A i . We next recall how it is possible to obtain information about the blocks of an algebra without actually nding the explicit algebra decomposition. Since simple modules are indecomposable, each simple lies in some block. The blocks then determine a partition (or equivalence relation) on the set of simple modules of A based on which block they lie in. This next proposition, taken from [Alp86] shows that this partition of simple modules can be determined in a few dierent ways. Proposition 1.2.3. [Alp86, 4.3] If S and T are simple A-modules then the fol- lowing are equivalent: 1) S and T lie in the same block; 2) There are simple A-modules S =S 1 ;S 2 ;:::;S m =T 8 such that S i ;S i+1 ; 1 i < m, are composition factors of an indecomposable pro- jective A-module; 3) There are simple A-modules S =T 1 ;T 2 ;:::;T n =T such that T i ;T i+1 ; 1 i < n, are equal or there is a non-split extension of one of them by the other. Remark 1.2.4. There is also a notion of blocks for the module category of an arbitrary algebraic group, which does not correspond in general to the module category of a nite dimensional algebra. In this case, the denition of blocks comes from the third equivalence relation above. 1.3 Blocks of Group Algebras We now will turn our attention to blocks of the group algebrakG of a nite group schemeG. While for arbitrary algebras the blocks have no obvious relation to one another, with a Hopf algebra there is some interdependence. This is due to the fact that the maps ;, and , and the diagrams they satisfy, put extra requirements on how the blocks must t together. In a moment we will give a proposition which illustrates this notion. We rst observe thatk has at least one structure as akG-module coming from the co-unit , and with this action is called the trivial module. It is a simple module, and it follows from the previous section that it must lie in some block, which is referred to as the principal block. We will usually write the block decomposition of kG as 9 kG =B 0 B r whereB 0 denotes the principal block. Another way to think of the principal block is to observe that the co-unit takes each central idempotent ofkG to either 0 or 1, and takes their sum to 1, so there is a unique such idempotent e 0 with (e 0 ) = 1, and the principal block is the block corresponding to this idempotent. The principal block is in many respects the most important block ofkG. Among other things, the cohomology ring of kG is determined by its principal block. Additionally, the theory of defect groups dictates that for group algebras of nite groups the principal block always has the most complicated representation type. While this is not always the case for more general nite group schemes, we do have the following well-known result which is true for any nite dimensional Hopf algebra. Proposition 1.3.1. Let H be a nite dimensional Hopf algebra with principal block B 0 . Then H is semi-simple i B 0 is semi-simple. Proof. If H is semi-simple, then its blocks are all matrix algebras, hence they are all semi-simple. Conversely, if B 0 is semi-simple, then the trivial module k is a projective H-module. If M is an arbitrary H-module, M = k M which is projective due to the fact that the tensor product over k of any H-module with a projective H-module is projective (see [Ben91a][3.1.5]). Thus every H-module is projective and so H is semi-simple. To illustrate the earlier comment about the way the blocks t together into a Hopf algebra, this proposition, combined with the Artin-Wedderburn structure theorem, says that the principal block is isomorphic to k only if every other block is a matrix algebra over k (again we assume k is algebraically closed). 10 We will next show how the Hopf algebra structure ofkG can be used to obtain a \left-adjoint" action onkG and its blocks, a structure which will be utilized many times throughout this thesis. First we recall that for any algebra A, we can form its enveloping algebra A e := A A op . Left A e -modules correspond to AA bi-modules, and in particular A is a left A e -module by dening the action of a simple tensor a b on an element x2A as (a b):x =axb. The two-sided ideals of A are then the A e -submodules, and the blocks of A are the indecomposable summands. For any nite dimensional Hopf algebra H, there is a map of algebras from H to H e given by the composition of maps (id ) . Following the notation of [PW09], we will denote this composite algebra map as . The action of H e on H then pulls back via to give a module structure of H on itself (and also its blocks), called the left-adjoint action. We will sometimes denote this structure by (H) and (B) for a block, but often will write it simply as H or B when the context makes the action clear. We note that for nite groups, the left-adjoint action arises from the group acting on itself via conjugation, and similarly for nite group schemes, where G(A) acts on itself by conjugation for every commutative k-algebra A. We have the following important lemma, proved in [PW09], which later will be recalled when we relate the cohomology ring of a group algebra to its Hochschild cohomology ring. Lemma 1.3.2. [PW09] Let H be a nite dimensional Hopf algebra. The map :H!H e is injective, and H e is free as a right and left module over H via this embedding. Furthermore, there is an isomorphism of left H e -modules: H e (H) k =H. 11 Another result we will use is based on the following lemma, whose proof we give here for lack of a good reference: Lemma 1.3.3. Let k be algebraically closed, and A;C be nite dimensional k- algebras. The isomorphism classes of simple A C-modules are the modules S i T j , where S i and T j run over the iso-classes of simple A-modules and C- modules respectively. Proof. Sincek is algebraically closed, the algebraA=Rad(A) C=Rad(C) is semi- simple ([Lan02, XVII 6.3]). It follows that Rad(A C) Rad(A) C+A Rad(C). But the ideal on the right is easily shown to be nilpotent, and as the radical is the largest nilpotent ideal, we must then have that the previous set inclusion is an equality. Thus the simple A C-modules are the simple modules of A=Rad(A) C=Rad(C), proving the claim. Corollary 1.3.4. LetkG be the group algebra of a nite group schemeG, andB a block of kG. Then as a module under the left adjoint action, B has a ltration of modules whose factors are of the form S T , where S and T lie in B. Proof. It is clear that as akG kG op -module,B lies in the blockB B op . It is also not hard to see that the simple B op -modules come from taking the linear duals of simpleB-modules (and vice-versa). By the previous lemma, this implies that as a kG e -module, B has a composition series with factors isomorphic to S T . This composition series remains a ltration when restricting to (kG) kG e , whose factors are still the same (though possibly not simple upon restriction). Although B is indecomposable as a kG e -module, it is not necessarily so when restricted to (kG)kG e . The following straight-forward proposition relates the p-divisibility of the dimensions ofB-modules to its structure under the left-adjoint action. 12 Proposition 1.3.5. Let B be a block of kG, with module structure coming from the left-adjoint action. If the trivial module is not a direct summand of B, thenp divides the dimension of every B-module. Proof. First we observe that if N is a B-module, then the algebra map B ! Hom k (N;N) is a map of G-algebras when B has left-adjoint action. Also, for any kG-module M, the trace map tr : Hom k (M;M)! k is a kG-module homomor- phism, since if a2kG;C2 Hom k (M;M), we have tr(a:C) = tr(a (1) C(a (2) )) = (tr(a (1) C(a (2) )) = (tr((a (2) )a (1) C) = tr(((a (2) )a (1) )C) = tr((a)C) =a:tr(C) Suppose now that p - dim k (N) for some module N lying in B. Then the composite of maps k i ,!B !Hom k (N;N) tr !k 1=dimN ! k is the identity on k. Thus, k is a summand of B. Hence if k is not a summand it must be that p divides the dimension of every B-module. 13 Chapter 2 Cohomological Support Varieties In this chapter we will introduce cohomological support varieties for nite group schemes. The rst section will provide a recap of the general theory, while in the second we will focus on a subvariety determined by the modules in a block. As we will note again later, some of proofs of results used in this chapter depend on the theory of p-points which we will wait to introduce until the following chapter. 2.1 Recollections It is well-known that for a nite dimensional Hopf algebra H, the cohomology ring H (H;k) := Ext H (k;k) is graded-commutative, where the product structure can be given either by the cup product, or by Yoneda composition, as the two agree. If the Hopf algebra is co-commutative, it corresponds to the group algebra of a nite group scheme G, and in this case we write H (G;k) := Ext kG (k;k). A fundamental theorem of E. Friedlander and A. Suslin shows that H (G;k) is a nitely generated k-algebra, and for any nite dimensionalG-moduleM, H (G;M) is a nitely generated H (G;k)-module. If the characteristic of k6= 2, then the graded-commutativity implies that everything in odd degree squares to 0, so we lose none of these nite generation results by considering the commutative ring: H (G;k) := M i0 H 2i (G;k) 14 In characteristic 2, we just set H (G;k) := H (G;k). We will denote the maximal ideal spectrum of H (G;k) by V G . One can then dene, as rst done by Jon Carlson for nite groups, the notion of a cohomological support variety for a module M of a nite group scheme. The basic idea is the following: for any pair of nitely generated G-modules M;N, Ext kG (M;N) = Ext kG (k; Hom k (M;N)) is a nite Ext kG (k;k)-module. Following standard notation (see [Ben91b, 5.7]), we let I G (M;N) denote the annihilator in Ext kG (k;k) of Ext kG (M;N). We then dene dene the relative support variety V G (M;N) as the sub- variety of V G which is the maximal ideal spectrum of H (G;k)=I G (M;N). Of particular interest is the case V G (M;M), which we simply denote as V G (M) and call the support variety of M. Note that in this case, the annihilator I G (M;M), which we will write asI G (M), is given as the kernel of the algebra homomorphism M : Ext kG (k;k)! Ext kG (M;M), or equivalently as the kernel of the map i : Ext kG (k;k)! Ext kG (k; Hom k (M;M)), where i is induced by the inclusion of G-algebras k,! Hom k (M;M). One of the reasons that support varieties are a useful piece of data to attach to modules is that they behave well with respect to various module operations. The following theorem lists many of the nice properties of support varieties for modules of nite group schemes. For nite groups this information can be found in [Ben91b], however we again note that the validity of this theorem for arbitrary nite group schemes relies on the work of Friedlander and Pevtsova and their notion of a p-point, which we will introduce in the next chapter where we can investigate them more thoroughly. Theorem 2.1.1. ([Ben91b, 5.1.1], [FP05]) Let G be a nite group scheme, and assume all modules referenced below are nite dimensional. Then support varieties satisfy the following properties: 15 (1) V G (k) =V G . (2) V G (M 1 M 2 ) =V G (M 1 )[V G (M 2 ). (3) V G (M 1 M 2 ) =V G (M 1 )\V G (M 2 ). (4) V G (M) = 0 if and only if M is projective. (5) If 0! M 1 ! M 2 ! M 3 ! 0 is exact, then for any permutation of f1; 2; 3g V G (M (1) ) (V G (M (2) )[V G (M (3) )) (6) dim V G (M) =cx(M), where cx(M) is the complexity of M. (7) If i :HG is a closed embedding of nite group schemes, then V H (M) = (i ) 1 (V G (M)). Since the algebra Ext kG (k;k) is graded, and the idealsI G (M) are homogeneous, we can also consider the projective varieties Proj V G (M). These projective support varieties will come up in the next section, as well as in the following chapter. 2.2 Blocks and Subvarieties of V G LetB be a block of a nite group schemeG. The intersection of all closed subsets of V G which contain the supports of all B-modules is itself a closed subset of V G , and is the smallest closed set with this property. We will denote this subvariety by V G (B). Since every module lying in B has a composition series made up of simpleB-modules, properties (2) and (5) from 2.1.1 imply that for any B-module M, V G (M) S V G (S i ), where S i ranges over all simple B-modules. From this we easily get the equality 16 V G (B) = [ S i 2B V G (S i ) The interest in this subvariety is demonstrated by the following theorem due to Farnsteiner which relates the dimension of V G (B) to its representation theory. Theorem 2.2.1. [Far07b] Let B be a block of kG. If dimV G (B) 3, then B has wild representation type. Besides trying to compute the dimension, we can consider other topological aspects of this space. For example, when is it irreducible, or at least connected (in the projectivization)? It will not in general be irreducible, as that would imply that the entire support space of G, which corresponds to the support variety of the principal block, is always irreducible, and there are many examples in which this is false. However, the weaker condition of connectedness does hold. Proposition 2.2.2. Let B be a block of a nite group scheme G. Then the projective variety Proj V G (B) is connected. Proof. According to Carlson's theorem [Car84] (stated for nite group schemes in [CFP07, 3.4]), Proj V G (U) is connected for any indecomposable kG-moduleU. In particular this is true for all simple modules. We also know by [Ben91b, 5.7.12] that ifM 1 ;M 2 arekG-modules such that Ext i kG (M 1 ;M 2 )6= 0 for somei> 0, then Proj V G (M 1 )\ Proj V G (M 2 )6=;. Now, suppose that Proj V G (B) was the union to two non-intersecting closed sets. For a simple module S, it follows by the property of connectedness that Proj V G (S) must be contained entirely in one of these two closed sets. This par- titions the simple modules into two groups S 1 ;S 2 with simple modules in one group not having support intersecting the support of any module in the other 17 (the projectivized supports that is). However, for these modules to all be in the same block, there must be simple modules S 1 2 S 1 ;S 2 2 S 2 with either Ext 1 kG (S 1 ;S 2 )6= 0 or Ext 1 kG (S 2 ;S 1 )6= 0. But as shown above, this would imply that Proj V G (S 1 )\ Proj V G (S 2 )6=;, thus Proj V G (B) must be connected. If we considerB as a left-module in the usual way (as a summand of the regular representation), then it will only be the case that V G (B) =V G (B) =f0g if B is a matrix algebra. However, it does turn out that if we look at B as a module under the left-adjoint action, then we do get equality withV G (B). We thank J. Pevtsova for pointing out that the statement of this proposition is analogous to a result proved by A. Premet in the context of support varieties of reduced enveloping algebras ([Pre98, 2.2]), though the method of proof is quite dierent. Proposition 2.2.3. LetB be a block of a nite group scheme G, and considerB as a module under the left-adjoint action. Then V G (B) =V G (B). Proof. Let U be any module lying in B and consider the module B U. We have a k-linear map f : B U ! U given by f(b u) = bu. If a2 kG, then f(a(b u)) =f(a (1) b a (2) u) =f(a (1) b(a (2) ) a (3) u) = a (1) b(a (2) )a (3) u. It follows from the coassociativity of and the property of the antipode that this last sum collapses to a (1) b(a (2) )u =abu. Thus f is a homomorphism of kG modules. We also have a maph :U!B U dened by sendingu toe u wheree is the central idempotent for B. It's not hard to see that this is also a kG-module homomorphism, and since e acts as the identity on U, fh = id U , hence U is a summand ofB U. This last statement immediately implies thatV G (U)V G (B) for all U lying in B, hence we can conclude that V G (B)V G (B) To get the other containment, by Corollary 1.3.4 we know that B has a has a ltration by modules of the form S T with S and T lying in B, and so 18 V G (B) [ (V G (S)\V G (T ))V G (B) Finally, we observe that for blocks of nite groups, the support variety of a block equals the inclusion of the support variety of a defect group, as would be expected. Proposition 2.2.4. Let G be a nite group, and B a block of kG having defect group D. Then V G (B) =i (V D ). Proof. IfM is aB-module, then it is a summand of an inducedD-moduleN, hence V G (M) i (V D (N)) (see [FPS07]), and so V G (B) i (V D ). On the other hand, there is an indecomposable B-module U which has vertex D and trivial source, thus it follows that V D (U) =V D . Hence i (V D )V G (U)V G (B). 19 Chapter 3 p-Points In this chapter we look at the theory of p-points, introduced by Friedlander and Pevtsova in [FP05]. As always, we will introduce the relevant material, and pay particular attention to the interaction with blocks. As we will see, part of the technical nature of the denition of a p-point is due to the block structure of a nite group scheme. 3.1 Denitions and Theorems Before stating the following denition, we recall that a group schemeU is unipotent if the trivial module is the only simple U-module. Equivalently, U is unipotent if it has a closed embedding into the subgroup schemeU n of strictly upper triangular matrices of GL n , for some n. Denition 3.1.1. [FP05] Let G be a nite group scheme. A p-point of kG is an algebra map :k[t]=(t p )!kG such that is left- at (i.e. (kG) is a projective k[t]=(t p )-module), and factors through an abelian unipotent subgroup scheme. Two p-points ; are said to be equivalent if for all kG-modules M: (M) proj. () (M) proj. 20 We denote by P (G) the set of all equivalence classes of p-points of kG. If M is a nite dimensional kG-module, we will denote by P (G) M the set of all equivalence classes of p-points of G on which M restricts to a non-projective k[t]=(t p )-module. There is a topology on P (G) given by declaring that the closed sets are all subsetsP (G) M . That this actually denes a topology onP (G) requires some work, and a critical step involved is proving that for kG-modulesM andN, P (G) M N = P (G) M \P (G) N . This is not at all obvious since p-points are not maps of Hopf algebras. Friedlander and Pevtsova prove further that this intrinsic topology on P (G) is a Noetherian topology. We highlight the following lemma as it will be used to prove a theorem in the next section. This lemma, proved in [FP05] for p-points, and observed in [Far07a] to hold for arbitrary at maps to kG (i.e. those not factoring through unipotent abelian subgroup schemes), provides a critical step in mapping the space ofp-points to the cohomological variety of G. Lemma 3.1.2. [FP05, 3.4] LetG be a nite group scheme and :k[t]=(t p )!kG a at map of algebras, then the kernel of the restriction map : H (G;k)! H (k[t]=(t p );k) is not equal to the augmentation ideal of H (G;k). The following key theorem of [FP05] states: Theorem 3.1.3. [FP05] Let G be a nite group scheme. Then sending a p-point to ker determines a well-dened map G :P (G) ! Proj V G such that for every nitely generated G-module M 1 G (Proj V G (M)) =P (G) M 21 In particular, thep-support space of a module satises, with appropriate modi- cations, the statements listed in Theorem 2.1.1 (and in fact, the validity of some of the properties listed in Theorem 2.1.1 come from establishing the related property for the p-support space and then applying Theorem 3.1.3 [FP05]). 3.2 The Factorization Requirement In this section we will look at the requirement thatp-points factor through unipo- tent abelian subgroup schemes, with our main result showing that Theorem 3.1.3 remains valid so long as the maps factor through unipotent (not necessarily abelian) subgroup schemes. We rst will illustrate why p-points must factor through unipotent subgroup schemes, roughly following the presentation given by Farnsteiner in [Far07a]. We start with the following denition, which introduces notation that will also be of use in the next section. Denition 3.2.1. [Far07a] LetA be a k-algebra. A at point ofA is an algebra map :k[t]=(t p )!A such that is left- at (i.e. (A) is a projective module). Two at points; are said to be equivalent if for all A-modules M: (M) proj. () (M) proj. We denote by Fl(A) the set of all equivalence classes of at points of A. 22 Just as is done withp-points, let us dene for any nitely generated A-module M the set Fl(A) M which is the subset of at points of A on which M restricts to a non-projective k[t]=(t p )-module. A fundamental dierence between at points and p-points is the way they behave with respect to the block decomposition of a group algebra, a dierence that is most clearly illustrated in the case of abelian nite group schemes. Farnsteiner shows that for any nite group scheme (in fact any algebra), ifkG =B 0 B n , then Fl(kG) = Fl(B 0 )Fl(B n ) (at the moment we have no additional structure on Fl(kG), so this product is as sets only). Now if G is abelian, then G =UD, whereU is a unipotent group scheme andD is a diagonalizable group scheme. We have then thatkG =kU kD, and askU is local, there is a one-to-one correspondence between the blocks of kG and the blocks of kD. Since kD is the dual of the group algebra of a nite abelian group H, we have kD = M h2H ke h Thus the block decomposition of kG is given as kG = M h2H kUe h This then implies thatFl(kG) =Fl(kU)Fl(kU). On the other hand, the inclusionU ,!G induces an isomorphism in cohomology: H (G;k) = H (U;k). By Theorem 3.1.3 this isomorphism implies thatP (G) =P (U) (as topological spaces). If p-points were not required to factor through unipotent subgroup schemes, then whenG is abelian we would have that as setsP (G) =Fl(kG), which we have just observed cannot be the case. Thus we see why it is necessary that p-points factor though unipotent subgroups. 23 We will next show that Theorem 3.1.3 still remains valid when only requiring the maps to factor through unipotent subgroup schemes. This will be accomplished by showing that for any nite unipotent group scheme U, there is a bijection P (U) = Fl(kU). That is to say, every at map to kU is equivalent (under the equivalence dened earlier) to a p-point. To show that this is not trivial, we rst observe the existence of at points of unipotent group schemes which are not p- points. Example 3.2.2. LetG be a non-abelian p-group, and chooseg 1 2Z(G) such that jg 1 j =p. Set x = 1g 1 ; N = X g2G g The elementx+N is p-nilpotent, and is not contained in any subgroup algebra as it is a linear combination with nonzero coecients of every element inG except forg 1 . We know that the map sendingt tox makeskG into a freek[t]=(t p )-module. SincexN = 0, there is some elementy2kG such thatN =x p1 y. It then follows by [FP05, 2.2] that the map sending t to x +N =x +x(x p2 y) also determines a left- at map from k[t]=(t p ) to kG, and thus is a at point which is not a p-point. Theorem 3.2.3. If G is a unipotent group scheme, then the natural inclusion P (G),!Fl(kG) is a bijection. Proof. Let ; be any two at maps such that the maps they induce in cohomol- ogy have the same kernel. That is, ker = ker . Since k is the only simple kG-module, then for any moduleM, we can calculateV G (M) according to the anni- hilator of Ext kG (k;M). Now consider the modulek* G := Hom (k[t]=(t p )) (kG;k) (i.e. the coinduced or induced module, depending on terminology, from the subalgebra 24 of kG that is the image of k[t]=(t p ) under ). The identication of n-fold exten- sions under the Eckmann-Shapiro isomorphism (see [Ben91a, 2.8]), shows that the action ofH (G;k) on Ext kG (k;k* G ) via the Yoneda product is the same as is given by H (G;k) acting on Ext k[t]=(t p ) (k;k) via followed by Yoneda product. Thus, ker = ker implies that V G (k* G ) = V G (k* G ). If M is any kG-module, we then have that V G (k* G M ) =V G (k* G M ), so that in particular: (k* G M ) proj. () (k* G M ) proj. We have the following isomorphisms: Ext n kG (k;k* G M ) = Ext n kG (M;k* G ) = Ext n k[t]=(t p ) ( (M);k) Since both kG and k[t]=(t p ) have k as their only simple module, then in both module categories any non-projective must have a non-trivial n-fold extension by k for all n. This implies that M restricts via to a projective k[t]=(t p )-module if and only if (k* G M ) is a projective kG-module. This argument shows that at maps to unipotent group schemes having the same kernel in cohomology are equivalent. The nal step is given by Lemma 3.1.2 and Theorem 3.1.3, which together say that for every at map to kG, there is a p-point which induces the same kernel in cohomology. Remark 3.2.4. Although this theorem shows that in terms of giving a representation-theoretic description of support varieties the denition of a p-point could simply be that it is a left- at map factoring through a unipotent subgroup scheme, we note that there are more rened invariants coming fromp-points which involve Jordan types, and it is unclear at this moment how these invariants would behave under such an altered denition. 25 We end this section with an observation about maps from k[t]=(t p ) to another algebra which are at. First o, we recall that in k[t]=(t p )-mod at = projective = free. Now, it is a general property of Hopf algebras that they are free both as left and right modules over any Hopf subalgebra. Since p-points factor through abelian subgroup schemes, it follows that given a p-point , kG is right-free over (k[t]=(t p ) if and only if it is left-free. We show in the next proposition that this is true more generally. Lemma 3.2.5. SupposeA is a nite dimensional Frobenius algebra over k. Then the map of k-algebras :k[t]=(t p )!A is left- at if and only if it is right- at. Proof. Since A is Frobenius, there is an isomorphism of left-modules A A = (A A ) . The action of t on (A A ) is given by t:f(x) = f(x(t)). If t:f = 0, then f = 0 on the subspace A(t). The dimension of all linear functionals with t:f = 0 is thus equal to the dimension of the subspace of A killed by right multiplication by (t). On the other hand, by A A = (A A ) , it is also equal to the dimension of the subspace of A killed from left multiplication by (t). Thus, the rank of the linear map given by right multiplication of (t) is equal to the rank of the linear map given by left multiplication, which proves the claim. 3.3 Invariants of Blocks The factorization requirement of p-points is the only part of the denition which involves the Hopf algebra structure of kG, since it is forces the at maps to land in particular Hopf subalgebras. Thus, for a module M lying in a block B, the p-support spaceP (G) M may not be a true invariant of the block B, as it relies on the total structure of kG as a Hopf algebra. A natural question to then ask is, if 26 two blocks of dierent group algebras are isomorphic as algebras, then after identi- fying their module categories, is it true that the support spaces for corresponding modules are homeomorphic? To investigate the question described above, in this section we will look at comparing the sets P (G) M and Fl(B) M . Let denote the projection map from kG ontoB. This is a at map and thus induces a map :P (G)!Fl(B), where where ([]) = []. This map is well-dened since two equivalentp-points must be equivalent on allkG-modules, so in particular are equivalent on allB-modules. By restricting the domain, we also have maps M :P (G) M !Fl(B) M . In general, the map will not be injective. This would be the case, for example, if B is a simple block of kG, and P (G) consists of more than a single point. This next proposition shows however that the maps M are injective for all modules lying in B. Proposition 3.3.1. Let B be a block of a nite group scheme kG. Then for any module M lying in B, the map M :P (G) M !Fl(B) M , which is induced by the projection of kG onto B, is injective. Proof. Suppose that , represent distinct equivalence classes in P (G) M . We simply must show that there is a moduleN lying inB on which the inequivalence of and can be detected. By [FP05, 5.1], there exists some cohomology class in degree 2n such that the Carlson module L restricts via to a non-projective k[t]=(t p )-module, while the restriction via is projective, or equivalently, []2 P (G) L and []62P (G) L . Thus by [FP05, 5.6] []2P (G) L M ; []62P (G) L M (3.1) 27 Now let e be the central idempotent of B. Following the proof from [Far07b, 2.2], since M is a B-module we have an equality: P (G) L M =P (G) e:(L M) Thus the statement in (3.1) holds for the module e:(L M), which is then a B-module which detects the inequivalence of and . As to the bijectivity of these maps, Theorem 3.2.3 establishes such a result for unipotent group schemes (which have a trivial block decomposition). The next proposition also establishes bijectivity for certain principal blocks. Proposition 3.3.2. Let G be a nite group scheme, and suppose that the trivial module is the only simple module lying in the principal block B 0 . Then every equivalence class of at points of B 0 has a representative which factors as a p- point ofkG followed by projection ontoB 0 . Consequently the maps M :P (G) M ! Fl(B 0 ) M are bijective for all M in B 0 . Proof. If the trivial module is the only simple module lying in the principal block, then e 0 e 0 is the unique non-zero idempotent of B 0 B 0 . Suppose now that kG =B 0 + +B r , and letB k ; k6= 0, be a block with central idempotente k . We know that (e k ) is an idempotent inkG kG, an algebra whose blocks are of the form B i B j , and so we must have (e k ) = X 0i;jr x ij ; x 2 ij =x ij 2B i B j Ifx 00 =e 0 e 0 , then (m(Id ))(e k )6=e k , thusx 00 = 0. This then implies that (B k ) kG (B 1 + +B r ) + (B 1 + +B r ) kG. From this we can deduce thatB 1 ++B r , which is the kernel of the projection map 0 :kG!B 0 , 28 is a Hopf ideal of kG, and B 0 is isomorphic as an algebra to the group algebra of a unipotent group scheme. Now let be a at map fromk[t]=(t p ) toB 0 . We have a map : Ext B 0 (k;k)! Ext k[t]=(t p ) (k;k). By [Far07a, 2.2], the kernel of this map is not the entire augmen- tation ideal of Ext B 0 (k;k). As 0 induces the isomorphism 0 : Ext B 0 (k;k) ! Ext kG (k;k), 0 (ker ) is not the augmentation ideal Ext kG (k;k), thus there is a p-point such that : Ext kG (k;k)! Ext k[t]=(t p ) (k;k) satises ker = ker ( 0 ) Thus, and ( 0 ) are two at maps to B 0 inducing the same kernel in cohomology, and sinceB 0 is isomorphic to the algebra of a unipotent group scheme, the equivalence of and 0 follows from Theorem 3.2.3. Question 3.3.3. Is it true for an arbitrary block B of kG, and module M lying in B, that M :P (G) M !Fl(B) M is bijective? Is it at least true for all principal blocks? 29 Chapter 4 Hochschild Cohomology In this chapter we compare the support varieties for modules introduced in Chapter 2 with an analogous construction derived from the Hochschild cohomology ring of a nite group scheme. Such Hochschild varieties have been developed for general k-algebras by Snashall and Solberg in [SS04], however the line of inquiry in this chapter goes back several years prior to the work of M. Linckelmann in [Lin99]. Linckelmann, working in the setting of nite groups, found a way to create a true invariant for modules lying in a block through the use of Hochschild cohomology (why this is a \true" invariant of the block will become clear later on). Linckelmann dened for a module its \block variety", a denition that involved Hochschild cohomology together with another ring constructed from the cohomology ring of a defect group of the block. It later was shown that Linckelmann's varieties are homeomorphic to the Hochschild support varieties dened by Snashall and Solberg. This was rst proved in several cases by J. Pakianathan and S. Witherspoon in [PW03], and then proved in full generality by Linckelmann in [Lin10]. It also must be noted that S. Siegel had developed, in unpublished work, a parallel theory of Hochschild varieties for nite groups, and it is his denition which we will be working with below. 30 4.1 Background In this section we will provide the background mathematics on Hochschild coho- mology, and dene Hochschild support varieties for blocks in terms in an approach similar to that of Siegel. We will then verify that this denition produces varieties homeomorphic to those dened by Snashall and Solberg. Let A be a k-algebra, which for our purposes will always be assumed to be nite dimensional overk. Recall from Chapter 1 the algebraA e , whose category of left-modules is equivalent to the category ofAA bi-modules. For anyA e -module N, there is a notion of the Hochschild cohomology groups with coecients in N. These groups can be computed as the cohomology groups of an explicitely dened cochain complex (see [Ben91b, 2.11]), however since k is a eld in this case, these cohomology groups are isomorphic to the groups Ext n A e(A;N) When the coecients are inA, we will also denote Ext n A e(A;A) as HH n (A). Just as there is for the cohomology groups of a Hopf algebra, there is a cup product for the Hochschild cohomology groups of an arbitrary k-algebra A which makes HH (A) into a graded-commutative algebra. Suppose now that B is a block of kG. By taking the left-adjoint action, B becomes aG-algebra (i.e. the multiplication map is a G-module homomorphism), and so Ext kG (k;B) is ak-algebra under the cup-product followed by multiplication inB. The inclusion of algebrask,!B gives rise to an algebra map Ext kG (k;k)! Ext kG (k;B). By the Friedlander-Suslin theorem, Ext kG (k;B) is a nite module over the nitely generated subalgebra which is the image of Ext kG (k;k), thus the larger algebra is itself nitely generated over k. 31 Applying Lemma 1.3.2, it follows thanks to the Eckmann-Shapiro isomorphism that, as least as graded vector spaces: Ext kG (k;B) = Ext B e(B;B) It turns out that the above isomorphism respects the product structures, as rst shown for nite groups in [SW99], and stated for nite dimensional Hopf algebras in [PW09]. Thus it follows that Ext kG (k;B) is a graded-commutative Noetherian algebra overk, and so Ext kG (k;B) is a commutative algebra. IfM is aB-module, then with the left-adjoint action on B the algebra map ' M : B! Hom k (M;M) is a map of G-algebras which induces a map of algebras (' M ) : Ext kG (k;B)! Ext kG (k; Hom k (M;M)). We now can make the following denition: Denition 4.1.1. IfB is a block of a nite group schemeG, then the Hochschild variety of B is X B := MaxSpecfExt kG (k;B)g. If M is any nite dimensional module lying in B, then the Hochschild support variety of M is X B (M) := MaxSpecfExt kG (k;B)=ker (' M ) g. Snashall and Solberg dene the Hochschild support variety M as the maximal ideal spectrum of the ring HH (B) modulo the kernel of the map: B M : HH (B)! Ext B (M;M) this map taking a self-extension ofB as aB e -module to a self-extension ofM as a B-module. The following lemma, which is a variant of [PW09, Lemma 13], shows that these two denitions of Hochschild varieties are equivalent. Lemma 4.1.2. The following diagram commutes: 32 Ext kG (k;kG) = (' M ) // Ext kG (k; Hom k (M;M)) = Ext kG e(kG;kG) kG M // Ext kG (M;M) Proof. Fix a kG projective resolution P !k, and let 2 Hom kG (P n ;kG) repre- sent an element in Ext n kG (k;kG). The composite of the leftmost downward map and the bottom horizontal map sends 7! e 7! e kG id, where e kG id2 Hom kG ((kG e (kG) P n ) kG M;kG kG M) As leftkG-modules (kG e (kG) P n ) kG M =P n M, andkG kG M =M. Tracing through the denitions it can be shown that ( e kG id)(1 1 p m) =(p)m. On the other hand, the top horizontal map followed by the rightmost downward map takes 7!7! () id, where () id2 Hom kG (P n M; Hom k (M;M) M) which, by then composing with the map : Hom k (M;M) M!M; (f m) = f(m), produces an element in Hom kG (P n M;M). This element sends p m to ()(p)(m), which when tracing through the denitions is precisely(p)m. Thus the diagram commutes. 4.2 Comparing The Varieties We will now investigate the relationship between Hochschild varieties and the support varieties of Chapter 2. The rst proposition is essentially immediate given 33 the above denition of Hochschild support varieties. This fact was already observed by Snashall and Solberg in [SS04], where they give a proof for modules lying in the principal block of a nite group, and their argument holds for arbitrary blocks of any nite group scheme. Here we will present the two-line proof which follows from our denition. Proposition 4.2.1. LetB be a block of a nite group scheme G, and letM be a B-module. There is nite surjective morphism of varieties X B (M)!V G (M). Proof. Since M is a B-module, the map of G-algebras k ,! Hom k (M;M) fac- tors through B, and consequently the map Ext kG (k;k)! Ext kG (k; Hom k (M;M)) factors through Ext kG (k;B). We next look at the principal block B 0 of kG. Under the left-adjoint action B 0 =ke 0 I B 0 . The composite of algebra maps k,!B 0 !k is the identity on k, and so there is a split surjective morphism of algebras Ext kG (k;B 0 )! Ext kG (k;k) with kernel Ext kG (k;I B 0 ). Linckelmann's work [Lin99] and [Lin10] proves that in the case of nite groups, this kernel is nilpotent. Theorem 4.2.2. (Linckelmann, [Lin99], [Lin10]) LetkG be the group algebra of a nite groupG, and letB 0 denote the principal block. Then the algebras HH (B 0 ) and H (G;k) are isomorphic modulo nilpotent elements. Consequently, for any module M lying in B 0 , X B (M) =V G (M). For principal blocks of an arbitrary nite group scheme, this result at least remains true if the hypotheses of Prop 3.3.2 are satised. We note that the argu- ment presented in our proof is analogous to the one used by Siegel and Witherspoon in [SW99, 10.1], which compares the Hochschild and ordinary cohomology rings of a nite p-group. 34 Theorem 4.2.3. Let kG be a nite group scheme, and suppose that the trivial module is the only simple module lying in the principal block B 0 . Then HH (B 0 ) and H (G;k) are isomorphic modulo nilpotent elements. Proof. The assumption that the trivial module is the only simple module lying in B 0 is equivalent to saying that the augmentation ideal of B 0 is also the Jacobson radical, hence a nilpotent ideal. Since the multiplication in the ideal Ext kG (k;I B 0 ) is given by the cup product followed by multiplication inI B 0 , the nilpotence ofI B 0 implies the nilpotence of Ext kG (k;I B 0 ). Combined with earlier observations, this says that the surjective map Ext kG (k;B 0 )! Ext kG (k;k) has a nilpotent kernel. Since HH (B 0 ) = Ext kG (k;B 0 ), the claim is proved. Corollary 4.2.4. Let G be an abelian nite group scheme, B an arbitrary block of kG. Then HH (B) and H (G;k) are isomorphic modulo nilpotents. Proof. We recall from Chapter 3 that G = UD, where U is a unipotent group scheme andD is a diagonalizable group scheme, and every block ofkG is isomorphic as an algebra tokU. The principal block corresponds to the factorkUe h 0 , whereh 0 denotes the identity element inH (see [Far07a, 1.2] for more on this decomposition of algebras). If B corresponds to the factor kUe h , then the map f : B 0 ! B, where f(ue h 0 ) = ue h , explicitly gives the isomorphism of algebras. It is also an isomorphism ofG-algebras, as in this case all the blocks are isomorphic under the left-adjoint action to dim k (kU) copies of the trivial module. Thus HH (B 0 ) = HH (B) as algebras, and the result now follows by the previous theorem. Question 4.2.5. Does Theorem 4.2.2 remain true for arbitrary principal blocks of nite group schemes? 35 Considering now arbitrary blocks (not necessarily principal), we have that for a block B the map Ext kG (k;k)! Ext kG (k;B) induced by the inclusion k ,! B will in general not be injective. The following lemma calculates this kernel. Lemma 4.2.6. The kernel of the map Ext kG (k;k) ! Ext kG (k;B), denoted I G (k;B), is equal to I G (B), the annihilator of the ring Ext kG (B;B). Proof. For any kG-module M, by [Ben91b, 5.7] there is an inclusion I G (M) I G (k;M) (notation as in Chapter 2, section 1). Conversely, the map Ext kG (k;k) B ! Ext kG (B;B) factors as the composite of maps Ext kG (k;k) i ! Ext kG (k;B) B ! Ext kG (B;B B) m ! Ext kG (B;B) proving the reverse inclusion. Proposition 4.2.7. Let B be a block of kG. The Krull Dimension of HH (B) is equal to the dimension of the variety V G (B). Proof. Follows from the previous lemma together with Proposition 2.2.3. This equality of dimensions can also be observed directly from the following fact. Proposition 4.2.8. Let B be a block of kG, andfS i g be a complete set of non- isomorphic simple B-modules. Then X B = S X B (S i ). Proof. The short exact sequence of modules 0! Rad(B) i !B !B=Rad(B)! 0 36 gives rise to a long exact sequence ! Ext n (k; Rad(B)) i ! Ext n (k;B) ! Ext n (k;B=Rad(B))! Suppose now that is a homogeneous element in T ker (' S i ) . SinceB=Rad(B) = L Hom k (S i ;S i ), this immediately says that () = 0, and hence = i ( ) for some 2 Ext (k; Rad(B)). As Rad(B) is a nilpotent ideal, the ideal i (Ext (k; Rad(B))) is a nilpotent ideal in Ext (k;B), and hence is a nilpotent element. Thus X B = S X B (S i ). Remark 4.2.9. In fact the last few results give another way to prove Propo- sition 2.2.3. First, we have that I G (B) = I G (k;B), the latter ideal clearly being contained in T I G (S i ). Conversely, by this last proof, any element in T I G (S i ) is sent to a nilpotent element in the map Ext (k;k) ! Ext (k;B). Thus T I G (S i ) p I G (k;B) = p I G (B), and hence the equality of varieties S V G (S i ) =V G (B). The next corollary was proved through a dierent approach by Linckelmann in his work in [Lin99]. He notes there that this result had also been observed in unpublished work by Siegel, who communicated a short direct proof to Linckel- mann at the time [Lin99] was written. Corollary 4.2.10. Let B be a block of a nite group. The Krull dimension of HH (B) is equal to the Krull dimension of the cohomology ring of a defect group of B. Proof. Follows from Proposition 2.2.4 and Corollary 4.2.7. 37 Chapter 5 Reductive Algebraic Groups A good source for examples of blocks of nite group schemes is given by studying the Frobenius kernels of a reductive algebraic group G. Additionally, results of Jantzen, Nakano-Parshall-Vella, and Suslin-Friedlander-Bendel allow the support variety (rank variety) of blocks to be calculated in particular situations. In this chapter we will quickly run through the relevant background material on reductive algebraic groups, almost all of which comes from Part II of [Jan03], as well as the necessary results on rank varieties found in [SFB97a] and [SFB97b]. We will then give a bit of detail about how the results referred to above can be used to compute the variety of a block, frequently referring to arguments and results from [NPV02]. 5.1 Notation and Background Results LetG be a reductive algebraic group, dened and split overF p . LetT be a maximal torus ofG with character groupX(T ), let be a root system forG with respect to T , and x a set of simple roots =f 1 ;:::; ` g. Denote by + the set of positive roots with respect to this choice, and let B + and B denote the Borel subgroups corresponding to + and + , with their unipotent radials denoted asU + ;U. The Weyl group of will be denoted W , and the dot action of w2W on2X(T ) is dened by w: =w( +), where is the half sum of the positive roots. The root datum forG consists of a quadruple (X(T ); ;Y (T ); _ ), along with a maph;i :X(T )Y (T )!Z. Additionally, _ Y (T ), and there is a specied 38 bijection ! _ . In order to apply the results of Nakano-Parshall-Vella, we will need to assume throughout that the prime p is good for . This requires p6= 2 if G has a component of type B,C, or D; p6= 2; 3 if G has a component of type G 2 , F 4 , E 6 , or E 7 ; and p6= 2; 3; 5 if G has a component of type E 8 . With the root datum given, the dominant integral weights ofX(T ) are dened as the set X(T ) + :=f2X(T )j0h; _ i i; 1i`g For aG-moduleM, denote byM the subspace ofM on whichT acts according to the weight. Every simpleB-module is one-dimensional, and thus is determined by its weight. Such a B-module will be written as k . We can now describe the simple modules for G as follows. For each 2 X(T ) + , there is a unique simple module L() with the following properties [Jan03, II.2]: 1. Every simple G-module is isomorphic to some L(). 2. L() U + =L() . 3. dim k L() U + = 1. 4. L() = soc G Ind G B (k ). ThusL() is the unique simple module of highest weight . We also point out that the induction in property 3 is with respect to the negative Borel subgroup. Suppose now that G r is the r-th Frobenius kernel of G. From the inclusion T r ,!T it is not hard to see that the respective character groups t into the short exact sequence 0!p r X(T )!X(T )!X(T r )! 0 39 We can thus speak of a character for T as being a character for T r , keeping in mind the isomorphism of abelian groups X(T r ) = X(T )=p r X(T ). To introduce a little more terminology, we dene the set of p r -restricted weights to be X r (T ) :=f2X(T )j0h; _ i i<p r ; 1i`g The set X r (T ) is a subset of X(T ), and contains a set of representatives of X(T )=p r X(T ). WhenG is semi-simple, thenX r (T ) is a set of coset representatives without repetition (it is easy to see why this is not the case for general reductive groups by looking at what happens when G is a torus). The description of simpleG r -modules is similar to the one above forG-modules. In this case, for each2X(T ) there is a simpleG r -moduleL r () with the following properties [Jan03, II.3]: 1. Every simple G r -module is isomorphic to some L r (). 2. L r () U + r =L r () . 3. dim k L r () U + r = 1. 4. L r () = soc Gr Ind Gr Br (k ). 5. L r () =L r () () 2p r X(T ). 6. If 2X r (T ), then as G r -modules L() =L r (). The modules Ind G B (k ) are commonly denoted as H 0 (). We will follow [NPV02] in denoting the modules Ind Gr Br (k ) by Z r () (rather than Z 0 r () as in [Jan03]). 40 5.2 Rank Varieties In Chapter 3 we introduced the notion ofp-points, which are in some sense the ulti- mate generalization of various representation-theoretic constructions which provide an alternate way to view cohomological support varieties. Since this chapter deals with innitesimal group schemes, we will instead use the language of 1-parameter subgroups developed by Suslin-Friedlander-Bendel in [SFB97b]. We recall that if G is innitesimal of heightr, then a 1-parameter subgroup of G is a map of group schemes G a(r) ! G. This is equivalent to a map of Hopf algebras kG a(r) !kG. As k[G a(r) ] =k[X]=X p r , we have that kG a(r) =k[u 0 ;:::u r1 ]=(u p 0 ;:::;u p r1 ) whereu i (X j ) = 1 ifj =p i and 0 otherwise. The following theorem, taken from [FP05], summarizes some of the relevant results from [SFB97b]. Theorem 5.2.1. Let G be an innitesimal group scheme of height r. The functor on nitely generated commutative k-algebras sending an algebra A to the set of all morphisms of group schemes over A, G a(r);A ! G A , is representable by an ane scheme whose k points are the 1-parameter subgroups of G. We denote the set of closed points of this scheme byV G . There is a natural homeomorphism G :V G !V G with the property for any nite dimensionalG-moduleM that 1 (V G (M)) = V G (M), the subset consisting of those 1-parameter subgroups : G a(r) ! G for which the pull-back of M via is not free over the subalgebra k[u r1 ]=(u p r1 ) ,! kG a(r) . 41 Now letG be an algebraic group satisfying the conditions listed at the beginning of this chapter. The Lie algebra of G, Lie(G), has the structure of a p-restricted Lie algebra. We denote the p-power map by [p] : Lie(G) ! Lie(G), and for x2 Lie(G) we denote its image under this map byx [p] . Given certain assumptions about G, which are satised in the situations dealt with in this chapter, Lemma 1.7 of [SFB97a] then gives the following concrete description of the 1-parameter subgroups of G r : V Gr =f( 0 ;:::; r1 )j i 2 Lie(G); [p] i = 0; [ i ; j ] = 08i;jg See [SFB97a] for more details on this equality. 5.3 Block Decompositions LetB r () denote the block of kG r which contains the simple module L r (). By denition thenB r () =B r () if and only ifL r () andL r () lie in the same block. The following result of Jantzen explicitly determines when this happens. Proposition 5.3.1. [Jan03, II 9.22] Let 2 X(T ), and let m be the smallest integer such that there is 2 + withh +; _ i = 2Zp m . Then L r ()2B r () if and only if 2W +p m Z +p r X(T ). For simplicity, we will also write 2B r () to indicate that L r ()2B r (). It is observed in [NPV02] that theG-modulesH 0 () when restricted toG r lie in the blockB r (). This next proposition summarizes various facts either directly proved or implied by the results of [NPV02]. Proposition 5.3.2. LetB r () be a block ofkG r . The support variety ofB r () is equal to all of the following: 42 1. S 2Br () V Gr (L()) 2. S 2Br () V Gr (H 0 ()) 3. S 2Br () GV Gr (Z r ()) Proof. In each case the modules in question lie in the blockB r (), so it suces to show that the support of every simple module inB r () lies in the given union. (1) Every L r () is isomorphic to some L r ( 0 ) with 0 2 X r (T ). Thus V Gr (L r ()) =V Gr (L( 0 )). (2) In light of (1), we just need to show that for every 2B r (), V Gr (L()) [ 2Br () V Gr (H 0 ()) The proof of this follows from the same type of induction argument used in the proof of Theorem 4.6.1 in [NPV02], replacing the support of a single H 0 () with S 2Br () V Gr (H 0 ()). (3) This comes directly from Theorem 4.4.1(b) of [NPV02]. 5.4 G 1 of Reductive Groups The results of [NPV02], particularly Theorem 6.2.1, now determine the support variety of any block of a reductive algebraic group with the assumptions stated in the beginning of this chapter. For 2X(T ) + , dene :=f2 jh +; _ i2pZg. Since we are working under the assumption that the primep is good, Nakano-Parshall-Vella observe that in this case there exists w2 W and a subset I such that w( ) =ZI\ . We then dene the Lie subalgebra u I of Lie(G) to be the subalgebra generated by 43 all x2 Lie(U ), where is a positive root not contained in ZI. Theorem 6.2.1 of [NPV02] then shows that the rank variety is given by: V G 1 (H 0 ()) =Gu I From this we get the following immediate corollary: Corollary 5.4.1. For 2X 1 (T ), the rank variety of the blockB 1 () is Gu I . Proof. If 2B 1 ()\X 1 (T ), then by Proposition 5.3.1 we have that 2 W + pX(T ) (this special case of the proposition was actually rst observed by Humphreys in [Hum71]). It then follows that if w( ) = ZI\ , then for some w 0 2W we have w 0 ( ) =ZI\ . Thus V G 1 (H 0 ()) =Gu I =V G 1 (H 0 ()) By Proposition 5.3.2 the rank variety of the block is then also given by this orbit. Combining this corollary with the results of Chapter 4 we obtain some infor- mation about the Krull dimension of the Hochschild cohomology ring of a block. Proposition 5.4.2. LetG 1 be the Frobenius kernel of a reductive algebraic group with the above assumptions, and letB 1 () be a block. Then the Krull dimension of HH (B 1 ()) = 2dim u I . Proof. Follows from Corollary 4.2.7 and the previous corollary, together with the observation in [NPV02] that the dimension of Gu I when p is good. 44 5.5 Blocks of SL 2(r) In [SFB97b], Suslin-Friedlander-Bendel compute for the Frobenius kernels of SL 2 , and every dominant weight , the rank variety of the module H 0 (), which as before can be analyzed to give the support of any block. Assume that a maximal torus, root system, etc. have been chosen forSL 2 , and set as the lone simple root. We have that ==2, and every dominant weight is given as =n, where n is a non-negative integer. Denote by C r (N p (sl 2 )) the set V SL 2(r) =f( 0 ;:::; r1 )j i 2 Lie(SL 2 ); [p] i = 0; [ i ; j ] = 08i;jg Proposition 5.5.1. [SFB97b, 7.8(b)] Let = n be a dominant weight of SL 2 . Consider the p-adic expansion of n: n =n 0 +n 1 p + +n q p q (0n i <p). Then for all r 1: V SL 2(r) (H 0 ()) =f( 0 ;:::; rs1 ; 0;:::; 0)2C r (N p (sl 2 ))g where s is the least integer such that n s 6=p 1. From this result, we can now prove the following theorem which gives the support variety of a block. Theorem 5.5.2. Let = n be a dominant weight of SL 2 . Consider the p-adic expansion of n: n = n 0 +n 1 p + +n q p q (0 n i < p). Then for all r 1, the rank variety of the blockB r () is equal to: f( 0 ;:::; rs1 ; 0;:::; 0)2C r (N p (sl 2 ))g =C rs (N p (sl 2 )) where s is the least integer such that n s 6=p 1. 45 Proof. On the one hand, as noted earlier the moduleH 0 () lies in the blockB r (), thus the previous proposition shows that the rank variety of the block contains C rs (N p (sl 2 )). To get the reverse inclusion, by Proposition 5.3.2 it suces to show that if2B r (), thenV SL 2(r) (H 0 ())V SL 2(r) (H 0 ()). If is not dominant, then H 0 () = 0, so it is clearly true in this case. If is dominant, then =t;t 0. By the previous proposition we need only prove that if the p-adic expansion t is t 0 +t 1 p + +t q p q (0t i <p), then we have t i =p 1 for all i<s. Sinceh +; _ i = 1 +n, then if s is the least integer such that n s 6= p 1, it follows that it is also the least integer such thath +; _ i = 2 p s Z. Thus, Proposition 5.3.1, together with the fact that Z = 2X(T ) implies that 2 W+2p s X(T )+bp r X(T ). Since the Weyl group has a single non-trivial element, W =( +). Working this out we see that t =(n + 1) 1 + 2ap s +bp r ; a;b2Z If m < s, then since p m j (n + 1), we also have p m j (t + 1). Thus the p-adic expansion oft is given as: t =p1++(p1)p s1 + , proving the claim. 46 Bibliography [Alp86] J.L. Alperin. Local Representation Theory. Cambridge University Press, 1986. [Ben91a] D.J. Benson. Representations and Cohomology, Vol. I. Cambridge Uni- versity Press, 1991. [Ben91b] D.J. Benson. Representations and Cohomology, Vol. II. Cambridge University Press, 1991. [Car84] J.F. Carlson. The variety of an indecomposable module is connected. Inventiones Mathematicae, 77:291{299, 1984. [CFP07] J. Carlson, E. Friedlander, and J. Pevtsova. Modules of constant jordan type. Journal f ur die reine und angewandte Mathematik, 614:1{44, 2007. [Far07a] R. Farnsteiner. Support spaces and auslander reiten components. Con- temp. Math., 442:61{87, 2007. [Far07b] R. Farnsteiner. Tameness and complexity of nite group schemes. Bull. London Math. Soc., 39:63{70, 2007. [FP05] E. Friedlander and J. Pevtsova. Representation-theoretic support spaces for nite group schemes. Amer. J. Math., 127:379{420, 2005. [FP07] E. Friedlander and J. Pevtsova. -supports for modules for nite group schemes. Duke Math. J., 139:317{368, 2007. [FPS07] E. Friedlander, J. Pevtsova, and A. Suslin. Generic and maximal jordan types. Invent. Math., 168:485{522, 2007. [FS02] R. Farnsteiner and A. Skowronski. Classication of restricted lie algebras with tame principal block. J. reine angew. Math., 546:1{45, 2002. [Hum71] J.E. Humphreys. Modular representations of classical lie algebras and semisimple groups. Journal of Algebra, 19:51{79, 1971. 47 [Jan03] J.C. Jantzen. Representations of Algebraic Groups. Number 107 in Mathematical Surveys and Monographs. American Mathematical Soci- ety, 2nd edition, 2003. [Lan02] S. Lang. Algebra. Grad. Texts in Math., vol. 211. Springer-Verlag, 2002. [Lin99] M. Linckelmann. Varieties in block theory. J. Algebra, 215:460{480, 1999. [Lin10] M. Linckelmann. Hochschild and block cohomology varieties are isomor- phic. J. Lond. Math Soc. (2), 81:389{411, 2010. [NPV02] D.K. Nakano, B.J. Parshall, and D.C. Vella. Support varieties for alge- braic groups. J. Reine Angew. Math., 547:15{49, 2002. [Pre98] A. Premet. Complexity of lie algebra representations and nilpotent ele- ments of the stabilizers of linear forms. Math. Z., 228(2):255{282, 1998. [PW03] J. Pakianathan and S. Witherspoon. Hochschild cohomology and linck- elmann cohomology for blocks of nite groups. J. Pure Appl. Algebra, 178:87{100, 2003. [PW09] J. Pevtsova and S. Witherspoon. Varieties for modules of quantum elementary abelian groups. Algebra Representation Theory, 12:567{595, 2009. [SFB97a] A. Suslin, E. Friedlander, and C. Bendel. Innitesimal 1-parameter subgroups and cohomology. Journal of the A.M.S., 10:693{728, 1997. [SFB97b] A. Suslin, E. Friedlander, and C. Bendel. Support varieties for innites- imal group schemes. Journal of the A.M.S., 10:729{759, 1997. [SS04] N. Snashall and . Solberg. Support varieties and hochschild cohomol- ogy rings. Pro. London Math. Soc. (3), 88:705{732, 2004. [SW99] S.F. Siegel and S.J. Witherspoon. The hochschild cohomology ring of a group algebra. Proc. London Math. Soc., 79:131{157, 1999. [Wat79] W. Waterhouse. Introduction to Ane Group Schemes. Grad. Texts in Math., vol. 66. Springer-Verlag, New York, 1979. 48

Abstract (if available)

Abstract We study the block theory of a finite group scheme G over an algebraically closed field of positive characteristic. Our primary interest will be in studying blocks by means of invariants for modules. These invariants include cohomological support varieties as well as p-point support spaces, the latter developed by E. Friedlander and J. Pevtsova.

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Creator Sobaje, Paul G., Jr. (author)

Core Title Blocks of finite group schemes

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Mathematics

Publication Date 06/27/2011

Defense Date 05/04/2011

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

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