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The geometry of motivic spheres

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The geometry of motivic spheres

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Content THE GEOMETRY OF MOTIVIC SPHERES by Adam Ericksen A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (MATHEMATICS) Aug 2013 Copyright 2013 Adam Ericksen Dedication To my family. This work was possible only because of your encouragement, love, and support. Thank you for all that you have provided for me. ii Acknowledgments I am grateful for my many excellent teachers’ contributions to my mathematical education, no- tably Francis Bonahon, Robert F. Brown, Eric Friedlander, Ko Honda, José Molina, and Alexan- der Merkurjev. Each has offered me valuable guidance, and left his own indelible mark on my intellectual development. I am happy to thank my fellow graduate students for useful mathematical conversations and moral support over the years, especially Russell Avdek, Taylan Bilal, and Guillaume Dreyer. During my graduate study at USC, my advisor has been an invaluable source of information and inspiration. If not for his expertise and patience, this dissertation might never have seen the light of day. With great admiration and respect, I thank Aravind Asok for guiding me through my Ph.D. studies. iii Table of Contents Dedication ii Acknowledgments iii Abstract vi Chapter 1 Introduction 1 Chapter 2 Cohomology of Sheaves 4 2.1 ˇ Cech Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 The Picard group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3 Vector bundles and torsors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Chapter 3 A 1 -homotopy theory 15 3.1 Simplicial model categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.2 The unstableA 1 -homotopy category . . . . . . . . . . . . . . . . . . . . . . . . 17 3.3 Unimodular rows and algebraic vector bundles . . . . . . . . . . . . . . . . . . . 21 Chapter 4 Algebraic de Rham cohomology 24 4.1 Hypercohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 4.2 Basics of algebraic de Rham cohomology . . . . . . . . . . . . . . . . . . . . . 25 4.3 The ˇ Cech-de Rham complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 4.4 The de Rham cohomology of affine motivic (2n 1;n)-spheres . . . . . . . . . 29 iv Chapter 5 Exotic affine (2n 1;n)-spheres 32 5.1 Existence of exotic affine (2n 1;n)-spheres . . . . . . . . . . . . . . . . . . . 32 5.2 Non-cancellative behavior of motivic spheres . . . . . . . . . . . . . . . . . . . 41 References xliii v Abstract We study a class of smooth algebraic varieties which are, in the sense of Morel and V oevod- sky’sA 1 -homotopy theory, homotopy equivalent to spheres. These varieties belong to a class of objects called vector bundle torsors, and we investigate their cohomological classification, and the isomorphism problem in general for them. In the affine case, we establish a criterion for non-isomorphy in terms of algebraic de Rham cohomology, as defined by Grothendieck. For all n 3, we then construct a pair of non-isomorphic smooth affine (2n 1)-dimensional varieties X andY with theA 1 -homotopy type ofA n n 0, with the property thatXA n1 'YA n1 . These results extend work of Dubouloz and Finston in the casen = 2, and illustrate that motivic spheres of higher dimension exhibit non-cancellative behavior. vi Chapter 1 Introduction In the context ofA 1 -homotopy theory as defined in [MV], the varieties considered herein are motivic spheres. A motivic sphere is any smooth k-scheme X which becomes isomorphic to a sphere in the unstableA 1 -homotopy categoryH (k). TheA 1 -homotopy category provides an interesting framework in which to study and refine the cancellation problem: if XA 1 is isomorphic toYA 1 , then isX isomorphic toY ? One of the phenomena we wish to illustrate in this work is that affine motivic spheres exhibit non-cancellative behavior. Motivic Sphere Isom. classification of affine models Non-cancellative behavior P 1 D n ,n2N;D n 6'D m forn6=m. D n A 1 'D m A 1 . A 2 n 0 At least two isom. classes,X6'Y . XA 1 'YA 1 . A n n 0,n 3 At least two isom. classes,X6'Y . XA n1 'YA n1 . The most famous unsolved variant of the cancellation problem is the Zariski cancellation problem: if XA 1 is isomorphic to A n , then is X is isomorphic to A n1 ? Topologically speaking, the examples considered here are remote from affine spaces; whenk =C, affine space A n is contractible in the Euclidean topology. A class of counterexamples to the cancellation problem was introduced in [Dan]. The Danielewski surfaces are the smooth affine surfaces D n = f(x;y;z)2A 3 : x n y +z 2 = 1g: 1 Over the complex numbers, the topological space underlyingD n is diffeomorphic (but not alge- braically isomorphic) to the total space ofO P 1(2n). Thus, eachD n is homotopy equivalent to P 1 'S 2 . However, eachO P 1(2n) has a different homotopy type at infinity, so the varietiesD n andD m are not isomorphic forn6= m. ButD n A 1 ' D m A 1 for alln andm. This is the content of the first row of the table above. The results in the second row of the table above were established in [DF]. The authors address the classification problem forG a -bundles overA 2 n0 whenk =C is the field of complex numbers, and the isomorphy problem for their total spaces as abstract varieties. One of their main results is that there exist a pair of affineG a -bundlesX andY which are non-isomorphic, but for whichXA 1 'YA 1 . The results in the third row of the above table are established in the present work, and are a direct generalization of the work in [DF]. The objects under consideration in this paper arise as torsors under the vector bundles on punctured affine spaceA n n 0 defined by E f = ( (x;y)2A n n 0A n : n X i=1 f i (x)y i = 0 ) ; wheref 1 ; ;f n 2 k[x 1 ; ;x n ] are polynomials over a fieldk withZ(f 1 ; ;f n ) =f0g in A n . Forg2k[x 1 ; ;x n ], the torsor underE f defined by V f (g) = ( (x;y)2A n n 0A n : n X i=1 f i (x)y i =g(x) ) ; is affine ifg(0)6= 0, and its topological space over the complex numbers has the homotopy type of S 2n1 . The main result of this paper provides a method for distinguishing varieties of the formV f (g) in terms of algebraic de Rham cohomology via the cohomology class of a volume form. With this criterion we are able to produce a pair of non-isomorphicE f torsorsX andY for whichXA n1 'YA n1 . It would be interesting to determine the smallestr for which 2 XA r 'YA r . Affine models for anA 1 -homotopy type are of particular importance inA 1 -homotopy theory because interesting objects, such as isomorphism classes of vector bundles on a schemeX, are often homotopy invariant only for affine schemes. Insight into the structure of the moduli of smooth affine varieties with a givenA 1 -homotopy type is valuable for concrete calculations, and the main result of this paper, Theorem 5.1.2, is a first attempt at understanding the isomorphism problem for smooth affine motivic spheres of higher dimension. We assume that the reader is familiar with the following concepts; precise definitions are not included. Definitions can be found in the references below. Sheaf, scheme [H, Ch. II]; Abelian category, derived functor [H, Ch. III], [V, Ch. 3], [W, Chs. 1,2]; Grothendieck topology, site [Art]; Simplicial object, homotopy theory, model category [Hov]; Spectral sequence of a double complex [W, Ch. 5][BT, Ch. 8]; de Rham cohomology [BT], [V]; We explain the following concepts; definitions are included, but proofs are not. Proofs can be found in the references below. ˇ Cech cohomology [H, Ch. 3.4] UnstableA 1 -homotopy category [MV] Hypercohomology [W, Ch. 5] Algebraic de Rham cohomology [G] 3 Chapter 2 Cohomology of Sheaves We begin by recalling some basic homological algebra. LetA be an abelian category. A (cochain) complex inA is a sequence of objectsC i together with morphismsd : C i ! C i+1 (called differentials, or coboundary maps) satisfying d 2 = 0. The ith cohomology object of (C ;d) is the quotient object h i ((C ;d)) = ker(d : C i ! C i+1 )=im(d : C i1 ! C i ). We will suppress the differentials when no confusion can arise. A complexC is said to be exact if h i (C ) = 0 for alli. Recall that for any objectA inA , the functor Hom A (;A) is left-exact. An objectI inA is called injective if the functor Hom A (;I) is also right exact. If A is any object inA and 0! A! I 0 ! I 1 ! is an exact complex withI k injective for eachk, we will say thatI is an injective resolution ofA. The category of sheaves of abelian groups onX is an abelian category. The global sections functor assigns to a sheafF the abelian group (X;F ) = F (X). In general, the global sections functor is left-exact but not right-exact. The ith cohomology group ofF is the value of the ith derived functor of . Namely H i (X;F ) = R i (X;F ) = h i ((X;I )), where F ,! I is an injective resolution ofF . This definition does not depend on the choice of injective resolution (cf. [H, Ch. 3, Thm 1.1(A)(a)]). A sheafF is called acyclic if H i (X;F ) = 0 for alli> 0. Injective sheaves are acyclic (cf. [H, Ch. 3, Thm 1.1A(e)]). 4 Proposition 2.0.1. [H, Ch. 3, Prop 1.2A] LetF be a sheaf of abelian groups onX and suppose there is an exact sequence of sheaves 0!F!C 0 !C 1 ! ; where eachC k is an acyclic sheaf (also called an acyclic resolutionF ! C ). Then for all i 0, there are natural isomorphisms H i (X;F ) ' h i ((X;C ))): Injective resolutions are very useful for developing theory, but the main computational tools for sheaf cohomology are long exact sequences and the ˇ Cech method. Proposition 2.0.2. [H, Ch. 3, Thm 1.1A(c)] LetX be a topological space and let 0!F 0 ! F ! F 00 ! 0 be a short exact sequence of sheaves on X. There is a natural long exact sequence 0! (X;F 0 )! (X;F )! (X;F 00 )! H 1 (X;F 0 )! H 1 (X;F )! H 1 (X;F 00 )! : Another fundamental result in algebraic geometry is Serre’s cohomological criterion for affineness of a noetherian scheme. Theorem 2.0.3. [H, Ch. 3, Thm 3.7] Let X be a noetherian scheme. Then the following conditions are equivalent: 1. X is affine; 2. H i (X;F ) = 0 for all quasicoherent sheavesF and alli> 0; 3. H 1 (X;I ) = 0 for all coherent sheaves of idealsI . 5 Serre’s theorem shows that on a noetherian affine scheme, all quasicoherent sheaves are acyclic. We will use this fact repeatedly in the sequel. 2.1 ˇ Cech Cohomology LetX be a topological space and letU =fU i g i2I be an open covering ofX indexed by a totally ordered setI. Leti 0 <<i p be an ordered (p+1)-tuple inI. For simplicity of notation we will denote ap-fold intersection of sets inU by U i 0 ip = U i 0 \\U ip : LetF be a sheaf of abelian groups on X. The ˇ Cech complex ofF with respect to the coveringU, is the complex of abelian groupsC (U;F ) whosepth term is C p (U;F ) = Y i 0 <<ip F (U i 0 ip ): The ˇ Cech coboundary map :C p !C p+1 is defined by the formula () i 0 i p+1 = p+1 X l=0 i 0 b i l i p+1 j U i 0 i p+1 ; (2.1.1) where i 0 b i l i p+1 2 F (U i 0 b i l i p+1 ) and b i l indicates that the i l term is omitted. One easily checks that 2 = 0, so that (C (U;F );) forms a cochain complex of abelian groups. Theith ˇ Cech cohomology group ofF with respect to the coveringU, is defined to be H i (U;F ) = h i ((C (U;F );)): 6 The ˇ Cech cohomology of a sheaf with respect to an arbitrary covering need not agree with the derived functor cohomology, but H 1 (X;F ) ' lim ! U H 1 (U;F ); (2.1.2) where the limit is taken over all open coverings ordered by refinement (cf. [H, Ch. 3, Exercise 4.4]). The next proposition provides conditions under which the ˇ Cech cohomology with respect to a single open covering agrees with the derived functor cohomology. Proposition 2.1.1. ([H, Ch. 3, Prop 4.5]) LetX be a noetherian scheme, and letU be a covering ofX by affines. For any quasi-coherent sheafF onX, H i (U;F ) ' H i (X;F ): In order to illustrate the utility of Proposition 2.1.1, we now give a calculation which we will need later. Proposition 2.1.2. Let k be a field and let A n be affine n-space over k, with coordinates x 1 ; ;x n . Let 0 = (0; ; 0) 2 A n be the origin, and let U = A n n 0 denote the open complement. Then H 1 (U;O U ) = 8 > > < > > : 0 n = 1; n 3; span k n 1 x i 1 x j 2 ji;j > 0 o n = 2: Proof. The case n = 1 is an immediate consequence of Theorem 2.0.3. The case n = 2 is more interesting. LetU i = D(x i ) be the basic open set upon whichx i is nonvanishing. Then U =fU 1 ;U 2 g is a covering ofU by affines, so Proposition 2.1.1 guarantees that H 1 (U;O U )' 7 H 1 (U;O U ). The ˇ Cech complex ofO U onU is 0!O U (U 1 )O U (U 2 ) !O U (U 1 \U 2 )! 0; so every element g 2 C 1 (U;O U ) is a ˇ Cech 1-cocycle (there is no cocycle condition). The ˇ Cech cohomology ofO U onU is then given by H 0 (U;O U ) =O U (U) (by the sheaf axiom) and H 1 (U;O U ) =O U (U 1 \U 2 )= (O U (U 1 )O U (U 2 )). Since(g 1 ;g 2 ) = g 2 g 1 , this reduces the calculation of H 1 to the determination of which elements inO U (U 1 \U 2 ) = k[x 1 1 ;x 1 2 ] are not of the formg 2 g 1 for (g 1 ;g 2 )2O U (U 1 )O U (U 2 ) = k[x 1 1 ;x 2 ]k[x 1 ;x 1 2 ]. From this description it is clear that H 1 (U;O U ) = span k n 1 x i 1 x j 2 ji;j > 0 o since no monomial of the form 1 x i 1 x j 2 is a difference of Laurent polynomialsg 2 g 1 withg i 2O U (U i ). Whenn 3, we again coverU by the basic open setsD(x i ). The ˇ Cech 1-cocycle condition g i 0 i 2 =g i 1 i 2 +g i 0 i 1 precludesg i 0 i 1 from containing any monomials of the form 1 x i 0 i 0 x i 1 i 1 with 0 , 1 > 0, and similarly forg i 0 i 2 andg i 1 i 2 . As such, it is not hard to show that every 1-cocycle is a coboundary, whence H 1 (U;O U ) = 0. The ˇ Cech complex ofF with respect toU can be made into a resolution, as follows. The sheafified ˇ Cech complex ofF with respect to U is the complex of sheaves whose pth term is given by C p (U;F ) = Y i 0 <<ip j (Fj U i 0 ip ); (2.1.3) where each j : U i 0 ip ,! X denotes the inclusion map. The differential : C p ! C p+1 is defined locally by formula (2.1.1). Upon taking global sections of the sheafified ˇ Cech complex, one obtains the usual ˇ Cech complex. Lemma 2.1.3. ([H, Ch. 3, Lemma 4.2]) For any sheaf of abelian groupsF onX, the complex C (U;F ) is a resolution ofF , i.e., there is a natural map" :F!C 0 such that the sequence 8 of sheaves 0!F " !C 0 (U;F )!C 1 (U;F )! ; is exact. In light of Theorem 2.0.3, the sheafified ˇ Cech complex can be used to produce acyclic reso- lutions of quasicoherent sheaves. Proposition 2.1.4. LetX be a noetherian, separated scheme and supposeU =fU i g is a covering ofX by open affines. IfF is a quasicoherent sheaf onX, thenF !C (U;F ) is an acyclic resolution ofF . Proof. SinceF is quasicoherent, so isFj U i 0 ip . Any intersection of affines in a separated scheme is affine [H, Ch. 2, Exercise 4.3], so by Theorem 2.0.3,Fj U i 0 ip is acyclic. For each inclusion j : U i 0 ip ! X, the direct image j carries acyclic sheaves to acyclic sheaves, so j (Fj U i 0 ip ) is acyclic. Since X is quasi-compact we may assume thatU is a finite covering, so the direct product in 2.1.3 is a direct sum. Direct sums of acyclic sheaves are acyclic, so the proposition follows. 2.2 The Picard group LetL andL 0 be rank one locally freeO X -modules (also known as invertible sheaves, or line bundles). Then both the tensor productL O X L 0 and the dual sheafL _ =Hom X (L;O X ) are locally free of rank one, and the perfect pairingL O X L _ !O X is an isomorphism. One deduces that the set of isomorphism classes of invertible sheaves onX forms an abelian group under O X with identity given by the class ofO X . This group is known as the Picard group of X, and is denoted Pic(X). 9 The Picard group admits the following cohomological interpretation (cf. [H] Chapter 3, Exercise 4.5). LetL be a line bundle, and suppose that on the open covering U =fU i g we have trivializations' i :Lj U i !O X j U i . The transition functions' ij =' j ' 1 i :O X j U i \U j ! O X j U i \U j areO X -linear automorphisms, and therefore given by unitsg ij 2O X (U i \U j ). In light of the compatibility requirement' ik =' jk ' ij for transition functions, the collection of units g = (g ij )2 Q i;j O X (U i \U j ) necessarily satisfy the ˇ Cech 1-cocycle conditiong ik =g jk g ij . As seen above, the transition data (g ij ) forL determines a cohomology class in H 1 (U;O X ), which we denotec(L ). One can show that this correspondence defines an isomorphism Pic(X)' lim U H 1 (U;O X )' H 1 (X;O X ): We will see further examples of this idea, which can be stated briefly: elements of H 1 classify locally trivial spaces by means of transition data. When U is a covering upon which all line bundles become trivial,L7!c(L ) defines an isomorphism Pic(X)' H 1 (U;O X ). Example 2.2.1. Let k be a field and let P n be projective n space over k with homogeneous coordinatesx 0 ; ;x n . The Picard group ofP n is isomorphic toZ, and is generated by the class of the bundleO P 1(1) The next proposition shows that the Picard group remains unchanged after removing a closed subset of codimension at least two. Proposition 2.2.2. Let X be a noetherian, integral, separated locally factorial scheme, and let Z X be a proper closed subset of codimension at least two. Then the restriction map Pic(X)! Pic(XnZ) is an isomorphism. Proof. Combine Proposition 6.5 and Corollary 6.16 in chapter II of [H]. Example 2.2.3. Let k be a field and let n 2 be an integer. Proposition 2.2.2 shows that Pic(A n n 0)' Pic(A n ) = 0 since line bundles on affine spaces are trivial. 10 2.3 Vector bundles and torsors Let X be a scheme over a field k. A (geometric) vector bundle on X is a scheme E together with a morphism : E! X such that there exists an open coveringfU i g ofX together with isomorphisms' i : 1 (U i )!U i A r overU i , such that each transition function' ij =' j ' 1 i is given by a linear automorphism ofO X (U i \U j )[x 1 ; ;x r ]. Moreover, the transition functions satisfy the cocycle condition ' ik = ' jk ' ij . The sheaf of sections of E over X, written S (E=X), is a rankr locally free sheaf ofO X -modules. Conversely, given a locally free sheafE , one can construct a geometric vector bundle E whose sheaf of sectionsS (E=X) isE . More precisely,E is the relative spectrum of theO X - algebra Sym(E _ ), i.e. E = Spec X Sym(E _ ). These constructions provide an equivalence of categories: Vector bundles of rankr onX ! Locally freeO X -modules of rankr; E 7! S (E=X); Spec X Sym(E _ ) [ E: In light of this equivalence, locally free sheaves are often referred to in the literature as vector bundles. WhenX = Spec(R), locally free sheaves are equivalent to projectiveR-modules, and so the study of vector bundles on affine schemes is equivalent to the study of projective modules. For more details, consult [H, Ch. 2, Exercises 5.17, 5.18]. Example 2.3.1. (Main example) Letk be a field and consider the polynomial ringR =k[x 1 ; ;x n ]. Letf 1 ; ;f n 2R be polynomials, and letZ =Z(f 1 ; ;f n ) denote their common vanishing locus. On the open complementU =A n nZ, thef i determine a surjective morphism of sheaves f :O n U !O U , defined by the formula (g 1 ; ;g n )7! P n i=1 f i g i . LetE f denote the kernel of this morphism; by construction,E f is locally free of rankn 1. The associated geometric vector 11 bundle is E f = ( (x;z)2UA n : n X i=1 f i z i = 0 ) ; with projection map : E f ! U given by projection onto the x coordinates. Over the basic open set D(f i ) upon which f i is nonvanishing, one can rewrite the defining equation as z i = 1 f i P j6=i f j z j . Consequently there is an isomorphism' i : 1 (D(f i ))!D(f i )A n1 defined by omittingz i , and the transition maps' ij satisfy' ik =' jk ' ij . LetG be a group. AG-torsor is a nonempty set upon whichG acts freely and transitively. One can define torsors for group objects in more general categories. The present focus will be on torsors for group objects in the categorySch=X of schemes over a fixed base schemeX. In the category of schemes overX, vector bundles are group objects. Indeed, given a vector bundleE with sheaf of sectionsE , the diagonal morphismE!EE induces a morphism of X-schemesE X E!E. On closed points, this morphism is fiberwise addition. Torsors under a vector bundle are the relative analogue of affine spaces associated with a vector space. Definition 2.3.2. LetX be a scheme, and letE be a geometric vector bundle onX. A torsor underE is anX-schemeV ! X together with an action mapE X V ! V which is locally isomorphic to the additive action ofV on itself. Let : E! X be a vector bundle, and let : V ! X be anE-torsor. SupposeU =fU i g is an open affine covering ofX upon which the action ofE onV is isomorphic to the action of V on itself. Another way of stating this is to say that there exist isomorphisms' i :Vj U i !Ej U i which are equivariant with respect to the action ofEj U i on each side. In turn, this implies that the transition functions' ij =' j ' 1 i j U i \U j areEj U i \U j -equivariant automorphisms, and therefore given by a translation determined by a section s ij 2 (U i \ U j ;E ). These sections satisfy the ˇ Cech 1-cocycle condition s ik = s jk +s ij and so determine a cohomology class which we denote byc(V )2 H 1 (U;E ). By analogy with the case of the Picard group, one can show that 12 the correspondence V 7! c(V ) determines a bijection between the set of isomorphism classes of E-torsors and the first cohomology group H 1 (X;E ). The trivial torsor (namely, E itself) corresponds to 0. Example 2.3.3. Letk be a field and letE be ann-dimensional vector space overk. Regarding E as a vector bundleE! Spec(k) we may think of affinen-spaceA n as a torsor underE. By Theorem 2.0.3, H 1 (Spec(k);E ) = 0, so all torsors underE are trivial, but not canonically so. This explains the adage, “an affine space is like a vector space which has forgotten its origin." Example 2.3.4. Let U = A 2 n 0. The vector bundles E f for n = 2 described in Example 2.3.1 are necessarily trivial since Pic(U) = 0 by Example 2.2.3. In [DF], the authors conduct an extensive study of torsors underO U = G a . They prove that aG a -bundle onU is nontrivial if and only if its total space is affine [DF, Prop 1.2]. Let : U!P 1 denote the quotient map. There is an equivalence of categories betweenG m -linearized bundles onU and vector bundles onP 1 [DF, Remark 1.9]. The structure sheafO U isG m -linearized, and O U = d O P 1(d). By Proposition 2.1.2, torsors under the line bundleO U are classified by elements of the infinite- dimensional vector space H 1 (U;O U ) = d H 1 (P 1 ;O P 1(d)). Let : V ! U be a torsor under O U . If the isomorphy class ofV belongs to H 1 (P 1 ;O P 1(d)) for somed 2,V is said to be d-homogeneous. For a fixed d 2 the total spaces of alld-homogeneousG a -bundles are isomorphic [DF, Thm 2.3]. Example 2.3.5. (Main example) Forn 3, letf 1 ; ;f n 2R =k[x 1 ; ;x n ] be polynomials with common vanishing locusZ =f0g and letU = A n nZ. LetE f be the locally free sheaf defined in Example 2.3.1, with associated geometric vector bundle :E f !U. In contrast with the casen = 2 described above,E f is usually a nontrivial bundle. The short exact sequence 0!E f !O n U f !O U ! 0; 13 gives rise to a long exact sequence in cohomology 0! H 0 (U;E f )!R n !R! H 1 (U;E f )! 0; since H 1 (U;O U ) = 0 forn 3 (Proposition 2.1.2). Consequently, H 1 (U;E f )'R=(f 1 ; ;f n ). The connecting homomorphismR! H 1 (U;E f ) takes an elementg2R to the cohomology class corresponding to the torsor V f (g) = ( (x;y)2UA n : n X i=1 f i y i =g ) ; (2.3.1) with projection map : V f (g)! U again given by projection onto the x coordinates. It is clear thatV f (g) is affine wheng has a nonzero constant term. Another departure from the case n = 2 described by [DF] is that when n 3, there are examples of nontrivial torsors over U with strictly quasi-affine total spaces; anyV f (g) whereg = 2 (f 1 ; ;f n ) but for whichg(0) = 0 is strictly quasi-affine and nontrivial. An important special case of this construction which we will use later is the case when all f i =x a i for some natural numbersa i ; in this case, we will writeV a (g) forV f (g). The (2n 1)- dimensional quadric isQ 2n1 =fx 1 y 1 + +x n y n = 1g; in the notation defined above we may writeQ 2n1 = V 1 (1). It would be interesting to extend cohomological results from [DF] to the torsorsV a (g) in a continuation of this work. 14 Chapter 3 A 1 -homotopy theory Letk be a field, and letSm=k denote the category of smooth schemes of finite type overk. Throughout this section, unless otherwise specified, all schemes will be smooth and of finite type overk. Morel and V oevodsky have developed a homotopy theory for schemes overk in which the affine lineA 1 = Spec(k[t]) plays the role of the unit interval, calledA 1 -homotopy theory. By analogy with classical homotopy theory, Morel and V oevodsky construct the (unstable)A 1 - homotopy category by embeddingSm=k into a larger category of “spaces"Spc=k, and formally inverting a class of morphisms calledA 1 -weak equivalences. We denote theA 1 -homotopy cat- egory of schemes over k byH (k). It is in this context that motivic spheres arise, so we now review the basics of the theory. In order to state the definition ofH (k) precisely, we use the language of model categories. A model category is a categoryC with all small limits and colimits, together with three classes of specified morphisms called fibrations, cofibrations, and weak equivalences satisfying certain axioms [Hov, Ch. 1]. The homotopy category ofC is obtained by formally inverting the class of weak equivalences inC [Hov, Ch. 2]. Loosely speaking, a model category is a model for homotopy theory inC . We will need neither fibrations nor cofibrations, so when we describe model categories below, we will only refer to the weak equivalences. The construction of theA 1 -homotopy categoryH (k) involves a choice of model structure for the category of simplicial sets, the category of simplicial sheaves on a siteT , the category 15 of simplicial Nisnevich sheaves on (Sm=k), and their pointed variants. The required categories, with their weak equivalences, are summarized in the table below. Model category Weak equivalences Homotopy category op (Sets) jX j weakly equivalent tojY j H ( op (Sets)) op (Shv(T )) x (X) ! x (Y ) is a bijection for all pointsx H s ( op (Shv(T ))) op (Shv((Sm=k) Nis )) Hom(Y;Z )! Hom(X;Z ) bijections forA 1 -localZ H (k) 3.1 Simplicial model categories Let denote the simplex category whose objects are the finite ordered sets n =f0; ;ng for n = 0; 1; , and whose morphisms are order preserving maps. A simplicial object in a categoryC is a contravariant functor X : ! C . Explicitly, a simplicial object consists of a collection of objects X n together with face morphisms @ i : X n ! X n1 and degeneracy morphisms i : X n ! X n+1 , satisfying relations corresponding to those between the order preserving maps in . We refer the reader to [W, Ch. 8] for the precise conditions. We denote the category of simplicial objects inC by op (C ). WhenC =Sets is the category of sets, objects in op (C ) are called simplicial sets. LetK denote the category of compactly generated Hausdorff topological spaces. Simplicial sets are similar to compactly generated Hausdorff spaces in the following sense. There are a pair of adjoint functors K ! op (Sets); X 7! Sing (X); op (Sets) ! K; X 7! jX j: 16 Further details aboutSing andjj (geometric realization) can be found in [W, Ch. 8] and [Hov, Ch. 3]. The most important consequence of this relationship is that basic notions of homotopy theory can be phrased in the language of simplicial sets. This is the starting point for the transport of homotopy-theoretic ideas into more general categories than that of topological spaces. The model structure on op (Sets) has as its weak equivalences the morphismsX ! Y for which the induced map on geometric realizationjX j!jY j is a topological homotopy equivalence. An important cosimplicial object in op (Sets) is the simplex object : ! op (Sets), which assigns to each object n the simplicial set n defined by n i = 8 > > < > > : I n;jIj =i in; ; i>n; with all the usual face and degeneracy maps. LetT be a site, i.e. a category together with a choice of Grothendieck topology. For back- ground information on Grothendieck topologies and sites, we refer the reader to [Art]. Recall that a point of a site is a functor x : T ! Sets which commutes with all limits and colimits. The category of simplicial sheaves on T , op (Shv(T )), carries a model structure in which a morphismX !Y is a weak equivalence if for all pointsx : T ! Sets the induced map at the stalkx : x(X )! x(Y ) is a bijection. The corresponding homotopy category is denoted H ( op (Shv(T ))). There is a cosimplicial object : ! op (Shv(T )) which assigns to each n2 the constant sheaf of simplicial sets n . 3.2 The unstableA 1 -homotopy category Recall that a morphism of schemes is étale if it is smooth of relative dimension zero [H, Ch. 3, Exercise 10.3]. A collection of morphismsfj i :U i !Xg is called a Nisnevich cover if eachj i is étale and for each pointx2X, there existsy2U i withj i (y) =x such that the induced map 17 on residue fields k(x)! k(y) is an isomorphism. Nisnevich covers generate a Grothendieck topology on Sm=k called the Nisnevich topology. The Nisnevich topology is finer than the Zariski topology but coarser than the étale topology. The site of smooth k-schemes with the Nisnevich topology will be denoted (Sm=k) Nis ; our present purpose is to embed this category into a larger category of “spaces"Spc=k in which we can perform the necessary constructions for homotopy theory. The aforementioned categorySpc=k is defined as the category of simplicial sheaves of sets on the site of smooth schemes overk endowed with the Nisnevich topology, namelySpc=k = op (Shv(Sm=k) Nis ). There is a fully faithful embedding Sm=k! Spc=k which associates with each schemeX a simplicial sheafX . We will abuse notation and writeA 1 for the simplicial sheaf associated with the affine lineA 1 , and likewise forG m ,P 1 , and other well-known schemes. We denote the simplicial homotopy categoryH ( op (Shv((Sm=k) Nis )) by simplyH s (k). A simplicial sheafX on (Sm=k) Nis is calledA 1 -local if for any simplicial sheafY the map Hom Hs(k) (Y;X )! Hom Hs(k) (Y A 1 ;X ); induced by the projectionY A 1 !Y , is a bijection. Given two simplicial sheavesX and Y , there is a simplicial set of morphismsS(X;Y ) =Hom Spc=k (X ;Y ). A morphism of simplicial sheavesf :X !Y is called anA 1 -weak equivalence if for anyA 1 -local, simplicially fibrant sheafZ , the map of simplicial sets S(Y;Z )!S(X;Z ); induced byf is a weak equivalence. The homotopy category obtained by inverting the class of A 1 -weak equivalences is called the (unstable)A 1 -homotopy category. We denote this category byH (k). 18 Example 3.2.1. [MV, 3, Examples 2.2 and 2.3] Any vector bundle : E ! X induces an A 1 -weak equivalence on the level of simplicial sheaves, and so does anyE-torsor :V!X. Let (Spc=k) be the category of pointed simplicial sheaves. A morphism of pointed sim- plicial sheaves is a fibration, cofibration, or simplicial weak equivalence if it belongs to the corresponding class as a morphism of unpointed simplicial sheaves. The corresponding simpli- cial homotopy category is denotedH s ((Sm=k) Nis ). The left adjoint to the forgetful functor (Spc=k) !Spc=k is the functorX 7!X + , whereX + is the pointed simplicial sheafX ` pt pointed by the natural embedding pt!X ` pt. These functors preserve weak equivalences and therefore induce a pair of adjoint functors betweenH s ((Sm=k) Nis ) andH s ((Sm=k) Nis ). The wedge product of two pointed simplicial sheaves (X;x) and (Y;y) is the pointed sim- plicial sheaf (X;x)_ (Y;y) = (X ` pt Y;x =y). The smash product of (X;x) and (Y;y) is (X;x)^ (Y;y) = (X Y=(X;x)_ (Y;y);xy). Motivic spheres are defined using the smash product as follows. There are two natural circle objects in (Spc=k) . First, the simplicial circle isS 1 s =A 1 =(0 1), pointed by the image of 1 (note: this quotient is formed in the category Spc=k, not in the category ofk-schemes). The simplicial circle is not represented by any smoothk-scheme. Sec- ondly, there is the Tate circleS 1 t = G m = A 1 n 0, pointed by the group identity 1. Due to the presence of the two aforementioned circle objects in (Spc=k) , spheres are naturally bigraded. The (p;q)-sphere in (Spc=k) is S p;q = S pq s ^S q t ; where S n s = (S 1 s ) ^n and S n t = (S 1 t ) ^n . By a motivic (p;q)-sphere, we mean any smooth k- schemeX which becomes isomorphic toS p;q inH (k). Example 3.2.2. InH (k), there is a canonical isomorphism (P 1 ;)'S 2;1 by Lemma 2.15 and Corollary 2.18 in [MV, 3]. 19 Example 3.2.3. [MV, 3, Example 2.20] Punctured affinen-space overk is a motivic (2n1;n)- sphere. There is a canonical isomorphism inH (k) A n n 0 ' (S 1 s ) ^n1 ^ (S 1 t ) ^n =S 2n1;n : By Example 3.2.1, any vector bundle : E! A n n 0 is a motivic (2n 1;n)-sphere, and so is any E-torsor : X! A n n 0. In particular, the vector bundles E f and their torsors V f (g) described in Examples 2.3.1 and 2.3.5 are also motivic (2n 1;n)-spheres. Associated with the simplicial and Tate circles, there are suspension functors s and t defined by s (X;x) = (S 1 s ; 1)^ (X;x) and t (X;x) = (S 1 t ; 1)^ (X;x). A presheaf of setsF onSm=k is said to beA 1 -invariant if the projection mapXA 1 !X induces a bijectionF (X)! F (XA 1 ) for all X 2 Sm=k. The A 1 -homotopy category was originally constructed to facilitate the study of A 1 -invariant presheaves such as algebraic K-theory and motivic cohomology. The next theorem implies that the Picard group and units of a scheme areA 1 -invariant. Theorem 3.2.4. [MV, 4, Thm 3.8] For any smooth schemeX overk, one has Hom H(k) ( n s m t X + ; (P 1 ;)) = 8 > > > > > > > > > > < > > > > > > > > > > : Pic(X) m;n = 0 O (X) m = 0;n = 1 H 0 (X;Z) m = 1;n = 1 0 otherwise: Affine vector bundle torsors are of particular importance in the study of A 1 -invariant co- homology theories, such as motivic cohomology and algebraic K-theory. In [Wei], the author exploits the existence of affine vector bundle torsors to construct a homotopy-invariant version of algebraic K theory. The precise conditions on a base scheme X for the existence of affine 20 vector bundle torsorsX were first noted by Jouanolou and then extended by Thomason. Theorem 3.2.5. (Jouanolou’s Device) Let X be a quasicompact, separated scheme with an ample line bundle. ThenX admits an affine vector bundle torsor. Proof. The proof given here is taken from [Wei, Prop 4.3], and is due to Thomason. LetL be an ample line bundle overX. ReplacingL byL n if need be, we may assume that there exist global sectionss 0 ; ;s N ofL such that the subschemes X s i = fx2X : s i (x)6= 0g; are affine and form a cover ofX. (See the proof of [H, Chapter II, Proposition 7.7]; Hartshorne assumes X to be Noetherian.) These s i determine an affine morphism s : X ! P n Z [H, Ch. II, Exercise 5.17]. Jouanolou noted [J, 1.5] that the Stiefel scheme W 0 of rank 1 idempotent matrices inM N+1 (Z) forms an affine vector bundle torsor overP n Z . Pulling backW 0 toX, we get the affine vector bundle torsorW =s W 0 overX. Remark 3.2.6. Theorem 3.2.5 is true more generally for schemes admitting an ample family of line bundles, but we will not make use of this more general statement. 3.3 Unimodular rows and algebraic vector bundles LetR be a commutative ring. A row of ring elements (f 1 ; ;f n )2R n is a unimodular row if there existy 1 ; ;y n 2R so thatf 1 y 1 + +f n y n = 1. A unimodular row (f 1 ; ;f n ) defines a split surjection of R-modules R n ! R, and its kernel P is therefore a stably free module satisfyingPR'R n . In particular,P is a projectiveR-module of rankn 1. Geometrically, P defines a rankn 1 vector bundle on Spec(R), which is trivial if and only ifP is free. A unimodular row (f 1 ; ;f n ) is said to be completable if there is a matrixA2 GL n (R) with first row equal to (f 1 ; ;f n ). A row is completable if and only if the corresponding pro- 21 jective moduleP is a freeR-module. Completability of a unimodular row is therefore equivalent to triviality of the corresponding vector bundle. An important theorem in the study of unimodular rows is Suslin’s (n 1)! theorem. Theorem 3.3.1. [S, Thm. 2] LetR be a commutative ring and suppose (x 1 ; ;x n ) is a unimod- ular row inR. Leta 1 ; ;a n be natural numbers. If (n1)! divides Q n i=1 a i , then (x a 1 1 ; ;x an n ) is completable. The connection between unimodular rows andA 1 homotopy theory is detailed in [AF, 3]. We review the essentials here. Two morphisms of k-schemes f;g : X ! Y are naivelyA 1 - homotopic if there exists a morphismF :XA 1 !Y withF (0) =f andF (1) =g. The group GL n (R) acts on the set of unimodular rows of lengthnUm n (R), by multiplication on the right. In particular, the subgroup of elementary matricesE n (R) also acts onUm n (R), and the quotient spaceUm n (R)=E n (R) identifies with the set of naiveA 1 -homotopy classes of morphisms from Spec(R) toA n n 0,Hom A 1(Spec(R);A n n 0) (assumingR is smooth). Moreover, in this special case, the natural mapHom A 1(Spec(R);A n n 0)! [Spec(R);A n n 0] is a bijection. WhenR =k[x 1 ; ;x n ;y 1 ; ;y n ]=h P x i y i = 1i, Spec(R) =Q 2n1 is the smooth (2n 1) dimensional quadric. The quadricQ 2n1 is often called the unimodular affine scheme because any unimodular row (r 1 ; ;r n ) inR gives rise to a morphism Spec(R)!Q 2n1 overA n n 0. By a careful analysis of the long exact sequence in homotopy arising from the fiber sequence SL n1 !SL n !A n n 0; Asok and Fasel are able to classify algebraic vector bundles of rank n 1 on Q 2n1 , and any other smooth affine schemeX with the sameA 1 -homotopy type. Their main results are below. Theorem 3.3.2. [AF, Thm 3.4] Ifn 2 is an even integer, then there is an isomorphism between the group of isomorphism classes of rankn 1 vector bundles onQ 2n1 and the groupZ=(n 22 1)!. Moreover, each isomorphism class admits a representative given by the unimodular row (x m 1 ;x 2 ; ;x n ) for 1m (n 1)!. Theorem 3.3.3. [AF, Thm 3.5] Ifn 3 is an odd integer, then there is an isomorphism between the group of isomorphism classes of rankn 1 vector bundles onQ 2n1 and the groupZ=(n 1)! Z=2 W (k), whereW (k) is the Witt group of the fieldk. 23 Chapter 4 Algebraic de Rham cohomology Algebraic de Rham cohomology was first formulated as a purely algebraic version of de Rham cohomology for smooth complex varieties, and then defined for more general fields. Since the Poincaré Lemma fails for the algebraic de Rham complex, one must define the de Rham cohomology in terms of hypercohomology, which we review now. 4.1 Hypercohomology Let X be a topological space. LetF andC be complexes of sheaves of abelian groups on X. A morphism :F !C is called a quasi-isomorphism if it induces isomorphisms i : h i (F )! h i (C ) of cohomology sheaves for alli. Hypercohomology extends sheaf cohomology to the category of complexes of abelian sheaves onX. LetF be a left-bounded complex of sheaves of abelian groups onX. SupposeF ,!I is an injective quasi-isomorphism of left-bounded complexes withI k injective for eachk. The ith hypercohomology group ofF : is defined as H i (X;F ) = h i ((X;I )): A priori, this depends on the choice of a quasi-isomorphismF ! I , but as with sheaf cohomology, one can show that the choice is irrelevant. 24 Proposition 4.1.1. [V, Prop 8.6] The hypercohomologyH (X;F ) is well defined up to canon- ical isomorphism, independently of the choice of quasi-isomorphismF !I . To illustrate how hypercohomology extends cohomology to the category of complexes of sheaves, consider a single sheafF as a complex concentrated in degree zero. A choice of injective resolutionF ,!I gives an inclusion of complexes as pictured below. F // 0 // 0 // I 0 //I 1 //I 2 // SinceI is exact except in degree zero, the cohomology sheaves of both complexes are justF in degree zero and 0 everywhere else. The inclusionF ,! I is therefore a quasi- isomorphism. It follows by the definitions that H i (X;F ) = H i (X;F ): The next result generalizes Proposition 2.0.1, and shows that we can calculate hypercoho- mology from a complex of acyclic sheaves. Proposition 4.1.2. [V, Prop 8.12] A quasi-isomorphismF !C of left bounded complexes of abelian sheaves, withC k acyclic for eachk, induces an isomorphism H i (X;F )' h i ((X;C )): 4.2 Basics of algebraic de Rham cohomology Letk be a field, letX be a smooth scheme over a fieldk and let p X denote the sheaf of regular differentialp-forms onX. The algebraic de Rham complex ofX is the complex X with differ- entiald : p X ! p+1 X given by the exterior derivative. The algebraic de Rham cohomology of 25 X is defined to be the hypercohomology of this complex, H (X) = H (X; ): Remark 4.2.1. LetM be a smooth manifold, and consider the usual de Rham complex M . The sheaves p M are fine, meaning that they admit partitions of unity, and as a consequence they are acyclic for the global sections functor [V, p. 104]. The Poincaré Lemma states thatR! M is an exact sequence of sheaves, so by Proposition 4.1.2, the hypercohomology H (M; ) is precisely the classical de Rham cohomology H dR (M;R) = h ((M; )). WhenX is a smooth scheme of finite type overC, one can construct the associated analytic spaceX an . The points ofX an are exactly the closed points ofX, endowed with the Euclidean topology. There is a natural map X an ! X defined by the identity on points, but it is not a homeomorphism because Euclidean open sets need not be Zariski open. The following compari- son theorem shows that the algebraic de Rham cohomology coincides with the classical de Rham cohomology. Theorem 4.2.2. [G, Thm 1’] LetX be a smooth scheme of finite type overC. The mapX an !X induces an isomorphism H (X) ' H dR (X an ;C): Algebraic de Rham cohomology behaves well under base change, so for varieties defined over subfields ofC, one can change base toC and compare with the associated analytic space to calculate the cohomology. Consequently, the results we discuss in Section 5 are valid over k =Q. Theorem 4.2.3. Let L=k be a field extension where both k and L admit embeddings into the complex numbers. If X is a smooth scheme of finite type over k, then H (X) is a k-vector 26 subspace of H (X Spec(k) Spec(L)) and H (X Spec(k) Spec(L)) ' H (X) k L: Our main result relies on Grothendieck’s theorem on the cohomology of affine varieties. Just as the classical de Rham cohomology of a smooth manifold is calculable from the complex of globally defined differential forms, Grothendieck’s theorem asserts that the algebraic de Rham cohomology of a smooth affine variety is also calculable from global sections of the de Rham complex. Theorem 4.2.4. [G, Thm 1] Let k be a field which admits an embedding into the complex numbers, and letX be a smooth affinek-scheme of finite type. Then H i (X) ' h i ((X; )): The proof uses resolution of singularities in an essential way, so the proof does not work for fields of characteristicp > 0. This is why we restrict to subfields ofC. Another important property of algebraic de Rham cohomology is homotopy invariance. Theorem 4.2.5. Algebraic de Rham cohomology is homotopy invariant. IfX andY are smooth complex varieties whose associated analytic spaces are homotopy equivalent, then H (X)' H (Y ). Example 4.2.6. Let k = C, and let X denote an algebraic n-torus, namely X = G n m , with coordinates x 1 ; ;x n . Let R = k[x 1 1 ; ;x 1 n ] denote the coordinate algebra of X. One can show that the global sections of the algebraic de Rham complex form a complex quasi- isomorphic to the complex V R with trivial differentials. In other words, H (X)' V R as graded-commutative algebras. 27 4.3 The ˇ Cech-de Rham complex Let X be a smooth scheme of finite type over a field k and let ( X ;d) be the de Rham complex ofX. LetU =fU i g be an open covering ofX, and consider the double complexC = C (U; ). The row differentials :C p ( U; q )!C p+1 ( U; q ) are the usual ˇ Cech coboundary maps as defined by formula 2.1.1, and the column differentials are signed exterior derivatives (1) p d :C p ( U; q )!C p ( U; q+1 ). We will callC (U; ) the ˇ Cech-de Rham complex of with respect to the open coveringU. Now assume that the open subschemes U i are affine. Then by Proposition 2.1.4, the qth row of the ˇ Cech-de Rham complexC (U; q ) is a resolution of q by acyclic sheaves. The total complex Tot (C ) consists of -acyclic sheaves, so the induced quasi-isomorphism r ! Tot (C ) is suitable for calculating hypercohomology by Proposition 4.1.2. In other words, we have H i (X) ' h i ((X; Tot (C )): (4.3.1) The augmented ˇ Cech-de Rham complexK =K (U; ) is defined by augmenting theqth rowC (U; q ) with the restriction map r : q !C 0 (U; q ) from Lemma 2.1.3. This double complex is pictured below. . . . . . . . . . 0 // q+1 X OO r //C 0 ( U; q+1 ) OO //C 1 ( U; q+1 ) OO // 0 // q X d OO r //C 0 ( U; q ) d OO //C 1 ( U; q ) d OO // . . . d OO . . . d OO . . . d OO In the next section, we will make use of the following reformulation of the isomorphism 28 given in Equation 4.3.1. Theorem 4.3.1. Let X be a smooth k-scheme of finite type and assume that U =fU i g is a covering ofX by open affines. Then the augmented ˇ Cech-de Rham complexK (U; ) has zero total hypercohomology. Proof. Regarding X as a double complex concentrated in degree1, we have a short exact sequence of double complexes 0! X r !C!K ! 0: The associated short exact sequence of total complexes 0! X ! Tot (C )! Tot (K )! 0; induces a long exact sequence in hypercohomology ! H i (X) r !H i (X; Tot (C ))!H i (X; Tot (K ))! H i+1 (X)! : As noted in 4.3.1 above, each mapr : H i (X)! H i (X; Tot (C )) is an isomorphism, therefore H i (X; Tot (K )) = 0 for alli. 4.4 The de Rham cohomology of affine motivic (2n 1;n)-spheres If X is an affine motivic (2n 1;n)-sphere, then so is XA 1 . Herein, we consider the isomorphism problem for affine motivic (2n1;n)-spheres of minimal dimension. Whenk =C the analytic space associated withA n n 0 isC n n 0, and deformation retracts ontoS 2n1 , so the 29 classical de Rham cohomology ofX an is H i (X an ;C) ' 8 > > < > > : C i = 0; 2n 1; 0 otherwise: By Theorem 4.2.4, the classical cohomology ofX an agrees with the algebraic de Rham coho- mology ofX, whence is computable from the complex X . In other words, H dR (X an ;C)' H (X)' h ((X; X )): Since the classical de Rham cohomology ofX an is nonzero in degree 2n 1, the complex (X; X ) must also be nonzero in degree 2n 1. It follows immediately that the dimension of X must be at least 2n 1. Fork a field which admits an embedding into the complex numbers, Theorem 4.2.3 together with the calculation above show that the algebraic de Rham cohomology of an affine motivic (2n 1;n)-sphereX overk is H i (X) = h i ((X; )) = 8 > > < > > : k i = 0; 2n 1; 0 otherwise: (4.4.1) The main result of the next section is a generalization of [DF, Thm 2.5]. The cohomology class of a nowhere-vanishing 2n 1-form (henceforth, volume form) can be used to distinguish affine motivic (2n 1;n)-spheres. We now reformulate the critical observations made in [DF, 2.3] as a lemma and corollary. Lemma 4.4.1. LetX be a smoothk-variety of dimensionm with trivial canonical bundle m X and trivial units,O X (X) = k . If! 1 and! 2 2 m X (X) are volume forms onX, then there is a nonzero constantc2k such that! 2 =c! 1 . 30 Proof. Choose an isomorphism m X 'O X . Upon taking global sections of such an isomorphism, O X -linearity guarantees that volume forms correspond precisely to units. Corollary 4.4.2. Supposen > 1, letX andY be affine motivic (2n-1,n)-spheres of dimension 2n 1. Let! X 2 (X; 2n1 X ) and! Y 2 (Y; 2n1 Y ) be volume forms. If! X is exact and! Y is not, thenX is not isomorphic toY . Proof. We proceed by contrapositive. An isomorphism ' : X! Y carries volume forms on Y to volume formsX, so' ! Y is a volume form onX. The canonical bundle 2n1 X is trivial, since Pic(X) = 0 by Example 2.2.3 and Theorem 3.2.4. The units of X are also trivial, so Lemma 4.4.1 guarantees that! X =c' ! Y for somec2k . Since ' : X ! Y is an isomorphism, it induces an isomorphism ' : H 2n1 (Y ) ! H 2n1 (X) and so ! Y is exact if and only if ' ! Y is exact. Since ! X and ' ! Y differ by a nonzero scalar multiple,! X is exact if and only if! Y is exact. Remark 4.4.3. The conditionn > 1 is essential. The punctured affine lineA 1 n 0 has volume forms which are exact as well as volume forms which are not exact. Both! 1 = dt t and! 2 = dt t 2 are nowhere vanishing, but! 2 is exact while! 1 is not. 31 Chapter 5 Exotic affine (2n 1;n)-spheres Throughout this section we assume thatk admits an embedding into the complex numbers, and thatX =V f (g) is defined as in Example 2.3.5, with :X!A n n0 the projection map onto thex coordinates. We now show how to decide whether a volume form! is exact in terms of f 1 ; ;f n andg. Using Corollary 4.4.2, this allows us to prove the existence of non-isomorphic affine motivic (2n 1;n)-spheres of minimal dimension. By Lemma 4.4.1, any two volume forms onX differ by a constant, so it suffices to work with a single representative. 5.1 Existence of exotic affine (2n 1;n)-spheres Consider the open coveringU =fU i g ofX, whereU i = 1 (D(f i )) andD(f i ) =A n nZ(f i ). Note that whenf i 6= 0, we can solve the equation definingX fory i : y i = 1 f i g X j6=i f j y j ! : (5.1.1) Equation 5.1.1 expressesy i as an affine-linear combination of the othery j , so there is an isomor- phismU i ' D(f i )A n1 defined by omittingy i . Define volume forms! i over eachU i by the 32 formula ! i = (1) i f i dx 1 ^^dx n ^dy 1 ^^ c dy i ^^dy n : It is easy to check that ! i 0 = ! i 1 on U i 0 i 1 , so the ! i glue together to produce a globally defined volume form!2 2n1 (X). In terms of the ˇ Cech complex of 2n1 onU, this means that (! i ) forms a ˇ Cech 0-cocyle, i.e. ((! i )) = 0. To decide whether! is exact, we now use the augmented ˇ Cech-de Rham complexK =K (U; ) described in Section 4.3. By Theorem 4.3.1, upon taking global sections ofK , we obtain a double complex K = (X;K ; ) with zero total cohomology, i.e. h (Tot (K)) = 0. We now investigate the vertical- horizontal spectral sequence associated withK. Since eachp-fold intersectionU i 0 ip is isomor- phic to a productD(f i 0 f ip )A n1 and eachD(f i 0 f ip ) is smooth affine and of dimension n, Theorem 4.2.4 together with Theorem 4.2.5 imply that H q (U i 0 U ip ) = 0 forq>n: Consequently, the only nonzero terms on theE 1 page live in either the1 columnE 1; 1 , or the rectangle 0pn 1, 0qn: E p;q 1 = 8 > > > > > > < > > > > > > : H q (X) p =1; 0q 2n 1; Q i 0 <<ip H q (U i 0 ip ) 0pn 1; 0 all other (p;q): TheE 1 page of the vertical-horizontal spectral sequence associated withK is pictured below. 33 H 2n1 (X) 0 0 . . . 0 0 0 Q i 0 H n (U i 0 ) d 1 // d 1 // H n (U 1n ) . . . . . . . . . H 0 (X) d 1 // Q i 0 H 0 (U i 0 ) d 1 // d 1 // H 0 (U 1n ): Suppose 1<r<n. The differentialsd r :E 1;2n1 r !E r1;2nr r , are all zero, and therefore E 1;2n1 n =E 1;2n1 1 = H 2n1 (X): Likewise, the differentials arriving atE n1;n r are all zero, so we must have E n1;n r =E n1;n 2 : SinceE p;q 1 vanishes for all (p;q), the differentiald n :E 1;2n1 n !E n1;n n induces an isomor- phism of one dimensionalk-vector spaces (which we continue to denote byd n ) d n : H 2n1 (X) ! H n (U 1n )=d 1 0 @ Y i 0 <<i n2 H n (U i 0 i n2 ) 1 A : (5.1.2) Remark 5.1.1. Whenn = 2, the isomorphism is the inverse of the connecting homomorphism in the Mayer-Vietoris exact sequence forX associated with the coveringX =U 1 [U 2 . Theorem 5.1.2. LetX = V f (g) be defined as in Example 2.3.5 and let!2 2n1 (X) be the 34 nowhere vanishing (2n 1)-form defined over eachU i = 1 (D(f i )) by the formula ! i = (1) i f i dx 1 ^^dx n ^dy 1 ^^ c dy i ^^dy n : Then under the isomorphism (5.1.1), the cohomology class [!] corresponds to d n [!] = (1) n g n1 dx 1 ^^dx n (n 1)!f 1 f n : (5.1.3) Proof. By definition of the differentials in the spectral sequence of a double complex, forp = 0; ;n1, the classd p+1 [!] p+1 is represented by a collection of cohomology classes [! i 0 ip ] r 2 E p;2np1 r = Q i 0 <<i p+1 H 2np1 (U i 0 ip ) where 1 i 0 < < i p n varies over all ordered p + 1-tuples inf1; ;ng. The! i 0 ip are defined inductively as follows. Assuming! i 0 ip is defined for eachp-fold intersection, choose primitives i 0 ip . In our case these are guaranteed to exist since H 2np1 (U i 0 ip ) = 0 forp = 0; ;n 2. Then the differentiald p+1 takes! to the cohomology classes represented by ! i 0 i p+1 = () i 0 i p+1 . The diagram below illustrates the movement of the representative for! within the double complex: ! r // (! i 0 ) ( i 0 ) d OO // (! i 0 i 1 ) ( i 0 i 1 ) d OO // // (! i 0 i n2 ) ( i 0 i n2 ) (1) n2 d OO // ! 1n : In order to arrive at the formula for d n [!] stated above in 5.1.3, we systematically choose primitives so as to eliminate all thedy j terms. LetI =fi 0 < <i p g be an ordered subset of f1; ;ng, and define 35 dX = dx 1 ^^dx n ; dY I = dy 1 ^^ d dy i 0 ^^ d dy ip ^^dy n : Furthermore, forj = 2I or fori l 2I the symbolsdY I[j anddY Ini l denote the forms obtained fromdY I by deletingdy j and insertingdy i l respectively. To avoid a proliferation of sums, we define (I) =i 0 + +i p . We will prove by induction onp that forp = 0; ;n 1, ! I = (np 1)! (n 1)! (1) (I)+p g p f i 0 f ip dX^dY I ; where p satisfies the recurrence relation p = p1 +p and 0 = 0. The casep = 0 is trivial sinced 1 [!] is represented by the classes of! i , which fit the description by construction. Before proceeding with the induction step, we introduce some sign changes that occur nat- urally as we consider possible primitives for! I . It is easy to movedy j from its initial position in dY to the leading position with this notation; the relation is dY = (1) j+1 dy j ^dY j . We introduce the quantity" j I = #fi2 I : i < jg in order to describe the sign change that occurs when movingdy j from its initial position indY I to the leading position. The formula reads dY I = (1) " j I +j+1 dy j ^dY I[j : (5.1.4) In light of Equation (5.1.4), thed-primitive of! i 0 ip with respect toy j forj = 2I is j I = (np 1)! (n 1)! (1) (I)+p g p f i 0 f ip dX^ (1) j+" j I +1 y j ^dY I[j : (5.1.5) In order to avoid giving preferential treatment to any particular y j as we consider all possible 36 p-fold intersections, we elect to average the j I forj = 2I. We define I = (1) p np 1 X j= 2I j I ; with the sign (1) p in place to ensure that I maps to! I via the vertical differential (1) p d inK. To begin the induction step, we setI =fi 0 ; ;i p+1 g and note that for eachl = 0; ;p + 1, the induction hypothesis gives ! Ini l = (np 1)! (n 1)! (1) (Ini l )+p g p f i 0 c f i l f i p+1 dX^dY Ini l : The primitives Ini l can be written as Ini l = (1) p np 1 0 @ i l Ini l + X j= 2I j Ini l 1 A ; and so by definition of the ˇ Cech coboundary map, we have ! i 0 i p+1 = (1) p np 1 0 @ p+1 X l=0 (1) l i l Ini l + X j= 2I p+1 X l=0 (1) l j Ini l 1 A : (5.1.6) We evaluate the first term in (5.1.6) by summing the equation (5.1.5) withIni l in place ofI andj =i l . Using the identities (I) = (Ini l ) +i l and" i l Ini l =l, this expression simplifies to p+1 X l=0 (1) l i l Ini l = (np 1)! (n 1)! (1) (I)+p+1 g p dX^ p+1 X l=0 y i l f i 0 c f i l f i p+1 dY I :(5.1.7) By analogy with Equation 5.1.1, over each (p + 1)-fold intersectionU i 0 i p+1 , we can rewrite 37 the defining equation as p+1 X l=0 y i l f i 0 c f i l f i p+1 = g f i 0 f i p+1 X j= 2I f j y j f i 0 f i p+1 : (5.1.8) Applying (5.1.8) to (5.1.7) shows that P p+1 l=0 (1) l i l Ini l is equal to (np 1)! (n 1)! (1) (I)+p+1 g p dX^ 0 @ g f i 0 f i p+1 X j= 2I f j y j f i 0 f i p+1 1 A dY I : (5.1.9) It remains to see that the double sum in (5.1.6) cancels the sum in (5.1.9). The double sum P j= 2I P p+1 l=0 (1) l j Ini l is equal to (np 1)! (n 1)! (1) p+1 g p dX^ X j= 2I (1) j y j p+1 X l=0 (1) l+(Ini l )+" j Ini l f i 0 c f i l f i p+1 dY I[jni l : Keeping track of the sign change using (5.1.4) as we movedy i l to the front ofdY I[jni l , we manipulate the expression using the following simple relations: (I) = (Ini l ) +i l ; " i l I[jni l = " i l Ini l +" i l j =l +" j i l + 1; " j I = " j Ini l +" j i l : The end result of this calculation is: (np 1)! (n 1)! (1) (I)+p+1 g p dX^ X j= 2I (1) j+" j I y j p+1 X l=0 dy i l f i 0 c f i l f i p+1 ^dY I[j : (5.1.10) 38 OnU i 0 i p+1 , the defining 1-form relation P n i=1 y i df i +f i dy i =dg takes the form p+1 X l=0 dy i l f i 0 c f i l f i p+1 = 1 f i 0 f i p+1 0 @ X j= 2I f j dy j +dg n X i=1 y i df i 1 A : (5.1.11) Using (5.1.11), and then (5.1.4), the expression for the double sum (5.1.10) becomes (np 1)! (n 1)! (1) (I)+p+1 g p dX^ X j= 2I f j y j dy j f i 0 f i p+1 ^dY I : (5.1.12) This is exactly the opposite of the sum in (5.1.9) that we wish to cancel. Combining (5.1.6), (5.1.9), (5.1.12) and setting p+1 = p +p + 1, we arrive at the desired formula: ! I = (np 2)! (n 1)! (1) (I)+ p+1 g p+1 f i 0 f i p+1 dX^dY I : This completes the induction step. The sign in the formula for! 1n is (1) (f1;;ng)+ n1 = (1) 2(1++n1)+n = (1) n ; therefore d n [!] = (1) n (n 1)! g n1 f 1 f n dX : Remark 5.1.3. With notation as in Example 2.3.5, we note that this calculation is valid for : V f (g)!A n nZ for anyZ defined byn equationsf 1 = =f n = 0. In particular, the formula is valid for a volume form on a bouquet of motivic spheres, whenZ(f 1 ; ;f n ) is a finite set of points. In order to make use of this theorem we now make the simplifying assumption thatf i =x a i i . 39 In the notation of Example 2.3.5, we are assuming thatX =V a (g). This severely constrains the topology of the basic open setsD(f i ) and their intersectionsU i 0 ip . Eachp-fold intersection is isomorphic to U i 0 ip ' D(x i 0 x ip )A n1 'G p+1 m A np1 A n1 : Consequently, we haveH q (U i 0 ip ) = 0 forq>p + 1, and in particularE n2;n 1 = 0, so that E n1;n n = E n1;n 1 = H n (U 1n )' H n (G n m ): Thus, the differentiald n induces the following isomorphism, refining that in (5.1.1): d n : H 2n1 (X)! H n (G n m ): (5.1.13) Corollary 5.1.4. LetX =V a (g) be defined as in Example 2.3.5. If the coefficient ofx a 1 1 1 x an1 n ing n1 is zero, then all volume forms onX are exact. Proof. The n-form ! 1n = g n1 x a 1 1 x an n dx 1 ^^dx n descends to the torus G n m . All n-forms on the torusG n m are of the form h(x 1 ; ;x n )dx 1 ^^dx n for a Laurent polynomial h2 C[x 1 1 ; ;x 1 n ]. By Example 4.2.6, the top cohomology H n (G n m ) is generated by the class of = dx 1 ^^dxn x 1 xn , and so! 1n is exact if and only if the coefficient of in g n1 x a 1 1 x an n is zero. Corollary 5.1.5. Let X = V a (g) and Y = X b (h) be affine motivic (2n 1;n)-spheres as defined in Example 2.3.5. If the coefficient ofx a 1 1 x an1 ing n1 is zero, while the coefficient ofx b 1 1 x bn1 inh n1 is nonzero, thenX is not isomorphic toY . Proof. Combine Corollary 4.4.2 and Corollary 5.1.4. 40 5.2 Non-cancellative behavior of motivic spheres Corollary 5.1.5 can be construed as the statement that there exist “exotic" affine (2n 1;n)- spheres of minimal dimension. Of course, the use of the word “exotic" is not fully justified, but the (2n1)-dimensional quadricQ 2n1 is a candidate for a “standard" affine motivic (2n1;n)- sphere. It would be interesting to develop further criteria for non-isomorphy between total spaces of vector bundle torsors onA n n0. We now apply Suslin’s (n1)! theorem to prove the following generalization of Corollary 2.6 in [DF]. Corollary 5.2.1. Forn 3, there exist a pair of affine motivic (2n 1;n)-spheresX andY of dimension 2n 1 such thatXA n1 'YA n1 , butX6'Y . Proof. Let (f 1 ; ;f n ) = (x a 1 1 ; ;x an n ) wherea 1 = (n1)! anda 2 = =a n = 1. Consider the motivic spheres X =V a (1) = n x (n1)! 1 y 1 +x 2 y 2 + +x n y n = 1 o A n n 0A n ; Y =V a (x a 1 1 1 + 1) = n x (n1)! 1 y 1 +x 2 y 2 + +x n y n =x (n1)!1 1 + 1 o A n n 0A n ; as defined in Example 2.3.5. Let :X!A n n0 and 0 :Y!A n n0 denote the projections onto thex coordinates. Then bothX andY are torsors under the vector bundleE = E f described in Example 2.3.1. Since X is affine, all vector bundle torsors on X are trivial. In particular, X A n n0 Y is a E-torsor, and therefore isomorphic to E. Arguing symmetrically onY we see thatX A n n0 Y is isomorphic to 0 E. We now argue that E is a trivial vector bundle onX and 0 E is a trivial vector bundle on Y . Upon pulling back the short exact sequence 0!E!O n A n n0 f !O A n n0 ! 0; 41 we obtain a short exact sequence ofO X -modules 0! E!O n X f !O X ! 0: Note that since is projection onto thex-coordinates, the morphismf :O n X !O X is described by # (f 1 ; ;f n ) = (f 1 ; ;f n ), and so after taking global sections of this sequence we obtain a short exact sequence ofR = (X;O X )-modules 0!P!R n !R! 0: By definition, this entails that E is the vector bundle corresponding to the unimodular row (f 1 ; ;f n ) overR. Now Theorem 3.3.1 shows that (f 1 ; ;f n ) is completable, which entails thatP is free. Consequently, E is a trivial vector bundle onX. The same argument applied to 0 E onY shows that 0 E is trivial onY , again essentially because 0# (f 1 ; ;f n ) = (f 1 ; ;f n ). In summary, we have established that XA n1 ' E'X A n n0 Y' 0 E'YA n1 : But by Corollary 5.1.5,X is not isomorphic toY . xlii References [Art] M. Artin, Grothendieck topologies. Harvard Math. Dept. Lecture Notes, 1962. [AF] A. Asok and J .Fasel, Algebraic vector bundles on spheres. Preprint, available at http://arxiv.org/pdf/1204.4538v2.pdf [BT] R. Bott and L. Tu, Differential forms in algebraic topology. Springer, 1982. [DY] B. Doran and J. Yu, Algebraic vector bundles on punctured affine spaces and smooth quadrics. Preprint, available at http://arxiv.org/pdf/1303.0575v1.pdf [Dan] V . Danielewski, On a cancellation problem and automorphism groups of affine algebraic varieties. Preprint, Warsaw, 1989. [DF] A. Dubouloz and D. Finston, On exotic affine 3-spheres, Preprint, available at http://arxiv.org/pdf/1106.2900v1.pdf [G] A. Grothendieck, On the de Rham cohomology of algebraic varieties, Publications mathe- matiques de l’I.H.E.S.. tome 29 (1956), p 95-103. [H] R. Hartshorne, Algebraic Geometry. Springer, 1977. [Hov] M. Hovey, Model categories. American Mathematical Society, 2007. [J] J.-P. Jouanolou, Un suite exacte de Mayer-Vietoris enK-théorie algebrique, Lecture Notes in Math. No. 341 (1973), Springer-Verlag. [May] J. P. May, Simplicial objects in algebraic topology, Chicago Lectures in Mathematics, 1967. [MV] F. Morel and V . V oevodsky, A 1 -homotopy theory of schemes. Inst. Hautes Études Sci. Publ. Math., (90):45-143 (2001), 1999. [S] A.A. Suslin, Stably free modules, Mat. Sbornik. 102 (1977), no. 4, 537â550. [W] C. Weibel, An introduction to homological algebra, Cambridge University Press, 1994. [Wei] C. Weibel, Homotopy algebraic K-theory, AMS Contemp. Math. 83 (1989), 461-488. [V] C. V oisin, Hodge Theory and Complex Algebraic Geometry I. Cambridge University Press, 2002. xliii

Abstract (if available)

Abstract We study a class of smooth algebraic varieties which are, in the sense of Morel and Voevodsky's A¹-homotopy theory, homotopy equivalent to spheres. These varieties belong to a class of objects called vector bundle torsors, and we investigate their cohomological classification, and the isomorphism problem in general for them. In the affine case, we establish a criterion for non-isomorphy in terms of algebraic de Rham cohomology, as defined by Grothendieck. For all n ≥ 3, we then construct a pair of non-isomorphic smooth affine (2n-1)-dimensional varieties X and Y with the A¹-homotopy type of Aⁿ\0, with the property that X × Aⁿ⁻¹ ≃ Y × Aⁿ⁻¹. These results extend work of Dubouloz and Finston in the case n=2, and illustrate that motivic spheres of higher dimension exhibit non-cancellative behavior.

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Creator Ericksen, Adam (author)

Core Title The geometry of motivic spheres

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Mathematics

Publication Date 07/18/2013

Defense Date 05/24/2013

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

Tag A¹-homotopy theory,cancellation,motivic sphere,OAI-PMH Harvest

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Asok, Aravind (committee chair), Friedlander, Eric M. (committee member), Johnson, Clifford V. (committee member)

Creator Email adamliniscus@gmail.com,aerickse@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c3-292749

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