aa r X i v : . [ m a t h . K T ] N ov ENUMERATING NON-STABLE VECTOR BUNDLES
PENG DU
Abstract.
In this article, we establish a motivic analog of a result of James-Thomas [25]. Usingthis, we obtain results on enumeration of vector bundles of rank d over a smooth affine k -algebra A ofdimension d , recovering in particular results of Suslin and Bhatwadekar on cancellation of such vectorbundles. Admitting a conjecture of Asok and Fasel, we also prove cancellation of such modules of rank d − if the base field k is algebraically closed. Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12. Some fundamental results and constructions for simplicial sets . . . . . . . . . . . . . . 33. Tools from motivic homotopy theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64. A -homotopy study of vector bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115. Motivic approach to enumerating non-stable vector bundles . . . . . . . . . . . . . . . . 136. Application to vector bundles of critical rank . . . . . . . . . . . . . . . . . . . . . . . . . . . 187. Application to vector bundles below critical rank . . . . . . . . . . . . . . . . . . . . . . . . 26References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Introduction
We begin with some classical problems in commutative ring and module theory. Let R be a com-mutative ring, and P a finitely generated projective R -module (of constant rank n ). We say that theprojective module P is cancellative if for any m > , P ⊕ R m ∼ = Q ⊕ R m for some R -module Q implies P ∼ = Q . Note that if R is a local ring, then any finitely generated projective R -module is cancellative(since such modules are always free and the isomorphism class of a finitely generated free module isuniquely determined by its rank in that case).We have the Serre splitting theorem , which says that for a finitely generated projective module P over a commutative noetherian ring R of Krull dimension d , if its rank is at least d + 1 , then it has afree summand of rank ; and Bass cancellation theorem : any finitely generated projective module P over R of rank at least d + 1 is cancellative.So it’s interesting to understand further cancellation properties of projective modules of lower rank.In order to use nice results in algebraic geometry, we restrict the case when the commutative ring A isa k -algebra, k is a field. Even for k algebraically closed ( k = ¯ k ), Mohan Kumar [27] gives a negativeanswer in the case when the rank n = d − is prime ( d = dim A ): there exists a rank n = d − stablyfree module which is not free. We are thus left to deal with n = d and n = d − . For the case n = d ,Suslin [37] confirms cancellation in the case k = ¯ k . This result was later extended to the case wherethe base field is a C -field by Bhatwadekar [10]. For the case n = d − , Fasel-Rao-Swan [18] confirmcancellation for stably free modules in the case k = ¯ k (with other mild conditions).The goal of this article is to restudy the cancellation problem for projective modules over algebras,using modern tools. Precisely, we explore cancellation properties of projective modules using ob-struction theory in A -homotopy theory (hence we need to assume our affine k -algebra A is smooth),following some ideas in [3, 5].Concretely, we extend a topological result of James-Thomas [25] to the motivic homotopy setting,namely, we identify certain set of homotopy lifting classes with nice source and target with the cokernelof a certain map of abelian groups associated to some A -homotopy classes. We then give a formula forthat map, following the method of [25]; similar methods were also explored in [38]. This is essentially Peng DUa study of the derived mapping space in A -homotopy theory (see Corollary 5.4), taking advantage ofthe fact that stable things are abelian group objects even in the unstable motivic homotopy category.Combining with the Suslin matrix construction and some known results on the cohomology of certainmotivic spaces, we are able to show that the image of that map contains n ! multiple of the target abeliangroup, which is the cohomology of the variety Spec A with coefficients in some Milnor (or Milnor-Witt)K-theory sheaf, by manipulating some characteristic classes. Hence the set of homotopy lifting classesis a singleton if the target abelian group is n ! -divisible. It’s worth to remark that Suslin’s matrixconstruction plays a very important role here—it provides sufficiently many elements in the targetabelian group in question.Further, using recent results—the Rost-Schmid complex and the motivic Bloch-Kato conjecture, onecan show that the target cohomology group is indeed n ! -divisible in many cases (e.g. when all elementsin the residue fields at all closed points of Spec A are n ! -th power).In the n = d case, we will mostly focus on the odd rank case. For the even rank case, there aresome further difficulties. On the one hand, as the A -fundamental group of BGL n is non-trivial (i.e. BGL n is not A -simply connected), the relevant A -homotopy fiber sequence we get via A -homotopicalobstruction theory is not principal hence do not fall into our framework as in [25]; so we make thecompromise that we restrict to enumeration for oriented vector bundles. On the other hand, thefirst nontrivial homotopy sheaf of the relevant space is more difficult to study, even in the case oforiented bundles. Nevertheless, it’s still quite useful to get results using BSL n etc. instead of BGL n ,hence studying enumeration problem for non-stable oriented vector bundles. So we will mostly discussenumeration problem for oriented vector bundles.The main A -homotopy foundation comes from Morel’s work [31] where the base fields are all as-sumed to be perfect (and infinity, but later, Hogadi-Kulkarni [21] gave a published version of Gabber’spresentation lemma for finite fields, which confirms that Morel’s results are indeed also true for fi-nite fields), but every field k contains a perfect (prime) subfield k , and since the k -group schemes GL n /k, SL n /k are extended from k , our arguments in the text hold for a general base field k as well;cf. [7, Comments on the proof of Theorem 3.1.7]. To make the statements more concise, we just assumeeverywhere that k is perfect.Our main result in the n = d case is the following (for more details, see Theorems 6.1 and 6.2). Theorem.
Let k be a field, A a smooth affine k -algebra of odd Krull dimension d > , and X =Spec( A ) , let ξ be a stable vector bundle over X , whose classifying map is still denoted ξ : X → BGL .Let ϕ ∗ : [ X, BGL d ] A → [ X, BGL] A be the stabilizing map. Then there is a bijection ϕ − ∗ ( ξ ) ∼ = coker (cid:0) K ( X ) ∆( c d +1 ,ξ ) −−−−−−→ H d ( X ; K M d +1 ) (cid:1) , where the homomorphism ∆( c d +1 , ξ ) is given as follows: for β ∈ K ( X ) = [ X, ΩBGL] A = [ X, GL] A , ∆( c d +1 , ξ ) β = (Ω c d +1 )( β ) + d X r =1 ((Ω c r )( β )) · c d +1 − r ( ξ ) . Here, (Ω c i )( β ) ∈ H i − ( X ; K M i ) for i = 1 , · · · , d + 1 are the Chern classes of β and c i ( ξ ) are the ordinaryChern classes of ξ .In other words, given a vector bundle ξ of rank d over X , then the set of isomorphism classes vectorbundles ν such that ν ⊕ O X ∼ = ξ ⊕ O X is in bijection with coker∆( c d +1 , ξ ) .In case A is of even dimension d > and the base field is of cohomological dimension at most , thesame results as above hold for oriented bundles (see Theorem 6.14).As a corollary, we see that if k is algebraically closed or of cohomological dimension at most thenall rank d vector bundles are cancellative (Theorem 6.12), hence recovering the cancellation theoremsof Suslin and Bhatwadekar. Similar result also holds for d even and oriented vector bundles (seeTheorems 6.14 and 6.16).It’s also possible to explore the idea in this article to get further cancellation results in the rank d − case, which is more difficult. In §7, we obtain a conditional cancellation result by a study of a2-stage Moore-Postnikov factorization, in which the first stage is very similar to the results in rank d case and give a condition on the characteristic of the base field k (namely, char( k ) = 0 or char( k ) > d ;see Theorem 7.5) to ensure that the isomorphism class of a stable bundle has at most one lifting tothe first stage in the tower; for the second stage, we invoke a conjecture of Asok-Fasel (Conjecture 7.6)describing π A d ( A d \ , which gives a vanishing result on some cohomology group and ensures that theNUMERATING NON-STABLE VECTOR BUNDLES 3isomorphism class of the lifting map from the first stage to the second stage in the tower (exists and)is unique, which again involves studying the derived mapping space in A -homotopy theory, takingadvantage of the abelian group object structures.The (conditional) cancellation result in the n = d − case is the following (see Theorem 7.8). Theorem.
Assume the base field k is algebraically closed. Let A be a smooth k -algebra of Krulldimension d > and assume char( k ) = 0 or char( k ) > d . Let X = Spec A . If Conjecture 7.6 holds,then every oriented rank d − vector bundle over X is cancellative.The structure of this article is briefly as follows. Firstly, as a warm-up, we start in §2 with recallingsome fundamental results and constructions for simplicial sets, some already familiar from classicalhomotopy theory. In §3, we present one of the most important input of A -homotopy theory, themotivic Moore-Postnikov tower (in a form slightly more general than that in the existing literature),which is our obstruction-theoretic tool; and (relative) cohomology groups in our motivic homotopyworld, this is useful and important in analyzing various maps in the sequel. In §4, we identify the firstnon-trivial A -homotopy sheaf of the A -homotopy fiber of the map from non-stable to stable classifyingspaces. In §5, we set up the framework of enumeration of lifting A -homotopy classes, following ideasdeveloped in James-Thomas [25] in the classical homotopy setting. In §6, we apply the previousresults to enumeration problems for non-stable (oriented) vector bundles by some computations andidentifying some characteristic classes (with mild conditions); along the way, we also describe (inTheorem 6.6) more concretely and more conceptually the computing formula we give by studying thederived mapping space in A -homotopy theory, which says that the formula we provide essentiallydescribes the homomorphism of the fundamental group of a general component of the derived mappingspace in question to another derived mapping space given by the k -invariant in question, surjectivityof which gives uniqueness of the lifting class. In §7, we give a (conditional) cancellation result in the n = d − case, provided the Asok-Fasel Conjecture 7.6 holds. Acknowledgements.
First of all, I will express my deep indebtedness to my Ph.D. advisor, professorJean Fasel, for many discussions on mathematics and for sharing numerous mathematical ideas withme, many of which are used in this work, e.g. the idea of using Suslin matrices. I also thank TariqSyed for discussing many things on A -homotopy theory and on vector bundles, and Aravind Asok forpointing out the paper of James and Thomas and his comment that our result extends to general basefields. Special thanks go to Daniel R. Grayson for useful discussions.2. Some fundamental results and constructions for simplicial sets
Let s S et be the category of simplicial sets, with the usual (Kan-Quillen) model structure. In thissection, we focus on the associated pointed category s S et ∗ of pointed simplicial sets, with the inducedmodel structure (see [20]). We first recall the following well-known result ([19, Chapter I, Lemma 7.3]). Theorem 2.1.
Let q : E → B be a Kan fibration of Kan complexes, with fiber F over ∗ ∈ B ; fixa vertex v : ∆ → F ⊂ E as base point of F and E . There is the boundary map ∂ : π n ( B, ∗ ) → π n − ( F, v ) , which is a group homomorphism for each n > , fitting into a long exact sequence ofpointed sets ( π are groups and π n for n > are abelian groups) · · · → π n ( F, v ) i ∗ −→ π n ( E, v ) q ∗ −→ π n ( B, ∗ ) ∂ −→ π n − ( F, v ) → · · ·· · · → π ( B, ∗ ) ∂ −→ π ( F, v ) i ∗ −→ π ( E, v ) q ∗ −→ π ( B, ∗ ) . This sequence is natural in the fiber sequence F → E q −→ B .There is a (left) action of π ( B, ∗ ) on π ( F, v ) given by [ γ ] · [ a ] = [˜ γd ] (i.e. [ γ ] · [˜ γd ] = [˜ γd ] ),where ˜ γ is any lifting map in the diagram ∆ a / / d (cid:15) (cid:15) E q (cid:15) (cid:15) ∆ γ / / ˜ γ > > B. Moreover, the exactness at π ( F, v ) can be strengthened as follows: for any [ a ] , [ a ′ ] ∈ π ( F, v ) , i ∗ ([ a ]) = i ∗ ([ a ′ ]) ∈ π ( E, v ) iff [ a ] and [ a ′ ] are in the same orbit under this π ( B, ∗ ) -action, i.e., Peng DU i − ∗ ( i ∗ ([ a ])) = π ( B, ∗ ) · [ a ] ⊂ π ( F, v ) . In words, two points in the fibre are in the same path componentof the total space iff they are in the same orbit of the action by the fundamental group of the base.The stabilizer of any [ u ] ∈ π ( F, v ) is Stab([ u ]) = q ∗ ( π ( E, u )) ⊂ π ( B, ∗ ) and so there is a bijection i − ∗ ( i ∗ ([ u ])) ∼ = π ( B, ∗ ) /q ∗ ( π ( E, u )) . The map i ∗ : π ( F, v ) → π ( E, v ) is injective on the orbit π ( B, ∗ ) · [ u ] iff q ∗ ( π ( E, u )) = π ( B, ∗ ) . Remark 2.2.
Exactness for pointed sets in the theorem means that the image of a map equals the preimageunder the next map of the class of the base point.
Remark 2.3.
Let [ α ] ∈ π n ( B, ∗ )( n > be represented by an n -simplex α of B , which fits into a commutativediagram ∂ ∆ n / / (cid:127) _ (cid:15) (cid:15) ∆ v (cid:15) (cid:15) ∆ n α / / B. The boundary of [ α ] is then given by ∂ [ α ] = [ d θ ] = [ θ ◦ d ] ∈ π n − ( F, v ) determined by the lifting diagram Λ n ,v, ··· ,v ) / / (cid:127) _ (cid:15) (cid:15) E q (cid:15) (cid:15) ∆ n α / / θ B. For n = 1 , this means we lift the loop α to a path in E ending at v , then ∂ [ α ] is the path component of thestarting point. Thus ∂ : π ( B, ∗ ) → π ( F, v ) is given by the action of π ( B, ∗ ) on the class [ v ] (component ofthe base point v in F ). We also state the following easily checked fact, it’s a good illuminating exercise to draw a pictureof lifting paths to see that the relevant map is indeed well-defined and has the claimed equivarianceproperty.
Theorem 2.4.
Let q : E → B be a Kan fibration of Kan complexes. Assume that the two vertices b , b ∈ B are in the same path component and write F j := q − ( b j ) , j = 0 , for the fibers. Choose apath ω : ∆ → B with ω d = b , ω d = b . This gives a group isomorphism ( ω ) ∗ : π ( B, b ) → π ( B, b ) given by [ ω ] [¯ ω ] · [ ω ] · [ ω ] , where ¯ ω is the “path” reversing the direction of ω .We also define a map of sets ( ω ) ∗ : π F → π F by mapping the class [ a ] of a vertex a in F to the class [ f ω d ] in F , where f ω is any lifting map inthe following diagram ∆ a / / d (cid:15) (cid:15) E q (cid:15) (cid:15) ∆ ω / / f ω > > B. Then with the action given by Theorem 2.1, the map ( ω ) ∗ : π F → π F is a bijection and isequivariant with respect to π ( B, b ) ( ω ) ∗ −−−→ π ( B, b ) and induces a bijection of orbit sets (dependingon the choice of [ ω ] ) ( ω ) ∗ : π F /π ( B, b ) → π F /π ( B, b ) . For [ u ] ∈ π ( F ) , let [ u ] = ( ω ) ∗ [ u ] ∈ π ( F ) , then Stab([ u ]) = ( ω ) ∗ Stab([ u ]) ⊂ π ( B, b ) .The map ( ω ) ∗ : π F → π F restricts to a bijection ( i ) − ∗ (( i ) ∗ [ u ]) ∼ = ( i ) − ∗ (( i ) ∗ [ u ]) , where i : F ֒ → E, i : F ֒ → E are the fiber inclusions. In particular, if ( i ) − ∗ (( i ) ∗ [ u ]) = { [ u ] } , then ( i ) − ∗ (( i ) ∗ [ u ]) = { [ u ] } . We now generalize the usual path fibration to the relative case. Let q : ( E, v ) → ( B, b ) be a Kanfibration of pointed Kan complexes, with fiber F over b ∈ B . Assume q admits a section s : B → E (we don’t require s ( b ) = v ). We denote this by the diagram(2.1) F E B. qs NUMERATING NON-STABLE VECTOR BUNDLES 5The commutative diagrams E ∆ E ∂ ∆ B ∆ B ∂ ∆ ( ∂ ∆ ֒ → ∆ ) ∗ q ∆1 q ∂ ∆1 ( ∂ ∆ ֒ → ∆ ) ∗ and E E ∂ ∆ = E × EB ∆ B ∂ ∆ = B × B ( sq, q ∂ ∆1 = q × q ( ∂ ∆ ֒ → ∆ ) ∗ induce maps α : E ∆ → B ∆ × B ∂ ∆1 E ∂ ∆ and β : E → B ∆ × B ∂ ∆1 E ∂ ∆ , where the vertical map E → B ∆ is the composite E q −→ B = B ∆ → B ∆ (the latter arrow is givenby taking “constant paths”, namely it is the map (∆ → ∆ ) ∗ ).We make E ∆ and B ∆ pointed by taking as base points the composites ∆ → ∆ v −→ E and ∆ → ∆ b −→ B respectively; the other spaces also have similar base points. Then the map α : E ∆ → B ∆ × B ∂ ∆1 E ∂ ∆ is a fibration (see [20, Proposition 9.3.8]) with fibre over the above chosen base pointthe loop space Ω F . Now we define a map u via the pull-back diagram(2.2) P B E E ∆ E B ∆ × B ∂ ∆1 E ∂ ∆ , u · y αβ in turn, a fibre sequence Ω F P B E E. u Decoding the definition, we see that, geometrically, the relative path space P B E is the space whosepoints (vertices) are pairs ( e, σ ) with e ∈ E and σ a path in E from the point sq ( e ) to e , totally lyinginside the fibre of q : E → B over q ( e ) ; the q ∆ : E ∆ → B ∆ component of α in the pull-back istaking care of the paths lying totally inside some fibers. The map u ( e, σ ) = e is just the ending pointsof such paths. We also have a map ν : B → P B E by taking “constant paths” at points of B (rigorously, one defines it by a commutative diagram andusing the universal property of the pull-back in eq. (2.2)). One then checks that uν = s and ( qu ) ν = id B , ν ( qu ) ∼ id P B E , hence B is a retract of P B E and there is a homotopy equivalence pair qu : P B E ⇄ B : ν ; geometrically,the homotopy equivalence can be given by shrinking every path to its starting point.We also have the relative loop space (2.3) Ω B E P B EB E, Ω q · y us yielding another fibration Ω q : Ω B E → B with fibre Ω F , the total space Ω B E has points (vertices)the loops σ in E based at points in B , totally lying inside the fibre of q : E → B ; Ω q has a canonicalsection s ′ : B → Ω B E (given by taking “constant loops”).Note that if B is just a point, then all these constructions specialize to the usual path fibration Ω E → P E → E .Now we assume G is a (discrete) group and acts on an abelian group M via a group homomorphism ρ : G → Aut( M ) . Then G acts on the Eilenberg-Mac Lane space K( M, n ) , n > . We have the twistedEilenberg-Mac Lane space K G ( M, n ) := E G × G K( M, n ) , fitting into a fibre sequence K( M, n ) → K G ( M, n ) q −→ B G when n > , which admits a section s = s G : B G → K G ( M, n ) Peng DUinduced by the inculsion of the base point of K( M, n ) into K( M, n ) . It’s easy to find that for n > , π j (K G ( M, n )) = π B G = G, j = 1; π j (K( M, n )) =
M, j = n ;0 , j = 1 , n. We are thus in the situation of eq. (2.1):(2.4) K( M, n ) K G ( M, n ) B G. qs The previous constructions thus yield a fibre sequence(2.5)
ΩK(
M, n ) ≃ K( M, n −
1) P G ( M, n ) K G ( M, n ) , u where P G ( M, n ) := P B G K G ( M, n ) in our previous notation. Recall that there is a pair of homotopyequivalences qu : P G ( M, n ) ⇄ B G : ν. Thus we get a homotopy fibre sequence(2.6) K( M, n −
1) B G K G ( M, n ) , s The following result is a consequence of the discussions in [19, Chapter VI].
Theorem 2.5.
The fibration u = u G : P G ( M, n + 1) → K G ( M, n + 1) , n > in eq. (2.5) is a universal K( M, n ) -fibration with G as fundamental group of the total space (and the base space), i.e. for any K( M, n ) -fibration p : E → B with G as fundamental group of the total space and the base space, thereis a unique element [ k ] ∈ [ B, K G ( M, n + 1)] such that the fibration p : E → B is equivalent to thepull-back of u : P G ( M, n + 1) → K G ( M, n + 1) along k . Thus we have homotopy cartesian squares(2.7) E P G ( M, n + 1) B K G ( M, n + 1) p uk and E B GB K G ( M, n + 1) . p sk Remark 2.6.
From this result, together with the existence result of Moore-Postnikov systems for maps in s S et ,one deduces a rather complete Moore-Postnikov decomposition for s S et as in [19, ChapterVI, §3]. There is asimilar result in the motivic homotopy category, see Theorem 3.2 in the next section (of similar form). Tools from motivic homotopy theory
We fix a noetherian finite dimensional scheme S . Write S m S for the category of S -schemes whichare finitely presented and smooth over S ; it is essentially small. We make it a Grothendieck site byequipping it with the Nisnevich topology (see [32]), called the Nisnevich site of S .Let s P re ( S m S ) and s S hv ( S m S ) be respectively the category of simplicial presheaves and simplicialsheaves on this site. We use ∗ to denote the final object in s P re ( S m S ) , which is represented by S viaYoneda lemma. Both the sheaf and presheaf categories have the Joyal-Jardine model structure, whichare Quillen equivalent; A -localizing we get the Morel-Voevodsky (unstable) motivic model structure on s P re ( S m S ) and s S hv ( S m S ) . The weak equivalences in the motivic model categories are called A -weakequivalences .The two are again Quillen equivalent, hence lead to the same homoptoy category H A ( S ) —the A -homotopy category or the unstable motivic homotopy category of S , see [32]. The homotopy classeswill be denoted by [ − , − ] A (or even [ − , − ] ).The motivic model category has all nice properties one could imagine: it’s a cofibrantly generated,(left and right) proper, cellular, combinatorial, simplicial model category (i.e. it’s a module over thesymmetric monodial simplicial model category s S et ), etc., see [32, 22]. In particular, the general niceresults of [20, 23] apply here.There is an obvious pointed version and we obtain the pointed A -homotopy category H A ∗ ( S ) of S .The homotopy classes will be denoted by [ − , − ] A , ∗ (or even [ − , − ] ). For any X ∈ s P re ( S m S ) , Y ∈ s P re ( S m S ) ∗ , we have canonical isomorphism [ X, Y ] A ∼ = [ X + , Y ] A , ∗ , where X + := X ∐ ∗ is X with adisjoint base point added.NUMERATING NON-STABLE VECTOR BUNDLES 7We fix an A -fibrant replacement functor L A with respect to the Morel-Voevodsky model structureon s P re ( S m k , Nis) ∗ .If S = Spec( k ) , k a commutative ring, we simply write S m k for S m S and write H A ( ∗ ) ( k ) for H A ( ∗ ) ( S ) .In fact, except for the Joyal-Jardine model structure in the first step (before performing A -localization), there are other Quillen equivalent model structures—one can choose the local projectivemodel structure for either the presheaf or sheaf category (or even some intermediate model structure,between the local projective model structure and the local injective model structure). They haveequivalent homotopy categories and each has some advantages in certain situations. See e.g. [14, 26].Now assume k is a perfect field. We write G r A k for the category of strongly A -invariant sheaves ofgroups and A b A k for the category of strictly A -invariant sheaves of abelian groups on the Nisnevichsite ( S m k , Nis) (see Morel [31, Definition 1.7]), the latter is an abelian category. A strictly A -invariantsheaf of abelian groups is a strongly A -invariant sheaf of groups. Conversely, after quite a lot of efforts,Morel shows that, if A is a sheaf of abelian groups on ( S m k , Nis) and A ∈ G r A k , then A ∈ A b A k (see[31]).For a sheaf of abelian groups A , we define its contraction A − by A − ( U ) := ker( A ( U × G m ) → A ( U )) induced by the map U = U × { } ֒ → U × G m , or equivalently, A − ( U ) := coker( A ( U ) → A ( U × G m )) induced by the projection U × G m → U . The contraction A − is also a sheaf of abelian groups. Andwe define inductively A − ( n +1) = ( A − n ) − , n ∈ N (by convention, A = A − = A ). The contractiondefines an exact functor ( − ) − : A b A k → A b A k .See also [7] for a systematic study of the group-theoretic properties of such sheaves.For any pointed motivic space X ∈ s P re ( S m k ) ∗ , we have π A ( X ) ∈ G r A k and π A n ( X ) ∈ A b A k for n > , where for n ∈ N , π A n ( X ) is its A -homotopy sheaf , i.e. the Nisnevich sheafification of thepresheaf U [ S n ∧ U + , X ] A , ∗ . We say that X ∈ s P re ( S m k ) ∗ is A -connected if π A ( X ) = 0 , A -simply connected if π A ( X ) = π A ( X ) = 0 , and A - n -connected if π A i ( X ) = 0 , for any i with i n .Results in the classical topological setting are always the guiding model for the study of abstracthomotopy theory, and indicate directions where to go, give (counter)examples, etc. The followingresult about long exact sequence of sheaves of motivic homotopy groups associated to an A -homotopyfibre sequence is of fundamental importance in motivic homotopy theory. Theorem 3.1.
Let k be a perfect field, let F → E q −→ B be an A -homotopy fibre sequence in themotivic model category s P re ( S m k ) ∗ . There are the boundary map maps ∂ : π A n B → π A n − F , which arehomomorphism of sheaves of groups for each n > , fitting into a long exact sequence · · · → π A n F i ∗ −→ π A n E q ∗ −→ π A n B ∂ −→ π A n − F → · · ·· · · → π A B ∂ −→ π A F i ∗ −→ π A E q ∗ −→ π A B. This sequence is natural in the A -fibre sequence F → E q −→ B . There is a (left) action of π A B on π A F .Moreover, the exactness at π A F can be strengthened as follows: for any local sections [ u ] , [ u ′ ] of π A F , i ∗ ([ u ]) = i ∗ ([ u ′ ]) ∈ π A E iff [ u ] and [ u ′ ] are in the same orbit of some local sections of the π A B -action. Now we are ready to state (without proof) the following result about Moore-Postnikov towers inthe motivic model categories, which, as in the classical topological setting, is a fundamental tool inobstruction theory; where we use ( C ↓ B ) to denote the category of objects over an object B in acategory C (some authors may prefer the notation C /B ) and the notion of fibre sequences is in thesense of [23]. For an exposition of this kind of result, see [31, Appendix B] and [1, Theorem 2.12];see also [3, 5] for related discussions. We only state the result for s P re ( S m k ) ∗ but it also works for s S hv ( S m k ) ∗ . Theorem 3.2 (Moore-Postnikov systems in motivic model category) . Let k be a perfect field, let F → E q −→ B be an A -fibre sequence in the motivic model category s P re ( S m k ) ∗ , with all of F, E, B
Peng DU being A -connected and A -fibrant. Denote π A E =: G ∈ G r A k . Then there is the Moore-Postnikovtower E → · · · q n +1 −−−→ E n q n −→ E n − q n − −−−→ · · · q −→ E q −→ E p −→ B, in s P re ( S m k ) ∗ , and maps i n : E → E n , p n : E n → B with the following properties: (1) All the spaces E n are A -fibrant, the maps q n , n > are A -fibrations, and p : E → B is an A -weak equivalence. (2) i n − = q n i n , p n − q n = p n , ∀ n > . (3) p n i n = q, ∀ n ∈ N . E n +1 E E n B q n +1 p n +1 i n +1 i n p n (4) The A -homotopy fibre F ( q n ) of q n ( n > is A -weakly equivalent to K( π A n F, n ) , hence foreach n > we have an A -homotopy fibre sequence K( π A n F, n ) → E n q n −→ E n − . (5) There are homotopy pullback diagrams in ( s P re ( S m k ) ↓ B G ) (with model structure induced fromthe Morel-Voevodsky motivic model structure on s P re ( S m k ) ) E n / / q n (cid:15) (cid:15) B G s n (cid:15) (cid:15) E n − k n +1 / / K G ( π A n F, n + 1) , for a unique [ k n +1 ] ∈ [ E n − , K G ( π A n F, n + 1)] A , for all n > . (6) The map E → holim n ∈ N op E n is a local weak equivalence hence an A -weak equivalence.Specializing to the case E = ∗ or B = ∗ , one gets respectively the Whitehead tower and
Postnikovtower . Moreover, by the above existence results, we have (7)
For any j n , ( i n ) ∗ : π A j E ∼ = −→ π A j E n . (8) For any j > n + 2 , ( p n ) ∗ : π A j E n ∼ = −→ π A j B . (9) There are exact sequences → π A n +1 E n ( p n ) ∗ −−−→ π A n +1 B ∂ −→ π A n F, n > , where ∂ is the connecting homomorphism in the homotopy long exact sequence of the homotopyfibre sequence F → E q −→ B . (10) There are A -homotopy fibre sequences F [ n ] → E n p n −→ B,F ( q n ) → F [ n ] q ′ n −→ F [ n −
1] ( n > , where the map q ′ n is induced from q n : E n → E n − and F [ n ] is the n -th stage of the Postnikovtower for F . Remark 3.3.
In practice, we usually consider the liftings of the homotopy class of a map from a smoothscheme X , regarded as a simplicial presheaf, to a motivic space B , to elements in [ X, E ] A ∼ = [ X + , E ] A , ∗ . Inthis situation, the above results on Moore-Postnikov systems is very useful in finding when a given A -homotopyclass can be lifted, especially when the A -homotopy fibre F of E q −→ B is highly connected. We also need the following results about (relative) cohomologies. We fix a strictly A -invariant sheafof abelian groups M on the Nisnevich site ( S m k , Nis) (see [31]). Let Y ∈ s P re ( S m k ) ∗ be a pointedsimplicial presheaf, we define its reduced cohomology with coefficients in M to be e H n ( Y ; M ) := [ Y, K( M , n )] A , ∗ . So e H n (Σ Y ; M ) = e H n − ( Y ; M ) and for X ∈ s P re ( S m k ) , we have H n ( X ; M ) = e H n ( X + ; M ) .NUMERATING NON-STABLE VECTOR BUNDLES 9When we write H n ( Y ; M ) , we will mean the (non-reduced) cohomology of Y (forgetting the basepoint).If X ⊂ Z , we define H n ( Z, X ; M ) , the cohomology of the pair ( Z, X ) with coefficients in M , tobe the reduced cohomology of Z/X (note that it is naturally pointed), the (homotopy) cofiber of theinclusion
X ֒ → Z : H n ( Z, X ; M ) := e H n ( Z/X ; M ) = [ Z/X, K( M , n )] A , ∗ . So H n ( X, ∅ ; M ) = e H n ( X + ; M ) = H n ( X ; M ) and e H n ( Y ; M ) = H n ( Y, ∗ ; M ) . All these are abeliangroups. Remark 3.4.
Here we are taking homotopy classes in the A -homotopy category. But since we are workingwith strictly A -invariant sheaves of abelian groups (which is the right coefficient sheaves for A -homotopytheory), the homotopy classes equal those in Jardine’s local model structure (no A -localization involved). Thelatter works for arbitrary sheaves of abelian groups. The following result, familiar from classical topology, follows easily from the general abstract for-malism of [23, Chapter 6]. The long exact sequence for cohomology of a pair follows from that for thetriple ∅ ⊂ Y ⊂ Z . In the following, we say that the inclusion i : Y ֒ → Z splits if there exists a map p : Z → Y such that pi = id Y . Theorem 3.5 (Long exact sequence for cohomology of a triple) . Given simplicial presheaves X ⊂ Y ⊂ Z , we have the long exact sequence of cohomology groups → H ( Z, Y ; M ) → H ( Z, X ; M ) → H ( Y, X ; M ) → H ( Z, Y ; M ) → · · ·· · · → H n ( Z, Y ; M ) → H n ( Z, X ; M ) → H n ( Y, X ; M ) → H n +1 ( Z, Y ; M ) → · · · , natural in the triple X ⊂ Y ⊂ Z .In particular, there is the long exact sequence for cohomology of a pair ( Z, Y ) : → H ( Z, Y ; M ) → H ( Z ; M ) → H ( Y ; M ) → H ( Z, Y ; M ) → · · ·· · · → H n ( Z, Y ; M ) → H n ( Z ; M ) → H n ( Y ; M ) → H n +1 ( Z, Y ; M ) → · · · . If the inclusion
Y ֒ → Z splits, then the long exact sequence reduces to split short exact sequences → H n ( Z, Y ; M ) → H n ( Z ; M ) → H n ( Y ; M ) → , n > . Applying the split case to the inclusion y : ∗ ֒ → Y for a pointed simplicial presheaf Y ∈ s P re ( S m k ) ∗ and noting that H n ( ∗ ; M ) = 0 for n > (since k is a field), H ( ∗ ; M ) = M ( ∗ ) = M ( k ) is the abeliangroup of global sections of the sheaf M , we get the following relation, as in classical algebraic topology. Corollary 3.6.
There are split short exact sequence → e H ( Y ; M ) → H ( Y ; M ) → M ( ∗ ) → , and isomorphisms e H n ( Y ; M ) ∼ = −→ H n ( Y ; M ) for n > , natural in Y ∈ s P re ( S m k ) ∗ and M ∈ A b A k . For a pointed simplicial presheaf Y ∈ s P re ( S m k ) ∗ and n > , we have the suspension homomorphism σ : e H n ( Y ; M ) = [ Y, K( M , n )] A , ∗ → [Ω Y, ΩK( M , n )] A , ∗ = e H n − (Ω Y ; M ) induced by the (derived) loop functor Ω ; it’s induced by the counit map ΣΩ Y → Y (as e H n (ΣΩ Y ; M ) = e H n − (Ω Y ; M ) ). So σ ([ θ ]) = [Ω θ ] . Recall also that in the category s P re ( S m k ) ∗ of pointed simplicial presheaves on S m k , we have the smash product ∧ . The smash product ( X, x ) ∧ ( Y, y ) is defined as the section-wise smash product ofsimplicial sets U ( X, x )( U ) ∧ ( Y, y )( U ) —the smash product ( X, x ) ∧ ( Y, y ) corepresents maps from X × Y that are base-point-preserving separately in each variable. So for X, Y ∈ s P re ( S m k ) , we have ( X × Y ) + = X + ∧ Y + , and for any K ∈ s P re ( S m k ) , we have ( X, x ) ∧ K + ∼ = X × K/x × K as pointedsimplicial presheaves.On the other hand, there is the pointed internal function complex Hom ∗ (( X, x ) , ( Y, y )) ∈ s P re ( S m k ) ∗ ,given by the equalizer diagram(3.1) Hom ∗ (( X, x ) , ( Y, y )) Hom(
X, Y ) Hom( ∗ , Y ) = Y, x ∗ y ∗ Hom is the internal hom of simplicial presheaves, with respect to the cartesian closed structureon s P re ( S m k ) and y ∗ is the composite Hom(
X, Y ) → Hom( X, ∗ ) = ∗ y −→ Hom( ∗ , Y ) = Y . We have Hom ∗ ( A + , ( Y, y )) ∼ = Hom( A, Y ) as (pointed) simplicial presheaves, for A ∈ s P re ( S m k ) , ( Y, y ) ∈ s P re ( S m k ) ∗ . There is the followingadjunction ( X, x ) ∧ ( − ) : s P re ( S m k ) ∗ ⇄ s P re ( S m k ) ∗ : Hom ∗ (( X, x ) , − ) . It passes to a Quillen adjunction for the motivic model structure.The operations ∧ , Hom ∗ make s P re ( S m k ) ∗ a closed symmetric monoidal category with unit S = ∂ ∆ = ∆ (viewed as a constant pointed simplicial presheaf). The simplicial suspension is given by Σ X := S ∧ X and the simplicial looping is given by Ω X := Hom ∗ ( S , X ) , where S := ∆ /∂ ∆ (viewed as a constant pointed simplicial presheaf).Now let X, Y, Z ∈ s P re ( S m k ) ∗ be pointed simplicial presheaves, let u : ( X × Y, ∗ × Y ) → ( Z, ∗ ) be a map of pairs, which can be identified with a pointed map u : X ∧ Y + → Z (whenever we write Y + ,we forget the base point of Y and add a new base point). Then we have the maps Y + → Hom ∗ ( X, Z ) → Hom ∗ (Ω X, Ω Z ) , which induces a map Ω X ∧ Y + → Ω Z by adjunction, hence a map of pairs v : (Ω X × Y, ∗ × Y ) → (Ω Z, ∗ ) . Since ∗ × Y is a deformation retract of P X × Y , by the long exact sequence for cohomology of thepair (P X × Y, ∗ × Y ) , we see that H ∗ (P X × Y, ∗ × Y ; M ) = 0 . So the connecting homomorphism δ : H q (Ω X × Y, ∗ × Y ; M ) → H q +1 (P X × Y, Ω X × Y ; M ) for the triple (P X × Y, Ω X × Y, ∗ × Y ) is an isomorphism (by the long exact sequence for cohomologyof a triple above).The given map u also induces a map of triples U : (P X × Y, Ω X × Y, ∗ × Y ) → (P Z, Ω Z, ∗ ) which fits into a commutative diagram (P X × Y, Ω X × Y, ∗ × Y ) U / / p × id Y (cid:15) (cid:15) (P Z, Ω Z, ∗ ) p (cid:15) (cid:15) ( X × Y, ∗ × Y, ∗ × Y ) u / / ( Z, ∗ , ∗ ) , where p = p X : P X → X denotes the “path fibration”. Naturality of the connecting homomorphismfor U gives δv ∗ = U ∗ δ , or v ∗ δ − = δ − U ∗ . So v ∗ δ − p ∗ = δ − U ∗ p ∗ = δ − ( p × id Y ) ∗ u ∗ . Denote ρ := δ − ( p × id Y ) ∗ : H q +1 ( X × Y, ∗ × Y ; M ) → H q (Ω X × Y, ∗ × Y ; M ) , so the map ρ is natural and σ = δ − p ∗ Z . Alternatively, ρ is the map [ X ∧ Y + , K( M , q + 1)] A , ∗ = [ Y + , Hom ∗ ( X, K( M , q + 1))] A , ∗ → [ Y + , Hom ∗ (Ω X, ΩK( M , q + 1))] A , ∗ = [Ω X ∧ Y + , K( M , q )] A , ∗ induced by the simplicial looping. Note that X × Y / ∗ × Y = X ∧ Y + and H q +1 ( X × Y, ∗ × Y ; M ) =[ X × Y / ∗ × Y, K( M , q + 1)] A , ∗ = [ X ∧ Y + , K( M , q + 1)] A , ∗ .Now let ι : ( X × Y, ∅ ) → ( X × Y, ∗ × Y ) and ι : (Ω X × Y, ∅ ) → (Ω X × Y, ∗ × Y ) be the inclusionsof pairs, we get the induced map ι : X + ∧ Y + = ( X × Y ) + = X × Y / ∅ → X × Y / ∗ × Y = X ∧ Y + (or smashing the cofiber sequence S → X + → X with Y + we get a cofiber sequence Y + = S ∧ Y + → X + ∧ Y + ι −→ X ∧ Y + and hence also X + ∧ Y + ι −→ X ∧ Y + → S ∧ Y + = Σ( Y + ) ), hence the induced map ι ∗ : H q +1 ( X × Y, ∗ × Y ; M ) → H q +1 ( X × Y ; M ) (note that H q +1 ( X × Y ; M ) = e H q +1 (( X × Y ) + ; M ) = e H q +1 ( X + ∧ Y + ; M ) = [ X + ∧ Y + , K( M , q + 1)] A , ∗ ).Clearly, the inclusion ∗ × Y ֒ → X × Y splits (by the projection), ι ∗ is an injection (by Theorem 3.5).Similarly, we have the induced map ι ∗ : H q (Ω X × Y, ∗ × Y ; M ) → H q (Ω X × Y ; M ) , NUMERATING NON-STABLE VECTOR BUNDLES 11which is an injection as well.We obtain the following commutative diagram:(3.2) e H q +1 ( Z ; M ) H q +1 ( X × Y, ∗ × Y ; M ) im( ι ∗ ) H q +1 ( X × Y ; M ) e H q (Ω Z ; M ) H q (Ω X × Y, ∗ × Y ; M ) im( ι ∗ ) H q (Ω X × Y ; M ) , u ∗ σ ρ ι ∗ σ ′ v ∗ ι ∗ where the partial suspension homomorphism σ ′ : im( ι ∗ ) → im( ι ∗ ) is defined by letting the squareon the right commutative (thanks to the fact that ker( ι ∗ ) ⊂ ker( ι ∗ ρ ) ); note that it’s not a map H q +1 ( X × Y ; M ) → H q (Ω X × Y ; M ) . Commutativity of the square on the left follows from naturalityof our construction.Assume K = M i ∈ Z K i is a graded sheaf of rings, each K i being a strictly A -invariant sheaf of abeliangroups. Then we have the following commutative diagram:(3.3) e H r +1 ( X ; K i ) ⊗ H s ( Y ; K j ) H r +1+ s ( X × Y, ∗ × Y ; K i + j ) e H r (Ω X ; K i ) ⊗ H s ( Y ; K j ) H r + s (Ω X × Y, ∗ × Y ; K i + j ) . µσ ⊗ ρµ Here for example, the top arrow µ is the external product [ X, K( K i , r + 1)] A , ∗ ⊗ [ Y, K( K j , s )] A → [ X × Y / ∗ × Y, K( K i + j , r + 1 + s )] A , ∗ induced by the multiplication in the graded sheaves of rings K (for more details, see [26, §8.4], whereit’s called cup product pairing ; Jardine’s assumption of commutativity of the coefficient sheaf of rings K is not essential). We denote a × b := µ ( a ⊗ b ) . Then by definition (when X = Y ), the cup product is given by a · b = a ⌣ b := ∆ ∗ µ ( a ⊗ b ) = ∆ ∗ ( a × b ) , where ∆ : X → X × X is the diagonal. With cupproduct, the cohomology e H ∗ ( X ; K ) becomes a ring (bigraded if K is a graded sheaf of rings), which mayor may not be anti-commutative (depending on commutativity property of K ). The commutativityof the diagram of eq. (3.3) means that ρ ( a × b ) = ( σa ) × b for a ∈ e H ∗ ( X ; K ) , b ∈ H ∗ ( Y ; K ) ( σ is thesuspension homomorphism for X ), which follows from our alternative definition of ρ (and in fact, wemay totally discard the first definition of ρ ; we give it only to respect the work of James-Thomas [25]).Similarly, we have σ ′ ( a × b ) = ( σa ) × b , for a ∈ e H ∗ ( X ; K ) , b ∈ e H ∗ ( Y ; K ) in eq. (3.2) (here σ is thesuspension homomorphism for X ).4. A -homotopy study of vector bundles We first state the following affine representability results in A -homotopy theory, which is (a specialcase of) [8, Theorem 5.2.3], where we use V r ( U ) to denote the set of isomorphism classes of rank r vector bundles over a scheme U and S m aff k denotes the category of (absolutely) affine k -schemes thatare smooth over k . By Gr r , we mean the ind-scheme of all finite Grassmannians Gr r,n , n > r ; BGL r isthe simplicial classifying space for the linear k -group scheme GL r . Theorem 4.1 (Affine representability for vector bundles) . Let k be a perfect field, let U ∈ S m aff k , thenthere is a bijection V r ( U ) ∼ = −→ [ U, Gr r ] A fitting into the following commutative diagram Hom( U, Gr r ) ' ' PPPPPPPPPPP (cid:15) (cid:15) V r ( U ) ∼ = / / [ U, Gr r ] A , where the other two arrows are the canonical ones, both of which are surjective.Thus we have a natural isomorphism of functors V r ( − ) ∼ = [ − , Gr r ] A : ( S m aff k ) op → S et . As there is an A -weak equivalence BGL r ≃ Gr r , we can use BGL r in place of Gr r . A -homotopy fiber sequence (see [31, §8.2]): A n +1 \ → BGL n → BGL n +1 . From this and the computation of the first few A -homotopy sheaves of the motivic sphere A n +1 \ (see [31, Corollary 6.43]) π A i ( A n +1 \
0) = ( , i < n, K MW n +1 , i = n, we see that the induced map of A -homotopy sheaves π A j BGL n → π A j BGL n +1 is an isomorphism if j < n and a surjection if j = n .Thus also, the induced map π A j BGL n → π A j BGL is an isomorphism if j < n and an epimorphism if j = n .Let F n be the A -homotopy fiber of the canonical map ϕ n : BGL n → BGL , so we have an A -homotopy fiber sequence F n → BGL n → BGL , and from the above results, we see that π A j F n = 0 , j < n. We now compute the next A -homotopy sheaf of F n . By [23, §6.1-§6.5], there is a commutativediagram A n +1 \ i / / F n / / (cid:15) (cid:15) F n +1 (cid:15) (cid:15) A n +1 \ / / BGL n / / (cid:15) (cid:15) BGL n +1 (cid:15) (cid:15) BGL BGL where the four 3-term rows and columns are A -homotopy fiber sequences, with some extra equivarianceproperties (which we omit to state here). From the A -homotopy fiber sequence in the first row we getan exact sequence π A n +1 F n +1 → π A n ( A n +1 \ → π A n F n → π A n F n +1 = 0 , so the map K MW n +1 = π A n ( A n +1 \ → π A n F n and hence also π A n +1 ( A n +2 \ → π A n +1 F n +1 are epimorphisms. And so π A n F n ∼ = coker( π A n +1 F n +1 → π A n ( A n +1 \ ∼ = coker( π A n +1 ( A n +2 \ → π A n ( A n +1 \ . The composite π A n +1 ( A n +2 \ → π A n +1 F n +1 → π A n ( A n +1 \ is the map K MW n +2 → K MW n +1 discussed in [4, Lemma 3.5], which is if n is even and is multiplication by η if n is odd; so im( π A n +1 F n +1 → π A n ( A n +1 \ ∼ = ( , n even ; η K MW n +2 , n odd . Thus π A n F n ∼ = coker( K MW n +2 → K MW n +1 ) ∼ = ( K MW n +1 , n even ; K MW n +1 /η K MW n +2 = K M n +1 , n odd . Moreover, by the A -homotopy fiber sequence A n +1 \ → F n → F n +1 we get exact sequences π A n +1 ( A n +1 \ → π A n +1 F n → π A n +1 F n +1 = K M n +2 0 −→ π A n ( A n +1 \ , n even , NUMERATING NON-STABLE VECTOR BUNDLES 13 π A n +1 ( A n +1 \ → π A n +1 F n → π A n +1 F n +1 = K MW n +2 η −→ K MW n +1 = π A n ( A n +1 \ , n odd , and since ker( K MW n +2 η −→ K MW n +1 ) ∼ = 2 K M n +2 , we get exact sequences(4.1) ( π A n +1 ( A n +1 \ → π A n +1 F n → K M n +2 = π A n +1 F n +1 → , n even ; π A n +1 ( A n +1 \ → π A n +1 F n → K M n +2 → , n odd . As π A j F n = 0 for j < n , by the Moore-Postnikov decomposition in A -homotopy theory stated inTheorem 3.2, we can factorize the map BGL n → BGL as BGL n → E = E n → BGL and the map E = E n → BGL fits into a homotopy cartesian diagram E = E n / / (cid:15) (cid:15) B G m (cid:15) (cid:15) BGL = E n − k n +1 / / K G m ( π A n F n , n + 1) , for a unique [ k n +1 ] ∈ [ E n − , K G m ( π A n F n , n +1)] A , if n > . For any j n , we have π A j BGL n ∼ = π A j E n .Now assume n > is odd, then G m acts trivially on π A n F n = K M n +1 , so the above homotopy cartesiandiagram reduces to an A -homotopy fiber sequence(4.2) E = E n → BGL = E n − θ −→ K( K M n +1 , n + 1) . whence a principal A -homotopy fiber sequence K( K M n +1 , n ) → E = E n → BGL = E n − . The A -homotopy class of the map θ is the universal ( n +1) -st Chern class c n +1 ∈ H n +1 (BGL; K M n +1 ) =CH n +1 (BGL) (see [6, Example 5.2 and Proposition 5.8]).In the algebro-geometric situation, let k be a perfect field, A a smooth affine k -algebra of Krulldimension d > , and X = Spec( A ) . Then by the A -homotopy long exact sequence and crawling upthe Moore-Postnikov tower of the map BGL n → BGL , one easily finds that the map [ X, BGL n ] A → [ X, E ] A is surjective if n > d − , and is bijective if n > d .5. Motivic approach to enumerating non-stable vector bundles
In this section, we develop a motivic homotopy theoretic approach to the enumeration problemfor non-stable vector bundles, following the ideas of I. M. James and E. Thomas [25] in the classicalhomotopy theoretic setting.Again we consider the algebro-geometric situation: k is a perfect field, A a smooth affine k -algebraof Krull dimension d > , and X = Spec( A ) . Recall that we have the A -homotopy fiber sequence F n → BGL n ϕ = ϕ n −−−−→ BGL . The problem is to study the induced map on A -homotopy classes ϕ ∗ : [ X, BGL n ] A → [ X, BGL] A . Suppose K ∈ s P re ( S m k ) ∗ is a pointed simplicial presheaf on the Nisnevich site ( S m k , Nis) . We haveits free path space P ∗ K := K ∆ and free loop space Ω ∗ K ⊂ P ∗ K given by the equalizer diagram Ω ∗ K P ∗ K = K ∆ K ∆ = K, ( d ) ∗ ( d ) ∗ where the two parallel arrows are induced by the coface maps d , d : ∆ ⇒ ∆ . Denote by r : Ω ∗ K → K the composite map in the above equalizer diagram; it’s easy to see that we have a cartesian diagram Ω ∗ K P ∗ K = K ∆ K K × K = K ∂ ∆ , r · y (( d ) ∗ , ( d ) ∗ ) δ K δ K : K → K × K is the diagonal map (we can use this cartesian diagram to define Ω ∗ K and r : Ω ∗ K → K ). If K is a section-wise Kan complex, then the right vertical map is a section-wiseKan fibration (since it’s induced by the inclusion ∂ ∆ ֒ → ∆ ; cf. [20, Proposition 9.3.9]), so the map r : Ω ∗ K → K is also a section-wise Kan fibration.The map ( s ) ∗ : K ∆ = K → P ∗ K = K ∆ defines a map c : K → Ω ∗ K which is a section of themap r (as s d = id = s d , by the cosimplicial identities).The usual (based) loop space Ω K ⊂ Ω ∗ K fits into the cartesian square Ω K Ω ∗ K ∗ K, i · y r where the bottom arrow is the base point of K . Thus there is a section-wise homotopy fiber sequence Ω K i −→ Ω ∗ K r −→ K. Now consider the following situation: we are given a strictly A -invariant sheaf of abelian groups M on ( S m k , Nis) and a principal A -homotopy fiber sequence K( M , n ) → E q −→ BGL classified by a map θ : BGL → K( M , n + 1) , n > . So the above A -homotopy fiber sequence extendsone step to the right (so that each 3-term forms an A -homotopy fiber sequence): K( M , n ) → E q −→ BGL θ −→ K( M , n + 1) . Moreover, by [9, Theorem 2.2.5] or [7, Lemma 3.1.3], we have an A -homotopy fiber sequence K( M , n ) = ΩK( M , n + 1) i −→ Ω ∗ K( M , n + 1) r −→ K( M , n + 1) . Later we will specialize to the case of M being the typical and naturally-arising strictly A -invariantsheaves like K M n +1 , K MW n +1 to give enumeration results for non-stable vector bundles. Proposition 5.1.
The induced map i ∗ : H n ( X ; M ) = [ X, K( M , n )] A = [ X, ΩK( M , n + 1)] A → [ X, Ω ∗ K( M , n + 1)] A is injective. If moreover dim X = d n , then H n +1 ( X ; M ) = [ X, K( M , n + 1)] A = 0 and hence i ∗ : H n ( X ; M ) = [ X, ΩK( M , n + 1)] A → [ X, Ω ∗ K( M , n + 1)] A is an isomorphism.Proof. The abelian group H n ( X ; M ) = [ X, ΩK( M , n + 1)] A ∼ = [ X, ΩL A K( M , n + 1)] A is the sameas the set of simplicial homotopy classes of maps X → ΩL A K( M , n + 1) by the strict A -invariance as-sumption of M (which ensures that K( M , n ) is A -local for all n ∈ N ), which is in turn π (ΩL A K( M , n +1)( X )) = [∆ , ΩL A K( M , n + 1)( X )] s S et ∗ . Note that all the constructions interplay well with the A -fibrant replacement functor L A , so the result follows from the classical result [25, Theorem 2.6](applying it to A = S with X replaced by our L A K( M , n +1)( X ) ). Caution: In [25], the authors workwith pointed connected CW-complexes from the outset, but one easily sees that the relevant results arestill true for A = ∆ = S .The last statement follows easily from the A -homotopy fiber sequence K( M , n ) = ΩK( M , n + 1) i −→ Ω ∗ K( M , n + 1) r −→ K( M , n + 1) . (cid:3) Giving a stable vector bundle over X is the same as giving a class ξ ∈ [ X, BGL] A . We will firststudy the lifting set q − ∗ ( ξ ) following the treatment of [25].For θ ∈ H n +1 (BGL; M ) = [BGL , K( M , n + 1)] A , let θ ′ = Ω ∗ θ : Ω ∗ BGL → Ω ∗ K( M , n + 1) . NUMERATING NON-STABLE VECTOR BUNDLES 15Given ξ ∈ [ X, BGL] A as above, we say γ ∈ H n ( X, M ) = [ X, K( M , n )] A = [ X, ΩK( M , n + 1)] A is θ -correlated to ξ if there exists an element ψ ∈ [ X, Ω ∗ BGL] A such that r ∗ ψ = ξ, θ ′∗ ψ = i ∗ γ .(5.1) ∃ ψ ∈ [ X, Ω ∗ BGL] A [ X, BGL] A ∋ ξγ ∈ [ X, ΩK( M , n + 1)] A [ X, Ω ∗ K( M , n + 1)] A [ X, K( M , n + 1)] A θ ′∗ r ∗ c ∗ θ ∗ i ∗ r ∗ c ∗ We denote the set of such elements γ ∈ [ X, ΩK( M , n + 1)] A by C θ ( ξ ) ⊂ [ X, ΩK( M , n + 1)] A =H n ( X, M ) . It’s easy to check that(5.2) C θ ( ξ ) = ∅ ⇐⇒ θ ∗ ( ξ ) = 0 ∈ H n +1 ( X ; M ) = [ X, K( M , n + 1)] A ⇐⇒ q − ∗ ( ξ ) = ∅ , in which case, one can see that for ψ = c ∗ ξ , we have θ ′∗ ψ ∈ i ∗ [ X, ΩK( M , n + 1)] A (= ker r ∗ ) .The following corresponds to [25, Theorem 3.4]. Theorem 5.2.
Consider the action of H n ( X ; M ) = [ X, K( M , n )] A on the set [ X, E ] A associated tothe fiber sequence (given by “concatenation of paths” in the simplicial direction by suitably applying L A ).For each η ∈ [ X, E ] A , this action restricts to a transitive action of H n ( X ; M ) = [ X, K( M , n )] A on theset q − ∗ ( q ∗ η ) ⊂ [ X, E ] A , and for any η ′ ∈ q − ∗ ( q ∗ η ) , its stabilizer under this action is C θ ( q ∗ η ) (whichonly depends on q ∗ η ′ = q ∗ η ∈ [ X, BGL] A ; also note that θ ∗ ( q ∗ η ) = 0 , compatible with the fact that C θ ( q ∗ η ) = ∅ , being a group). So each orbit set forms a fiber of the map q ∗ : [ X, E ] A → [ X, BGL] A .In particular, if q − ∗ ( ξ ) = ∅ , then q − ∗ ( ξ ) ∼ = H n ( X ; M ) /C θ ( ξ ) and hence it has an abelian groupstructure.Proof. As in the proof of Proposition 5.1, by using the functorial A -fibrant replacement functor L A ,we are reduced to the classical topological situation via simplicial homotopy. Then the result followsfrom Theorem 2.1 and [25, Lemma 3.1, Theorems 3.2 and 3.3] (applying them to A = S with C replaced by our L A K( M , n + 1)( X ) ); notice that eq. (5.1) translates well to a diagram of simplicialhomotopy classes in s S et ∗ . (cid:3) Note that K( M , n + 1) is a sheaf of simplicial abelian groups, and BGL has a binary operation m induced by the maps GL r × GL s → GL r + s , ( A, B ) (cid:18) A B (cid:19) . Strictly speaking, these maps don’tgive a map GL × GL → GL on colimits, but will give a map ` n > BGL n × ` n > BGL n → ` n > BGL n .By [32, Proposition 4.3.10], we have an A -weak equivalence BGL × Z ≃ −→ R ΩB( ` n > BGL n ) , we get amap m : (BGL × Z ) × (BGL × Z ) → BGL × Z in the A -homotopy category, which restrict to the desiredmap (operation) m : BGL × BGL → BGL . Similar considerations appear in Quillen’s construction ofalgebraic K-theory space using a choice of a bijection N × N → N and shows the choice doesn’t matterup to homotopy on the + -construction (see also [24, §15.2, Remark 2.3] in the topological situation);in the sequel, we just write BGL instead of the correct (but awkward) form L A BGL . Below, we provethat
BGL is an abelian group object in H A ∗ ( S ) . Theorem 5.3.
Let H = ( H , H , H , · · · ) be a motivic T -spectrum ( T = P ≃ S ∧ G m ) for a basescheme S . Define H ′ = ( H ′ , H ′ , H ′ , · · · ) by H ′ n := R Hom ∗ ( G ∧ nm , H n ) . Then (1) H ′ = ( H ′ , H ′ , H ′ , · · · ) is a motivic S -spectrum, and H ′ ≃ L A H . (2) H is an abelian group object in the pointed A -homotopy category H A ∗ ( S ) . (3) For any X ∈ s P re ( S m S ) ∗ , the derived mapping space RMap ∗ ( X, H ) ∈ s S et ∗ is an ∞ -loop spaceand hence all its components are weakly equivalent.Proof. We have H ′ = R Hom ∗ ( G ∧ m , H ) = R Hom ∗ ( S , H ) ≃ Hom ∗ ( S , L A H ) = L A H .By performing a fibrant replacement, we may assume that H is a fibrant motivic T -spectrum, sothat each H n ∈ s P re ( S m S ) ∗ is A -fibrant and the adjoint bonding maps H n → Hom ∗ ( P , H n +1 ) are A -weak equivalences (even local weak equivalences). Thus H ′ n = Hom ∗ ( G ∧ nm , H n ) ≃ Hom ∗ ( G ∧ nm , Hom ∗ ( S ∧ G m , H n +1 )) ∼ = Hom ∗ ( S ∧ G ∧ n +1 m , H n +1 ) ∼ = Hom ∗ ( S , Hom ∗ ( G ∧ n +1 m , H n +1 )) = Ω H ′ n +1 , showing that H ′ = ( H ′ , H ′ , H ′ , · · · ) is a motivic S -spectrum, (1) is proved.6 Peng DUFor (2), just note that for any X ∈ s P re ( S m S ) ∗ , the set [ X, H ] A , ∗ = [ X, H ′ ] A , ∗ = [ X, Ω H ′ ] A , ∗ isan abelian group (any term in a motivic S -spectrum is an abelian group object in H A ∗ ( S ) ).For (3), note that for all K ∈ s S et ∗ , X ∈ s P re ( S m S ) ∗ , there are canonical isomorphisms [ K, RMap ∗ ( X, H )] s S et ∗ ∼ = [ K ∧ X, H ] A , ∗ , the right hand side are abelian groups by (2). (cid:3) Corollary 5.4.
The spaces
BGL is an abelian group object in the pointed A -homotopy category H A ∗ ( S ) and for any X ∈ S m S , all the components of the derived mapping space RMap( X, BGL) ∈ s S et ∗ areweakly equivalent.If S = Spec( k ) for a perfect field k , then BSL is an abelian group object in the pointed A -homotopycategory H A ∗ ( k ) and all the components of the derived mapping space RMap( X, BSL) ∈ s S et ∗ areweakly equivalent.Proof. Since the motivic T -spectrum representing algebraic K-theory has term BGL × Z in each level,by the previous result we see BGL × Z is an abelian group object in H A ( S ) . We conclude by notingthat the projection BGL × Z → Z is a homomorphism of abelian group objects with kernel BGL .If S = Spec( k ) , note that there is an A -homotopy fiber sequence BSL → BGL → B G m = K( K M1 , (thanks to the fact that the Picard group of a normal scheme is A -invariant, yielding that G m ∈ G r A k ),realizing BSL as the kernel of
BGL → K( K M1 , (note that the second arrow splits), thus BSL is anabelian group object as well. (cid:3)
This is to be compared with the fact in classical topology that for “non-stable” groups, the path-components of
Map( X, B G ) —whose homotopy types are closed related with gauge groups—may rep-resent (infinitely) many distinct homotopy types (see e.g. [35]). Proposition 5.5.
Let ( B, ∈ s S et ∗ be an abelian group object in the pointed homotopy category Ho( s S et ∗ ) (i.e. an abelian H -group) with a binary operation m : ( B, × ( B, → ( B, , ( b, b ′ ) bb ′ .Denote its path components by B ξ , ξ ∈ π B . Then ( B, ∼ = ( B , × π ( B, as abelian H -groups.Proof. Fix a base point b ξ in each component B ξ with b = 0 . Then we easily find that the maps B → B × π ( B, , ( b ∈ B ξ ) ( bb − ξ , ξ ) and B × π ( B, → B, ( b, ξ ) bb ξ are homomorphisms of H -groups and are homotopy inverse to each other. (cid:3) The map m : BGL × BGL → BGL induces for any X ∈ S m k the addition on e K ( X ) = [ X, BGL] A : m ∗ ( ξ, ξ ′ ) = ξ + ξ ′ . Using this, for a given pointed map θ : (BGL , e ) → (K( M , n + 1) , , we can define a map θ : BGL × BGL → K( M , n + 1) which on sections is given by ( x, y ) θ ( m ( x, y )) − θ ( y ) and further a map θ : ΩBGL × BGL → ΩK( M , n + 1) which on a loop is given by θ (see [25, p. 486]) since θ ( e, y ) = 0 . Formally, let θ ♭ : BGL → Hom(BGL , K( M , n + 1)) be the adjoint of θ : BGL × BGL → K( M , n + 1) , then θ is the restrictionof the adjoint of the composite BGL θ ♭ −→ Hom(BGL , K( M , n + 1)) → Hom(Ω ∗ BGL , Ω ∗ K( M , n + 1)) . It’s obtained in the same way as we get v from u in §3.We thus get a map ( θ ) ∗ : [ X, ΩBGL] A × [ X, BGL] A → [ X, ΩK( M , n + 1)] A = [ X, K( M , n )] A . For a class ξ ∈ [ X, BGL] A , we obtain the map(5.3) ∆( θ, ξ ) : K ( X ) = [ X, ΩBGL] A → [ X, K( M , n )] A = H n ( X ; M ) NUMERATING NON-STABLE VECTOR BUNDLES 17given by β ( θ ) ∗ ( β, ξ ) . Then ∆( θ, ξ ) = ( θ ) ∗ ( − , ξ ) is homomorphism of abelian groups, whose effectis given by Proposition 5.7 below.Note further that if τ : M → M ′ is a homomorphism of sheaves of abelian groups, then it’s easy tosee that τ ∗ ∆( θ, ξ ) = ∆( τ ∗ θ, ξ ) . K ( X ) H n ( X ; M )H n ( X ; M ′ ) ∆( θ,ξ )∆( τ ∗ θ,ξ ) τ ∗ The binary operation m : BGL × BGL → BGL also determines a map m ′ : ΩBGL × BGL → Ω ∗ BGL which on a loop is given by m (see [25, p. 493]). Formally, let m ♭ : BGL → Hom(BGL , BGL) be theadjoint of m : BGL × BGL → BGL , then m ′ is the restriction of the adjoint of the composite BGL m ♭ −−→ Hom(BGL , BGL) ( − ) ∆1 −−−−→ Hom(BGL ∆ , BGL ∆ ) . On local sections ( λ, y ) , m ′ “translates” a loop λ at the base point of BGL to a loop based at y . Proposition 5.6.
The map m ′ : ΩBGL × BGL → Ω ∗ BGL is an isomorphism in H A ∗ ( S ) .So the induced map m ′∗ : [ X, ΩBGL] A × [ X, BGL] A → [ X, Ω ∗ BGL] A is an isomorphism of abelian groups and satisfies m ′∗ ( β, α ) = i ∗ β + c ∗ α, where c : BGL → Ω ∗ BGL , i : ΩBGL → Ω ∗ BGL are the maps constructed before. So r ∗ m ′∗ ( β, α ) = α .Proof. As in the proof of Proposition 5.1, for X ∈ S m k , using [ X, ΩBGL] A ∼ = [ X, ΩL A BGL] A = [∆ , ΩL A BGL( X )] s S et ∗ etc., and in general, for X ∈ s P re ( S m S ) ∗ , we have [ X, ΩBGL] A , ∗ ∼ = π RMap ∗ ( X, ΩL A BGL) etc., we are reduced to the classical topological situation, which is given in [25, Theorem 2.7]. (cid:3)
Denoting the addition on K( M , n + 1) by a , then we have a similar construction of an isomorphism a ′ : ΩK( M , n + 1) × K( M , n + 1) → Ω ∗ K( M , n + 1) , which induces an isomorphism a ′∗ : [ X, ΩK( M , n + 1)] A × [ X, K( M , n + 1)] A → [ X, Ω ∗ K( M , n + 1)] A . In fact, the map a ′ itself is an isomorphism by the degree-wise split fiber sequence ΩK( M , n + 1) Ω ∗ K( M , n + 1) K( M , n + 1) . i rc Proposition 5.7.
We have (5.4) θ ′∗ m ′∗ ( β, ξ ) = a ′∗ (∆( θ, ξ )( β ) , θ ∗ ξ ) = a ′∗ (( θ ) ∗ ( β, ξ ) , θ ∗ ξ ) . [ X, ΩBGL] A × [ X, BGL] A [ X, ΩK( M , n + 1)] A × [ X, K( M , n + 1)] A [ X, Ω ∗ BGL] A [ X, Ω ∗ K( M , n + 1)] A (( θ ) ∗ ,θ ∗ pr )=(( θ ) ∗ ,θ ∗ r ∗ m ′∗ ) ∼ = m ′∗ a ′∗ ∼ = θ ′∗ If θ ∗ ξ = 0 , then (5.5) θ ′∗ m ′∗ ( β, ξ ) = i ∗ (∆( θ, ξ )( β )) . Proof.
As in the proof of Proposition 5.1, we are reduced to the classical topological situation, whichis given in [25, eq. (2.8), p. 495]. (cid:3)
The following corresponds to [25, Theorems 2.9 and 1.2].8 Peng DU
Theorem 5.8.
Let θ ∈ H n +1 (BGL; M ) = [BGL , K( M , n + 1)] A and ξ ∈ [ X, BGL] A with θ ∗ ( ξ ) = 0 ∈ H n +1 ( X, M ) . Then for γ ∈ [ X, K( M , n )] A , we have γ ∈ C θ ( ξ ) ⇐⇒ γ ∈ im (cid:0) K ( X ) = [ X, ΩBGL] A ∆( θ,ξ ) −−−−→ [ X, K( M , n )] A = H n ( X ; M ) (cid:1) . Thus if θ ∗ ( ξ ) = 0 ∈ H n +1 ( X ; M ) , we have (5.6) im ∆( θ, ξ ) = C θ ( ξ ) , q − ∗ ( ξ ) ∼ = coker ∆( θ, ξ ) . Proof.
Assume θ ∗ ( ξ ) = 0 ∈ H n +1 ( X ; M ) . If γ = ∆( θ, ξ ) β for some β ∈ K ( X ) = [ X, ΩBGL] A , wetake ψ = m ′∗ ( β, ξ ) , then r ∗ ψ = ξ and by eq. (5.5), θ ′∗ ψ = i ∗ γ , i.e. γ ∈ C θ ( ξ ) .Conversely, let γ ∈ C θ ( ξ ) and assume r ∗ ψ = ξ, θ ′∗ ψ = i ∗ γ for some ψ ∈ [ X, Ω ∗ BGL] A . As m ′∗ is a bijection, we can take β ∈ [ X, ΩBGL] A and α ∈ [ X, BGL] A such that m ′∗ ( β, α ) = ψ , then α = r ∗ m ′∗ ( β, α ) = r ∗ ψ = ξ and again by eq. (5.5), i ∗ γ = θ ′∗ ψ = θ ′∗ m ′∗ ( β, ξ ) = i ∗ ∆( θ, ξ ) β. Since i ∗ is injective by Proposition 5.1, we see that γ = ∆( θ, ξ ) β ∈ im ∆( θ, ξ ) . (cid:3) Application to vector bundles of critical rank
Recall that we have the A -homotopy fiber sequence F n → BGL n ϕ = ϕ n −−−−→ BGL . We have the induced map on A -homotopy classes ϕ ∗ : [ X, BGL n ] A → [ X, BGL] A . We want to describe the inverse-image under ϕ ∗ of a class in the right hand side.The following corresponds to [25, Theorem 1.6]. Theorem 6.1.
Let n = d be odd, let ξ be a stable vector bundle over X , whose classifying map is stilldenoted by ξ : X → BGL . Then there is a bijection ϕ − ∗ ( ξ ) ∼ = coker (cid:0) K ( X ) = [ X, ΩBGL] A ∆( c n +1 ,ξ ) −−−−−−→ [ X, K( K M n +1 , n )] A = H n ( X ; K M n +1 ) (cid:1) . Proof.
This is an easy consequence of Theorem 5.8: for n = d , the condition c n +1 ( ξ ) = c d +1 ( ξ ) = 0 issatisfied as Chern classes vanish above the rank of the vector bundle. (cid:3) The universal j -th Chern class c j ∈ H j (BGL; K M j ) = CH j (BGL) for j ∈ N satisfy(6.1) m ∗ c n = n X r =0 ( c r × c n − r ) ∈ [BGL × BGL , K( K M n , n )] A , where c r × c n − r := µ ( c r ⊗ c n − r ) , µ being the obvious map [BGL , K( K M r , r )] A ⊗ [BGL , K( K M n − r , n − r )] A → [BGL × BGL , K( K M n , n )] A induced by the multiplication in the graded sheaves of Milnor K-groups K M ∗ .Indeed, for any U ∈ S m k and ( ξ, ξ ′ ) ∈ [ U, BGL] A × [ U, BGL] A , as m ∗ ( ξ, ξ ′ ) = ξ + ξ ′ , by Whitneysum formula for Chern classes, ( m ∗ c n )( ξ, ξ ′ ) = c n m ∗ ( ξ, ξ ′ ) = c n ( ξ + ξ ′ ) = n X r =0 c r ( ξ ) · c n − r ( ξ ′ ) = n X r =0 ( c r × c n − r )( ξ, ξ ′ ) , hence the result.We have the suspension homomorphism σ : CH j (BGL) = [BGL , K( K M j , j )] A , ∗ → [ΩBGL , ΩK( K M j , j )] A , ∗ = e H j − (GL; K M j ) for j > , induced by the loop functor Ω which is given by σ ([ θ ]) = [Ω θ ] . Note that in the above, it doesn’t matter whether we use pointed or unpointed A -homotopy classesprovided j > , see [4, Lemma 2.1]. In fact this is also the case for j = 2 by Corollary 3.6.NUMERATING NON-STABLE VECTOR BUNDLES 19 Theorem 6.2.
For β ∈ K ( X ) = [ X, ΩBGL] A = [ X, GL] A , we have ∆( c n +1 , ξ ) β = β ∗ σc n +1 + n X r =1 ( β ∗ σc r ) · c n +1 − r ( ξ )= (Ω c n +1 )( β ) + n X r =1 ((Ω c r )( β )) · c n +1 − r ( ξ )= ∆( c n +1 , β + n X r =1 ((Ω c r )( β )) · c n +1 − r ( ξ )= ∆( c n +1 , β + n X r =1 (∆( c r , β ) · c n +1 − r ( ξ ) . (6.2) H j (BGL; K M j ) σ −→ [GL , K( K M j , j − A β ∗ −→ [ X, K( K M j , j − A = H j − ( X ; K M j ) To prove this, we first prove the following more general result, which corresponds to [25, Theorem1.3] whose proof is given in [25, §5].
Proposition 6.3.
Let K be a sheaf of rings, strictly A -invariant as a sheaf of abelian groups. Let β ∈ K ( X ) = [ X, ΩBGL] A = [ X, GL] A and α ∈ e H ∗ (K( M , n + 1); K ) . Assume a ∗ α = α × × α, m ∗ θ ∗ α ∈ µ ( e H ∗ (BGL; K ) ⊗ ) ; more precisely, let m ∗ θ ∗ α = θ ∗ α × × θ ∗ α + P u i × v i = µ ( θ ∗ α ⊗ ⊗ θ ∗ α + P u i ⊗ v i ) , where u i , v i ∈ e H ∗ (BGL; K ) . (1) We have θ ∗ α = m ∗ θ ∗ α − × θ ∗ α . (2) Denote γ = ∆( θ, ξ ) β , then γ ∗ σα = ( β ∗ σ ) · ( θ ∗ α ) + X ( β ∗ σu i ) · ( ξ ∗ v i ) .α ∈ e H ∗ (K( M , n + 1); K ) e H ∗− (K( M , n ); K ) H ∗− ( X ; K ) α ∈ e H ∗ (K( M , n + 1); K ) e H ∗ (K( M , n + 1) × K( M , n + 1); K ) e H ∗ (K( M , n + 1); K ) ⊗ α ′ = θ ∗ α ∈ e H ∗ (BGL; K ) e H ∗ (BGL × BGL; K ) e H ∗ (BGL; K ) ⊗ σ γ ∗ a ∗ θ ∗ θ ∗ µm ∗ µ Proof.
Write α ′ = θ ∗ α .(1) Let g be the composite BGL × BGL × ∆ −−−→ BGL × BGL × BGL m × −−−→ BGL × BGL and let h bethe composite K( M , n + 1) × K( M , n + 1) × w −−−→ K( M , n + 1) × K( M , n + 1) a −→ K( M , n + 1) ,where ∆ : BGL → BGL × BGL is the diagonal and w : K( M , n + 1) → K( M , n + 1) is theinverse map. We have θ = h ( θ × θ ) g. As w ∗ α = − α , we have h ∗ α = (1 × w ) ∗ a ∗ α = (1 × w ) ∗ ( α × × α ) = α × − × α , so ( θ × θ ) ∗ h ∗ α = α ′ × − × α ′ . We conclude by noting that g ∗ ( α ′ ×
1) = (1 × ∆) ∗ ( m × ∗ ( α ′ ×
1) = (1 × ∆) ∗ ( m ∗ α ′ ×
1) = m ∗ α ′ (here we need the fact that m ∗ α ′ = P a i × b i ∈ µ ( e H ∗ (BGL; K ) ⊗ ) and the observation that (1 × ∆) ∗ (( a × b ) ×
1) = a × b ) and g ∗ (1 × α ′ ) = (1 × ∆) ∗ ( m × ∗ (1 × α ′ ) = (1 × ∆) ∗ (1 × (1 × α ′ )) = 1 × α ′ . (2) Consider the maps of pairs u : (BGL × BGL , ∗ × BGL) → (K( M , n + 1) , and v : (ΩBGL × BGL , ∗ × BGL) → (ΩK( M , n + 1) , determined by the maps θ , θ respectively.0 Peng DUBy (1), we can write θ ∗ α = µ ( α ′ ) with α ′ = α ′ ⊗ P u i ⊗ v i ∈ e H ∗ (BGL; K ) ⊗ . Let ι : (BGL , ∅ ) → (BGL , ∗ ) , ι : (BGL × BGL , ∅ ) → (BGL × BGL , ∗ × BGL) and ι : (ΩBGL × BGL , ∅ ) → (ΩBGL × BGL , ∗ × BGL) be the inclusions of pairs, so θ = uι , θ = vι . Let α := (1 ⊗ ι ∗ )( α ′ ) = α ′ ⊗ P u i ⊗ ι ∗ v i ∈ e H ∗ (BGL; K ) ⊗ H ∗ (BGL; K ) . α ∈ e H ∗ (K( M , n + 1); K ) H ∗ (BGL × BGL; K ) e H ∗ (BGL; K ) ⊗ ∋ α ′ α ∈ e H ∗ (K( M , n + 1); K ) H ∗ (BGL × BGL , ∗ × BGL; K ) e H ∗ (BGL; K ) ⊗ H ∗ (BGL; K ) ∋ α θ ∗ µ ⊗ ι ∗ u ∗ ι ∗ µ Then ι ∗ u ∗ α = θ ∗ α = µ ( α ′ ) = ι ∗ µ ( α ) , u ∗ α = µ ( α ) + ε, ε ∈ ker ι ∗ . So by eqs. (3.2) and (3.3), θ ∗ σα = ι ∗ v ∗ σα = ι ∗ ρu ∗ α = ι ∗ ρµ ( α ) + ι ∗ ρε,ι ∗ ρµ ( α ) = ι ∗ µ ( σ ⊗ α = ι ∗ µ ( σα ′ ⊗ X σu i ⊗ ι ∗ v i ) = ι ∗ ( σα ′ × X σu i × ι ∗ v i ) ,ι ∗ ρε = σ ′ ι ∗ ε = 0 , hence θ ∗ σα = σ ′ θ ∗ ( α ) . Of course, as im θ ∗ ⊂ im ι ∗ , one can get this relation more directly from the “outer contour” ofeq. (3.2), without using eq. (3.3).By (1) and the fact that σ ′ ( a × b ) = ( σa ) × b in eq. (3.2), we get θ ∗ σα = σ ′ ( m ∗ α ′ − × α ′ ) = σα ′ × X ( σu i ) × v i . Since γ = ∆( θ, ξ ) β = ( θ ) ∗ ( β, ξ ) is represented by the composite X ∆ −→ X × X β × ξ −−→ ΩBGL × BGL θ −→ ΩK( M , n + 1) , we have γ ∗ σα = ∆ ∗ ( β × ξ ) ∗ θ ∗ σα = ∆ ∗ ( β × ξ ) ∗ ( σα ′ × X ( σu i ) × v i )= ∆ ∗ ( β ∗ σα ′ × X ( β ∗ σu i ) × ( ξ ∗ v i )) = ( β ∗ σ ) · ( θ ∗ α ) + X ( β ∗ σu i ) · ( ξ ∗ v i ) . (cid:3) Proof of
Theorem 6.2 . We apply Proposition 6.3 to the situation when M = K M n +1 , K = K M ∗ , θ = c n +1 : BGL → K( K M n +1 , n + 1) and α ∈ e H n +1 (K( K M n +1 , n + 1); K M n +1 ) = [K( K M n +1 , n + 1) , K( K M n +1 , n +1)] A , ∗ being represented by the identity map of K( K M n +1 , n + 1) , in which case, θ ∗ α = θ, γ ∗ σα = γ .We need to prove that [ a ] = a ∗ α = α × × α holds. But α × × α is also represented bythe addition map a (Eckmann-Hilton property). (cid:3) We give the following two obvious corollaries of Theorems 6.1 and 6.2.
Corollary 6.4.
Assume dim X = d > is odd. Let ξ, ξ ′ ∈ [ X, BGL] A . If c j ( ξ ) = c j ( ξ ′ ) , j d ,then ξ and ξ ′ have the same number of representatives in V d ( X ) . In particular, we have the following.
Corollary 6.5.
Assume dim X = d > is odd. Let ξ, ξ ′ ∈ V d ( X ) , sharing the same total Chern class.If ξ is cancellative, then so is ξ ′ . The following result says that the map ∆( c n +1 , ξ ) : [ X, ΩBGL] A → [ X, K( K M n +1 , n )] A = H n ( X ; K M n +1 ) is essentially the map induced by c n +1 : BGL → K( K M n +1 , n + 1) ; so the complicated operations wemade just transfer things on a general component RMap( X, BGL) ξ of the derived mapping space RMap( X, BGL) ∈ s S et ∗ to the weakly equivalent component RMap( X, BGL) (component of thecanonical base point). In light of Corollary 5.4, this is not very surprising.NUMERATING NON-STABLE VECTOR BUNDLES 21 Theorem 6.6.
Given ξ ∈ [ X, BGL] A , let T ξ := m ( − , ξ ) ∗ : π (RMap( X, BGL) , → π (RMap( X, BGL) , ξ ) be the isomorphism induced by m ( − , ξ ) : RMap( X, BGL) → RMap( X, BGL) ξ introduced in Proposi-tion 5.5. Assume c n +1 ( ξ ) = 0 . Then we have c n +1 ◦ T ξ = ∆( c n +1 , ξ ) : K ( X ) = [ X, ΩBGL] A → [ X, K( K M n +1 , n )] A = H n ( X ; K M n +1 ) . Similar statements hold with
BSL in place of
BGL .Proof.
By Proposition 5.5, the map T ξ is an isomorphism.Since m ∗ c n +1 ( − , ξ ) = c n +1 m ( − , ξ ) : RMap( X, BGL) → RMap( X, K( K M n +1 , n + 1)) , we see ( m ∗ c n +1 ( − , ξ )) ∗ = ( c n +1 ) ∗ T ξ as maps π (RMap( X, BGL) , → π (RMap( X, K( K M n +1 , n + 1)) ,
0) = H n ( X ; K M n +1 ) .On the other hand, since m ∗ c n +1 = n +1 X r =0 c r × c n +1 − r ∈ [BGL × BGL , K( K M n +1 , n + 1)] A , by taking RMap( X, − ) to the two sides we see the following two maps in s S et ∗ are homotopic:(6.3) m ∗ c n +1 ( − , ξ ) ≃ n +1 X r =0 c r ( − ) · c n +1 − r ( ξ ) : RMap( X, BGL) → RMap( X, K( K M n +1 , n + 1)) . Here the map c r ( − ) · c n +1 − r ( ξ ) is the composite RMap( X, BGL) ( c r ( − ) ,c n +1 − r ( ξ )) −−−−−−−−−−−→ RMap( X, K( K M r , r )) × RMap( X, K( K M n +1 − r , n + 1 − r )) → RMap( X × X, K( K M n +1 , n + 1)) ∆ ∗ −−→ RMap( X, K( K M n +1 , n + 1)) , where the second coordinate of the first arrow is given by RMap( X, BGL) → ∆ c n +1 − r ( ξ ) −−−−−−→ RMap( X, K( K M n +1 − r , n + 1 − r )) , the second arrow is the obvious map and the third is induced by the diagonal of X . From thisdescription it’s clear that Ω( c r ( − ) · c n +1 − r ( ξ )) ≃ (Ω c r ( − )) · c n +1 − r ( ξ ) : ΩRMap( X, BGL) → RMap( X, K( K M n +1 , n )) . Additions are also preserved: for any [ a ] , [ b ] ∈ [RMap( X, BGL) , RMap( X, K( K M n +1 , n + 1))] s S et ∗ , wehave [ a ] ∗ + [ b ] ∗ ≃ ([ a ] + [ b ]) ∗ : ΩRMap( X, BGL) → RMap( X, K( K M n +1 , n )) (the subscript ∗ refers theeffects on loop spaces). Indeed, by facts about H -spaces, the sum [ a ] + [ b ] is represented the composite RMap( X, BGL) ∆ −→ RMap( X, BGL) × RMap( X, BGL) ( a,b ) −−−→ RMap( X, K( K M n +1 , n + 1)) × RMap( X, K( K M n +1 , n + 1)) + −→ RMap( X, K( K M n +1 , n + 1)) , where ∆ refers the diagonal map and “ + ” is the H -group operation of RMap( X, K( K M n +1 , n +1)) ; taking Ω everywhere gives a similar composite for [ a ] ∗ + [ b ] ∗ ∈ [ΩRMap( X, BGL) , RMap( X, K( K M n +1 , n ))] s S et ∗ ,which says exactly that [ a ] ∗ + [ b ] ∗ = ([ a ] + [ b ]) ∗ ∈ [ΩRMap( X, BGL) , RMap( X, K( K M n +1 , n ))] s S et ∗ . Soapplying π Ω = π to eq. (6.3) we find ( m ∗ c n +1 ( − , ξ )) ∗ = n +1 X r =0 (Ω c r )( − ) · c n +1 − r ( ξ ) = ∆( c n +1 , ξ ) by Theorem 6.2. So c n +1 ◦ T ξ = ( c n +1 ) ∗ T ξ = ∆( c n +1 , ξ ) .The BSL case is also valid by Corollary 5.4. (cid:3)
Remark 6.7.
Theorems 2.1 and 6.6 together give an independent proof of Theorems 6.1 and 6.2, so we couldtotally avoid using the method of [25]. We still present both, as the method of [25] is the motivation of ourwork. A n +1 := k [ x , · · · , x n , y , · · · , y n ] / ( n X i =0 x i y i − and Q n +1 := Spec( A n +1 ) . The projection to the first n + 1 coordinates gives a morphism Q n +1 → A n +1 \ A n +1 k \ , which is an A n -fibration hence an A -weak equivalence.Let A be a k -algebra. To any A -point ( a, b ) of Q n +1 , Suslin associated (systemically and inductively)a matrix α n +1 ( a, b ) ∈ SL n ( A ) ⊂ GL n ( A ) , yielding a k -morphism α n +1 : Q n +1 → GL n ֒ → GL . Moreover, Suslin gave a systemical way to reduce the matrix α n +1 ( a, b ) ∈ SL n ( A ) ⊂ GL n ( A ) via ele-mentary matrix operations (though non-explicitly) to an element β n +1 ( a, b ) ∈ SL n +1 ( A ) ⊂ GL n +1 ( A ) ,whose first row can be designated to be ( a n !0 , a , · · · , a n ) if a = ( a , a , · · · , a n ) —(a special case of) thefamous Suslin’s n ! theorem (see [36, 28]). This gives a k -morphism β n +1 : Q n +1 → SL n +1 ⊂ GL n +1 . Note that we have [ α n +1 ( a, b )] = [ β n +1 ( a, b )] ∈ GL( A ) / E( A ) = K ( A ) by construction. GL n +1 X Q n +1 GL n ( a,b ) α n +1 β n +1 With the A -weak equivalences A n +1 \ ≃ Q n +1 , we obtain a morphism α n +1 : A n +1 \ → GL ≃ ΩBGL in H A ∗ ( k ) (base points suitably chosen). Taking adjunction we get a morphism α ♯n +1 : ( P ) ∧ ( n +1) ≃ Σ( A n +1 \ → BGL in H A ∗ ( k ) . Under the canonical isomorphism [ A n +1 \ , K( K M r , r − A , ∗ ∼ = −→ [( P ) ∧ ( n +1) , K( K M r , r )] A , ∗ , the class (Ω c r )( α n +1 ) corresponds to c r ( α ♯n +1 ) . A n +1 \ α n +1 −−−→ ΩBGL Ω c r −−→ K( K M r , r − ! ( P ) ∧ ( n +1) α ♯n +1 −−−→ BGL c r −→ K( K M r , r ) As c r ( α ♯n +1 ) ∈ e H r (( P ) ∧ ( n +1) ; K M r ) and by [4, Lemma 4.5], e H r (( P ) ∧ ( n +1) ; K M r ) = e H r − ( A n +1 \ K M r ) ∼ = ( , r = n + 1;( K M r ) − ( n +1) (Spec k ) = K M0 (Spec k ) = Z , r = n + 1 (where M − i denotes the i -th contraction of a sheaf M and we have used [4, Lemma 2.7]), we obtainthe following. Proposition 6.8. (Ω c r )( α n +1 ) = (Ω c r )( β n +1 ) = 0 for r = n + 1 . Remark 6.9.
By eq. (6.5) below, we see that (Ω c n +1 )( α n +1 ) is ± n ! in ( K M0 )(Spec k ) = Z . It’s an easy exercise of scheme theory that for any k -algebra A , we have Hom k (Spec A, A n +1 \ ∼ =Um n +1 ( A ) (the set of unimodular row in A of length n + 1 ). Suppose now that X := Spec A is smoothover k . With some effort, one can show the following (see [15, Theorem 2.1] for a proof). Proposition 6.10.
Assume dim A = d n , then H n (Spec A ; K MW n +1 ) ∼ = [Spec A, A n +1 \ A ∼ =Um n +1 ( A ) / E n +1 ( A ) . Here E n +1 ( A ) denotes the group of elementary matrices of size n + 1 , with itsnatural action on rows of length n + 1 .Moreover, let pr : GL n +1 → A n +1 \ be the projection to the first row, then we obtain a morphism ψ n +1 : A n +1 \ → A n +1 \ , ( x , x , · · · , x n ) ( x n !0 , x , · · · , x n ) NUMERATING NON-STABLE VECTOR BUNDLES 23 in H A ∗ ( k ) , being the composite A n +1 \ ≃ Q n +1 β n +1 −−−→ GL n +1 pr −−→ A n +1 \ . Then the induced map ( ψ n +1 ) ∗ : [Spec A, A n +1 \ A → [Spec A, A n +1 \ A on the cohomotopy set is given by ( ψ n +1 ) ∗ ([ a , a , · · · , a n ]) = [ a n !0 , a , · · · , a n ] = n !2 h · [ a , a , · · · , a n ] under the above isomorphism, for ( a , · · · , a n ) ∈ Um n +1 ( A ) (where [ a , · · · , a n ] denotes its class in theorbit set Um n +1 ( A ) / E n +1 ( A ) and h = h , − i ). Remark 6.11.
Let ω : A n +1 \ → K( K MW n +1 , n ) be the morphism determined by the projection to the firstnon-trivial ( n -th) Postnikov tower, and let [ ω ] ∈ H n ( A n +1 \ K MW n +1 ) be the cohomology class it represents.Then H n ( A n +1 \ K MW n +1 ) ∼ = K MW0 ( k ) is a rank- free K MW0 ( k ) -module generated by [ ω ] and ψ ∗ n +1 [ ω ] = n !2 h · [ ω ] . Then we get the last statement as follows: Let a = ( a , · · · , a n ) ∈ Um n +1 ( A ) , then ω ∗ ( ψ n +1 ) ∗ ([ a ]) = [ ω ◦ ψ n +1 ◦ a ] = ( ψ ∗ n +1 [ ω ])[ a ] = ( n !2 h · [ ω ])[ a ] = ω ∗ ( n !2 h [ a ]) , and so ( ψ n +1 ) ∗ ([ a ]) = n !2 h [ a ] (using Postnikov tower argumentto see that ω ∗ : [ X, A n +1 \ A → H n ( X ; K MW n +1 ) is a bijection, since d n ). X = Spec A a −→ A n +1 \ ψ n +1 −−−→ A n +1 \ ω −→ K( K MW n +1 , n ) In fact, let V ⊂ A n +1 \ be the closed subvariety defined by x = · · · = x n = 0 , so V ∼ = A \ , k ( V ) ∼ = k ( x ) .The homology of the portion M y ∈ ( A n +1 \ ( n − K MW2 ( κ y ) → M y ∈ ( A n +1 \ ( n ) K MW1 ( κ y ) → M y ∈ ( A n +1 \ ( n +1) K MW0 ( κ y ) of the Rost-Schmid complex computes H n ( A n +1 \ K MW n +1 ) , and one finds that [ x ] ∈ K MW1 ( k ( V )) is indeed acycle. Moreover, in the localization exact sequence n ( A n +1 ; K MW n +1 ) → H n ( A n +1 \ K MW n +1 ) ∂ −→ H n +1 { } ( A n +1 ; K MW n +1 ) = K MW0 ( k ) → we have ∂ [ x ] = h i ∈ K MW0 ( k ) , so H n ( A n +1 \ K MW n +1 ) = K MW0 ( k ) · [ x ] . By the relation [ a n ] = n ǫ [ a ] in K MW1 we see [ x n !0 ] = n !2 h [ x ] so ∂ [ x n !0 ] = n !2 h . Thus ψ ∗ n +1 [ x ] = n !2 h · [ x ] . For an abelian group H and an integer m , we write H/m := H/mH for the quotient abelian group.We say that H is m -divisible if H/m = 0 , i.e. if mH = H . Theorem 6.12.
Let k be a perfect field with char( k ) = 2 . Assume that A is a smooth k -algebra of oddKrull dimension d > . Let X = Spec A , ξ ∈ [ X, BGL] A . (1) We have d ! · H d ( X ; K M d +1 ) ⊂ im∆( c d +1 , ξ ) . (2) There is a surjective homomorphism H d ( X ; K M d +1 ) /d ! ։ coker∆( c d +1 , ξ ) . (3) If H d ( X ; K M d +1 ) is d ! -divisible, then coker∆( c d +1 , ξ ) = 0 . So in this case, any rank d vectorbundle is cancellative. Moreover, the map ( c d +1 ) ∗ : π (RMap( X, BSL) , ξ ) → π (RMap( X, K( K M d +1 , d + 1)) ,
0) = H d ( X ; K M d +1 ) is surjective for every ξ ∈ [ X, BGL] A .Proof. (1) By [4, Proposition 2.6], there is an exact sequence → I d +2 ∼ = η K MW d +2 → K MW d +1 → K M d +1 → , where I = ker( GW = K MW0 → K M0 ) = η K MW1 is the fundamental ideal and I d +2 its power.Since X has A -cohomological dimension at most dim X = d , we see H d +1 ( X ; I d +2 ) = 0 , hencethe map τ : H d ( X ; K MW d +1 ) → H d ( X ; K M d +1 ) given by reducing coefficients is a surjection. So we4 Peng DUcan write any element of H d ( X ; K M d +1 ) as τ ([ a, b ]) with ( a, b ) ∈ Q d +1 ( A ) , a = ( a , a , · · · , a d ) ∈ Um d +1 ( A ) . [ X, Q d +1 ] A = Um d +1 ( A ) / E d +1 ( A ) = H d ( X ; K MW d +1 ) H d ( X ; K M d +1 )[ β d +1 ( a, b )] ∈ [ X, GL d +1 ] A [ X, GL] A = K ( X ) ∋ [ β d +1 ( a, b )] τ (pr ) ∗ ( β d +1 ) ∗ ∆( c d +1 ,ξ ) We will show that the following equality holds:(6.4) d ! · τ ([ a, b ]) = ± ∆( c d +1 , ξ )([ β d +1 ( a, b )]) . Indeed, by Proposition 6.10 (and notice that h = η · [ −
1] + 2 becomes in K M ∗ , yielding that τ ( d !2 h ) = d ! ), d ! · τ ([ a, b ]) = τ ( d ! · [ a, b ]) = τ (( ψ d +1 ) ∗ [ a, b ]) = τ ((pr ) ∗ [ β d +1 ( a, b )]) = τ ([ a d !0 , a , · · · , a d ]) . While Proposition 6.8 tells ∆( c d +1 , ξ )([ β d +1 ( a, b )]) = (Ω c d +1 )([ β d +1 ( a, b )]) . We are thus reduced to showing that(6.5) τ ([ a d !0 , a , · · · , a d ]) = τ ((pr ) ∗ [ β d +1 ( a, b )]) = ± (Ω c d +1 )([ β d +1 ( a, b )]) . We will prove more generally that(6.6) τ ([(1 , , · · · , · β ]) = ± (Ω c d +1 )([ β ]) , for β ∈ SL d +1 ( A ) ⊂ SL( A ) . For this, note that by Theorem 3.2 (10), we have an A -homotopy fibre sequence (BSL d ) [ d ] p d −→ BSL d +1 e d +1 −−−→ K( π A d ( A d +1 \ , d + 1) = K( K MW d +1 , d + 1) (where e d +1 is the relevant k -invariant) hence also K( K MW d +1 , d ) → (BSL d ) [ d ] p d −→ BSL d +1 . As ( A d +1 \ d ] = K( K MW d +1 , d ) , by Theorem 3.2 (10) and naturality statement of (dual of) [23,Proposition 6.5.3], we get a map of A -homotopy fibre sequences SL d +1 A d +1 \ d BSL d +1 SL d +1 K( K MW d +1 , d ) (BSL d ) [ d ] BSL d +1 . pr τ ′ Ω e d +1 p d In this diagram, the map SL d +1 ≃ R ΩBSL d +1 pr −−→ A d +1 \ is indeed “projection to the firstrow”, since after applying [ U, − ] A , the induced map on homotopy is given by the natural actionof SL d +1 ( U ) / E d +1 ( U ) on the class of the base point (1 , , · · · , ∈ ( A d +1 \ k ) .Thus Ω e d +1 = τ ′ ◦ pr : SL d +1 → K( K MW d +1 , d ) (in H A ∗ ( k ) ), composing with the maps X β −→ SL d +1 and K( K MW d +1 , d ) → K( K M d +1 , d ) we find that, after reducing coefficients, Ω e d +1 ([ β ]) equals to τ ([(1 , , · · · , · β ]) . While by [6, Theorem 1 and Proposition 5.8], our k -invariant e d +1 is the (universal) Euler class (up to a unit in GW( k ) ), whose reduction coincides with theuniversal Chern class c d +1 (up to sign, as a unit in GW( k ) = K MW0 ( k ) is mapped to a unit in K M0 ( k ) ∼ = Z ), establishing our eq. (6.6).Statements (2) and (3) then follow easily, with the help of the last part of Theorem 2.1. (cid:3) Remark 6.13. If k is algebraically closed ( k = ¯ k ), then using the Rost-Schmid complex (see [13, 17] for somenice expositions on related notions and results), one easily finds that H d ( X ; K M d +1 ) is d ! -divisible, as it is aquotient of the direct sum of groups of the form K M1 ( κ x ) = κ × x = k × which is d ! -divisible ( x ranges over closedpoints of X ). Of course, for the same reason, we have the more refined result that H d ( X ; K M d +1 ) is d ! -divisibleif κ × x is d ! -divisible (i.e. κ x = ( κ x ) d ! ) for all closed points x ∈ X .More generally, H d ( X ; K M d +1 ) is d ! -divisible if k has cohomological dimension at most (see for instance [16,Theorem 2.2]), as a consequence of Voevodsky’s confirmation of the motivic Bloch-Kato conjecture (or normresidue isomorphism theorem ), a highly non-trivial result. NUMERATING NON-STABLE VECTOR BUNDLES 25Then we treat the case when n = d is even. To still get a principal A -homotopy fiber sequence, weneed to restrict ourselves to the case of oriented vector bundles—namely those classified by homotopyclasses of maps to BSL or BSL d (since π A BSL d = 0 , as opposed to the fact that π A BGL d = G m ).Note first that our discussion from §4 up to §6 here are still valid if we replace GL with SL everywhere,essentially because we also have the A -homotopy fiber sequence A n +1 \ → BSL n → BSL n +1 . The only difference is that the obstruction class (namely the k -invariant θ ) is different from the casewhen n = d is odd.We now identify the k -invariant θ . By the functoriality of the Moore-Postnikov tower, applied tothe square BSL d BSL d +1 BSL d BSL (factoring the rows up to the first non-trivial stage E ′ , E ) we have the following map of A -homotopyfibre sequences (when deleting the last column) K( K MW d +1 , d ) E ′ BSL d +1 K( K MW d +1 , d + 1) K( K M d +1 , d + 1)K( K MW d +1 , d ) E BSL K( K MW d +1 , d + 1) K( K M d +1 , d + 1) . e d +1 s d +1 τθ τ So τ e d +1 = c d +1 = τ θs d +1 (as Chern classes stabilize, we can write c d +1 = τ θ ).Note that as dim X = d , we have SK ( X ) = [ X, ΩBSL] A = [ X, SL] A ( s d +1 ) ∗ ==== [ X, SL d +1 ] A . We thushave a commutative diagram (since τ is induced by the homomorphism K MW d +1 → K M d +1 , we can move τ out; cf. [25, (1.1)]) [ X, SL] A H d ( X ; K MW d +1 )[ X, SL] A H d ( X ; K M d +1 ) . ∆( θ,ξ ) τ ∆( c d +1 ,ξ ) Again by the exact sequence → I d +2 → K MW d +1 → K M d +1 → , we see that the right vertical map τ : H d ( X ; K MW d +1 ) → H d ( X ; K M d +1 ) is surjective (since X has A -cohomological dimension at most d ).If we further assume that the -cohomological dimension of our base field k (perfect and char( k ) = 2 )is at most : c . d . ( k ) , then by (the proof of) [16, Theorem 2.1] (using Gersten-Witt complex of X = Spec A and assuming d > ), we have H d ( X ; I d +2 ) = 0 . So in this case, the right vertical map τ : H d ( X ; K MW d +1 ) → H d ( X ; K M d +1 ) is an isomorphism.Finally, we are able to give results similar to those in the odd dimension case (whose proof is almostverbatim the same after suitably changing notations).Similarly as before, we have the induced map on A -homotopy classes ϕ ∗ : [ X, BSL n ] A → [ X, BSL] A and want to enumerate its fibers. We also use V ◦ n ( X ) to denote the set of isomorphism classes of rank n oriented vector bundle over X . Theorem 6.14.
Assume that the base field k is perfect and char( k ) = 2 with c . d . ( k ) . Let n = d > be even, let ξ be a stable oriented vector bundle over X , whose classifying map is stilldenoted ξ : X → BSL . Then there is a bijection ϕ − ∗ ( ξ ) ∼ = coker (cid:0) SK ( X ) = [ X, ΩBSL] A ∆( c d +1 ,ξ ) −−−−−−→ [ X, K( K M d +1 , d )] A = H d ( X ; K M d +1 ) (cid:1) . Proposition 6.15.
Assume that the condition of Theorem 6.14 is satisfied. (1)
Let ξ, ξ ′ ∈ [ X, BSL] A . If c j ( ξ ) = c j ( ξ ′ ) , j d , then ξ and ξ ′ has the same number ofrepresentatives in V ◦ d ( X ) . Let ξ, ξ ′ ∈ V ◦ d ( X ) , sharing the same total Chern class. If ξ is cancellative, then so is ξ ′ . Theorem 6.16.
Assume that the base field k is perfect and char( k ) = 2 with c . d . ( k ) . Assumethat A is a smooth k -algebra of even Krull dimension d > . Let X = Spec A , ξ ∈ [ X, BSL] A . (1) We have d ! · H d ( X ; K M d +1 ) ⊂ im∆( c d +1 , ξ ) . (2) There is a surjective homomorphism H d ( X ; K M d +1 ) /d ! ։ coker∆( c d +1 , ξ ) . (3) If H d ( X ; K M d +1 ) is d ! -divisible, then coker∆( c d +1 , ξ ) = 0 . So in this case, any rank d orientedvector bundle is cancellative. Moreover, the map ( c d +1 ) ∗ : π (RMap( X, BSL) , ξ ) θ ∗ −→ π (RMap( X, K( K MW d +1 , d + 1)) ,
0) = H d ( X ; K MW d +1 ) τ −→ H d ( X ; K M d +1 ) is surjective for every ξ ∈ [ X, BSL] A . Remark 6.17.
For the even rank case, our assumption on the -cohomological dimension of the base field k cannot be omitted in order to get H d ( X ; I d +2 ) = 0 so that τ : H d ( X ; K MW d +1 ) → H d ( X ; K M d +1 ) is an isomorphism: c . d . ( R ) = ∞ , if we take A = R [ x, y, z ] / ( x + y + z − , then H d (Spec A ; I d +2 ) = 0 (note that c . d . ( R ) = ∞ ).On the other hand, quite a lot of fields satisfy our assumption, e.g. any finite field (with odd characteristic),any algebraically closed field, or any field of the form L ( t ) or L ( t , t ) for an algebraically closed field L with char( L ) = 0 . Application to vector bundles below critical rank
Assume that the base field k is perfect and char( k ) = 2 , A a smooth affine k -algebra of Krulldimension d > , and X = Spec( A ) . Let ξ be a stable oriented vector bundle over X , whose classifyingmap is still denoted ξ : X → BSL . We will now investigate the isomorphism classes of rank n = d − (oriented) vector bundles that are stably equivalent to the given ξ (if it exists).If d > is even, we have the A -homotopy fiber sequence F d − → BSL d − ϕ = ϕ d − −−−−−→ BSL . We consider the two-stage Moore-Postnikov factorization (Theorem 3.2) of the map ϕ : BSL d − → BSL ,(7.1) F = F d − BSL d − K( π A d F d − , d ) E ′ K( K M d , d − E K( π A d F d − , d + 1) X BSL K( K M d , d ) . q ′ q θ ′ pξξ E θ By the properties listed in Theorem 3.2, it’s easy to see that the map q ′ : BSL d − → E ′ induces abijection q ′∗ : [ X, BSL d − ] A → [ X, E ′ ] A . We thus need only to find under what conditions, the maps p ∗ : [ X, E ] A → [ X, BSL] A and q ∗ : [ X, E ′ ] A → [ X, E ] A are injections.As before, we have the following commutative diagram BSL d K( K MW d , d )BSL K( K M d , d ) . e d s d τθ NUMERATING NON-STABLE VECTOR BUNDLES 27Thus θs d = τ e d = c d by [6, Example 5.2 and Proposition 5.8] as before.On the other hand, by Serre’s splitting theorem, we can write ξ = ( s d ) ∗ ([ ξ d ]) for some [ ξ d ] ∈ [ X, BSL d ] A . So θ ∗ ([ ξ ]) = ( θs d ) ∗ [ ξ d ] = ( τ e d ) ∗ [ ξ d ] = c d ( ξ d ) = c d ( ξ ) (as Chern classes stabilizes), we canwrite θ = c d .Note that whenever ξ is represented by a rank d − vector bundle, we will have c d ( ξ ) = 0 . Soby Theorem 5.8 with Theorem 6.2 and Proposition 6.3 applied to the case n = d − , we obtain thefollowing description of p − ∗ ([ ξ ]) . Proposition 7.1.
Let k be a perfect field with char( k ) = 2 , A a smooth affine k -algebra of even Krulldimension d > , and X = Spec( A ) , let ξ be a stable oriented vector bundle over X , whose classifyingmap is still denoted ξ : X → BSL . If ξ is represented by a rank d − vector bundle, there is a bijection p − ∗ ( ξ ) ∼ = coker (cid:0) SK ( X ) = [ X, ΩBSL] A ∆( c d ,ξ ) −−−−→ [ X, K( K M d , d − A = H d − ( X ; K M d ) (cid:1) . The homomorphism ∆( c d , ξ ) is given as follows: for β ∈ SK ( X ) = [ X, SL] A , ∆( c d , ξ ) β = (Ω c d )( β ) + d − X r =1 ((Ω c r )( β )) · c d − r ( ξ ) . Below, for a sheaf of abelian groups K and an integer m , we denote by K /m := K /m K for themod- m quotient sheaf and m K := ker( K m −→ m K ) for the m -torsion subsheaf. Since the contractionfunctor ( − ) − : A b A k → A b A k is exact, the constructions of the mod- m quotient and the m -torsionsubsheaves are preserved by (iterated) contractions. We also write µ m for the étale sheaf of m -th rootsof , and µ ⊗ nm for its n -th tensor power. Denote ¯ I j := I j / I j +1 ∼ = K M j / . The proof of the followingresult is adapted from [18, Proposition 6.1]. Proposition 7.2.
Assume the base field k is algebraically closed, X = Spec( A ) is a connected smoothaffine k -scheme of dimension d . Then the group H d − ( X ; K M d ) is divisible prime to char( k ) :H d − ( X ; K M d ) /m = 0 for any m ∈ Z > with char( k ) ∤ m . In particular, if char( k ) = 0 or char( k ) > d , then H d − ( X ; K M d ) is ( d − -divisible : H d − ( X ; K M d ) / ( d − .Proof. Writing m = ℓ r · · · ℓ r s s , ℓ i prime, ℓ i = char( k ) and r i ∈ N , we see that it suffices to consider thecase m = ℓ r . Let ℓ be a prime number and ℓ = char( k ) , let r ∈ N . Consider the short exact sequences → ℓ r K M d → K M d ℓ r −→ ℓ r K M d → and → ℓ r K M d → K M d → K M d /ℓ r → . Using the Rost-Schmid complex we see that H d ( X ; ℓ r K M d ) is a quotient of a direct sum of groups of theform ℓ r K M0 ( κ x ) ∼ = ℓ r Z = 0 (over all closed points x ∈ X ( d ) ) as Z is torsion-free, thus H d ( X ; ℓ r K M d ) = 0 .We then have exact sequences H d − ( X ; K M d ) → H d − ( X ; ℓ r K M d ) → H d ( X ; ℓ r K M d ) = 0 and H d − ( X ; ℓ r K M d ) → H d − ( X ; K M d ) → H d − ( X ; K M d /ℓ r ) . Splicing together we get an exact sequence H d − ( X ; K M d ) ℓ r −→ H d − ( X ; K M d ) → H d − ( X ; K M d /ℓ r ) and hence → H d − ( X ; K M d ) /ℓ r → H d − ( X ; K M d /ℓ r ) . Therefore to prove H d − ( X ; K M d ) /ℓ r = 0 , it suffices to prove that H d − ( X ; K M d /ℓ r ) = 0 .For j, n ∈ N , let H j ( n ) = ( R j i ∗ ) µ ⊗ nℓ r be the Zariski sheaf associated to the presheaf U H j ´et ( U ; µ ⊗ nℓ r ) (where i is the inclusion of the Zariski site into the étale site). We have the biregular Bloch-Ogus spectral sequence ([12]; it’s one incarnation of the
Leray spectral sequence ) E ij = H i Zar ( X ; H j ( n )) = ⇒ H i + j ´et ( X ; µ ⊗ nℓ r ) . E ij = H i Zar ( X ; H j ( n )) can be computed as the i -th cohomology of the Gersten complex H j ´et ( κ η ; µ ⊗ nℓ r ) → · · · → M x ∈ X ( i ) H j − i ´et ( κ x ; µ ⊗ n − iℓ r ) → · · · , where η ∈ X (0) is the generic point of X .Since k = ¯ k , the cohomological dimension c . d . ( κ x ) d − dim( O X,x ) ([34, §4.2, Proposition 11]), wesee H j − i ´et ( κ x ; µ ⊗ n − iℓ r ) = 0 for x ∈ X ( i ) . Thus E ij = H i Zar ( X ; H j ( n )) = 0 if i > d = dim X or j > d or i > j . Hence in the filtration of the converging term H d − ( X ; µ ⊗ nℓ r ) , the only (possibly) non-trivialterm is E d − ,d = H d − ( X ; H d ( n )) . While by [30, Chapter VI, Theorem7.2], H d − ( X ; µ ⊗ nℓ r ) = 0 since X is affine over k = ¯ k . Thus E d − ,d = H d − ( X ; H d ( n )) = 0 as well.There is a commutative diagram ([11, Theorem 2.3]) L x ∈ X ( d − K M2 ( κ x ) /ℓ r L x ∈ X ( d − K M1 ( κ x ) /ℓ r L x ∈ X ( d ) K M0 ( κ x ) /ℓ r L x ∈ X ( d − H ( κ x ; µ ⊗ ℓ r ) L x ∈ X ( d − H ( κ x ; µ ℓ r ) L x ∈ X ( d ) H ( κ x ; Z /ℓ r ) , where the vertical maps are isomorphisms by [29] (or Voevodsky’s confirmation of the motivic Bloch-Kato conjecture). The homology of the middle terms in the two rows compute H d − ( X ; K M d /ℓ r ) andrespectively H d − ( X ; H d ( d ))(= 0) . Thus H d − ( X ; K M d /ℓ r ) = H d − ( X ; H d ( d )) = 0 . We are done. (cid:3)
Let’s now treat the case when the dimension of X is odd. Still assume k = ¯ k . By the exact sequence → I d +1 → K MW d τ −→ K M d → we get an exact sequence H d − ( X ; K MW d ) τ −→ H d − ( X ; K M d ) → H d ( X ; I d +1 ) . Rost-Schmid complex for I d +1 says that H d ( X ; I d +1 ) is a subquotient of M x ∈ X ( d ) I ( κ x ) = M x ∈ X ( d ) I (¯ k ) = 0 ,thus H d ( X ; I d +1 ) = 0 and τ is surjective as well. In fact, more is true: by Voevodsky’s confirmation ofthe Milnor conjecture, we have an isomorphism of sheaves of abelian groups ¯ I d + j ∼ = H d + j ( d + j ) ( j > ,where H d + j ( d + j ) is the Zariski sheaf associated to the presheaf U H d + j ´et ( U ; µ ⊗ d + j ) ; by reason ofcohomological dimension, H d + j ( d + j ) | X = 0 ( j > (restricting to the Zariski site of X ). Thus wehave ¯ I j | X = 0 , j > d, I d +1 | X = I d +2 | X = · · · .The Arason-Pfister Hauptsatz gives T j > I d + j = 0 , thus I d +1 | X = 0 and so τ : K MW j | X → K M j | X is infact an isomorphism for every j > d . This suffices to conclude that the induced maps on cohomologies τ : H i ( X ; K MW j ) → H i ( X ; K M j ) for j > d = dim X are isomorphisms, since these sheaves are strictly A -invariant, Nisnevich and Zariski cohomologies of X coincide (and are computed by Rost-Schmidcomplexes).The exact sequence → K M j → K M j → ¯ I j → gives isomorphisms H i ( X ; 2 K M j ) ∼ = −→ H i ( X ; K M j ) , j > d. We summarize the results as follows:(7.2) ( H i ( X ; I j ) = 0 , H i ( X ; 2 K M j ) ∼ = H i ( X ; K M j ) , j > d ; τ : H i ( X ; K MW j ) ∼ = −→ H i ( X ; K M j ) , j > d. Proposition 7.3.
Let k be an algebraically closed field with char( k ) = 2 , A a smooth affine k -algebraof Krull dimension d > , and X = Spec( A ) , let ξ be a stable oriented vector bundle over X , whoseclassifying map is still denoted ξ : X → BSL . If ξ is represented by a rank d − vector bundle, thenthere is a bijection p − ∗ ( ξ ) ∼ = coker (cid:0) SK ( X ) = [ X, ΩBSL] A ∆( c d ,ξ ) −−−−→ [ X, K( K M d , d − A = H d − ( X ; K M d ) (cid:1) . NUMERATING NON-STABLE VECTOR BUNDLES 29
The homomorphism ∆( c d , ξ ) is given as follows: for β ∈ SK ( X ) = [ X, SL] A , ∆( c d , ξ ) β = (Ω c d )( β ) + d − X r =1 ((Ω c r )( β )) · c d − r ( ξ ) . So ∆( c d , ξ )([ β d +1 ( a, b )]) = 0 for all A -point ( a, b ) of Q d +1 .Proof. We already treated in Proposition 7.1 the case when d is even. For d odd, since π A d − F d − ∼ = K MW d , we have a similar two-stage Moore-Postnikov factorization as in eq. (7.1), with K M d replaced by K MW d there.There are the following commutative diagrams: BSL d K( K MW d , d )BSL K( K M d , d ) , e d s d τc d θ and hence SK ( X ) H d − ( X ; K MW d )H d − ( X ; K M d ) . ∆( θ,ξ )∆( c d ,ξ ) τ Since τ : K MW d | X → K M d | X is an isomorphism, so is the right vertical map. Thus ∆( θ, ξ ) and ∆( c d , ξ ) are essentially the same. Therefore the result for the d odd case holds as with the case when d is evenin Proposition 7.1.The last statement follows from Proposition 6.8. (cid:3) Remark 7.4.
Since c d ( ξ ) = τ θ ( ξ ) , we see that ξ lifts to a class in [ X, E ] A iff c d ( ξ ) = 0 . While θ ′∗ maps [ X, E ] A to , hence no further obstruction. We thus get Murthy’s splitting result [33] for oriented rank d vector bundles:Let X be a smooth affine variety of dimension d over an algebraically closed field k , then an oriented rank d vector bundle ξ over X splits off a trivial line bundle iff c d ( ξ ) = 0 . Theorem 7.5.
Assume k = ¯ k and char( k ) = 2 . Let A be a smooth k -algebra of Krull dimension d > .Let X = Spec A , ξ ∈ [ X, BSL] A which is represented by a rank d − vector bundle. (1) We have ( d − · H d − ( X ; K M d ) ⊂ im∆( c d , ξ ) . (2) There is a surjective homomorphism H d − ( X ; K M d ) / ( d − ։ coker∆( c d , ξ ) . (3) If char( k ) = 0 or char( k ) > d , then the lifting set p − ∗ ( ξ ) ⊂ [ X, E ] A is a singleton. So the map θ ∗ : π (RMap( X, BSL) , ξ ) → π (RMap( X, K( K M d , d )) ,
0) = H d − ( X ; K M d ) is surjective.Proof. This is along the same line as the proof of Theorem 6.12. We only briefly write down somepoints. Using the Postnikov tower of A d \ , it’s easy to see that we have a surjective map [ X, A d \ A ։ H d − ( X ; K MW d ) . Thus the composite Um d ( A ) / E d ( A ) = [ X, Q d − ] A = [ X, A d \ A ։ H d − ( X ; K MW d ) τ −→ H d − ( X ; K M d ) is surjective. So every element in H d − ( X ; K M d ) is the image of some [ a, b ] with ( a, b ) ∈ Q d − ( A ) , a =( a , · · · , a d ) ∈ Um d ( A ) which we write as τ ([ a, b ]) . SL d X Q d − SL d − ( a,b ) α d β d We will show ( d − · τ ([ a, b ]) = ± ∆( c d , ξ )([ β d ( a, b )]) . As in the proof of Theorem 6.12 (from eq. (6.5) to the end, where X essentially plays no role), wehave τ ((pr ) ∗ [ β ]) = τ ([(1 , , · · · , · β ]) = ± (Ω c d )([ β ]) , for β ∈ SL d ( A ) ⊂ SL( A ) and (Ω c r )( α d ) = ( , r = d ; ± ( d − ∈ K M0 ( k ) = Z , r = d. ∆( c d , ξ )([ β d ( a, b )]) = (Ω c d )([ β d ( a, b )]) . Thus ( d − · τ ([ a, b ]) = τ (( d − · [ a, b ]) = τ (( ψ d ) ∗ [ a, b ]) = τ ((pr ) ∗ [ β d ( a, b )])= ± (Ω c d )([ β d ( a, b )]) = ± ∆( c d , ξ )([ β d ( a, b )]) . This finishes proving (1). Statements (2) and (3) then follow easily from (1), the divisibility result inProposition 7.2 and the last part of Theorem 2.1. (cid:3)
Finally we study the map q ∗ : [ X, E ′ ] A → [ X, E ] A . By the discussion in Section 4, we get exactsequences(7.3) ( π A d ( A d \ → π A d F d − → K M d +1 = π A d F d → , d odd ; π A d ( A d \ → π A d F d − → K M d +1 → , d even , where in the d even case, the term K M d +1 sits in an exact sequence → K M d +1 → K MW d +1 = π A d F d → I d +1 → , and I d +1 | X = 0 if k = ¯ k , telling that the canonical homomorphism K M d +1 → K MW d +1 = π A d F d induces an isomorphism H d ( X ; 2 K M d +1 ) ∼ = H d ( X ; π A d F d ) . Using the fact that the Nisnevichcohomological dimension of X is bounded above by dim( X ) = d , we get exact sequences for highestdegree cohomology: ( H d ( X ; π A d ( A d \ → H d ( X ; π A d F d − ) → H d ( X ; K M d +1 ) → , d odd ;H d ( X ; π A d ( A d \ → H d ( X ; π A d F d − ) → H d ( X ; 2 K M d +1 ) → , d even . If k = ¯ k , then these two exact sequences become one:(7.4) H d ( X ; π A d ( A d \ → H d ( X ; π A d F d − ) → H d ( X ; K M d +1 ) ∼ = H d ( X ; π A d F d ) → , d > . We now invoke the following conjecture of Asok-Fasel describing π A d ( A d \ . Conjecture 7.6 (Asok-Fasel) . Let k be a perfect field with char( k ) = 2 , then there is an exact sequence (7.5) K M d +2 / → π A d ( A d \ → GW dd +1 → in A b A k . This gives an exact sequence (7.6) H d ( X ; K M d +2 / → H d ( X ; π A d ( A d \ → H d ( X ; GW dd +1 ) → if X is a k -scheme of dimension d . If k = ¯ k , then any generator of the group K M2 ( k ) can be written in the form { a , b } = 24 { a, b } bythe group law of Milnor K-theory. Thus K M2 ( k ) /
24 = 0 . Since H d ( X ; K M d +2 / is a subquotient of M x ∈ X ( d ) K M2 ( κ x ) / ∼ = M x ∈ X ( d ) K M2 ( k ) /
24 = 0 , we see H d ( X ; K M d +2 /
24) = 0 and so if Conjecture 7.6 holds,then(7.7) H d ( X ; π A d ( A d \ ∼ = −→ H d ( X ; GW dd +1 ) . We have the sheafified Karoubi periodicity sequences K Q d +1 H −→ GW dd +1 η −→ GW d − d f −→ K Q d in A b A k , which is exact. Let A := im( H ) , B := im( η ) , then we have exact sequences K Q d +1 H −→ A → , → A → GW dd +1 η −→ B → , yielding exact sequences on cohomologies: H d ( X ; K M d +1 ) ∼ = H d ( X ; K Q d +1 ) → H d ( X ; A ) → , H d ( X ; A ) → H d ( X ; GW dd +1 ) → H d ( X ; B ) → and hence H d ( X ; K M d +1 ) → H d ( X ; GW dd +1 ) → H d ( X ; B ) → . NUMERATING NON-STABLE VECTOR BUNDLES 31Contracting the sheafified Karoubi periodicity sequence d -times we get an exact sequence K M1 H −→ GW η −→ B − d → . While the composite K M1 H −→ GW f −→ K M1 is multiplication by , we see thecomposite K M1 ֒ → K M1 H −→ GW is , so we have an exact sequence K M1 / H −→ GW η −→ B − d → ,which splits into two: K M1 / H −→ A − d → , → A − d → GW η −→ B − d → . Using again Rost-Schmid complexes we find an exact sequence H d ( X ; K M d +1 / → H d ( X ; A ) → , with H d ( X ; A ) → H d ( X ; GW dd +1 ) → H d ( X ; B ) → we obtain an exact sequence(7.8) d ( X ; K M d +1 / → H d ( X ; GW dd +1 ) → H d ( X ; B ) → . By [5, Lemma 3.6.3], we have H d ( X ; B ) ∼ = Ch d ( X ) , where Ch d ( X ) = H d ( X ; K M d / ∼ = CH d ( X ) / is the group of mod- codimension- d cycle classes on X . Since k = ¯ k , we have Ch d ( X ) = 0 . Thus H d ( X ; GW dd +1 ) = 0 and so H d ( X ; π A d ( A d \ (assuming Conjecture 7.6). And by eq. (7.4),(7.9) H d ( X ; π A d F d − ) ∼ = −→ H d ( X ; π A d F d ) ∼ = H d ( X ; K M d +1 ) ∼ = [ X, A d +1 \ A ∼ = [ X, Q d +1 ] A , d > . Theorem 7.7.
Assume k = ¯ k and char( k ) = 2 . Let A be a smooth k -algebra of Krull dimension d > ,let X = Spec A . Assume Conjecture 7.6 holds.
Then the map q ∗ : [ X, E ′ ] A → [ X, E ] A is a bijection.Proof. Since there is an A -homotopy fibre sequence K( π A d F d − , d ) → E ′ q −→ E θ ′ −→ K( π A d F d − , d + 1) , this shows q ∗ is surjective, and gives a homotopy fibre sequence in s S et ∗ : RMap(
X, E ′ ) q ∗ −→ RMap(
X, E ) θ ′∗ −→ RMap( X, K( π A d F d − , d + 1)) . By the last part of Theorem 2.1, to show injectivity of q ∗ , we need to show: for any ξ ∈ [ X, BSL] A which is represented by a rank d − vector bundle (or equivalently, c d ( ξ ) = 0 ), let ξ E ∈ [ X, E ] A bethe unique lifting of ξ as in eq. (7.1) (so θ ′∗ ( ξ E ) = 0 ∈ H d +1 ( X ; π A d F d − ) ), then the map θ ′∗ : π (RMap( X, E ) , ξ E ) → π (RMap( X, K( π A d F d − , d + 1)) ,
0) = H d ( X ; π A d F d − ) is surjective.Consider now the comparison diagram of Moore-Postnikov towers F d − / / (cid:15) (cid:15) BSL d − / / (cid:15) (cid:15) E ′ q / / (cid:15) (cid:15) E p / / p (cid:15) (cid:15) BSL F d / / BSL d / / ˜ E / / BSL BSL , where ˜ E is the first stage in the Moore-Postnikov tower of the map BSL d → BSL . By functoriality,the first stage k -invariants (that of the column of E ) gives a commutative square(7.10) E K( π A d F d − , d + 1)BSL K( π A d F d , d + 1) , θ ′ p ˜ θ where ˜ θ is the k -invariant “ θ ” in the rank d case, we write it as ˜ θ to distinguish it from the k -invariant“ θ ” in the rank d − case here) and the right vertical map is induced by the map F d − → F d . Applying RMap( X, − ) we obtain a commutative diagram(7.11) RMap(
X, E ) RMap( X, K( π A d F d − , d + 1))RMap( X, BSL) RMap( X, K( π A d F d , d + 1)) , θ ′ p ˜ θ c ′′ ∈ π (RMap( X, E ) , ξ E ) π (RMap( X, K( π A d F d − , d + 1)) , ∋ cT ξ [ β d +1 ( a, b )] ∈ π (RMap( X, BSL) , ξ ) π (RMap( X, K( π A d F d , d + 1)) , ∋ c [ β d +1 ( a, b )] ∈ π (RMap( X, BSL) ,
0) H d ( X ; K M d +1 ) ∋ c [ X, Q d +1 ] A ∼ = H d ( X ; K MW d +1 ) ∋ c ′ = [ a, b ] , θ ′∗ p ∗ ∼ =˜ θ ∗ T ξ ∆( c d +1 ,ξ ) β d +1 ± d ! where the arrow ˜ θ ∗ is surjective by Theorems 6.12 and 6.16, and the right vertical maps are iso-morphisms by eq. (7.9); the middle square commutes by Theorem 6.6, and the lower triangle isgiven by eq. (6.4), the arrow labeled by ± d ! is surjective (see Remark 6.13) . So for any c ∈ π (RMap( X, K( π A d F d − , d + 1)) ,
0) = H d ( X ; K M d +1 ) , we can find c ′ = [ a, b ] ∈ [ X, Q d +1 ] A with c = ± d ! · [ a, b ] = ∆( c d +1 , ξ )([ β d +1 ( a, b )]) .Theorem 6.6 (again) and the last statement of Proposition 7.3 tell that θ ∗ ( T ξ [ β d +1 ( a, b )]) = ( c d ) ∗ ( T ξ [ β d +1 ( a, b )]) = 0 . On the other hand, the fiber sequence
RMap(
X, E ) ξ E p −→ RMap( X, BSL) ξ θ −→ RMap( X, K( K M(W) d , d )) in s S et ∗ , where the subscripts refer the corresponding components, induces another Ω ξ E RMap(
X, E ) p −→ Ω ξ RMap( X, BSL) θ −→ RMap( X, K( K M(W) d , d − . Thus T ξ [ β d +1 ( a, b )] ∈ ker( θ ∗ ) = im( p ∗ ) , we see there exists c ′′ ∈ π (RMap( X, E ) , ξ E ) with p ∗ ( c ′′ ) = T ξ [ β d +1 ( a, b )] , which then satisfies θ ′∗ ( c ′′ ) = c , proving that the map θ ′∗ : π (RMap( X, E ) , ξ E ) → π (RMap( X, K( π A d F d − , d + 1)) ,
0) = H d ( X ; π A d F d − ) is surjective. Hence q ∗ is injective as well. (cid:3) We finally arrive at the following cancellation result for (oriented) rank d − vector bundles over asmooth affine variety of dimension d , admitting Asok-Fasel conjecture. (As before, we let ϕ = ϕ d − :BSL d − → BSL be the stabilizing map.)
Theorem 7.8.
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