aa r X i v : . [ m a t h . K T ] A p r E-MOTIVES AND MOTIVIC STABLE HOMOTOPY
NGUYEN LE DANG THI
Abstract.
We introduce in this work the notion of the category of pure E -Motives, where E is a motivic strict ring spectrum and construct twisted E -cohomology by using six functorsformalism of J. Ayoub. In particular, we construct the category of pure Chow-Witt motives CHW ( k ) Q over a field k and show that this category admits a fully faithful embedding intothe geometric stable A -derived category D A ,gm ( k ) Q . Contents
1. Introduction 12. A -homotopy category 52.1. Unstable A -homotopy category 52.2. Stable A -homotopy category 73. A -homological algebra 83.1. Effective A -derived category 83.2. P -stable A -derived category 104. E -Motives 124.1. E -Correspondences 124.2. Functoriality in motivic stable homotopy 164.3. Cohomology with supports 174.4. Relation to the category of twisted E -correspondences 185. Proof of theorem 1.2 545.1. Homotopy t -structure 545.2. Isomorphism between Hom -groups 576. Appendix 596.1. Model Categories 596.2. Localization 646.3. Symmetric motivic T -Spectra 65References 681. Introduction
One of the main motivations for this work is the embedding theorem of Voevodsky [Voe00],which asserts that there is a fully faithful embedding of the category of Grothendieck-Chow
Date : 19. 05. 2015.1991
Mathematics Subject Classification.
Key words and phrases. stable A -homotopy, E -motives, Milnor-Witt K -theory. ure motives Chow ( k ) into the category of geometric motives DM gm ( k ) , hence also into thecategory of motives DM − Nis ( k ) Chow ( k ) op → DM gm ( k ) , if k is a perfect field, which admits resolution of singularities (see e.g. [MVW06, Prop.20.1 and Rem. 20.2], the assumption on resolution of singularities can be removed by us-ing Poincaré duality). In this note, we construct a category CHW ( k ) Q , which we call thecategory of pure Chow-Witt motives over a field k and show that CHW ( k ) Q admits a fullyfaithful embedding into the geometric P -stable A -derived category D A ,gm ( k ) Q rationally.Our work can be viewed as an A -version for Voevodsky’s embedding theorem. The advan-tage here is that by using duality formalism for P -stable A -derived category D A ( k ) (see[Hu05, App. A] for stable A -homotopy categories) and the six operations formalism of J.Ayoub [Ay08], we do not have to assume the resolution of singularities. However, unlike inmotivic setting, one of the main problems here is that we don’t have cancellation theorem forthe effective A -derived category in general, see [AH11, Rem. 3.2.4], that is the reason whywe can prove the embedding result only for Q -coefficient. F. Morel conjectured in generalthat (see [Mor04]): Conjecture 1.1. [Mor04]
Let S be a regular Noetherian scheme of finite Krull dimen-sion. One has a direct decomposition in the rationally motivic stable homotopy category StHo A , P ( S ) : [ S i , G ∧ jm ] P ⊗ Q = H j − iB ( S, Q ( j )) ⊕ H − iNis ( S, W ⊗ Q ) , where H ∗ B ( − , Q ( ∗ )) denotes the Beilinson motivic cohomology, W is the unramified Wittsheaf and [ − , − ] P ⊗ Q denotes Hom
StHo A , P ( S ) ( − , − ) Q . Over a general base scheme S one can split the rational motivic sphere spectrum Q = Q + ∨ Q − . The identifcation of the plus part Q + = H B has been done in [CD10, Thm16.2.13] over any Noetherian scheme of finite Krull dimension S . The minus part Q − = H W ∗ Q , where H W ∗ Q denotes the Eilenberg-Maclane spectrum associated to the rationalWitt homotopy module W ∗ Q , is given in the work of A. Ananyeskiy, M. Levine and I. Panin([ALP15, §3, Thm. 5]) over fields S = Spec k . In general, the conjecture 1.1 over a regularNoetherian scheme S of finite Krull dimension is still widely open, as far as I know. Onthe other hand, our interest started originally from the study of the existence of -cycles ofdegree one on algebraic varieties. More precisely, Hélène Esnault asked (cf. [Lev10]): Givena smooth projective variety X over a field k , such that X has a zero cycle of degree one.Are there "motivic" explanations which give the (non)-existence of a k -rational point? In[AH11], A. Asok and C. Haesemeyer show that the existence of zero cycles of degree one overan infinite perfect field of char ( k ) = 2 is equivalent to the assertion that the structure map H st A ( X ) → H st A (Spec k ) is a split epimorphism, where H st A i ( X ) denotes the P -stable A -homology sheaves, while in an earlier work [AH11a] they also showed that the existenceof a k -rational point over an arbitrary field k is equivalent to the condition that the structuremap H A ( X ) → H A (Spec k ) is split surjective. So roughly speaking, the obstruction to thelifting of a zero cycle of degree one to a rational point arises by passing from S -spectra to P -spectra. As remarked by M. Levine, it is not to expect that the category of Chow-Wittmotives CHW ( k ) contains any information about the existence of rational points. Now westate our main theorem in this work: heorem 1.2. Let k be a field. There exists a category of pure Chow-Witt motives CHW ( k ) Q ,which admits a fully faithful embedding CHW ( k ) Q → D A ,gm ( k ) Q . In fact, one of the main steps in the work of [AH11] is to exhibit a natural isomorphism H st A ( X )( L ) → f CH ( X L ) for any separable, finitely generated field extension L/k . So onemay relate this step to our work as evaluating at a generic point, but much weaker thanexpected, since we can only prove the result for Q -coefficient. Now our paper is organizedas follows: we will review shortly A -homotopy theory in section §2. Section §3 is devotedfor A -derived categories, in fact we will define the geometric P -stable A -derived category D A ,gm ( k ) over a field k in 3.9 at the end of §3. In fact, this is the subcategory of compactobjects D A ,c ( k ) of D A ( k ) (see [CD10, Ex. 5.3.43]). In these §2 and §3 we simply stealeverything which is needed from the presentation of [AH11]. For a complete treatment westrongly recommend the reader to [Ay08], [CD10] and [Mor12]. In section §4 we introducethe notion of pure E -motives, where E is a motivic strict ring spectrum and relate severalcategories of E -correspondences with each other via the twisted E -cohomology. The twisted E -cohomology appears since we will not assume the motivic ring spectrum E to be orientable.In topology, if E is a multiplicative cohomology theory and V is an E -orientable vector bundleof rank r , then one has a Thom-Dold isomorphism E ∗ ( X ) ∼ = −→ e E ∗ + r ( T h ( V )) , where T h ( V ) is the Thom space of V and the right hand side is the reduced cohomology. If E is a ring spectrum, then one can intepret this isomorphism as following: Via the Thomdiagonal T h ( V ) → T h ( V ) ∧ X + , which is induced by the diagonal X + → X + ∧ X + one can express T h ( V ) as a comodule over X + and the comodule map is the natural map X + → T h ( V ) . The geometric Thom isomorphism is the homotopy equivalence E ∧ T h ( V ) → E ∧ T h ( V ) ∧ X + → E ∧ Σ n E ∧ X + µ E ∧ id −→ E ∧ Σ n X + . The composition is an E -module map, hence one may take function spectrum F E ( E ∧ Σ n X + , E ) ≃ F (Σ n X + , E ) ≃ −→ F E ( E ∧ T h ( V ) , E ) ≃ F ( T h ( V ) , E ) , which induces the Thom isomorphism on E -cohomology. In algebraic geometry one has asimilar result. For an oriented motivic ring spectrum E ∈ SH ( S ) , where S is a regular base,one has ([NSO09, Thm. 2.12]) E ∗ , ∗ ( X ) ∼ = −→ E ∗ +2 r, ∗ + r ( T h ( V )) , where V is a vector bundle of rank r on a smooth S -scheme X . The key point is that since E is oriented one can define the first Chern class and then prove the projective bundle theorem[NSO09, Thm. 2.11]. The situation becomes much more difficult, even in topology, if E is ot necessary oriented. One has to introduce twisted cohomology. Again in topology, byAtiyah duality one has a commutative diagram in the ( ∞ , -category S M od : T h ( − T X ) ≃ (cid:15) (cid:15) S P T : : ✈✈✈✈✈✈✈✈✈✈ / / X ∨ where P T : S → T h ( − T X ) is the Pontryagin-Thom collapse map. Let E be an E ∞ -ringspectrum. By taking − ∧ S E one obtains a map in the ( ∞ , -category E M od E → X ∨ ∧ S E . Taking function spectrum we have the (twisted) Umkehr map F E ( T h ( − T X ) ∧ S E , E ) ≃ F ( T h ( − T X ) , E ) → E . If E is non-oriented, there is no geometric Thom isomorphism. However, F E ( T h ( − T X ) ∧ S E , E ) will give the twisted cohomology. This is the motivation from topology for us, since inalgebraic geometry we also have the Atiyah-Spanier-Whitehead duality, but I do not knowany ∞ -categorical approach to twisted cohomology like the one in topology [ABGHR14]. SoI introduce in section §4 the twisted E -cohomology rather through the guide of the six func-tors formalism of J. Ayoub. The reader may recognize that the notion of E -correspondencesis similar to the construction of Jack Morava in topology. While it is very simple to definethe category of E -correspondences Corr E ( k ) , it is quite difficult to construct the cateogry ] Corr E ( k ) via twisted E -cohomology. This category exists only up to a number of natu-ral -isomorphisms. This phenomenon reflexes the fact that we rely on six functors for-malism, where Thom transformations are only -isomorphic to each other. The compo-sition in ] Corr E ( k ) is associative only up to a natural isomorphism induced by a natural -isomorphism. In §5 we give the proof of the main theorem. In the appendix we give aminimal list of well-known facts and definitions of model categories. We fix now some nota-tions throughout this work. For a pair of adjoint functors F : A → B and G : B → A , wewill adopt the notation in [CD10] F : A ⇄ B : G, where F is left adjoint to G and G is right adjoint to F . Sometime we will write ε ( F,G ) : F G → id , η ( F,G ) : id → GF for the counit and unit of the adjunction repsectively. For every morphism f : F Y → X in M or ( B ) , there is a unique morphism g : Y → GX in M or ( A ) such that the followingdiagram commutes: F Y F ( g ) (cid:15) (cid:15) f ' ' PPPPPPPPPPPPPPP
F G ( X ) ε ( F,G ) ( X ) / / X or every morphism g : Y → GX in M or ( A ) , there is a unique morphism f : F Y → X in M or ( B ) , such that the following diagram commutes: Y η ( F,G ) ( Y ) / / g ' ' PPPPPPPPPPPPPPP GF ( Y ) G ( f ) (cid:15) (cid:15) GX In a symmetric monoidal category ( C , ∧ , ) , an object A is called strongly dualizable if thereexists an object A ∨ and morphisms coev A : → A ∧ A ∨ , ev A : A ∨ ∧ A → , such that the following compositions A ∼ = ∧ A coev A ∧ id −→ A ∧ A ∨ ∧ A id ∧ ev A −→ A ∧ ∼ = A and A ∨ ∼ = A ∨ ∧ id ∧ coev A −→ A ∨ ∧ A ∧ A ∨ ev A ∧ id −→ ∧ A ∨ ∼ = A ∨ are the identities id A and id A ∨ . The natural isomorphism α : Hom C ( − , A ) ∼ = −→ Hom C ( A ∨ ∧ − , ) is given by α ( φ ) = ev A ∨ ◦ (id A ∨ ∧ φ ) , and its inverse α − is given by α − ( ϕ ) = (id A ∧ ϕ ) ◦ ( coev A ∨ ∧ id − ) . Given two smooth k -schemes X, Y ∈ Sm/k and two vector bundles E , E ′ over X resp. Y , wewrite E × E ′ /X × Y for the external sum over X × k Y . The P - stable homotopy categoryover a base scheme S will be denoted by StHo A , P ( S ) and we write StHo A ,S ( S ) for the S -stable homotopy category. Sometime when it is clear which category we are talking about,we just abbreviate our P -stable homotopy category by SH ( S ) .2. A -homotopy category Unstable A -homotopy category. Let
Sm/k denote the category of separated smoothschemes of finite type over a field k . We write Spc/k for the category ∆ op Sh Nis ( Sm/k ) consisting of simplicial Nisnevich sheaves of sets on Sm/k . An object in
Spc/k is simplycalled a k -space, which is usually denoted by calligraphic letter X . The Yoneda embedding Sm/k → Spc/k is given by sending a smooth scheme X ∈ Sm/k to the corresponding rep-resentable sheaf
Hom
Sm/k ( − , X ) then by taking the associated constant simplicial object,where all face and degeneracy maps are the identity. We will identify Sm/k with its es-sential image in
Spc/k . Denote by
Spc + /k the category of pointed k -space, whose objectsare ( X , x ) , where X is a k -space and x : Spec k → X is a distinguished point. One has anadjoint pair Spc/k ⇄ Spc + /k, which means that the functor Spc/k → Spc + /k sending X → X + = X ` Spec k is left-adjoint to the forgetful functor Spc + /k → Spc/k . The category
Spc/k can be equipped withthe injective local model structure ( C s , W s , F s ) , where cofibrations are monomorphisms, weak quivalences are stalkwise weak equivalences of simplicial sets and fibrations are morphismswith right lifting property wrt. morphisms in C s ∩ W s . Denote by Ho Niss ( k ) the resultingunpointed homotopy category as constructed by Joyal-Jardine (cf. [MV01, §2 Thm. 1.4]).We will write Ho Niss, + ( k ) for the pointed homotopy category. Definition 2.1. [MV01](1) A k -space Z ∈
Spc/k is called A -local if and only for any object X ∈
Spc/k , theprojection
X × A → X induces a bijection Hom Ho Niss ( k ) ( X , Z ) ≃ → Hom Ho Niss ( k ) ( X × A , Z ) . (2) Let
X → Y ∈
M or ( Spc/k ) be a morphism of k -spaces. It is an A -weak equivalenceif and only for any A -local object Z , the induced map Hom Ho Niss ( k ) ( Y , Z ) → Hom Ho Niss ( k ) ( X , Z ) is bijective. In [MV01, §2 Thm. 3.2], F. Morel and V. Voevodsky proved that
Spc/k can be endowedwith the A -local injective model structure ( C, W A , F A ) , where cofibrations are monomor-phisms, weak equivalences are A -weak equivalences. The associated homotopy category ob-tained from Spc/k by inverting A -weak equivalences is denoted by Ho A ( k ) def = Spc/k [ W − A ] .This category is called the unstable A -homotopy category of smooth k -schemes. Let Ho Niss, A − loc ( k ) ⊂ Ho Niss ( k ) be the full subcategory consisting of A -local objects. In fact,one has an adjoint pair (cf. [MV01]) L A : Ho Niss ( k ) ⇄ Ho Niss, A − loc ( k ) : i, where L A is the A -localization functor sending A -weak equivalences to isomorphisms. L A induces thus an equivalence of categories Ho A ( k ) → Ho Niss, A − loc ( k ) . This will imply that if X ∈
Spc/k is any object and Y is an A -local object, then one has a canonical bijection Hom Ho Niss ( k ) ( X , Y ) ≃ → Hom Ho A ( k ) ( X , Y ) . We will write Ho A , + ( k ) for the unstable pointed A -homotopy category of smooth k -schemes.Recall Definition 2.2.
Let X ∈ Sm/k and E be a vector bundle over X . The Thom space of E isthe pointed sheaf T h ( E/X ) =
E/E − s ( X ) , where s : X → E is the zero section of E . Let T ∈ Spc + /k be the quotient sheaf A / ( A − { } ) pointed by the image of A − { } .Then T ∼ = S t ∧ S s in Ho A , + ( k ) ([MV01, Lem. 2. 15]). For a pointed space X ∈
Spc + /k , wedenote by Σ T ( X , x ) = T ∧ ( X , x ) . Remark that P n / P n − ∼ = T n def = T ∧ n is an A -equivalence.In particular, we have ( P , ∗ ) ∼ = T ([MV01, Cor. 2.18]). Recall Proposition 2.3. [MV01, §3 Prop. 2. 17]
Let
X, Y ∈ Sm/k and
E, E ′ be vector bundleson X and Y respectively. One has (1) There is a canonical isomorphism of pointed sheaves
T h ( E × E ′ /X × Y ) = T h ( E/X ) ∧ T h ( E ′ /Y ) . There is a canonical isomorphism of pointed sheaves
T h ( O nX ) = Σ nT X + (3) The canonical morphism of pointed sheaves P ( E ⊕ O X ) / P ( E ) → T h ( E ) is an A -weak equivalence. The following theorem due to Voevodsky will play an essential role for our purpose. How-ever, as pointed out by M. Levine, the identities in K ( − ) are not enough for us to constructmaps between twisted E -cohomology. Following a suggestion by M. Levine, we will refinethis result of Voevodsky later (see 4.25). Theorem 2.4. [Voe03, Thm. 2.11]
Let X ∈ SmP roj/k a smooth projective variety of puredimension d X over a field k . There exists an integer n X and a vector bundle V X over X ofrank n X , such that V X ⊕ T X = O n X + d X X ∈ K ( X ) , where T X denotes the tangent bundle of X . Moreover, there exists a morphism T ∧ n X + d X → T h ( V X ) in Ho A , + ( k ) , such that the induced map H d X M ( X, Z ( d X )) → Z coincides with thedegree map deg : CH ( X ) → Z , where T = S s ∧ G m . Remark 2.5.
One can always add a trivial bundle to V X in Voevodsky’s theorem 2.4 toincrease n X appropriately.2.2. Stable A -homotopy category. Let
Spect Σ ( k ) be the category of symmetric spectrain k -spaces, which can be viewed as category of Nisnevich sheaves of symmetric spectra. Byapplying the construction in [Ay08, Def. 4.4.40, Cor. 4.4.42, Prop. 4.4.62], Spect Σ ( k ) has the structure of a monoidal model category. Let StHo S ( k ) be the resulting homotopycategory. The stable A -homotopy category of S -spectra StHo A ,S ( k ) is obtained from StHo S ( k ) by Bousfield localization. Equivalently, the category Spect Σ ( k ) can be equippedwith an A -local model structure (cf. [Ay08, Def. 4.5.12]). The homotopy category ofthis A -local model structure is StHo A ,S ( k ) , which is also known to be equivalent to thecategory StHo S A − loc ( Sm/k ) constructed by F. Morel in [Mor05, Def. 4.1.1]. The A -localsymmetric sphere spectrum is defined by taking the functor n L A ( S ∧ ns ) with an action of symmetric groups, where L A denotes the A -localization functor. For apointed space ( X , x ) , its A -local symmetric suspension spectrum is defined as the symmetricsequence n L A ( S ∧ ns ∧ X ) together with symmetric groups actions. Let E be an A -local symmetric spectrum in Spc/k .One defines ([AH11, Def. 2.1.11]) the i -th S -stable A -homotopy sheaf π st A ,S i ( E ) of E asthe Nisnevich sheaf on Sm/k associated to the presheaf U Hom
StHo A ,S ( k ) ( S ∧ is ∧ Σ ∞ s U + , E ) . Now we consider the symmetric T -spectra or P -spectra ([Jar00]). P is pointed with ∞ and P ∧ n has a natural action of Σ n by permutation of the factors, so the association n P ∧ n is a symmetric sequence. A symmetric P -spectrum is a symmetric sequence with a module tructure over the sphere spectrum S . Denote by Spect Σ P ( k ) the full subcategory of thecategory of symmetric sequence in k -spaces Fun ( Sym , Spc + /k ) consisting of symmetric P -spectra, which also has a model structure [Ay08, Def. 4.5.21]. Here we denote by Sym thegroupoid, whose objects are n and morphisms are given by bijections. Let StHo A , P ( k ) be the resulting homotopy category, which is called P -stable A -homotopy category. Fora pointed space ( X , x ) , we will write Σ ∞ P ( X , x ) for the suspension symmetric P -spectrum,i.e., it is given by the functor n P ∧ n ∧ X equipped with an action of symmetric group bypermuting the first n -factors. Let S i be the suspension symmetric P -spectrum of S is . If E isa symmetric P -spectrum, then the i -th P -stable A -homotopy sheaf π st A , P i ( E ) is definedas the Nisnevich sheaf on Sm/k associated to the presheaf (cf. [AH11, Def. 2.1.14]) U Hom
StHo A , P ( k ) ( S i ∧ Σ ∞ P U + , E ) . Theorem 2.6. [Mor05, Thm. 6.1.8 and Cor. 6.2.9]
Let E be an A -local symmetric S -spectrum. The homotopy sheaves π st A ,S i ( E ) are strictly A -invariant. One has a canonical isomorphism [AH11, Prop. 2.1.16] colim n Hom
StHo A ,S ( k ) (Σ ∞ s G ∧ nm ∧ Σ ∞ s ( U + ) , Σ ∞ s G ∧ nm ∧ Σ ∞ s ( X , x )) ∼ = → Hom
StHo A , P ( k ) (Σ ∞ P ( U + ) , Σ ∞ P ( X , x )) . So one may view that
StHo A , P ( k ) is obtained from StHo A ,S ( k ) by formally invertingthe A -localized suspension spectrum of G m . So from 2.6, we see that for a pointed k -space ( X , x ) , the homotopy sheaves π st A , P i ( X ) are also strictly A -invariant. By the computationof F. Morel ([Mor04], [Mor12]), one can identify the Milnor-Witt K -theory sheaves withstable homotopy sheaves of spheres K MWn ∼ = π st A , P (Σ ∞ P ( G ∧ nm )) . This identification allows us to conclude that K MWn are strictly A -invariant sheaves.3. A -homological algebra Effective A -derived category. Let Ch − ( A b k ) be the category of chain complexesover the category A b k of abelian Nisnevich sheaves. Denote by Ch ≥ ( A b k ) the category ofchain complexes of abelian Nisnevich sheaves on Sm/k , whose homoglocial degree ≥ . Thesheaf-theoretical Dold-Kan correspondence N : ∆ op A b k ⇄ Ch ≥ ( A b k ) : K, where ∆ op A b k is the cateogry of simplicial abelian Nisnevich sheaves, gives us via the inclu-sion functor Ch ≥ ( A b k ) ֒ → Ch − ( A b k ) , a functor ∆ op ( A b k ) → Ch − ( A b k ) . By applying this functor on the Eilenberg-Maclane spectrum H Z , we obtain a ring spectrum g H Z in Fun ( Sym , Ch − ( A b k )) . Let Spect Σ ( Ch − ( A b k )) be the full subcategory of the category Fun ( Sym , Ch − ( A b k )) consisting of modules over g H Z . On the other hand, by composing withthe free abelian group functor Z ( − ) : Spc/k → ∆ op ( A b k ) , ne obtains a functor Fun ( Sym , Spc + /k ) → Fun ( Sym , Ch − ( A b k )) , which sends the sphere symmetric sequence to g H Z . This induces then a functor betweencategories of symmetric spectra Spect Σ ( Spc/k ) → Spect Σ ( Ch − ( A b k )) . In fact, by [Hov01, Thm. 9.3], this induces a Quillen functor, which one refers as Hurewiczfunctor H ab : StHo S ( k ) → D − ( A b k ) . Now the effective A -derived category D eff A ( k ) is constructed by applying A -localization onthe category Spect Σ ( Ch − ( A b k )) . By the work of Cisinski and Déglise (cf. [CD10, §5]), thiscategory is equivalent to the A -derived category constructed by F. Morel in [Mor12]. Let ( X , x ) ∈ Spc + /k be a pointed space, and Σ ∞ s ( X , x ) its suspension symmetric spectrum. Weapply the Hurewicz functor on Σ ∞ s ( X , x ) and then L ab A ( − ) , so we may define a functor e C A ∗ : StHo S ( k ) → D eff A ( k ) , Σ ∞ s ( X , x ) L ab A ( H ab (Σ ∞ s ( X , x ))) . Here we write L ab A for the A -localization functor on chain complexes to distinguish fromthe A -localization L A on spaces. If X ∈
Spc/k is not pointed, then we write C A ∗ ( X ) def = e C A ∗ ( X + ) . Define Z [ n ] = H ab (Σ ∞ s S ns ) . Definition 3.1.
Let
X ∈
Spc/k be a k -space. Its i -th A -homology sheaf is the Nisnevichsheaf H A i ( X ) associated to the presheaf U Hom D eff A ( k ) ( C A ∗ ( U )[ i ] , C A ∗ ( X )) def = Hom D eff A ( k ) ( C A ∗ ( U ) ⊗ Z [ i ] , C A ∗ ( X )) . Consider ( P , ∞ ) pointed by ∞ . According to [MV01, Cor. 2.18], we have P = S s ∧ G m ,so we have an identification e C A ∗ ( P ) = e C A ∗ ( S s ∧ G m ) . We define the A -Tate complex (calledenhanced Tate (motivic) complex by A. Asok and C. Haesemeyer [AH11, Def. 2.1.25 andDef. 3.2.1 and Lem. 3.2.2]) as Z A ( n ) def = e C A ∗ ( P ∧ n )[ − n ] = Z A (1) ⊗ n . Definition 3.2.
Let
X ∈
Spc/k be a k -space. The bigraded unstable A -cohomology group H p,q A ( X , Z ) is defined as H p,q A ( X , Z ) = Hom D eff A ( k ) ( C A ∗ ( X ) , Z A ( q )[ p ]) . The relationship between unstable A -cohomology and Nisnevich hypercohomology withcoefficient Z A ( n ) is given by the following Proposition 3.3. [AH11, Prop. 3.2.5]
Let k be a field and X ∈
Spc/k be a k -space. Onehas (1) For any p, q , there is a canonical isomorphism H pNis ( X , Z A ( q )) ≃ → H p,q A ( X , Z ) . (2) The cohomology sheaves H p ( Z A ( q )) = 0 , if p > q . (3) There is a canonical isomorphism H p ( Z A ( p )) ∼ = K MWp , for all p > . emark 3.4. By construction the complex Z A ( n ) is A -local, hence by definition (cf.[Mor12, Def. 5.17]) one has immediately that the sheaves H p ( Z A ( q )) are strictly A -invariant.3.2. P -stable A -derived category. Having defined an A -Tate complex, the way thatwe stabilize the category D eff A ( k ) is to invert formally the A -Tate complex to obtain the P -stable A -derived category D A ( k ) . This can be done by following the construction de-tailed in [CD10, §5]. As before, we take D A ( k ) as the resulting homotopy category of themodel category Spect Σ P ( Ch − ( A b k )) consisting of modules over the A -localization of thenormalized chain complex of the free abelian group on the sphere symmetric P -spectrum.For a pointed space ( X , x ) ∈ Spc + /k , the stable A -complex e C st A ∗ ( X ) of ( X , x ) is defined as L ab A ( N Z (Σ ∞ P ( X , x ))) and if X ∈
Spc/k is an unpointed k -space, then we write C st A ∗ ( X ) for e C st A ∗ ( X + ) . The category D A ( k ) has an unit object, denoted by k , which is the complex e C st A ∗ ( S ) . Define k [ n ] = k ⊗ e C st A ∗ ( S ns ) and e C st A ∗ ( X )[ n ] = e C st A ∗ ( X ) ⊗ k [ n ] for a k -space ( X , x ) ∈ Spc + /k . Definition 3.5.
Let
X ∈
Spc/k be a k -space. The i -th P -stable A -homology sheaf H st A i ( X ) is the Nisnevich sheaf associated to the presheaf U Hom D A ( k ) ( C st A ∗ ( U )[ i ] , C st A ∗ ( X )) . Just like in case of stable A -homotopy categories, one has the following result Proposition 3.6. [AH11, Prop. 2.1.29]
Let U ∈ Sm/k and ( X , x ) ∈ Spc + /k . One has acanonical isomorphism (3.1) colim n Hom D eff A ( k ) ( C A ∗ ( U ) ⊗ Z A ( n )[ i ] , e C A ∗ ( X ) ⊗ Z A ( n )[ i ]) ∼ = → Hom D A ( k ) ( C st A ∗ ( U ) , e C st A ∗ ( X )) . The Hurewicz formalism induces the following functors, which one still calls Hurewiczfunctors (or abelianization functors)
StHo A ,S ( k ) → D eff A ( k ) , StHo A , P ( k ) → D A ( k ) , which give rise to morphisms of sheaves π st A ,S i (Σ ∞ s ( X + )) → H A i ( X ) ,π st A , P i (Σ ∞ P ( X + )) → H st A i ( X ) . Definition 3.7.
Let
X ∈
Spc/k be a k -space. The bigraded P -stable A -cohomology group H p,qst A ( X , Z ) is defined as H p,qst A ( X , Z ) = Hom D A ( k ) ( C st A ∗ ( X ) , Z A ( q )[ p ]) . The advantage of P -stable A -derived category D A ( k ) is that one has duality formalism.In the context of stable A -homotopy theory, it was done in [Hu05, App. A] and also[Rio05]. Firstly we recall that Deligne introduced in [Del87, §4] virtual categories. If f : X → Spec k is a smooth k -scheme, the category V ( X ) of virtual bundles on X is identifiedto the fundamental groupoid of K ( X ) where K is some A -fibrant genuine model of algebraic -theory. An actual vector bundle ξ defines an object ξ in V ( X ) whose isomorphism classcorresponds to [ ξ ] ∈ K ( X ) . A short exact sequence of vector bundles → ξ ′ → ξ → ξ ′′ → gives not just an equality [ ξ ] = [ ξ ′ ] + [ ξ ′′ ] in K ( X ) but also a specific isomorphism ξ ∼ = ξ ′ ⊕ ξ ′′ of objects in V ( X ) . By using universal property of V ( X ) as a Picard category, one can definean isomorphism (see [Rio10, §4]) T h ( ξ/X ) ∼ = T h ( ξ ′ /X ) ∧ T h ( ξ ′′ /X ) . We haven’t yet introduced in this section the operations formalism of J. Ayoub, howeverwe should mention that the construction of Thom spectrum extends to a functor (cf. [Rio10,Prop. 4.1.1, Def. 4.2.1] and [Ay08, Thm. 1.5.18]) T h X : V ( X ) → StHo A , P ( X ) f → StHo A , P ( k ) . We discuss a little bit more about the Thom spectrum of virtual bundles. If ξ is a virtualvector bundle on an affine variety U , then there exist an actual vector bundle ξ ′ on U andan integer n ≥ , such that ξ ⊕ O nX = ξ ′ . So one may define Σ ∞ P T h ( ξ/U ) = Σ ∞ P T h ( ξ ′ /U ) ∧ S − n, − n . If X is a projective variety, one can define an affine variety, which is A -weak equivalent to X (see [Hu05, p. 10]): Consider first of all the projective space P N . One defines U = P N × P N \ P roj k [ x , · · · , x N , y , · · · , y N ] / ( N X i =0 x i y i = 0) , which is an A N -bundle pr : U → P N . If X is a projective variety, one has i : X ֒ → P N andthe affine variety π : i ∗ U → X is an A -weak equivalence, where π is the pullback of pr along the closed immersion i : i ∗ U π (cid:15) (cid:15) / / U pr (cid:15) (cid:15) X (cid:31) (cid:127) i / / P N If − T X is the virtual normal bundle on X of the diagonal embedding ∆ X : X ֒ → X × k X ,which is the virtual tangent bundle, then its Thom spectrum is defined to be the Thomspectrum T h ( µ/i ∗ U ) , where µ is the complement of the pullback of the tangent bundle of X along π . We state the following result in D A ( k ) , although the proof in case of StHo A , P ( k ) is given in [Hu05, Thm. A1] or see [Rio05, Thm. 2.2] Proposition 3.8. [AH11, Prop. 3.5.2 and Lem. 3.5.3]
Let X ∈ SmP roj/k , then C st A ∗ ( X ) is a strong dualizable object in D A ( k ) and its dual is C st A ∗ ( X ) ∨ = e C st A ∗ ( T h ( − T X )) . Con-sequently, one has a canonical isomorphism (3.2) Hom D A ( k ) ( k , C st A ∗ ( X )) ∼ = → Hom D A ( k ) ( C st A ∗ ( X ) ∨ , k ) . Fortunately, we will use later duality via operations formalism of J. Ayoub, which isgood enough for our main purpose. We end up this section by a definition: efinition 3.9. Let k be a field. One defines the geometric stable A -derived category D A ,gm ( k ) over k as the thick subcategory of D A ( k ) generated by C st A ∗ ( X ) , where X ∈ Sm/k . E -Motives E -Correspondences. Let k be a field and we denote by SH ( k ) the motivic stablehomotopy category. Throughout this section we fix a motivic spectrum E ∈ SH ( k ) togetherwith a multiplication map µ E : E ∧ L S E → E and a unit map ϕ E : S → E , such that the fowlling diagrams commute E id ∧ ϕ E / / ❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖ ❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖ E ∧ L S E µ E (cid:15) (cid:15) E ϕ E ∧ id o o ♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦ EE ∧ L S E ∧ L S E µ E ∧ id (cid:15) (cid:15) id ∧ µ E / / E ∧ L S E µ E (cid:15) (cid:15) E ∧ L S E µ E / / E Such a triple ( E , µ E , ϕ E ) is called a motivic ring spectrum. Proposition 4.1.
Let
X, Y, Z, W ∈ SmP roj ( k ) . Let α ∈ SH ( k )[Σ ∞ T, + X, Σ ∞ T, + Y ∧ L E ] , β ∈ SH ( k )[Σ ∞ T, + Y, Σ ∞ T, + Z ∧ L S E ] and γ ∈ SH ( k )[Σ ∞ T, + Z, Σ ∞ T, + W ∧ L S E ] . Let’s denote β ◦ M α : Σ ∞ T, + X α → Σ ∞ T, + Y ∧ L S E β ∧ id E −→ Σ ∞ T, + Z ∧ L S E ∧ L S E id Z ∧ µ E −→ Σ ∞ T, + Z ∧ L S E and similarly for γ ◦ M β . Then ◦ M is associative and unital.Proof. Both γ ◦ M ( β ◦ M α ) and ( γ ◦ M β ) ◦ M α are equal to the following composition Σ ∞ T, + X α → Σ ∞ T, + Y ∧ L S E β ∧ id E −→ Σ ∞ T, + Z ∧ L S E ∧ L S E id Z ∧ µ E −→ Σ ∞ T, + Z ∧ L S E γ ∧ id E −→ Σ ∞ T, + W ∧ L S E ∧ L S E id W ∧ µ E −→ Σ ∞ T, + W ∧ L S E . (cid:3) Definition 4.2.
The category of E -correspondences Corr E ( k )) is defined as: Obj ( Corr E ( k )) = Obj ( SmP roj ( k )) and Corr E ( k )( X, Y ) = SH ( k )[Σ ∞ T, + X, Σ ∞ T, + Y ∧ L S E ] , where the composition ◦ M : Corr E ( k )( X, Y ) ⊗ Corr E ( k )( Y, Z ) → Corr E ( k )( X, Z ) , ( α, β ) β ◦ M α is defined as β ◦ M α : Σ ∞ T, + X α → Σ ∞ T, + Y ∧ L S E β ∧ id E −→ Σ ∞ T, + Z ∧ L S E ∧ L S E id Z ∧ µ E −→ Σ ∞ T, + Z ∧ L S E . roposition 4.3. There is a functor h : SmP roj ( k ) → Corr E ( k ) , X X, which sends a morphism f : X → Y of k -schemes to Σ ∞ T, + ( f ) ∧ ϕ E : Σ ∞ T, + X = Σ ∞ T, + X ∧ L S S → Σ ∞ T, + Y ∧ L S E . Proof.
The identity morphism in
Corr E ( k )( X, X ) is given by id X ∧ ϕ E : Σ ∞ T, + X = Σ ∞ T, + X ∧ L S S → Σ ∞ T, + X ∧ L S E . Let α ∈ Corr E ( k )( X, X ) be an arbitrary E -correspondence. By definition we have α ◦ M (id X ∧ ϕ E ) = (id X ∧ µ E ) ◦ ( α ∧ id E ) ◦ (id X ∧ ϕ E ) . Since E is a ring spectrum, we must have α ◦ M (id X ∧ ϕ E ) = α . Similarly, (id X ∧ ϕ E ) ◦ M α = α .We check the compatibility of the composition laws. Let X f → Y g → Z be morphisms of k -schemes. By definition we have h ( g ◦ f ) = Σ ∞ T, + ( g ◦ f ) ∧ ϕ E and h ( g ) ◦ M h ( f ) = (id Z ∧ µ E ) ◦ (Σ ∞ T, + ( g ) ∧ ϕ E ∧ id E ) ◦ (Σ ∞ T, + ( f ) ∧ ϕ E ) . The equality h ( g ◦ f ) = h ( g ) ◦ M h ( f ) follows from the fact that E is a ring spectrum. (cid:3) Let
Spect Σ T ( k ) be the model category of symmetric motivic T -spectra ([Jar00]). Following[CD10], [Deg13, §2.2] we call E ∈ Spect Σ T ( k ) a strict motivic ring spectrum, if E is acommutative monoid object in Spect Σ T ( k ) . An E -module spectrum is a pair ( M, γ M ) , where M ∈ Spect Σ T ( k ) and γ M : M ∧ E → M , such that the following diagrams commute: S ∧ M ϕ E ∧ id M / / ❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘ ❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘ E ∧ M γ M (cid:15) (cid:15) M E ∧ E ∧ M µ E ∧ id M / / id E ∧ γ M (cid:15) (cid:15) E ∧ M γ M (cid:15) (cid:15) E ∧ M γ M / / M Given two E -modules ( M, γ ) and ( N, γ N ) , an E -module map is a map f : M → N , suchthat the following diagram commutes: E ∧ M id E ∧ f / / γ M (cid:15) (cid:15) E ∧ N γ N (cid:15) (cid:15) M f / / N Given a strict motivic ring spectrum E one can form the model category E − M od Σ of E -modules with respect to the symmetric monoidal model category Spect Σ T ( k ) (see e.g SS00])and there is a Quillen adjunction of model categories (we will return to this point inthe last discussion in the Appendix): − ∧ E : Spect Σ T ( k ) ⇆ E − M od Σ : U, where U denotes the forgetful functor. This Quillen adjunction induces an adjunction be-tween homotopy categories:(4.1) − ∧ L S E : SH ( k ) ⇄ Ho k ( E − M od ) :
RU, where we denote by Ho k ( E − M od ) the homotopy category associated to the category ofstrict E -modules. Theorem 4.4.
Let k be a field and E ∈ Spect Σ T ( k ) be a strict motivic ring spectrum. Thereis a functor Corr E ( k ) → Ho ( E − M od ) , X Σ ∞ T, + X ∧ L S E . Proof.
Recall that we may regard Σ ∞ T, + X ∧ L S E as an E -modules via the map Σ ∞ T, + X ∧ L S E ∧ L S E id X ∧ µ E −→ Σ ∞ T, + X ∧ L S E . Let us denote the assocation above by F : Corr E ( k ) → Ho ( E − M od ) , X Σ ∞ T, + X ∧ L S E .F maps on morphisms as following: Given α : Σ ∞ T, + X → Σ ∞ T, + Y ∧ L S E , we associate Σ ∞ T, + X ∧ L S E α ∧ id E −→ Σ ∞ T, + Y ∧ L S E ∧ L S E id Y ∧ µ E −→ Σ ∞ T, + Y ∧ L S E . We have to check firstly, that (id Y ∧ µ E ) ◦ ( α ∧ id E ) is a morphism of E -modules. Since E isa ring spectrum, there is a commutative diagram Σ ∞ T, + X ∧ L S E ∧ L S E id X ∧ µ E (cid:15) (cid:15) α ∧ id E ∧ id E / / Σ ∞ T, + Y ∧ L S E ∧ L S E ∧ L S E id Y ∧ µ E ∧ id E / / Σ ∞ T, + Y ∧ L S E ∧ L S E id Y ∧ µ E (cid:15) (cid:15) Σ ∞ T, + X ∧ L S E (id Y ∧ µ E ) ◦ ( α ∧ id E ) / / Σ ∞ T, + Y ∧ L S E Now we have to check the compatibility of the composition laws. Given α : Σ ∞ T, + X → Σ ∞ T, + Y ∧ L S E and β : Σ ∞ T, + Y → Σ ∞ T, + Z ∧ L S E . Then F ( β ) ◦ F ( α ) is the following composition Σ ∞ T, + X ∧ L S E α ∧ id E −→ Σ ∞ T, + Y ∧ L S E ∧ L S E id Y ∧ µ E −→ Σ ∞ T, + Y ∧ L S E β ∧ id E −→ Σ ∞ T, + Z ∧ L S E ∧ L S E id Z ∧ µ E −→ Σ ∞ T, + Z ∧ L S E . The composition F ( β ◦ M α ) is Σ ∞ T, + X ∧ L S E ( β ◦ M α ) ∧ id E −→ Σ ∞ T, + Z ∧ L S E ∧ L S E id Z ∧ µ E −→ Σ ∞ T, + Z ∧ L S E , where β ◦ M α : Σ ∞ T, + X α → Σ ∞ T, + Y ∧ L S E β ∧ id E −→ Σ ∞ T, + Z ∧ L S E ∧ L S E id Z ∧ µ E −→ Σ ∞ T, + Z ∧ L S E . ence, F ( β ◦ M α ) is the following composition Σ ∞ T, + X ∧ L S E α ∧ id E −→ Σ ∞ T, + Y ∧ L S E ∧ L S E β ∧ id E ∧ id E −→ Σ ∞ T, + Z ∧ L S E ∧ L S E ∧ L S E id Z ∧ µ E ∧ id E −→ Σ ∞ T, + Z ∧ L S E ∧ L S E id Z ∧ µ E −→ Σ ∞ T, + Z ∧ L S E . Since E is a ring spectrum, we have a commutative diagram Σ ∞ T, + Y ∧ L S E ∧ L S E id Y ∧ µ E (cid:15) (cid:15) β ∧ id E ∧ id E / / Σ ∞ T, + Z ∧ L S E ∧ L S E ∧ L S E id Z ∧ µ E ∧ id E / / Σ ∞ T, + Z ∧ L S E ∧ L S E Σ ∞ T, + Y ∧ L S E β ∧ id E / / Σ ∞ T, + Y ∧ L S E This implies that F ( β ◦ M α ) = F ( β ) ◦ F ( α ) . (cid:3) Definition 4.5.
Let k be a field and E ∈ Spect Σ T ( k ) be a strict motivic ring spectrum. Wedefine the category Mot E ( k ) of pure E -motives over k to be the smallest pseudo-abeliansubcategory of Ho k ( E − M od ) generated as an additive category by { Σ ∞ T, + X ∧ L S E | X ∈ SmP roj ( k ) } . Remark 4.6.
We know that if char ( k ) = 0 then there is an equivalence of categories Ho k ( H Z − M od ) ∼ = DM ( k ) , where DM ( k ) denotes the category of big Voevodsky’s motives (cf. [RO08]). As the categoryof pure Grothendieck-Chow motives Chow ( k ) ֒ → DM ( k ) is embedded fully faithful into DM ( k ) , we raise a question: is Mot H Z ( k ) equivalent to Chow ( k ) via the equivalence above?We only know that Mot H Z ( k )( X, Y ) ∼ = SH ( k )[Σ ∞ T, + X, Σ ∞ T, + Y ∧ L S H Z ] ∼ = H n Y + d Y ) , ( n Y + d Y ) M ( X + ∧ T h ( V Y ) , Z ) ∼ = H d Y ,d Y M ( X × Y, Z ) ∼ = CH d Y ( X × Y ) , where the first isomorphism comes from the adjunction − ∧ L S H Z : SH ( k ) ⇆ Ho k ( H Z − M od ) . The second isomorphism comes from duality, the third isomorphism is the Thom isomor-phism for motivic cohomology and the last isomorphism is the comparison isomorphism ofVoevodsky ([MVW06, Cor. 19.2]). The question is, if these isomorphisms are compatiblewith the equivalence Ho k ( H Z − M od ) ∼ = DM ( k ) ? It seems the problem with the first threeisomorphisms is not difficult, however it seems that the problem with the last isomorphismis hard. Corollary 4.7.
Let k be a field and E be a strict motivic ring spectrum. There is a functor Mot E ( k ) → SH ( k ) . Proof.
This follows from the adjunction 4.1. (cid:3) .2. Functoriality in motivic stable homotopy.
Following [Ay08], we recall that thestable homotopy category of schemes defines a -functor from category of quasi-projectivesmooth schemes over a field QSP rojSm/k to the category of symmetric monoidal closedtriangulated categories. Remark that the six operations formalism works much more generalthan what we here require. However we restrict ourselves only to
QSP rojSm/k , since it isalready enough for our aim. We will list now a minimal list of properties of the six operationsformalism: for any morphism of schemes f : T → S , there is a pullback functor f ∗ : SH ( S ) → SH ( T ) , such that ( f ◦ g ) ∗ = g ∗ ◦ f ∗ . Moreover,(1) One has an adjunction for any morphism of schemes f : T → Sf ∗ : SH ( S ) ⇆ SH ( T ) : f ∗ . If f is smooth, then one has an adjunction f : SH ( T ) ⇆ SH ( S ) : f ∗ (2) Given a cartesian square Y q / / g (cid:15) (cid:15) X f (cid:15) (cid:15) T p / / S and assume f is smooth, then f p ∗ ∼ = → g q ∗ (3) Let f : Y → X be a smooth morphism, E ∈ SH ( Y ) and F ∈ SH ( X ) , the naturaltransformation f ( E ∧ f ∗ F ) ∼ = → f E ∧ F is an isomorphism.(4) Let i : Z ֒ → X be a closed immersion with complement j : U ֒ → X , then there is adistinguished triangle j j ∗ → Id → i ∗ i ∗ +1 → (5) For any closed immersion i : Z ֒ → X , one has an adjunction i ∗ : SH ( Z ) ⇆ SH ( X ) : i ! (6) Given a cartesian square T k / / g (cid:15) (cid:15) Y f (cid:15) (cid:15) Z i / / X where i : Z ֒ → X is a closed immersion, then one has an isomorphism f ∗ i ∗ ∼ = → k ∗ g ∗
7) Let i : Z ֒ → X be a closed immersion, E ∈ SH ( Z ) and F ∈ SH ( X ) , the naturaltransformation i ∗ ( E ∧ i ∗ F ) ∼ = → i ∗ E ∧ F is an isomorphism.(8) For any separated morphism of finite type f : Y → X , there is an adjunction f ! : SH ( Y ) ⇆ SH ( X ) : f ! . (9) For a smooth separated morphism of finite type f : Y → X with the relative tangentbundle T f there are canonical natural isomorphisms, which are dual to each other f ∼ = −→ f ! ( T h Y ( T f ) ∧ Y − ) , f ∗ ∼ = −→ T h Y ( − T f ) ∧ Y f ! . Moreover, for any separated morphism of finite type f : Y → X , there exist naturalisomorphisms Ex ( f ∗ ! , ∧ ) : ( f ! K ) ∧ X L ∼ = −→ f ! ( K ∧ Y f ∗ L ) , Hom X ( f ! L, K ) ∼ = −→ f ∗ Hom Y ( L, f ! K ) ,f ! Hom X ( L, M ) ∼ = −→ Hom Y ( f ∗ L, f ! M ) . (10) If f : Y → X is a smooth projective morphism then f ( Y ) is strongly dualizable in SH ( X ) with the dual D X ( f ( Y )) = f T h Y ( − T f ) . Furthermore, one has D X ( f ∗ K ) ∼ = f ∗ D Y ( K ∧ Y T h Y ( T f )) , ∀ K ∈ SH ( Y ) .We will need some facts about cohomology with supports in the next subsection.4.3. Cohomology with supports.
Let S = Spec k . We consider the category SH ( k ) . Fora ring spectrum E ∈ SH ( k ) and a closed pair ( X, Z ) , where π X : X → S is a smooth quasi-projective k -scheme and i : Z ֒ → X a smooth closed subscheme, one defines the cohomologywith support as E p,qZ ( X ) = SH ( S )[ X/X − Z, E ∧ S p,q ] ∼ = SH ( X )[ i ∗ ( Z ) , E X ∧ S p,q ] ∼ = SH ( Z )[ Z , i ! E X ∧ S p,q ] , where we write E X = π ∗ X E . As Σ ∞ T, + X/X − Z := π X i ∗ ( Z ) in SH ( k ) , so the first isomor-phism follows from the adjunction π X : SH ( X ) ⇆ SH ( S ) : π ∗ X and the last isomorphism comes from the adjunction i ∗ : SH ( Z ) ⇆ SH ( X ) : i ! . If f : Y → X is a smooth morphism of smooth quasi-projective S -schemes we have acanonical homomorphism f ∗ : E p,qZ ( X ) → E p,qT ( Y ) , where T = Y × X Z defined as following: Consider the commutative diagram T j / / g (cid:15) (cid:15) Y f (cid:15) (cid:15) Z i / / X or a morphism α : i ∗ Z → E X ∧ S p,q we can associate to a morphism f ∗ α : j ∗ T ∼ = j ∗ g ∗ Z ( Ex ∗∗ ) − ∼ = f ∗ i ∗ Z → f ∗ E X ∧ S p,q ∼ = → E Y ∧ S p,q . If T j ֒ → Z i ֒ → X are closed immersions, we can define a pushforward on cohomology withsupports j ! : E p,qT ( X ) → E p,qZ ( X ) as following: Given a morphism α : X/X − T → E X ∧ S p,q we associate j ! ( α ) = α ◦ ¯ j , where ¯ j : X/X − Z → X/X − T is the canonical morphism in SH ( X ) induced by the immersion X − Z ֒ → X − T . If α ∈ E p,qZ ( X ) and β ∈ E m,nZ ( X ) we define their product in E p + m,q + nZ ( X ) as a morphism α ∪ β : X/X − Z ∆ → X/ ( X − Z ) ∧ X/ ( X − Z ) α ∧ β −→ E ∧ L E ∧ S p + m,q + n µ E → E ∧ S p + m,q + n . If ξ/X is a vector bundle over a smooth k -scheme X with the zero section s : X → ξ , thenthe E -cohomology of the Thom spectrum T h ( ξ ) is E p,q ( T h ( ξ )) = E p,qX ( ξ ) . The pushforward defined as above works only for closed immersions. We will define laterpushforward on E -cohomology of Thom spectrum for projective smooth morphism usingduality.4.4. Relation to the category of twisted E -correspondences.Notation 4.8. For a quasi-projective smooth k -scheme π X : X → Spec k and a vectorbundle p ξ : ξ → X with -section s X : X → ξ we will write T h X ( ξ ) = p ξ s X ! ( X ) for theThom transformation T h ( s X , p ξ ) = p ξ s X ! applying on X . T h X ( ξ ) is an object in SH ( X ) and T h X ( − ξ ) = s ! X p ∗ ξ ( X ) for its inverse as the inverse Thom transformation T h − ( s X , p ξ ) applying on X . The Thom spectrum will be denoted by T h ( ξ/X ) , which means T h ( ξ/X ) = π X T h X ( ξ ) = π X p ξ s X ! ( X ) ∼ = π X p ξ s X ! π ∗ X k . Sometime we only write
T h ( ξ ) for the Thom spectrum, if it is clear which scheme X wetalk about. One can see easily that this definition coincides with the traditional definitionof Thom spectrum as follow: Let j : ξ − s X ( X ) ֒ → ξ be the open immersion with thecomplement s X : X → ξ . One has a localization sequence j j ∗ ( ξ ) → ξ → s X ∗ s ∗ X ( ξ ) . Applying π p ξ and as s X ∗ ∼ = s X ! is a natural -isomorphism one has a natural isomorphismin SH ( k ) : T h X ( ξ ) ∼ = Σ ∞ T, + ξ/ξ − s X ( X ) . Let E ∈ SH ( k ) be a ring spectrum and X/k a quasi-projective smooth k -scheme. Let p ξ : ξ → X be a vector bundle of rank r with the zero section s : X → ξ . We define E -cohomology of X twisted by a vector bundle as E p,q ( X, ξ ) = SH ( X )[ X , s ! p ∗ ξ E r,rX ∧ S p,q ] = SH ( X )[ X , T h X ( − ξ ) ∧ X E r,rX ∧ S p,q ] . here we write E r,rX = E X ∧ S r,r . We denote by E ∗ , ∗ ( X, ξ ) the bigraded ring E ∗ , ∗ ( X, ξ ) = ⊕ p,q E p,q ( X, ξ ) . Remark that E ∗ , ∗ ( X, ξ ) is bigraded ring. Even if E is a commutative ring spectrum, E ∗ , ∗ ( X, ξ ) is never bigraded commutative. If ξ ∈ V ( X ) is a virtual vector bundle of rank r < then p − ξ : − ξ → X is an actual vector bundle, so we define E p,q ( X, ξ ) = SH ( X )[ X , p − ξ s ! E r,rX ∧ S p,q ] . This group has the following interpretation by Jouanolou trick: As X is quasi-projective, sowe have an immersion i : X ֒ → P N . Via the Segre embedding P N × P N ֒ → P N +2 N , U is anaffine variety. Let U = P N × P N − P roj k [ x , · · · , x N , y , · · · , y N ] / N X i =0 x i y i .pr : U → P N is an A N -bundle. Consider the pullback diagram i ∗ U π (cid:15) (cid:15) / / U pr (cid:15) (cid:15) X i / / P N Then E p,q ( X, ξ ) ∼ = SH ( k )[ T h ( ζ /U ) , E r + n ) , ( r + n ) ∧ S p,q ] , where ζ is an actual vector bundle on U , such that π ∗ ξ ⊕ O n = ζ . Proposition 4.9.
Let f : ξ ∼ = −→ ξ ′ be an isomorphism of vector bundles on Xξ f ∼ = / / p ξ (cid:30) (cid:30) ❃❃❃❃❃❃❃❃ ξ ′ p ξ ′ (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧ X There is a natural isomorphism E p,q ( X, ξ ) ∼ = E p,q ( X, ξ ′ ) . Proof.
Consider the Cartesian squares X s / / ξ ∼ = f (cid:15) (cid:15) p ξ / / XX s ′ / / ξ ′ p ξ ′ / / X One has two -isomorphisms ([Ay08, §1.5.5]) T h X ( s, p ξ ) ∼ = −→ T h X ( s ′ , p ′ ξ ) , and T h − X ( s ′ , p ′ ξ ) ∼ = −→ T h − X ( s, p ξ ) , which prove the Proposition. (cid:3) roposition 4.10. If E is orientable in sense of [CD10, Def. 12.2.2] , then there is a naturalisomorphism E p,q ( X, ξ ) ∼ = −→ E p,q ( X ) . Proof.
Since E is orientable, one has by [CD10, Thm. 2.4.50 (3)] a canonical natural iso-morphism p ∗ ξ E X ∼ = → p ! ξ E X ∧ S − r, − r . This induces a natural isomorphism E p,q ( X, ξ ) = SH ( X )[ X , s ! p ∗ ξ E r,rX ∧ S p,q ] ∼ = −→ SH ( X )[ X , s ! p ! ξ E X ∧ S p,q ] = E p,q ( X ) . (cid:3) Proposition 4.11. (twisted Thom isomorphism) Let
X/k be a quasi-projective smooth k -scheme and p ξ : ξ → X be a vector bundle of rank r with the zero section s : X → ξ . Onehas a natural isomorphism th XE ( ξ ) : E p,q ( X, ξ ) ∼ = E p +2 r,q + r ( T h ( ξ )) , which we call the twisted Thom isomorphism.Proof. We have two adjunctions s ! : SH ( X ) ⇄ SH ( ξ ) : s ! , p ξ : SH ( ξ ) ⇄ SH ( X ) : p ∗ ξ . Hence, we have E p,q ( X, ξ ) = SH ( X )[ X , s ! p ∗ ξ E r,rX ∧ S p,q ] ∼ = ∼ = SH ( X )[ p ξ s ! ( X ) , E r,rX ∧ S p,q ] ∼ = E p +2 r,q + r ( T h ( ξ )) , where the last natural isomorphism is induced by the adjunction ( π X , π ∗ X ) , where π X : X → Spec k is the structure morphism. So for a morphism α : X → s ! p ∗ ξ E r,rX ∧ S p,q the twisted Thom isomorphism is explicitly given by th XE ( ξ )( α ) = ε ( π X ,π ∗ X ) ◦ π X ◦ ε ( p ξ ,p ∗ ξ ) ◦ p ξ ◦ ε ( s ! ,s ! ) ◦ s ! ( α ) . (cid:3) Example 4.12.
The twisted Chow-Witt group f CH p ( X, det ξ ) defined by J. Fasel (cf. [Fas07]and [Fas08]) and also by F. Morel ([Mor12]) is an example of twisted cohomology. One hasa natural isomorphism f CH p ( X, det ξ ) defn = H pNis ( X, K MWp (det ξ )) ∼ = H ( K MW ∗ ) p,p ( X, ξ ) , where H ( K MW ∗ ) denotes the Eilenberg-Maclane spectrum associated to the homotopy mod-ule K MW ∗ . We will discuss later about H K MW ∗ after introducing the homotopy t-structure.Before going further, we want to give a list of properties of Thom transformations that wewill need for our constructions. roposition 4.13. [Ay08, Prop. 2.3.19] Let X be a quasi-projective k -scheme and ξ/X bea vector bundle. Let f : Y → X be a morphism. Then one has two natural -isomorphisms f ∗ T h X ( ξ ) ∼ = −→ T h Y ( f ∗ ξ ) f ∗ , f ∗ T h X ( − ξ ) ∼ = −→ T h Y ( − f ∗ ξ ) f ∗ , which satisfy: For all ( K, L ) ∈ Obj ( SH ( X ) ) , there are two commutative diagrams f ∗ K ∧ Y ( f ∗ T h X ( ξ ) L ) ∼ = (cid:15) (cid:15) ∼ = / / f ∗ ( K ∧ X T h X ( ξ ) L ) ∼ = / / f ∗ T h X ( ξ )( K ∧ X L ) ∼ = (cid:15) (cid:15) f ∗ K ∧ Y T h Y ( f ∗ ξ ) f ∗ L ∼ = / / T h Y ( f ∗ ξ )( f ∗ K ∧ Y f ∗ L ) ∼ = / / T h Y ( f ∗ ξ ) f ∗ ( K ∧ X L ) and f ∗ K ∧ Y ( f ∗ T h X ( − ξ ) L ) ∼ = (cid:15) (cid:15) ∼ = / / f ∗ ( K ∧ X T h X ( − ξ ) L ) ∼ = / / f ∗ T h X ( − ξ )( K ∧ X L ) ∼ = (cid:15) (cid:15) f ∗ K ∧ Y T h Y ( − f ∗ ξ ) f ∗ L ∼ = / / T h Y ( − f ∗ ξ )( f ∗ K ∧ Y f ∗ L ) ∼ = / / T h Y ( − f ∗ ξ ) f ∗ ( K ∧ X L ) Proposition 4.14. [Ay08, Prop. 2.3.20]
Let f : Y → X be a k -morphism of quasi-projectiveschemes and ξ/X be a vector bundle. There are two natural -isomorphisms T h X ( ξ ) f ∗ ∼ = −→ f ∗ T h X ( f ∗ ξ ) , T h X ( − ξ ) f ∗ ∼ = −→ f ∗ T h Y ( − f ∗ ξ ) , such that the following diagrams commute for all ( K, L ) ∈ SH ( X ) × SH ( Y ) : K ∧ X T h X ( ξ ) f ∗ L ∼ = (cid:15) (cid:15) ∼ = / / K ∧ X f ∗ T h Y ( f ∗ ξ ) L ∼ = / / f ∗ ( f ∗ K ∧ Y T h Y ( f ∗ ξ ) L ) ∼ = (cid:15) (cid:15) T h X ( ξ )( K ∧ X f ∗ L ) ∼ = / / T h X ( ξ ) f ∗ ( f ∗ K ∧ Y L ) ∼ = / / f ∗ T h Y ( f ∗ ξ )( f ∗ K ∧ X L ) K ∧ X T h X ( − ξ ) f ∗ L ∼ = (cid:15) (cid:15) ∼ = / / K ∧ X f ∗ T h Y ( − f ∗ ξ ) L ∼ = / / f ∗ ( f ∗ K ∧ Y T h Y ( − f ∗ ξ ) L ) ∼ = (cid:15) (cid:15) T h X ( − ξ )( K ∧ X f ∗ L ) ∼ = / / T h X ( − ξ ) f ∗ ( f ∗ K ∧ Y L ) ∼ = / / f ∗ T h Y ( − f ∗ ξ )( f ∗ K ∧ X L ) Let f : Y → X be any morphism of finite type and separated of quasi-projective smooth k -schemes. In the following we define a pullback map on twisted E -cohomology E p,q ( X, ξ ) → E p,q ( Y, f ∗ ξ ) . Consider the functor f ∗ : SH ( X ) → SH ( Y ) . f ∗ induces a map E p,q ( X, ξ ) = SH ( X )[ X , s ! p ∗ ξ E r,rX ∧ S p,q ] → SH ( Y )[ f ∗ X , f ∗ s ! p ∗ ξ E r,rX ∧ S p,q ] == SH ( Y )[ Y , f ∗ s ! p ∗ ξ E r,rX ∧ S p,q ] . Let s Y be the -section of the vector bundle p f ∗ ξ : f ∗ ξ → Y and we write f ξ : f ∗ ξ → ξ . Onehas an exchange transformation (see [Ay08, Prop. 1.4.15]) Ex ∗ ! : f ∗ s ! → s ! Y f ∗ ξ , hich is the following composition ( s ∗ ∼ = s ! , s Y ! ∼ = s Y ∗ since s and s Y are closed immersion): f ∗ s ! η ( sY ∗ ,s ! Y ) −→ s ! Y s Y ∗ f ∗ s ! Ex ∗∗ ( − ) − −→ s ! Y f ∗ ξ s ∗ s ! ε ( s ∗ ,s !) −→ s ! Y f ∗ ξ , where f ξ : f ∗ ξ → ξ is the induced map on vector bundles. Note that the exchange trans-formation Ex ∗ ! is an isomorphism, when f is smooth ([Ay08, Cor. 1.4.17]). At this pointwe also notice that for an actual bundle ξ , the Thom transformation T h X ( ξ ) behaves wellunder pullback of a general morphism, since f ∗ T h X ( ξ ) = f ∗ p ξ s X ! ∼ = p f ∗ ξ f ∗ ξ s X ∗ ∼ = p f ∗ ξ s Y ∗ f ∗ ∼ = p f ∗ ξ s Y ! f ∗ = T h Y ( f ∗ ξ ) , and we have a natural transformation Ex ∗ ! : f ∗ T h X ( − ξ ) = f ∗ s ! X p ∗ ξ → s ! Y f ∗ ξ p ∗ ξ ∼ = s ! Y p ∗ f ∗ ξ f ∗ = T h Y ( − f ∗ ξ ) , which is an isomorphism, if f is smooth (see [Ay08, Lem. 1.5.4]). However, he showed that T h X ( ξ ) and T h X ( − ξ ) are inverse to each other [Ay08, Thm. 1.5.7], hence T h Y ( − f ∗ ξ ) ∼ = −→ f ∗ T h X ( − ξ ) (cf. [Ay08, Rem. 1.5.10]) for all morphism not necessary smooth f . That is avery crucial point. Now consider the pullback diagram f ∗ ξ p f ∗ ξ (cid:15) (cid:15) f ξ / / ξ p ξ (cid:15) (cid:15) Y f / / X We have a natural isomorphism f ∗ ξ p ∗ ξ ∼ = p ∗ f ∗ ξ f ∗ . Hence, we obtain a map E p,q ( X, ξ ) → E p,q ( Y, f ∗ ξ ) , which we define as pullback of twisted E -cohomology. Remark 4.15.
The composition of pullback on twisted E -cohomology g ∗ ◦ f ∗ is only definedup to the natural isomorphism ( f ◦ g ) ∗ ∼ = −→ g ∗ f ∗ . Remark 4.16.
Let a : ξ ∼ = −→ ξ be an automorphism of a vector bundle ξ of rank r on X .Then one has the cartesian squares X s ′ X / / id ξ a ∼ = (cid:15) (cid:15) p ′ ξ / / X id X s X / / ξ p ξ / / X As in [Ay08, §1.5.5 p. 84] a induces two -isomorphisms between the Thom transformations ω ( a ) : T h ( s ′ X , p ′ ξ ) = p ′ ξ s ′ X ! ∼ = −→ T h ( s X , p ξ ) = p ξ s X ! and ω − ( a ) : T h − ( s X , p ξ ) = s ! X π ∗ ξ ∼ = −→ T h − ( s ′ X , p ′ ξ ) = s ′ ! X p ′∗ ξ . This induces an isomorphism, which is not necessary identity ¯ ω ( a ) : SH ( X )[ X , s ! X p ∗ ξ E r,rX ∧ S p,q ] ∼ = −→ SH ( X )[ X , s ′ ! X p ′∗ ξ E r,rX ∧ S p,q ] . owever, the two pullbacks induced on twisted E -cohomology along a morphism f : Y → X must not be on the same target E p,q ( Y, f ∗ ξ ) , as there are two different pullback diagrams f ∗ ξ p f ∗ ξ (cid:15) (cid:15) / / ξ p ξ (cid:15) (cid:15) f ∗ ξ / / p ′ f ∗ ξ (cid:15) (cid:15) ξ p ′ ξ (cid:15) (cid:15) Y f / / X Y f / / ξ Consequently, there is no problem with maps between E -cohomology created by automor-phisms of ξ . Remark 4.17.
Thanks to the Proposition 4.13. The pullback of cohomology of virtualvector bundles is defined in the same way.
Proposition 4.18.
Let Z g → Y f → X be morphisms of quasi-projective smooth k -schemes.Let ξ/X be a vector bundle. Then one has up to a natural isomorphism induced by a natural -isomorphism ( f ◦ g ) ∗ = g ∗ ◦ f ∗ : E p,q ( X, ξ ) → E p,q ( Z, g ∗ f ∗ ξ ) . Proof.
Consider the chain of pullback bundles g ∗ f ∗ ξ p g ∗ f ∗ ξ (cid:15) (cid:15) g ξ / / f ∗ ξ p f ∗ ξ (cid:15) (cid:15) f ξ / / ξ p ξ (cid:15) (cid:15) Z g / / Y f / / X Let s X , s Y and s Z be the -sections of ξ, f ∗ ξ and g ∗ f ∗ ξ respectively. The functoriality up toa natural isomorphism follows easily from the natural -isomorphism ( f ◦ g ) ∗ s ! X ∼ = g ∗ f ∗ s ! X Ex ∗ ! −→ g ∗ s ! Y f ∗ ξ Ex ∗ ! −→ s ! Z g ∗ ξ f ∗ ξ ∼ = s ! Z ( f ξ ◦ g ξ ) ∗ . (cid:3) This motivates us to give the following definition:
Definition 4.19.
A twisted E -cohomology pre-theory is an association, which is contravari-ant in both variables: E ∗ , ∗ ( − , − ) : QSP rojSm ( k ) × V ⊃ A → Ring ∗ , where Ring ∗ denotes the category of bigraded rings and V is the 2-category, where objectsare categories of virtual vector bundles V ( X ) for X ∈ QSP rojSm ( k ) and − M or V ( V ( X ) , V ( Y )) = F un ( V ( X ) , V ( Y ))2 − M or V ( F, G ) =
Nat ( F, G ) . A is the full subcategory of QSP rojSm ( k ) × V consisting of those pairs ( X, ξ ) , where X ∈ QSP rojSm ( k ) and ξ ∈ V ( X ) . M or A (( X, ξ ) , Y ( , η )) consists of pair ( f, g ) , where : X → Y is a morphism of quasi-projective smooth k -schemes and g : ξ → η is a bundlemap ξ g / / (cid:15) (cid:15) η (cid:15) (cid:15) X f / / Y such that ξ → f ∗ η is a monomorphism in V ( X ) . E ∗ , ∗ ( − , − ) sends such a pair ( X, ξ ) to E ∗ , ∗ ( X, ξ ) . Given an A -morphism ( f, g ) : ( X, ξ ) → ( Y, η ) , E ∗ , ∗ ( − , − ) sends ( f, g ) to thefollowing composition E ∗ , ∗ ( Y, η ) f ∗ −→ E ∗ , ∗ ( X, f ∗ η ) −→ E ∗ , ∗ ( X, ξ ) , where f ∗ is the pullback map on twisted E -cohomology constructed as above and the lastmap is induced by T h X ( ξ ) → T h X ( f ∗ η ) , as ξ → f ∗ η is a monomorphism in V ( X ) . Proposition 4.20.
Let f : Y → X be a k -morphism of quasi-projective smooth k -schemesand p ξ : ξ → X be a vector bundle of rank r on X . There is a commutative diagram up to anatural isomorphism induced by a natural -isomorphism E p,q ( X, ξ ) f ∗ / / ∼ = th XE ( ξ ) (cid:15) (cid:15) E p,q ( Y, f ∗ ξ ) ∼ = th YE ( f ∗ ξ ) (cid:15) (cid:15) E p +2 r,q + r ( T h ( ξ )) f ∗ / / E p +2 r,q +2 r ( T h ( f ∗ ξ )) where f ∗ : E p +2 r,q + r ( T h ( ξ )) → E p +2 r,q + r ( T h ( f ∗ ξ )) is the pullback given by SH ( X )[ T h X ( ξ ) , E r,rX ∧ S p,q ] f ∗ → SH ( Y )[ f ∗ T h X ( ξ ) , f ∗ E r,rX ∧ S p,q ] Ex ! ∗ −→ SH ( Y )[ T h Y ( f ∗ ξ ) , E r,rY ∧ S p,q ] . Proof.
It is obvious by construction. Remark that for general morphism f we always havethe pullback SH ( X )[ T h X ( ξ ) , E r,rX ∧ S p,q ] → SH ( Y )[ T h Y ( f ∗ ξ ) , E r,rY ∧ S p,q ] . Since π X : X → Spec k and π Y : Y → Spec k are smooth, then one has the naturalisomorphisms via the adjunctions ( π X , π ∗ X ) and ( π Y , π ∗ Y ) : SH ( X )[ T h X ( ξ ) , E r,rX ∧ S p,q ] ∼ = SH ( k )[ π X T h X ( ξ ) , E r,r ∧ S p,q ] = E p,q ( T h ( ξ )) , and SH ( Y )[ T h Y ( f ∗ ξ ) , E r,rY ∧ S p,q ] ∼ = SH ( k )[ π Y T h Y ( f ∗ ξ ) , E r,r ∧ S p,q ] = E p,q ( T h ( f ∗ ξ )) . Explicitly, given a morphism α : X → s ! X p ∗ ξ E r,rX ∧ S p,q we have f ∗ th XE ( ξ )( α ) = ε ( s X ∗ ,s ! X ) ◦ Ex ∗∗ ( − ) − ◦ η ( s Y ∗ ,s ! Y ) f ∗ ◦ ε ( p ξ ,p ∗ ξ ) ◦ p ξ ◦ ε ( s X ! ,s ! X ) ◦ s X ! ( α ) nd th YE ( f ∗ ξ )( f ∗ α ) = ε ( p f ∗ ξ ,p ∗ f ∗ ξ ) ◦ p f ∗ ξ ε ( s Y ! ,s ! Y ) s Y ! ◦ ε ( s X ∗ ,s ! X ) ◦ Ex ∗∗ ( − ) − ◦ η ( s Y ∗ ,s ! Y ) ◦ f ∗ ( α ) . The two composition are natural isomorphism to each other, as we have the natural -isomorphisms: f ∗ s X ! ∼ = −→ s Y ! f ∗ ξ , p f ∗ ξ f ∗ ξ ∼ = −→ f ∗ p ξ . (cid:3) Let f : Y → X be a smooth projective morphism of projective smooth k -schemes ofrelative dimension d = dim( Y ) − dim( X ) and p ξ : ξ → X be a vector bundle of rank r with the zero section s : X → ξ . We define in the following a pushforward on twisted E -cohomology: Consider SH ( Y )[ Y , T h Y ( T f ) ∧ Y s ! Y p ∗ f ∗ ξ E r − d ) , ( r − d ) Y ∧ S p,q ] , where T f is the normal bundle of the diagonal immersion δ : Y → Y × X Y . SH ( Y )[ Y , T h Y ( T f ) ∧ Y s ! Y p ∗ f ∗ ξ E r − d ) , ( r − d ) Y ∧ S p,q ] ∼ = SH ( Y )[ T h Y ( f ∗ ξ ) ∧ Y T h Y ( − T f ) , E r − d ) , ( r − d ) Y ∧ S p,q ] , where T h Y ( − T f ) ∈ SH ( Y ) is the inverse of T h Y ( T f ) ∈ SH ( Y ) . Since E Y = f ∗ E X , theadjunction f : SH ( Y ) ⇆ SH ( X ) : f ∗ gives us a natural isomorphism SH ( Y )[ T h Y ( f ∗ ξ ) ∧ Y T h Y ( − T f ) , E r − d ) , ( r − d ) Y ∧ S p,q ] ∼ = SH ( X )[ f ( T h Y ( f ∗ ξ ) ∧ Y T h Y ( − T f )) , E r − d ) , ( r − d ) X ∧ S p,q ] By the projection formula
P r ∗ and since T h Y ( f ∗ ξ ) ∼ = f ∗ T h X ( ξ ) as ξ is an actual bundle,we have then a natural isomorphism f ( T h Y ( f ∗ ξ ) ∧ Y T h Y ( − T f )) ∼ = T h X ( ξ ) ∧ X f T h Y ( − T f ) . So we have then a natural isomorphism SH ( X )[ f ( T h Y ( f ∗ ξ ) ∧ Y T h Y ( − T f )) , E r − d ) , ( r − d ) X ∧ S p,q ] ∼ = SH ( X )[ T h X ( ξ ) ∧ X f T h Y ( − T f ) , E r − d ) , ( r − d ) X ∧ S p,q ] . By [CD10, Prop. 2.4.31] we have f T h Y ( − T f ) ∼ = D X ( f Y ) , where D X ( f Y ) means the dual of f Y in SH ( X ) . Hence there is a natural isomorphism SH ( X )[ T h X ( ξ ) ∧ X f T h Y ( − T f ) , E r − d ) , ( r − d ) X ∧ S p,q ] ∼ = SH ( X )[ T h X ( ξ ) , f Y ∧ X E r − d ) , ( r − d ) X ∧ S p,q ] . From the counit of the adjunction ( f , f ∗ ) f Y ∼ = f f ∗ X → X e have an induced map SH ( X )[ T h X ( ξ ) , f Y ∧ X E r − d ) , ( r − d ) X ∧ S p,q ] → SH ( X )[ T h X ( ξ ) , E r − d ) , ( r − d ) X ∧ S p,q ] . By the twisted Thom isomorphism, the later group is SH ( X )[ T h X ( ξ ) , E r − d ) , ( r − d ) X ∧ S p,q ] ∼ = E p − d,q − d ( X, ξ ) . Now we define formally:
Definition 4.21.
Let f : Y → X be a smooth projective morphism of projective smooth k -schemes of relative dimension d = dim( Y ) − dim( X ) and p ξ : ξ → X be a vector bundle.We define E p,q ( Y, f ∗ ξ − T f ) defn = SH ( Y )[ Y , T h Y ( T f ) ∧ Y s ! Y p ∗ f ∗ ξ E r − d ) , ( r − d ) Y ∧ S p,q ] , where π f ∗ ξ : f ∗ ξ → Y is the pullback bundle and s Y is its -section. The pushforward mapis the induced map constructed as above E p,q ( Y, f ∗ ξ − T f ) f ∗ → E p − d,q − d ( X, ξ ) . Remark that our definition of projective pushforward f ∗ doesn’t require E to be an orientedcohomology theory, however we need the assumption on smoothness of f . The reason thatwe choose the notation E p,q ( Y, f ∗ ξ − T f ) is that this group should behave like the so-calledcohomology twisted by formal difference of vector bundles. The shifting in the definition ( − d, − d ) reminds us that the inverse Thom transformation T h Y ( − T f ) should behave likethe Thom spectrum of the virtual bundle − T f after taking π Y , where π Y : Y → Spec k is the structure morphism of Y , as the rank of the virtual bundle − T f is − d . As alreadymentioned in § we refer the reader to [Rio10, §4] and [Del87] for the discussion on Picardcategory of virtual bundles. But we remind the reader again that we always work with anactual bundle ξ . Remark 4.22.
Let Z g −→ Y f −→ X be a sequence of composable morphisms. Then f ∗ ◦ g ∗ is not defined for a trivial reason: One has only a natural -isomorphism e ∨ σ : T h − Z ( s Z , p T fg ) ∼ = −→ T h − Z ( s Z , p T g ) ◦ g ∗ T h − Y ( s Y , p T g ) , which comes from the exact sequence → g ∗ T f → T fg → T g → . The -isomorphism e ∨ σ is not an identity. Consequently f ∗ ◦ g ∗ is only defined up to thisspecific natural -isomorphism. Remark 4.23.
Another variant to construct pushforward can be obtained as follows: Onehas a natural isomorphism via Thom transformation adjunctions Apply the functor f ∗ : SH ( Y ) → SH ( X ) we obtain a map SH ( Y )[ Y , T h Y ( T f ) ∧ Y T h ( − f ∗ ξ ) ∧ Y E r − d ) , ( r − d ) Y ∧ S p,q ] f ∗ ( − ) −→ SH ( X )[ f ∗ Y , f ∗ ( T h Y ( T f ) ∧ Y T h Y ( − f ∗ ξ ) ∧ E r − d ) , ( r − d ) Y ∧ S p,q )] . y projection formula P r ∗∗ ( f ) we have a canonical isomorphism Ex ∗ ! ( f, s X ) ◦ Ex ∗∗ ( f, ∧ ) − : f ∗ ( T h Y ( T f )) ∧ Y T h Y ( − f ∗ ξ ) ∧ Y E Y ) ∼ = −→ f ∗ ( T h Y ( T f )) ∧ X T h X ( − ξ ) ∧ X E X , which induces a canonical isomorphism Ex ∗ ! ( f, s X ) ◦ Ex ∗∗ ( f, ∧ ) − ◦ : SH ( X )[ f ∗ Y , f ∗ ( T h Y ( T f ) ∧ Y T h Y ( − f ∗ ξ ) ∧ E r − d ) , ( r − d ) Y ∧ S p,q )] ∼ = −→ SH ( X )[ f ∗ Y , f ∗ ( T h Y ( T f )) ∧ X T h X ( − ξ ) ∧ X E r − d ) , ( r − d ) X ∧ S p,q ] . From the unit η ( f ∗ ,f ∗ ) ( X ) : X → f ∗ f ∗ X ∼ = f ∗ Y of the adjunction ( f ∗ , f ∗ ) one obtains amap − ◦ η ( f ∗ ,f ∗ ) ( X ) : SH ( X )[ f ∗ Y , f ∗ ( T h Y ( T f )) ∧ X T h X ( − ξ ) ∧ X E r − d ) , ( r − d ) X ∧ S p,q ] −→ SH ( X )[ X , f ∗ ( T h Y ( T f )) ∧ X T h X ( − ξ ) ∧ X E r − d ) , ( r − d ) X ∧ S p,q ] . As f is projective we have f ∗ ∼ = f ! and since f is smooth we have the canonical purityisomorphism p f : f ( − ) ∼ = −→ f ∗ ( T h Y ( T f ) ∧ Y − ) , which induces a canonical isomorphism − ◦ p − f : SH ( X )[ X , f ∗ ( T h Y ( T f )) ∧ X T h X ( − ξ ) ∧ X E r − d ) , ( r − d ) X ∧ S p,q ] ∼ = −→ SH ( X )[ X , f Y ∧ X T h X ( − ξ ) ∧ X E r − d ) , ( r − d ) X ∧ S p,q ] . The counit ε ( f ,f ∗ ) ( X ) : f Y ∼ = f f ∗ X → X of the adjunction ( f , f ∗ ) induces then amap ε ( f ,f ∗ ) ( X ) ◦ : SH ( X )[ X , f Y ∧ X T h X ( − ξ ) ∧ X E r − d ) , ( r − d ) X ∧ S p,q ] −→ SH ( X )[ X , T h X ( ξ ) ∧ X E r − d ) , ( r − d ) X ∧ S p,q ] = E p − d,q − d ( X, ξ ) . So we obtain a map defined as the composition of the maps above SH ( Y )[ Y , T h Y ( T f ) ∧ Y s ! Y p ∗ f ∗ ξ E r − d ) , ( r − d ) Y ∧ S p,q ] → E p − d,q − d ( X, ξ ) . One can check the two constructions are equivalent. And again the Proposition 4.14 tellsus that the composition of pushforward maps f ∗ ◦ g ∗ is only defined up to a specific natural -isomorphism. Proposition 4.24.
Let Z g → Y f → X be smooth projective morphisms of projective smooth k -schemes of relative dimension e resp. d . Let ξ/X be a vector bundle of rank r . Then onehas up to a natural -isomorphism ( f ◦ g ) ∗ = f ∗ ◦ g ∗ : E p,q ( Z, g ∗ f ∗ ξ − T fg ) → E p − d + e ) ,q − ( d + e ) ( X, ξ ) . Proof.
We have an exact sequence of vector bundles on Z [EGA4, 17.2.3] → g ∗ T f → T fg → T g → . So we have an isomorphism (cf. [CD10, Rem. 2.4.52]) e σ : T h Z ( T fg ) ∼ = T h Z ( T g ) ∧ Z T h Z ( g ∗ T f ) ∼ = T h Z ( T g ) ∧ Z g ∗ T h Y ( T f ) , here − ∧ Z − means relative wedge product over Z . Since g ∗ is strong monoidal and sinceall f and g are smooth, which means that the ∧ Z -inverse object of T h Z ( T fg ) is T h Z ( − T fg ) and T h Z ( T g ) − = T h Z ( − T g ) and ( g ∗ T h Y ( T f )) − = g ∗ ( T h Y ( − T f )) (cf. [CD10, Thm. 2.4.50(3)]). Hence we have e ∨ σ : T h Z ( − T fg ) ∼ = T h Z ( − T g ) ∧ Z g ∗ T h Y ( − T f ) . Functoriality of pushforward follows from this isomorphism as follow: We write h = f ◦ g .Let s Z : Z → g ∗ f ∗ ξ be the -section of the vector bundle p g ∗ f ∗ ξ : g ∗ f ∗ ξ → Z . Let us recallthe notation now: For an adjunction between categories L : A ⇆ B : R, we denote ε ( L,R ) : LR → id , η ( L,R ) : id → RL the counit and unit of the adjunction ( L, R ) respectively. The composition f ∗ ◦ g ∗ is byconstruction the following composition: E p,q ( Z, g ∗ f ∗ ξ − T fg ) def = SH ( Z )[ Z , T h Z ( T fg ) ∧ Z s ! Z p ∗ g ∗ f ∗ ξ E r − d + e ) ,r − ( d + e ) Z ∧ S p,q ] (1) → SH ( Z )[ T h Z ( g ∗ f ∗ ξ ) ∧ Z T h Z ( − T fg ) , E r − d + e ) ,r − ( d + e ) Z ∧ S p,q ] (2) → SH ( Y )[ g ( T h Z ( g ∗ f ∗ ξ ) ∧ Z T h Z ( − T fg )) , E r − d + e ) ,r − ( d + e ) Y ∧ S p,q ] (3) → SH ( Y )[ T h Y ( f ∗ ξ ) ∧ Y g T h Z ( − T fg ) , E r − d + e ) ,r − ( d + e ) Y ∧ S p,q ] (4) → SH ( Y )[ T h Y ( f ∗ ξ ) ∧ Y g ( T h Z ( − T g ) ∧ Z g ∗ T h Y ( − T f )) , E r − d + e ) ,r − ( d + e ) Y ∧ S p,q ] (5) → SH ( Y )[ T h Y ( f ∗ ξ ) ∧ Y T h Y ( − T f ) ∧ Y g T h Z ( − T g ) , E r − d + e ) ,r − ( d + e ) Y ∧ S p,q ] (6) → SH ( Y )[ T h Y ( f ∗ ξ ) ∧ Y T h Y ( − T f ) ∧ D Y ( g Z ) , E r − d + e ) ,r − ( d + e ) Y ∧ S p,q ] (7) → SH ( Y )[ T h Y ( f ∗ ξ ) ∧ Y T h Y ( − T f ) , g ( Z ) ∧ Y E r − d + e ) ,r − ( d + e ) Y ∧ S p,q ] (8) → SH ( Y )[ T h Y ( f ∗ ξ ) ∧ Y T h Y ( − T f ) , E r − d + e ) ,r − ( d + e ) Y ∧ S p,q ] (9) → SH ( X )[ f ( T h Y ( f ∗ ξ ) ∧ Y T h Y ( − T f )) , E r − d + e ) ,r − ( d + e ) X ∧ S p,q ] (10) → SH ( X )[ T h X ( ξ ) ∧ X f T h Y ( − T f ) , E r − d + e ) ,r − ( d + e ) X ∧ S p,q ] (11) → SH ( X )[ T h X ( ξ ) ∧ X D X ( f Y ) , E r − d + e ) ,r − ( d + e ) X ∧ S p,q ] (12) → SH ( X )[ T h X ( ξ ) , f ( Y ) ∧ X E r − d + e ) ,r − ( d + e ) X ∧ S p,q ] (13) → SH ( X )[ T h X ( ξ ) , E r − d + e ) ,r − ( d + e ) X ∧ S p,q ] ∼ = E p − d + e ) ,q − ( d + e ) ( X, ξ ) , where (1) is the natural isomorphism given by the adjunction of the Thom transformations T h ( s Z , p g ∗ f ∗ ξ ) and T h Z ( T fg ) , (2) is the natural isomorphism given by the adjunction ( g , g ∗ )(2)( − ) = ε ( g ,g ∗ ) ( − ) ◦ g ( − ) , (3) is the natural isomorphism given by the projection formula P r ∗ ( g )(3)( − ) = ε ( g ,g ∗ ) ( − ) ◦ g ( η ( g ,g ∗ ( − ) ∧ Z id)( − ) , is the natural isomorphism given by e ∨ σ : T h Z ( − T fg ) ∼ = T h Z ( − T g ) ∧ Z g ∗ T h Y ( − T f ) , (5) is the natural isomorphism given by the projection formula P r ∗ ( g ) , (6) is the naturalisomorphism given by duality in SH ( Y ) : g T h Z ( − T g ) ∼ = D Y ( g Z ) , (7) is the natural isomorphism given by adjunction of duality in SH ( Y )(7)( − ) = (id g Z ∧ − ) ◦ ( coev D Y ( g Z ) ∧ id − ) , (8) is the pushforward induced by the counit g Z ∼ = g g ∗ Y → Y (8) = ε ( g ,g ∗ ) ( − ) ∧ Y − , (9) is the natural isomorphism given by the adjunction ( f , f ∗ )(9) = ε ( f ,f ∗ ) ( − ) ◦ f ( − ) , (10) is the natural isomorphism given by the projection formula P r ∗ ( f )(10)( − ) = ε ( f ,f ∗ ) ( − ) ◦ f ( η ( f ,f ∗ ) ∧ Y id)( − ) , (11) is the natural isomorphism given by duality in SH ( X ) : f T h Y ( − T f ) ∼ = D X ( f Y ) , (12) is the natural isomorphism given by the adjunction of duality in SH ( X )(12)( − ) = (id f Y ∧ − ) ◦ ( coev D X ( f Y ) ∧ id − ) , and finally (13) is the pushforward induced by the counit f Y ∼ = f f ∗ X → X : (13)( − ) = ε ( f ,f ∗ ) ( − ) ∧ X − .h ∗ = ( f ◦ g ) ∗ is the following composition: E p,q ( Z, g ∗ f ∗ ξ ) defn = SH ( Z )[ Z , T h Z ( T fg ) ∧ Z s ! Z p ∗ g ∗ f ∗ ξ E r − d + e ) ,r − ( d + e ) Z ∧ S p,q ] (1 ′ ) → SH ( Z )[ T h Z ( g ∗ f ∗ ξ ) ∧ Z T h Z ( − T fg ) , E r − d + e ) ,r − ( d + e ) Z ∧ S p,q ] (2 ′ ) → SH ( X )[ h ( T h Z ( g ∗ f ∗ ξ ) ∧ Z T h Z ( − T fg )) , E r − d + e ) ,r − ( d + e ) X ∧ S p,q ] (3 ′ ) → SH ( X )[ T h X ( ξ ) ∧ X h T h Z ( − T fg ) , E r − d + e ) ,r − ( d + e ) X ∧ S p,q ] (4 ′ ) → SH ( X )[ T h X ( ξ ) ∧ X D X ( h Z ) , E r − d + e ) ,r − ( d + e ) X ∧ S p,q ] (5 ′ ) → SH ( X )[ T h X ( ξ ) , h ( Z ) ∧ X E r − d + e ) ,r − ( d + e ) X ∧ S p,q ] (6 ′ ) → SH ( X )[ T h X ( ξ ) , E r − d + e ) ,r − ( d + e ) X ∧ S p,q ] ∼ = E p − d + e ) ,q − ( d + e ) ( X, ξ ) , where (1 ′ ) is the natural isomorphism given by the adjunction of the Thom transformations T h ( s Z , p g ∗ f ∗ ξ ) and T h Z ( T fg ) , (2 ′ ) is the natural isomorphism given by the adjunction ( h , h ∗ )(2 ′ )( − ) = ε ( h ,h ∗ ) ( − ) ◦ h ( − ) , (3 ′ ) is the natural isomorphism given by the projection formula P r ∗ ( h )(3 ′ )( − ) = ε ( h ,h ∗ ) ( − ) ◦ h ( η ( h ,h ∗ ) ( − ) ∧ Z id)( − ) , ′ ) is the natural isomorphism by duality in SH ( X ) : h T h Z ( − T fg ) ∼ = D X ( h Z ) , (5 ′ ) is the natural isomorphism given by the adjunction of duality in SH ( X ) : (5 ′ )( − ) = (id h Z ∧ X − ) ◦ ( coev D X ( h Z ) ∧ id − ) and finally (6 ′ ) is the pushforward induced by the counit h Z ∼ = h h ∗ X → X : (6 ′ )( − ) = ε ( h ,h ∗ ) ( − ) ∧ X − . The maps (1) and (1 ′ ) are identical. The diagram • (2) (cid:127) (cid:127) ⑦⑦⑦⑦⑦⑦⑦ (2 ′ ) (cid:31) (cid:31) ❅❅❅❅❅❅❅ • ( a ) / / • commutes, because h ∼ = f ◦ g , where ( a ) : SH ( Y )[ g ( T h Z ( h ∗ ξ ) ∧ Z T h Z ( − T fg )) , E r − d + e ) ,r − ( d + e ) Y ∧ S p,q ] ∼ = → SH ( X )[ h ( T h Z ( h ∗ ξ ) ∧ Z T h Z ( − T fg )) , E r − d + e ) ,r − ( d + e ) X ∧ S p,q ] is the natural isomorphism induced from the adjunction ( f , f ∗ ) . Indeed, let α : T h Z ( h ∗ ξ ) ∧ Z T h ( − T fg ) → E r − d + e ) ,r − ( d + e ) Z ∧ S p,q be a morphism. Then one has (2 ′ )( α ) = ε ( h ,h ∗ ) ( − ) ◦ h ( α ) , and ( a ) ◦ (2)( α ) = ( ε ( f ,f ∗ ) ◦ f ( − )) ◦ ( ε ( g ,g ∗ ) ◦ g ( − ))( α ) . So we have (2 ′ ) = ( a ) ◦ (2) . Consider the pentagon • ( a ) / / (3) (cid:127) (cid:127) ⑦⑦⑦⑦⑦⑦⑦ • (3 ′ ) (cid:31) (cid:31) ❅❅❅❅❅❅❅ • ( a ) ' ' ❖❖❖❖❖❖❖❖❖❖❖❖❖❖ •• ( a ) ♦♦♦♦♦♦♦♦♦♦♦♦♦♦ where ( a ) : SH ( Y )[ T h Y ( f ∗ ξ ) ∧ Y g T h Z ( − T fg ) , E r − d + e ) ,r − ( d + e ) Y ∧ S p,q ] ∼ = → SH ( X )[ f ( T h Y ( f ∗ ξ ) ∧ Y g T h Z ( − T fg )) , E r − d + e ) ,r − ( d + e ) X ∧ S p,q ] is the natural isomorphism induced by the adjunction ( f , f ∗ ) and ( a ) : SH ( X )[ f ( T h Y ( f ∗ ξ ) ∧ Y g T h Z ( − T fg )) , E r − d + e ) ,r − ( d + e ) X ∧ S p,q ] ∼ = → SH ( X )[ T h X ( ξ ) ∧ X h T h ( − T fg ) , E r − d + e ) ,r − ( d + e ) X ∧ S p,q ] s the natural isomorphism given by the projection formula P r ∗ ( f ) . We remind the readerthat the isomorphism in the projection formula P r ∗ ( f ) is given by the composition: f ( M ∧ Y f ∗ N ) → f ( f ∗ f ( M ) ∧ Y f ∗ N ) ≃ f f ∗ ( f M ∧ X N ) → f ∧ X N. The pentagon commutes since isomorphism induced by the projection formula
P r ∗ ( h ) is thecomposing of isomorphisms coming from projection formulas P r ∗ ( g ) and P r ∗ ( f ) . Indeed,let α : g ( T h Z ( h ∗ ξ ) ∧ Z T h Z ( − T fg )) → E r − d + e ) ,r − ( d + e ) Y ∧ S p,q be a morphism. We have (3 ′ )( − ) ◦ ( a )( α ) = (3 ′ )( − )( ε ( f ,f ∗ ) ◦ f ( α )) = P r ∗ ( h )( − )( ε ( f ,f ∗ ) ◦ f ( α ))= ε ( h ,h ∗ ) ( − ) ◦ h ( η ( f ,f ∗ ) ( − ) ∧ Z id)( ε ( f ,f ∗ ) ◦ f ( α )) , and ( a − ) ◦ ( a − ) ◦ (3)( α ) = ( a )( − ) ◦ ( a )( − ) ◦ P r ∗ ( g )( α ) =( a − ) ◦ ( a − ) ◦ ε ( g ,g ∗ ) ( − ) ◦ g ( η g ,g ∗ ) ( − ) ∧ Z id)( α )= ( a − ) ◦ ε ( f ,f ∗ ) ( − ) ◦ f ( − ) ◦ ε ( g ,g ∗ ) ( − ) ◦ g ( η g ,g ∗ ) ( − ) ∧ Z id)( α )= ε ( f ,f ∗ ) ( − ) ◦ f ( − )( η ( f ,f ∗ ) ( − ) ∧ Y id) ◦ ε ( f ,f ∗ ) ( − ) ◦ f ( − ) ◦ ε ( g ,g ∗ ) ( − ) ◦ g ( η g ,g ∗ ) ( − ) ∧ Z id)( α ) . So we have ( a − ) ◦ ( a − ) ◦ (3)( α ) = (3 ′ )( − ) ◦ ( a )( α ) . For any K ∈ SH ( Z ) one has commutative diagram (see [Ay08, §1.4.2, §1.5] and [CD10,Rem. 2.4.52]) h K (cid:15) (cid:15) f g K ∼ = (cid:15) (cid:15) f ! ( T h Y ( T f ) ∧ Y g ! ( T h Z ( T g ) ∧ Z K ) ∼ = (cid:15) (cid:15) f ! g ! ( g ∗ T h Y ( T f ) ∧ Z T h Z ( T g ) ∧ Z K ) e σ ∼ = (cid:15) (cid:15) h ! ( T h Z ( T fg ) ∧ Z K ) f ! g ! ( T h Z ( T fg ) ∧ Z K ) Now we take K = Z and dualize D X ( − ) the commutative diagram above. One has D X ( h ! ( T h Z ( T fg )) ∼ = D X ( h Z ) ∼ = h T h Z ( − T fg ) ,D X ( f ! T h Y ( T f )) ∼ = D X ( f Y ) ∼ = f T h Y ( − T f ) , and D Y ( g ! T h Z ( T g )) = D Y ( g Z ) ∼ = g T h Z ( − T g ) . So we can conclude that (6 ′ ) ◦ (5 ′ ) ◦ (4 ′ ) ◦ ( a ) ◦ ( a ) = (13) ◦ (12) ◦ · · · ◦ (5) ◦ (4) . Putting all together we have (6 ′ ) ◦ · · · ◦ (1 ′ ) = (13) ◦ · · · ◦ (1) , hich means the pushforward on twisted E -cohomology satisfies ( f ◦ g ) ∗ = f ∗ ◦ g ∗ up to anatural isomorphism induced by the natural -isomorphism e ∨ σ : T h − Z ( s Z , p T fg ) ∼ = −→ T h − Z ( s Z , p T g ) ◦ g ∗ T h − Y ( s Y , p T g ) . (cid:3) Now we follow a suggestion by M. Levine to make a refinement to the result of Voevodskyin 2.4, since as pointed out by M. Levine it is not enough to use the identities in K ( − ) toconstructs maps between twisted E -cohomology groups. Proposition 4.25. (A refinement of Voevodsky’s theorem) Let X ∈ SmP roj ( k ) of dimen-sion d X , where k is a field. After fixing an embedding X ֒ → P d there exists a vector bundle V X on X of rank d + 2 d − d X , such that one has a specific isomorphism between objects inthe Picard category of virtual bundles V ( X ) on X : V X ⊕ T X ∼ = O d +2 dX . Proof.
Case 1: X = P d . One has an exact sequence → O P d → O P d (1) ⊕ ( d +1) → T P d → . By taking dual one also has → Ω P d → O P d ( − ⊕ ( d +1) → O P d → . There are two isomorphisms between objects in V ( X ) : O P d ⊕ T P d ∼ = O P d (1) ⊕ ( d +1) and Ω P d ⊕ O P d ∼ = O P d ( − ⊕ ( d +1) . Define V P d defn = Ω P d ⊕ (Ω P d ⊗ T P d ) . As the Picard category V ( X ) = V ( V ect ( X )) (the category of virtual objects associated tothe exact category of vector bundles on X ) has not just ⊕ , but also a biexact functor − ⊗ − : V ( X ) × V ( X ) → V ( X ) , which is distributive ([Del87]), one has an isomorphism in V ( X ) : (Ω P d ⊕ O P d ) ⊗ ( O P d ⊕ T P d ) ∼ = Ω P d ⊕ O P d ⊕ (Ω P d ⊗ T P d ) ⊕ T P d ∼ = O ⊕ ( d +2 d +1) P d . This implies that we have an isomorphism in V ( X ) : V P d ⊕ T P d ∼ = O d +2 d P d . Case 2: X is smooth projective. Let i : X ֒ → P d be a closed embedding. One define V X defn = N X/ P d ⊕ i ∗ ( V P d ) , where N X/ P d denotes the normal bundle of X in P d . In V ( X ) one has an isomorphismbetween objects N X/ P d ⊕ i ∗ ( V P d ⊕ T P d ) ∼ = N X/ P d ⊕ O d +2 dX . From the exact sequence → T X → i ∗ T P d → N X/ P d → ne has an isomorphism in V ( X ) : T X ⊕ N X/ P d ∼ = i ∗ T P d . This implies that we have a isomorphism in V ( X ) : N X/ P d ⊕ i ∗ V P d ⊕ T X ⊕ N X/ P d ∼ = N X/ P d ⊕ O d +2 dX . This implies that we have a specific isomorphism in V ( X ) : V X ⊕ T X ∼ = O d +2 dX . (cid:3) P. Hu in [Hu05] didn’t check if her construction is the same as the construction of Vo-evodsky. We notice that the map constructed by Voevodsky [Voe03, Thm. 2.11] T ∧ n X + d X → T h ( V X ) is first of all only in Ho A , + ( k ) and secondly very difficult to follow. We will take therefinement V X ⊕ T X ∼ = −→ O d +2 dX in V ( X ) and construct the Pontryagin-Thom collapse map P T V : S → Σ ∞ T, + T h ( V X ) ∧ S − d +2 d ) , − ( d +2 d ) by unpacking Hu’s construction. Firstly, for a projective smooth k -variety i : X ֒ → P d , wehave by definition V X = N X/ P d ⊕ i ∗ V P d ∼ = N X/V P d . If P T V is already for P d constructed, then P T V for X is defined by the composition S −→ Σ ∞ T, + T h ( V P d ) ∧ S − d +2 d ) , − ( d +2 d ) q −→ Σ ∞ T, + T h ( V X ) ∧ S − d +2 d ) , − ( d +2 d ) , where q is the quotient map T h ( V P d ) → V P d / ( V P d − X ) ∼ = −→ T h ( N X/V P d ) = T h ( V X ) . The isomorphism V P d / ( V P d − X ) ∼ = −→ T h ( N X/V P d ) is the homotopy purity isomorphism([MV01, §3 Thm.2.23]). For X = P one has a commutative diagram in Ho A , + ( k ) ([Hu05,pp. 9]) (( X × X ) − ∆ X ) + / / pr ∼ = (cid:15) (cid:15) ( X × X ) + g / / T h ( T X ) X + f ✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐ because pr : ( X × X ) − ∆ X → X is an affine bundle. So one has a cofiber sequence in Ho A , + ( k ) X + f −→ ( X × X ) + g −→ T h ( T X ) . For a vector bundle ξ on X one has T h ( pr ∗ ξ/X × X ) = T h ( ξ/X ) ∧ X + . ne the other hand one has commutative diagram ([Hu05, (3.13)]) T h ( pr ∗ ξ/X × X − ∆ X ) ∼ = pr (cid:15) (cid:15) / / T h ( pr ∗ ξ/X × X ) pr (cid:15) (cid:15) T h ( ξ/X ) T h ( ξ/X ) So one obtains a cofiber sequence in Ho A , + ( k ) T h ( ξ/X ) f ξ −→ T h ( ξ/X ) ∧ X + g ξ −→ T h ( T X ⊕ ξ ) . Now we take ξ = V X and by the refinement V X ⊕ T X ∼ = −→ O d +2 dX in V ( X ) we have then acofiber sequence T h ( V X ) f VX −→ T h ( V X ) ∧ X + g VX −→ T h ( V X ⊕ T X ) ∼ = S d +2 d ) , ( d +2 d ) ∧ X + . This gives rise to a map in SH ( k ) : ε : Σ ∞ T, + T h ( V X ) ∧ X + ∧ S − d +2 d ) , − ( d +2 d ) g VX −→ Σ ∞ T, + X → S . To construct the Pontryagin-Thom collapse map
P T V : S → Σ ∞ T, + T h ( V X ) ∧ S − d +2 d ) , − ( d +2 d ) , such that the composition g V X ◦ ( P T V ∧ id) : S ∧ Σ ∞ T, + X P T V ∧ id −→ Σ ∞ T, + T h ( V X ) ∧ X + ∧ S − d +2 d ) , − ( d +2 d ) g VX −→ Σ ∞ T, + X is the identity id Σ ∞ T, + X in SH ( k ) , it is enough to construct a map P T V : S → Σ ∞ T, + T h ( V X ) ∧ S − d +2 d ) , − ( d +2 d ) , such that the composition ε ◦ ( P T V ∧ id) is the collapse map Σ ∞ T, + X → S , because g V X isthe composition Σ ∞ T, + T h ( V X ) ∧ X + ∧ S − d +2 d ) , − ( d +2 d ) id ∧ ∆ −→ Σ ∞ T, + T h ( V X ) ∧ X + ∧ X + ∧ S − d +2 d ) , − ( d +2 d ) ε ∧ id −→ Σ ∞ T, + X By adjunction ε gives us a map λ X : Σ ∞ T, + T h ( V X ) ∧ S − d +2 d ) , − ( d +2 d ) → D k (Σ ∞ T, + ( X )) = Hom(Σ ∞ T, + X, S ) . We remind the reader that P. Hu started with X = P before [Hu05, Lem. 3.8], since shewanted to prove some particular results for projective quadric. For general X and a vectorbundle ξ on X one still has the map g : ( X × X ) + → T h ( T X ) = T h ( N X/X × X ) and hence a map g ξ : T h ( pr ∗ ξ/X × X ) = T h ( ξ/X ) ∧ X + → T h ( T X ⊕ ξ ) nd hence by applying ξ = V X one has a map λ X : Σ ∞ T, + T h ( V X ) ∧ S − d +2 d ) , − ( d +2 d ) → Hom(Σ ∞ T, + X, S ) . Now we consider the linear embedding i : P d ֒ → P d +1 . By construction V P d +1 = N P d / P d +1 ⊕ i ∗ ( V P d ) and so the diagram(4.2) Σ ∞ T, + T h ( V P d +1 ) ∧ S − d +1) +2( d +1)) , − (( d +1) +2( d +1)) q P d +1 P d (cid:15) (cid:15) λ P d +1 / / D ( P d +1+ ) D ( i ) (cid:15) (cid:15) Σ ∞ T, + T h ( V P d ) ∧ S − d +2 d ) , − ( d +2 d ) λ P d / / D ( P d + ) commutes, since it is adjoint to the commutativity of the diagram T h ( V P d +1 ) ∧ P d + ∧ S ⋆,⋆q P d +1 P d ∧ id (cid:15) (cid:15) id ∧ i + / / T h ( V P d +1 ) ∧ P d +1 ∧ S ⋆,⋆g V P d +1 (cid:15) (cid:15) T h ( V P d ) ∧ P d + ∧ S ∗ , ∗ i + ◦ g V P d / / P d +1+ where we write q P d +1 P d for the quotient map and the last commutative diagram is obtained by V P d +1 -Thomification (i.e. we apply g V P d +1 on P d +1 × P d +1 → T h ( T P d +1 ) ) of the commutativediagram ( P d +1 × P d ) + i / / (cid:15) (cid:15) ( P d +1 × P d +1 ) + (cid:15) (cid:15) ( P d +1 × P d ) + / (( P d +1 × P d ) − ( P d × P d )) + (cid:15) (cid:15) ( P d +1 × P d ) + / (( P d +1 × P d ) − ∆ P d ) + / / ( P d +1 × P d +1 ) + / (( P d +1 × P d +1 ) − ∆ P d +1 ) + Consider the composition ( P d +1 − P d ) × P d +1 → P d +1 × P d +1 → ( P d +1 × P d +1 ) / ( P d +1 × P d +1 − ∆ P d +1 ) . ( P d +1 − P d ) × P d is mapped to ( P d +1 × P d +1 ) − ∆ P d +1 . So the composition above induces amap T h ( j ∗ V P d +1 ) ∧ ( P d +1 / P d ) + ∧ S ⋆,⋆ → P d +1+ , where j : ( P d +1 − P d ) ֒ → P d +1 denotes the open immersion. After composing with the collapsemap P d +1+ → S and taking adjoint one obtains a map λ P d +1 / P d : T h ( j ∗ V P d +1 ) ∧ S ⋆,⋆ → D (( P d +1 / P d ) + ) . y construction there is a commutative diagram in SH ( k ) :(4.3) T h ( j ∗ V P d +1 ) ∧ S ⋆,⋆ λ P d +1 / P d / / T h ( j ) (cid:15) (cid:15) D (( P d +1 / P d ) + ) D ( p ) (cid:15) (cid:15) T h ( V P d +1 ) ∧ S ⋆,⋆ λ P d +1 / / D ( P d +1+ ) Now the Claim in the proof of [Hu05, Lem. 3.8] implies that there is a morphism ofdistinguished triangles given by the commutative diagrams 4.2 and 4.3. S − d, − d (cid:15) (cid:15) ∼ = / / S − d, − d (cid:15) (cid:15) T h ( V P d +1 ) ∧ S ⋆,⋆ λ P d +1 / / (cid:15) (cid:15) D ( P d +1+ ) (cid:15) (cid:15) T h ( V P d ) ∧ S ∗ , ∗ λ P d / / D ( P d + ) So by induction on d one can conclude that λ P d : T h ( V P d ) ∧ S − d +2 d ) , − ( d +2 d ) → D ( P d + ) is an isomorphism in SH ( k ) for all d ≥ . Now we can construct the Pontryagin-Thomcollapse map P T V : S → D ( P d + ) λ − P d −→ T h ( V P d ) ∧ S − d +2 d ) , − ( d +2 d ) . If X ֒ → P d is a smooth projective k -variety, we define P T V for X as the composition of P T V for P d with the quotient map T h ( V d ) ∧ S − d +2 d ) , − ( d +2 d ) → T h ( V X ) ∧ S − d +2 d ) , − ( d +2 d ) . So by construction we have:
Proposition 4.26.
Let X be a smooth projective k -scheme. After fixing an embedding i : X ֒ → P d , there is a commutative diagram in SH ( k ) S P T V / / P T H * * ❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯ Σ ∞ T, + T h ( V X ) ∧ S − d +2 d ) , − ( d +2 d ) ∼ = (cid:15) (cid:15) Σ ∞ T, + T h ( − T X ) where P T H : S → Σ ∞ T, + T h ( − T X ) is the map constructed in [Hu05, Lem. 3.18] . Proposition 4.27.
Let f : Y → X be a projective smooth morphism of projective smooth k -schemes of relative dimension d = d Y − d X . After fixing an embedding X ֒ → P N , there isan isomorphism tt YE : E p,q ( Y, f ∗ V X − T f ) ∼ = E p +2 n Y ,q + n Y ( T h ( V Y )) , where V X and V Y are vector bundles on X and Y of rank n X and n Y as in theorem 2.4respectively with the refinement in the Proposition 4.25. Moreover, the isomorphism tt YE is ndependent from the choice of the projective embeddings up to a unique canonical isomor-phism.Proof. Let us denote by s Y the -section of the vector bundle p f ∗ V X : f ∗ V X → Y and by s ′ Y : Y → T f the -section of the relative tangent bundle. Let V ( X ) and V ( Y ) be thecategories of virtual bundles on X and Y respectively. We have E p,q ( Y, f ∗ V X − T f ) defn = SH ( Y )[ Y , T h Y ( T f ) ∧ Y s ! Y p ∗ f ∗ V X E n X + d X − d Y ) ,n X + d X − d Y Y ∧ S p,q ] ∼ = SH ( Y )[ p f ∗ V X s Y ! T h Y ( − T f ) , E n X + d X − d Y ) ,n X + d X − d Y Y ∧ S p,q ] ∼ = SH ( Y )[ T h Y ( f ∗ V X ) ∧ Y T h Y ( − T f ) , E n X + d X − d Y ) ,n X + d X − d Y Y ∧ S p,q ] ∼ = SH ( Y )[ T h Y ( f ∗ V X − T f ) , E n X + d X − d Y ) ,n X + d X − d Y Y ∧ S p,q ] , where we write T h Y ( f ∗ V X − T f ) for the Thom transformation T h Y ( f ∗ V X − T f ) = T h ( s Y , p f ∗ V X ) ◦ T h − ( s ′ Y , p T f )( Y ) . Let π Y : Y → Spec k be the structure morphism of Y . By the adjunction ( π Y , π ∗ Y ) andsince E Y = π ∗ Y E , we have then SH ( Y )[ T h Y ( f ∗ V X − T f ) , E n X + d X − d Y ) ,n X + d X − d Y Y ∧ S p,q ] ∼ = SH ( k )[ T h ( f ∗ V X − T f ) , E n X + d X − d Y ) ,n X + d X − d Y ∧ S p,q ] , which comes from the fact that (cf. [Ay08, Thm. 1.5.9] and [Ay08, Rem. 1.5.10]): π Y ( T h ( s Y , p f ∗ V X ) ◦ T h − ( s ′ Y , T f ))( Y ) ∼ = T h ( f ∗ V X − T f ) . Now we apply the Voevodsky’s theorem 2.4 with a refinement as in Proposition 4.25. Afterfixing an embedding
X ֒ → P N we have in V ( X ) : V X ⊕ T X ∼ = O N +2 NX . Since f is projective, there is a factorization Y f / / $ $ ■■■■■■■■■ X P M × k X pr : : ✉✉✉✉✉✉✉✉✉ where Y ֒ → P M × k X is a closed immersion. We take then the Segre embedding Y ֒ → P M × P N ֒ → P ( N +1)( M +1) − . and apply the Proposition 4.25, so we have in V ( Y ) a specific isomorphism V Y ⊕ T Y ∼ = O (( N +1)( M +1) − +2(( N +1)( M +1) − Y . One has a functor [Del87, §4] f ∗ : V ( X ) → V ( Y ) . Since f is smooth we have an exact sequence → f ∗ T X → T Y → T f → , hich gives rise to an isomorphism in V ( Y ) f ∗ T X ⊕ T f ∼ = T Y . So we have then in V ( Y ) an isomorphsim f ∗ V X − T f ∼ = −→ V Y + O ( n X + d X ) − ( n Y + d Y ) Y , where − means + the opposite object as explained in [Del87] and n X + d X = N + 2 Nn Y + d Y = (( N + 1)( M + 1) − + 2(( N + 1)( M + 1) − . Now since
T h defines a functor (cf. [Rio10, Def. 4.1.2])
T h : V ( Y ) → SH ( k ) , where V ( Y ) is the category of virtual bundles on Y , we can conclude that there is canonicalisomorphism T h ( f ∗ V X − T f ) ∼ = T h ( V Y ) ∧ S n X + d X ) − n Y + d Y ) , ( n X + d X ) − ( n Y + d Y ) , where the right hand side is by 4.26 canonical isomorphic to D ( Y + ) ∧ S n X + d X ) , ( n X + d X ) . Sowe can conclude that there is an isomorphism tt YE : E p,q ( Y, f ∗ V X − T f ) ∼ = SH ( k )[ T h ( V Y ) , E n Y ,n Y ∧ S p,q ] = E p +2 n Y ,q + n Y ( T h ( V Y )) . Now we have to show that this isomorphism is independent from the projective embeddingsup to a unique canonical isomorphism. Let
Y ֒ → P N ′ be any closed embedding. Then westill have a canonical isomorphism T h ( f ∗ V ′ X − T f ) ∼ = D ( Y + ) ∧ S − ∗ , −∗ As D ( Y + ) is unique up to a canonical isomorphism we can conclude that tt YE is independentfrom the choice of the embeddings up to a unique canonical isomorphism. (cid:3) Remark 4.28.
As pointed out by M. Levine, one can simplify the arguments in the Propo-sition above by using the maps S → S − d − d ∧ Σ ∞ T, + T h ( V X ) ∧ Σ ∞ T, + X and S − d − d ∧ Σ ∞ T, + T h ( V X ) ∧ Σ ∞ T, + X → S , which rigidify the situation considerably. Remark 4.29.
We will show later that with the refinement of
T h ( V X ) as in 4.26 the iso-morphism tt YE is natural in sense that it is compatible with duality. Remark 4.30.
The isomorphism tt YE ( V Y ) in 4.27 is a natural candidate for a replacementof the twisted Thom isomorphism th E in 4.11 in case of E -cohomology twisted by formaldifference of vector bundles. But we should remind the reader that we can only computefor a very particular case, namely ξ = V X , where V X is the vector bundle as in Voevodsky’stheorem 2.4.Now we can compare: orollary 4.31. Let f : Y → X be a smooth projective morphism of projective smooth k -schemes. There is an isomorphism up to a natural isomorphisms induced by the naturalcanonical isomorphism between duals E p,q ( Y, f ∗ V X − T f ) tt YE → E p +2 n Y ,q + n Y ( T h ( V Y )) th YE ( V Y ) − −→ E p,q ( Y, V Y ) . Proof.
This is a consequence of 4.11 and 4.27. (cid:3)
Remark 4.32.
The Corollary 4.31 is a surprising fact to us. At a first glance we have theimpression that E p,q ( Y, f ∗ V X − T f ) should depend relatively wrt. f and X . At the end itturns out that E p,q ( Y, f ∗ V X − T f ) is isomorphic to E p,q ( Y, V Y ) , which depends absolutelyonly on Y . But it is clear that this is not the case for a general vector bundle ξ .Let f : Y → X be a smooth projective morphism of smooth projective k -schemes of dimen-sion d Y and d X respectively. By Atiyah-Spanier-Whitehead duality and by the Proposition4.26 we obtain its dual morphism in SH ( k ) : f ∨ : X ∨ = Σ ∞ T, + T h ( V X ) ∧ S − n X + d X ) , − ( n X + d X ) → Y ∨ = Σ ∞ T, + T h ( V Y ) ∧ S − n Y + d Y ) , − ( n Y + d Y ) , where V X and V Y are vector bundles on X and Y of rank n X and n Y as in theorem 2.4with a refinement in 4.25 respectively. By taking pullback of this map on E -cohomology andappyling Thom isomorphism we obtain a pushforward E p +2 d Y ,q + d Y ( Y, V Y ) th E ∼ = E p +2( n Y + d Y ) ,q +( n Y + d Y ) ( T h ( V Y )) ( f ∨ ) ∗ −→ E p +2( n X + d X ) ,q +( n X + d X ) ( T h ( V X )) th E ∼ = E p +2 d X ,q + d X ( X, V X ) . We show that the two pushforwards are the same and the isomorphism tt YE is natural in thesense that it is compatible with the duality in the following: Proposition 4.33.
Let f : Y → X be a smooth projective k -morphism between smoothprojective k -schemes. One has a commutative diagram up to natural isomorphisms inducedby natural -isomorphisms and the natural canonical isomorphism between duals E p +2 d Y ,q + d Y ( Y, f ∗ V X − T f ) f ∗ / / ∼ = tt YE (cid:15) (cid:15) E p +2 d X ,q + d X ( X, V X ) ∼ = th XE ( V X ) (cid:15) (cid:15) E p +2( n Y + d Y ) ,q +( n Y + d Y ) ( T h ( V Y )) ( f ∨ ) ∗ / / E p +2( n X + d X ) ,q +( n X + d X ) ( T h ( V X )) Proof.
Let us denote by p V X : V X → X the duality vector bundle on X (cf. 2.4) withthe zero-section s X : X → V X and s Y : Y → f ∗ V X the -section of the pullback bundle f ∗ V X : f ∗ V X → Y . By construction, the first pushforward map is the following composition: f ∗ : E p +2 d Y ,q + d Y ( Y, f ∗ V X − T f ) defn = SH ( Y )[ Y , T h Y ( T f ) ∧ Y s ! Y p ∗ f ∗ V X E n X + d X ) , ( n X + d X ) Y ∧ S p,q ] (1) ∼ = SH ( Y )[ p f ∗ V X s Y ! T h Y ( − T f ) , E n X + d X ) , ( n X + d X ) Y ∧ S p,q ] (2) ∼ = SH ( Y )[ T h Y ( f ∗ V Y ) ∧ Y T h Y ( − T f ) , E n X + d X ) , ( n X + d X ) Y ∧ S p,q ] (3) ∼ = SH ( X )[ f ( T h Y ( f ∗ V Y ) ∧ Y T h Y ( − T f )) , E n X + d X ) , ( n X + d X ) X ∧ S p,q ] (4) ∼ = SH ( X )[ T h X ( V X ) ∧ X f T h Y ( − T f ) , E n X + d X ) X ∧ S p,q ] (5) ∼ = SH ( X )[ T h X ( V X ) ∧ X D X ( f Y ) , E n X + d X ) , ( n X + d X ) X ∧ S p,q ] (6) ∼ = SH ( X )[ T h X ( V X ) , f ( Y ) ∧ X E n X + d X ) , ( n X + d X ) X ∧ S p,q ] (7) → SH ( X )[ T h X ( V X ) , E n X + d X ) , ( n X + d X ) X ∧ S p,q ] th XE ( V X ) − ∼ = E p +2 d X ,q + d X ( X, V X ) , where (1) is the natural isomorphism induced by adjunction of Thom transformations, (2) is the natural isomorphism given by composing Thom transformations, (3) is the naturalisomorphism given by the adjunction ( f , f ∗ )(3)( − ) = ε ( f ,f ∗ ) ( − ) ◦ f ( − ) , (4) is the natural isomorphism given by projection formula P r ∗ ( f )(4)( − ) = ε ( f ,f ∗ ) ( − ) ◦ f ( η ( f ,f ∗ ) ( − ) ∧ Y id)( − ) , (5) is the natural isomorphism induced by f T h Y ( − T f ) ∼ = D X ( f Y ) , (6) is the naturalisomorphism induced by adjunction of duality in SH ( X ) : (6)( − ) = (id D X ( f Y ) ∧ X − ) ◦ ( coev f Y ∧ id − ) , and finally (7) is the pushforward induced by the counit η ( f ,f ∗ ) : f ( Y ) ∼ = f f ∗ ( X ) → X : (7)( − ) = ε ( f ,f ∗ ) ( − ) ∧ X − . The last isomorphism is the inverse of the twisted Thom isomorphism. So we have th XE ( V X ) ◦ f ∗ = (7) ◦ · · · ◦ (1) . he map tt YE is the following composition tt YE : E p +2 d Y ,q + d Y ( Y, f ∗ T X − T f ) defn = SH ( Y )[ Y , T h Y ( T f ) ∧ Y s ! Y p ∗ f ∗ V X E n X + d X ) , ( n X + d X ) Y ∧ S p,q ] (1 ′ ) ∼ = SH ( Y )[ p f ∗ V X s Y ! T h Y ( − T f ) , E n X + d X ) , ( n X + d X ) Y ∧ S p,q ] (2 ′ ) ∼ = SH ( Y )[ T h Y ( f ∗ V X ) ∧ Y T h Y ( − T f ) , E n X + d X ) , ( n X + d X ) Y ∧ S p,q ] (3 ′ ) ∼ = SH ( Y )[ T h Y ( f ∗ V X − T f ) , E n X + d X ) , ( n X + d X ) Y ∧ S p,q ] (4 ′ ) ∼ = SH ( Y )[ T h Y ( V Y ) ∧ S n X + d X ) − n Y + d Y ) , ( n X + d X ) − ( n Y + d Y ) , E n X + d X ) , ( n X + d X ) Y ∧ S p,q ] (5 ′ ) ∼ = SH ( Y )[ T h Y ( V Y ) , E n Y + d Y ) , ( n Y + d Y ) Y ∧ S p,q ] (6 ′ ) ∼ = E p +2( n Y + d Y ) ,q +( n Y + d Y ) ( T h ( V Y )) , where (1 ′ ) = (1) , (2 ′ ) = (2) , (3 ′ ) is the natural isomorphism induced by composing Thomtransformations, (4 ′ ) is induced by the isomorphism in V ( Y ) : f ∗ V Y − T f ∼ = V Y + O ( n X + d X ) − ( n Y + d Y ) Y , (5 ′ ) is the cancellation in SH ( Y ) and finally (6 ′ ) is the natural isomorphism induced by theadjunction ( π Y , π ∗ Y ) with π Y : Y → Spec k is the structure morphism of Y : (6 ′ )( − ) = ε ( π Y ,π ∗ Y ) ( − ) ◦ π Y ( − ) . Let us consider the diagram Y f / / π Y ❋❋❋❋❋❋❋❋❋ X π X { { ①①①①①①①①① Spec k We have [ SH ( Y )[ T h Y ( f ∗ V X − T f ) , E n X + d X ) , ( n X + d X ) Y ∧ S p,q ] ∼ = −→ SH ( X )[ f T h Y ( f ∗ V X − T f ) , E n X + d X ) , ( n X + d X ) X ∧ S p,q ] , where the natural isomorphism is induced by the adjunction ( f , f ∗ ) as f is smooth: ε ( f ,f ∗ ) ◦ f ( − ) . By the projection formula
P r ∗ ( f ) we have a natural isomorphism SH ( X )[ f T h Y ( f ∗ V X − T f ) , E n X + d X ) , ( n X + d X ) X ∧ S p,q ] ∼ = −→ SH ( X )[ T h X ( V X ) ∧ X f T h Y ( − T f ) , E n X + d X ) X ∧ S p,q ] , which is explicitly written as ε ( f ,f ∗ ) ( − ) ◦ f ( η ( f ,f ∗ ) ( − ) ∧ Y id)( − ) . But since f is smooth and projective p f : f T h Y ( − T f ) ∼ = −→ f ∗ ( Y ) ∼ = f ∗ f ∗ ( X ) . hanks to Proposition 4.26 the composition ( f ∨ ) ∗ ◦ tt YE on E -cohomology is nothing butjust the composition of natural isomorphisms above with the map induced by the unit X → f ∗ f ∗ X of the adjunction ( f ∗ , f ∗ ) : ◦ η ( f ∗ ,f ∗ ) : SH ( X )[ T h X ( V X ) ∧ X f ∗ f ∗ ( X ) , E n X + d X ) , ( n X + d X ) X ∧ S p,q ] −→ SH ( X )[ T h X ( V X ) , E n X + d X ) , ( n X + d X ) X ∧ S p,q ] , then followed by the natural isomorphism induced from the adjunction ( π X , π ∗ X ) : ε ( π X ,π ∗ X ) ◦ π X : SH ( X )[ T h X ( V X ) , E n X + d X ) , ( n X + d X ) X ∧ S p,q ] ∼ = −→ SH ( k )[ π X T h X ( V X ) , E n X + d X ) , ( n X + d X ) ∧ S p,q ] = E p +2( n X + d X ) ,q +( n X + d X ) ( T h ( V X )) . Indeed, by the very construction of the operations formalism [Ay08, Thm. 4.5.23], thestabilization functor Σ ∞ T, + : Sm/k → SH ( k ) induces a morphism in SH ( k ) : Σ ∞ T, + ( f ) : Σ ∞ T, + Y → Σ ∞ T, + X, which can be understood as a morphism π Y ( Y ) → π X ( X ) , which in turn is the compo-sition π X ◦ ε ( f ,f ∗ ) ( X ) . In terms of six operations and by the Proposition 4.26, the dualobjects in SH ( k ) are: T h ( V X ) ∧ S − n X + d X ) , − ( n X + d X ) ∼ = −→ X ∨ = D k ( π X X ) , and T h ( V Y ) ∧ S − n Y + d Y ) , − ( n Y + d Y ) ∼ = −→ Y ∨ = D k ( π Y Y ) . Hence the pullback map ( f ∨ ) ∗ is just the pullback SH ( k )[ − , E ∗ , ∗ ∧ S ∗ , ∗ ] of the map D k ( π X ◦ ε ( f ,f ∗ ) ) . We have to check that (7) ◦ · · · ◦ (3) = D k ( π X ◦ ε ( f ,f ∗ ) ) ◦ (6 ′ ) ◦ · · · ◦ (3 ′ ) , which means that we have to check D k ( π X ◦ ε ( f ,f ∗ ) ( − )) ◦ ε ( π Y ,π ∗ Y ) ◦ π Y ( − ) = ( ε ( f ,f ∗ ) ( − ) ∧ X − ) ◦ (id D X ( f Y ) ∧ X − ) ◦ ( coev f Y ∧ X id − ) ◦ ε ( f ,f ∗ ) ( − ) ◦ f ( η ( f ,f ∗ ) ( − ) ∧ Y id)( − ) ◦ ε ( f ,f ∗ ) ◦ f ( − ) . But this is clear, since for a smooth projective morphism π : T → S one has a natural -isomorphism D S ( π ∗ ( − )) ∼ = −→ π ∗ D T ( − ∧ T T h T ( T f )) , which is the composition D S ( f ∗ ( − )) = Hom S ( π ∗ ( − ) , S ) ∼ = −→ Hom S ( π ! ( − ) , S ) ∼ = −→ π ∗ Hom T (( − ) , π ! S ) ∼ = −→ π ∗ Hom T (( − ) , π ∗ S ∧ T T h T ( − T f )) ∼ = −→ π ∗ Hom T (( − ) ∧ T T h T ( T f ) , T ) = π ∗ D T ( −∧ T T h T ( T f )) . The equality, which we need to check above, follows simply from this fact and from the factthat, we have a natural -isomorphism f T h Y ( s Y , p f ∗ V X ) ∼ = −→ T h X ( s X , p V X ) f as f is assumed to be smooth (cf. [Ay08, Thm. 1.5.9]). (cid:3) et us construct the pullback for twisted E -cohomology of formal difference of vectorbundles along a cartesian square. Let Y ′ g (cid:15) (cid:15) v / / Y f (cid:15) (cid:15) X ′ u / / X be a cartesian square of projective smooth k -schemes, where f is smooth projective of relativedimension d = dim( Y ) − dim( X ) and u is any morphism. Let pi ξ : ξ → X be a vector bundleof rank r and denote by s Y : Y → f ∗ ξ the -section of the pullback bundle p f ∗ ξ : f ∗ ξ → Y .Consider E p,q ( Y, f ∗ ξ − T f ) defn = SH ( Y )[ Y , T h Y ( T f ) ∧ Y s ! Y p ∗ f ∗ ξ E r − d ) , ( r − d ) Y ∧ S p,q ] . By adjunction of Thom transformation we have SH ( Y )[ Y , T h Y ( T f ) ∧ Y s ! Y p ∗ f ∗ ξ E r − d ) , ( r − d ) Y ] ∼ = −→ SH ( Y )[ T h Y ( f ∗ ξ ) , T h Y ( T f ) ∧ Y E r − d ) , ( r − d ) Y ] , where the isomorphism is ev T h Y ( f ∗ ξ ) ◦ (id T h Y ( f ∗ ξ ) ∧ − ) By applying the functor v ∗ : SH ( Y ) → SH ( Y ′ ) we have an induced map E p,q ( Y, f ∗ ξ − T f ) → SH ( Y ′ )[ v ∗ T h Y ( f ∗ ξ ) , v ∗ ( T h Y ( T f ) ∧ Y E r − d ) , ( r − d ) Y ∧ S p,q )] . We have v ∗ T h Y ( f ∗ ξ ) ∼ = T h Y ′ ( v ∗ f ∗ ξ ) = T h Y ′ ( g ∗ u ∗ ξ ) as ξ is an actual bundle. Since v ∗ is amonoidal functor, so we have v ∗ ( T h Y ( T f ) ∧ Y E r − d ) , ( r − d ) Y ) ∼ = v ∗ T h Y ( T f ) ∧ Y ′ E r − d ) , ( r − d ) Y ′ . But we have v ∗ T h Y ( T f ) ∼ = T h Y ′ ( v ∗ T f ) , since T f is an actual bundle. By [EGA4, 16.5.12.2]one has v ∗ T f ∼ = T g , so v ∗ T h Y ( T f ) ∼ = T h Y ′ ( T g ) . So we obtain the pullback map for twisted E -cohomology of formal difference of vector bundle E p,q ( Y, f ∗ ξ − T f ) → E p,q ( Y ′ , v ∗ f ∗ ξ − T g ) = E p,q ( Y ′ , g ∗ u ∗ ξ − T g ) . Proposition 4.34.
Consider the composition of cartesian squares of smooth projective k -schemes Y ′′ v ′ / / h (cid:15) (cid:15) Y ′ v / / g (cid:15) (cid:15) Y f (cid:15) (cid:15) X ′′ u ′ / / X ′ u / / X where f is a smooth projective morphism, u and u ′ are morphisms. Let ξ be a vector bundleon X . Then up to natural isomorphisms induced by the natural -isomorphism induced by ( − ◦ − ) ∗ ∼ = −→ ( − ) ∗ ◦ ( − ) ∗ one has ( v ◦ v ′ ) ∗ = v ′∗ ◦ v ∗ : E p,q ( Y, f ∗ ξ − T f ) → E p,q ( Y ′′ , v ′∗ v ∗ f ∗ ξ − T f ) = E p,q ( Y ′′ , h ∗ u ′∗ u ∗ ξ − T g ) . Proof.
Obvious. (cid:3) roposition 4.35. (projective smooth base change) Consider a cartesian square of projectivesmooth k -schemes Y ′ g (cid:15) (cid:15) v / / Y f (cid:15) (cid:15) X ′ u / / X where f is smooth projective of relative dimension d = dim( Y ) − dim( X ) and u is a mor-phism. Let ξ/X be a vector bundle of rank r . One has a commutative diagram up to naturalisomorphisms induced by natural -isomorphisms E p,q ( Y, f ∗ ξ − T f ) f ∗ / / v ∗ (cid:15) (cid:15) E p − d,q − d ( X, ξ ) u ∗ (cid:15) (cid:15) E p,q ( Y ′ , g ∗ u ∗ ξ − T g ) g ∗ / / E p − d,q − d ( X ′ , u ∗ ξ ) Proof.
It is quite straightforward. We write s Y : Y → f ∗ ξ and s Y ′ : Y → v ∗ f ∗ ξ for the -sections of the vector bundles p f ∗ ξ : f ∗ ξ → Y and p v ∗ f ∗ ξ : v ∗ f ∗ ξ → Y ′ respectively. u ∗ f ∗ isthe following composition E p,q ( Y, f ∗ ξ − T f ) defn = SH ( Y )[ Y , T h Y ( T f ) ∧ Y s ! Y p ∗ f ∗ ξ E r − d ) , ( r − d ) Y ∧ S p,q ] α ∼ = SH ( Y )[ T h Y ( f ∗ ξ ) ∧ Y T h Y ( − T f ) , E r − d ) , ( r − d ) Y ∧ S p,q ] ε ( f ,f ∗ ) ◦ f ∼ = SH ( X )[ f ( T h Y ( f ∗ ξ ) ∧ Y T h Y ( − T f )) , E r − d ) , ( r − d ) X ∧ S p,q ] P r ∗ ( f ) ∼ = SH ( X )[ T h X ( ξ ) ∧ X f T h Y ( − T f ) , E r − d ) , ( r − d ) X ∧ S p,q ] α ∼ = SH ( X )[ T h X ( ξ ) , f ( Y ) ∧ X E r − d ) , ( r − d ) X ∧ S p,q ] −◦ ε ( f ,f ∗ ) −→ SH ( X )[ T h X ( ξ ) , E r − d ) , ( r − d ) X ∧ S p,q ] u ∗ ( − ) −→ SH ( X ′ )[ u ∗ T h X ( ξ ) , E r − d ) , ( r − d ) X ′ ∧ S p,q ] α −→ SH ( X ′ )[ T h X ′ ( u ∗ ξ ) , E r − d ) , ( r − d ) X ′ ∧ S p,q ] th X ′ E ( u ∗ ξ ) − ∼ = E p − d,q − d ( X ′ , u ∗ ξ ) , where α ( − ) = ev T h Y ( f ∗ ξ ) ◦ (id T h Y ( f ∗ ξ ) ∧ − ) ◦ ev T h Y ( T f ) ◦ (id T h Y ( T f ) ∧ − ) , and α ( − ) = (id f Y ∧ − ) ◦ ( coev f T h Y ( − T f ) ∧ id) .α is the natural isomorphism α = Ex ∗∗ (∆ b ) ◦ Ex ∗ (∆ a ) − ∆ a is the Cartesian square u ∗ ξ u ξ / / p u ∗ ξ (cid:15) (cid:15) ξ p ξ (cid:15) (cid:15) X ′ u / / X x ∗ (∆ a ) − : u ∗ p ξ ∼ = −→ p u ∗ ξ u ∗ ξ . ∆ b is the Cartesian square X ′ u (cid:15) (cid:15) s X ′ / / u ∗ ξ u ξ (cid:15) (cid:15) X s X / / ξEx ∗∗ (∆ b ) : u ∗ ξ s X ! ∼ = u ∗ ξ s X ∗ ∼ = −→ s X ′ ∗ u ∗ ∼ = s X ′ ∗ u ∗ .g ∗ v ∗ is the following composition E p,q ( Y, f ∗ ξ − T f ) defn = SH ( Y )[ Y , T h Y ( T f ) ∧ Y s ! Y p ∗ f ∗ ξ E r − d ) , ( r − d ) Y ∧ S p,q ] β −→ SH ( Y )[ T h Y ( f ∗ ξ ) , T h Y ( T f ) ∧ Y E r − d ) , ( r − d ) Y ∧ S p,q ] v ∗ ( − ) −→ SH ( Y ′ )[ v ∗ T h Y ( f ∗ ξ ) , v ∗ ( T h Y ( T f ) ∧ Y E r − d ) , ( r − d ) Y ∧ S p,q )] β ∼ = SH ( Y ′ )[ T h Y ′ ( g ∗ u ∗ ξ ) ∧ Y ′ T h Y ′ ( − T g ) , E r − d ) , ( r − d ) Y ′ ∧ S p,q ] ε ( g ,g ∗ ) ◦ g ∼ = SH ( X ′ )[ g ( T h Y ′ ( g ∗ u ∗ ξ ) ∧ Y ′ T h Y ′ ( − T g )) , E r − d ) , ( r − d ) X ′ ∧ S p,q ] P r ∗ ( g ) ∼ = SH ( X ′ )[ T h X ′ ( u ∗ ξ ) ∧ X ′ g T h Y ′ ( − T g )) , E r − d ) , ( r − d ) X ′ ∧ S p,q ] β ∼ = SH ( X ′ )[ T h X ′ ( u ∗ ξ ) , g ( Y ′ ) ∧ X ′ E r − d ) , ( r − d ) X ′ ∧ S p,q ] ◦ ε ( g ,g ∗ ) −→ SH ( X ′ )[ T h X ′ ( u ∗ ξ ) , E r − d ) , ( r − d ) X ′ ∧ S p,q ] th X ′ E ( u ∗ ξ ) − ∼ = E p − d,q − d ( X ′ , u ∗ ξ ) , where β is the natural isomorphism β ( − ) = ev T h Y ( f ∗ ξ ) ◦ (id T h Y ( f ∗ ξ ) ∧ − ) , and β ( − ) = β ′ ◦ Ex ∗∗ (∆ ) ◦ Ex ∗ (∆ ) − ◦ Ex ∗∗ (∆ ) ◦ Ex ∗ (∆ ) − . ∆ is the Cartesian square g ∗ u ∗ ξ = v ∗ f ∗ ξ v ξ / / p v ∗ f ∗ ξ = p g ∗ u ∗ ξ (cid:15) (cid:15) f ∗ ξ p f ∗ ξ (cid:15) (cid:15) Y ′ v / / YEx ∗ (∆ ) − : v ∗ p f ∗ ξ ∼ = −→ p v ∗ f ∗ ξ v ∗ ξ . ∆ is the Cartesian square Y ′ s Y ′ / / v (cid:15) (cid:15) v ∗ f ∗ ξ v ξ (cid:15) (cid:15) Y s Y / / f ∗ ξEx ∗∗ (∆ ) : v ∗ ξ s Y ! ∼ = v ∗ ξ s Y ∗ ∼ = −→ s Y ′ ∗ v ∗ ∼ = s Y ′ ! v ∗ . is the Cartesian square T g ∼ = v ∗ T f v Tf / / p v ∗ Tf = p Tg (cid:15) (cid:15) T fp Tf (cid:15) (cid:15) Y ′ v / / YEx ∗ (∆ ) − : v ∗ p T f ∼ = −→ p v ∗ T f v ∗ T f ∼ = p T g v ∗ T f . ∆ is the Cartesian square Y ′ s Y ′ /Tg / / v (cid:15) (cid:15) v ∗ T f ∼ = T gv Tf (cid:15) (cid:15) Y s Y /T f / / T f Ex ∗∗ (∆ ) : v ∗ T f s Y/T f ! ∼ = v ∗ T f s T f ∗ ∼ = −→ s Y ′ /T g ∗ v ∗ T f ∼ = s Y ′ /T g ! v ∗ T f .β ′ is the natural isomorphism β ′ ( − ) = ev T h Y ′ ( − T g ) ◦ (id T h Y ′ ( − T g ) ∧ − ) .β is the natural isomorphism β = (id g Y ′ ∧ − ) ◦ ( coev g T h Y ′ ( − T g ) ∧ id − ) . Gathering all together we have to check the following equality up to natural -isomorphisms: th X ′ E ( u ∗ ξ ) − ◦ Ex ∗∗ (∆ b ) ◦ Ex ∗ (∆ a ) − ◦ ǫ ( f ,f ∗ ) ◦ u ∗ ◦ (id f Y ∧ X − ) ◦ ( coev f T h Y ( − T f ) ∧ X id − ) ◦ P r ∗ ( f ) ◦ ε ( f ,f ∗ ) ◦ f ◦ ev T h Y ( f ∗ ξ ) ◦ (id T h Y ( f ∗ ξ ) ∧ Y − ) ◦ ev T h Y ( T f ) ◦ (id T h Y ( T f ∧ Y − ) = th X ′ E ( u ∗ ξ ) − ◦ ε ( g ,g ∗ ) ◦ (id g Y ′ ∧ X ′ − ) ◦ ( coev g T h Y ′ ( − T g ) ∧ X ′ id − ) ◦ P r ∗ ( f ) ◦ ε ( g ,g ∗ ) ◦ g ◦ ev T h Y ′ ( − T g ) ◦ (id T h Y ′ ( T g ) ∧ Y ′ − ) ◦ Ex ∗∗ (∆ ) ◦ Ex ∗ (∆ ) − ◦ Ex ∗∗ (∆ ) ◦ Ex ∗ (∆ ) − ◦ v ∗ ◦ ev T h Y ( f ∗ ξ ) ◦ (id T h Y ( f ∗ ξ ) ∧ − ) . This equality can be chased step by step by using the natural -isomorphism f u ∗ ∼ = ←− g v ∗ , which is the following composition g v ∗ η ( f ,f ∗ ) −→ g v ∗ f ∗ f ∼ = g ( f ◦ v ) ∗ f = g ( u ◦ g ) ∗ f ∼ = g g ∗ u ∗ f ε ( g ,g ∗ ) −→ u ∗ f and also the coherence of the exchange transformations. (cid:3) Now we construct the exceptional pullback for twisted E -cohomology. We keep the nota-tion as above and let i : T ֒ → Y be a regular embedding, where T is a smooth k -scheme. Let N T/Y be the normal bundle of T in Y . Let Bl T ( Y ) be the blow-up of X with the center Z .Similarly, Bl T ×{ } ( Y × A ) is the blow-up of Y × A with the center T ×{ } . The deformationspace is the k -scheme D T ( Y ) defn = Bl T ×{ } ( Y × A ) − Bl T ( Y ) . ote that D T ( T ) = T × A is a closed subscheme of D T ( Y ) . The scheme D T ( Y ) is fibredover A . The flat morphism π : D T ( Y ) → A has π − (1) = Y and π − (0) = N T/Y . One has a deformation diagram of closed pairs ( Y, T ) σ −→ ( D T ( Y ) , T × A ) σ ←− ( N T/Y , T ) . The homotopy purity theorem of Morel-Voevodsky [MV01, §3 Thm. 2.23] states
Y /Y − T σ ∗ −→ D T ( Y ) /D T ( Y ) − T × A σ ∗ ←− T h ( N T/Y ) are isomorphism in Ho A , + ( k ) , which is generalized to motivic categories in [CD10, Thm.2.4.35]. Consider now the adjunction i ! : SH ( T ) ⇄ SH ( Y ) : i ! . Let T i / / g (cid:15) (cid:15) Y f (cid:15) (cid:15) S k / / X be a cartesian square of smooth projective k -schemes, where f is smooth projective of relativedimension d = dim( Y ) − dim( X ) , k and i are regular embeddings. Let ξ be a vector bundleof rank r on X . We define the exceptional pullback of twisted E -cohomology along a regularembedding i : T ֒ → Y as the following composition: i ! : E p,q ( Y, f ∗ ξ − T f ) defn = SH ( Y )[ Y , T h Y ( T f ) ∧ Y s ! Y p ∗ f ∗ ξ E r − d ) , ( r − d ) Y ∧ S p,q ] i ! ( − ) −→ SH ( T )[ i ! Y , i ! T h Y ( T f ) ∧ Y s ! Y p ∗ f ∗ ξ E r − d ) , ( r − d ) Y ∧ S p,q ] ε ( i ! ,i !) ◦ i ! ∼ = SH ( Y )[ i ! i ! ( Y ) , T h Y ( T f ) ∧ Y s ! Y p ∗ f ∗ ξ E r − d ) , ( r − d ) Y ∧ S p,q ] ◦ Ex ∗ ! −→ SH ( Y )[ i ! i ∗ ( Y ) , T h Y ( T f ) ∧ Y s ! Y p ∗ f ∗ ξ E r − d ) , ( r − d ) Y ∧ S p,q ] ∼ = SH ( Y )[ i ∗ T , T h Y ( T f ) ∧ Y s ! Y p ∗ f ∗ ξ E r − d ) , ( r − d ) Y ∧ S p,q ] i ∗ ( − ) −→ SH ( T )[ i ∗ i ∗ T , i ∗ ( T h Y ( T f ) ∧ Y s ! Y p ∗ f ∗ ξ E r − d ) , ( r − d ) Y ∧ S p,q )] ∼ = SH ( T )[ T , i ∗ ( T h Y ( T f ) ∧ Y s ! Y p ∗ f ∗ ξ E r − d ) , ( r − d ) Y ∧ S p,q )] Ex ! ∗ ◦ Ex ! ∗ −→ SH ( T )[ T , T h T ( i ∗ T f ) ∧ T s ! T p ∗ i ∗ f ∗ ξ E r − d ) , ( r − d ) T ∧ S p,q ] = E p,q ( T, i ∗ f ∗ ξ − T g ) . Proposition 4.36.
Let T ′ h (cid:15) (cid:15) i ′ / / T i / / g (cid:15) (cid:15) Y f (cid:15) (cid:15) S ′ k ′ / / S k / / X be a chain of cartesian squares of smooth projective k -schemes, where f is smooth projective, i, i ′ , k, k ′ are regular embeddings. Then we have ( i ◦ i ′ ) ! = i ! ◦ i ′ ! up to natural isomorphismsinduced by natural -isomorphisms . roof. Obvious. (cid:3)
By using deformation to the cone as discussed above, one can prove the following result.However, we will not need this result, so we just omit the proof.
Proposition 4.37.
Consider a cartesian square of projective smooth k -schemes T i / / g (cid:15) (cid:15) Y f (cid:15) (cid:15) S k / / X where f is smooth projective of relative dimension d = dim( Y ) − dim( X ) and k and i areregular embeddings. Let p ξ : ξ → X be a vector bundle of rank r . One has a commutativediagram up to a natural isomorphism E p,q ( Y, f ∗ ξ − T f ) i ! (cid:15) (cid:15) f ∗ / / E p − d,q − d ( X, ξ ) k ! (cid:15) (cid:15) E p,q ( T, g ∗ k ∗ ξ − T g ) g ∗ / / E p − d,q − d ( S, k ∗ ξ ) Let p ξ : ξ → X and p ξ ′ : ξ ′ → X be two vector bundles of rank r and r ′ resp. on X withthe zero sections s : X → ξ and s ′ : X → ξ ′ respectively. Let s ′′ : X → ξ ⊕ ξ ′ to be the zerosection of the bundle ξ ⊕ ξ ′ . We define the cup product ∪ E : E p,q ( X, ξ ) ⊗ E p ′ ,q ′ ( X, ξ ′ ) → E p + p ′ ,q + q ′ ( X, ξ ⊕ ξ ′ ) as follow: Given morphisms in SH ( X ) α : X → s ! p ∗ ξ E r,rX ∧ S p,q and β : X → s ′ ! p ∗ ξ ′ E r ′ ,r ′ X ∧ S p ′ ,q ′ . Then α ∪ E β = µ E ◦ ( α ∧ X β ) : X = X ∧ X X α ∧ X β −→ s ! p ∗ ξ E r,rX ∧ X S p,q ∧ s ′ ! p ∗ ξ ′ E r ′ ,r ′ X ∧ S p ′ ,q ′ ∼ = ∼ = s ′′ ! p ∗ ξ ⊕ ξ ′ E X ∧ X E X ∧ S p + p ′ +2 r +2 r ′ ,q + q ′ + r + r ′ µ E → s ′′ ! p ∗ ξ ⊕ ξ ′ E r + r ′ ) ,r + r ′ X ∧ S p + p ′ ,q + q ′ . Remark 4.38. If f : T → S is a morphism of finite type between schemes, then we have f ∗ ( E ∧ L S F ) = f ∗ E ∧ L T f ∗ F. Proposition 4.39. (projection formula) Let f : Y → X be a smooth projective morphismof smooth projective k -schemes of relative dimension d = dim( Y ) − dim( X ) . Let ξ and ξ ′ betwo vector bundles on X . Let a ∈ E p,q ( X, ξ ) and b ∈ E p ′ ,q ′ ( Y, f ∗ ξ ′ − T f ) . Then one has upto natural isomorphisms f ∗ ( f ∗ a ∪ E b ) = a ∪ E f ∗ b in E p + p ′ − d,q + q ′ − d ( X, ξ ⊕ ξ ′ ) . roof. This follows from the projective smooth base change 4.35 by standard argument.Consider the commutative diagram Y f (cid:15) (cid:15) Γ f / / Y × k X f × id (cid:15) (cid:15) X ∆ X / / X × k X We have ∆ ∗ X ( f × id) ∗ = f ∗ Γ ∗ f = f ∗ ∆ ∗ X ( f × id) ∗ , where Γ f = ( f × id)∆ X . (cid:3) Proposition 4.40.
Let E ∈ SH ( k ) be a motivic ring spectrum. Let X, Y, Z ∈ SmP roj ( k ) .Let α ∈ E d Y ,d Y ( X × Y, pr XY ∗ Y V Y ) and β ∈ E d Z ,d Z ( Y × Z, pr
Y Z ∗ Z V Z ) , where V Y and V Z are thevector bundles given in theorem 2.4 with a refinement in 4.25. Then we have up to naturalisomorphisms pr XY ZXZ ∗ ( pr XY Z ∗ XY α ∪ E pr XY Z ∗ Y Z β ) ∈ E d Z ,d Z ( X × Z, pr XZ ∗ Z V Z ) . Proof.
This follows from our construction of pullback, pushforward and cup product and theprojections fit to the following commutative diagram X × Z pr XZZ , , pr XZX ! ! X × Y × Z pr XY ZXZ i i ❙❙❙❙❙❙❙❙❙❙❙❙❙❙ pr XY ZXY (cid:15) (cid:15) pr XY ZY Z / / Y × Z pr Y ZY (cid:15) (cid:15) pr Y ZZ / / ZX × Y pr XYX (cid:15) (cid:15) pr XYY / / YX (cid:3) Proposition 4.41.
Let E ∈ SH ( k ) be a motivic ring spectrum. Let X, Y, Z, W ∈ SmP roj ( k ) .Let α ∈ E d Y ,d Y ( X × Y, pr XY ∗ Y V Y ) , β ∈ E d Z ,d Z ( Y × Z, pr
Y Z ∗ Z V Z ) and γ ∈ E d W ,d W ( Z × W, pr ZW ∗ W V W ) . Let’s denote β ◦ α = pr XY ZXZ ∗ ( pr XY Z ∗ XY α ∪ E pr XY Z ∗ Y Z β ) , and similarly for γ ◦ β . Then ◦ is associative up to natural isomorphisms induced by -isomorphisms.Proof. We have γ ◦ ( β ◦ α ) (1) = pr XZWXW ∗ ( pr XZW ∗ XZ ( pr XY ZXZ ∗ ( pr XY Z ∗ XY α ∪ E pr XY Z ∗ Y Z β )) ∪ E pr XZW ∗ ZW γ ) (2) = pr XZWXW ∗ ( pr XY ZWXZW ∗ ( pr XY ZW ∗ XY Z ( pr XY Z ∗ XY α ∪ E pr XY Z ∗ Y Z β )) ∪ E pr XZW ∗ ZW γ ) (3) = pr XZWXW ∗ ( pr XY ZWXZW ∗ (( pr XY ZW ∗ XY α ∪ E pr XY W Z ∗ Y Z β ) ∪ E pr XY ZW ∗ XZW pr XZW ∗ ZW γ )) (4) = pr XY ZWXW ∗ ( pr XY ZW ∗ XY α ∪ E ( pr XY ZW ∗ Y Z β ∪ E pr XY ZW ∗ ZW γ )) , here (1) is the definition, (2) follows from smooth projective base change: pr XZWXW ∗ pr XY Z ∗ XZ = pr XY ZWXZW ∗ pr XY ZW ∗ XY Z , (3) follows from the compatibility of pullback and ∪ E (4.38), functoriality of pullback (4.18)and the projection formula (4.39), (4) follows from functoriality of pullback (4.18) and push-forward (4.24), (5) follows from the associativity of ∪ E , which is a consequence of our re-quirement that E is a motivic ring spectrum (see the beginning of §4.1). Symmetrically, thelast expression is exactly ( γ ◦ β ) ◦ α . (cid:3) Definition 4.42.
Let E ∈ SH ( k ) be a motivic ring spectrum. We define the category oftwisted E -correspondences ] Corr E ( k ) to be the category, whose objects are Obj ( ] Corr E ( k )) = Obj ( SmP roj ( k )) and morphisms are given by ] Corr E ( k )( X, Y ) = E d Y ,d Y ( X × Y, pr XY ∗ Y V Y ) , where V Y /Y is the vector bundle given in the theorem 2.4. Given α ∈ E d Y ,d Y ( X × Y, pr XY ∗ Y V Y ) and β ∈ E d Z ,d Z ( Y × Z, pr
Y Z ∗ Z V Z ) we define their composition to be β ◦ α = pr XY ZXZ ∗ ( pr XY Z ∗ XY α ∪ E pr XY Z ∗ Y Z β ) , which is associative up to natural isomorphisms. Proposition 4.43.
Let E ∈ SH ( k ) be a motivic ring spectrum. Let X, Y, Z ∈ SmP roj ( k ) .Let α ∈ E n Y + d Y ) , ( n Y + d Y ) ( X + ∧ T h ( V Y )) and β ∈ E n Z + d Z ) ,n Z + d Z ( Y + ∧ T h ( V Z )) . Then thefollowing composition β ◦ † α : Σ ∞ T, + X ∧ T h ( V Z ) coev Y −→ Σ ∞ T, + X ∧ Y ∧ T h ( V Y ) ∧ S − n Y + d Y ) , − ( n Y + d Y ) ∧ T h ( V Z ) τ → Σ ∞ T, + X ∧ T h ( V Y ) ∧ Y ∧ T h ( V Z ) ∧ S − n Y + d Y ) , − ( n Y + d Y ) α ∧ β −→ E ∧ L S E ∧ S n Z + d Z ) , ( n Z + d Z ) µ E −→ E ∧ S n Z + d Z ) , ( n Z + d Z ) lies in E n Z + d Z ) , ( n Z + d Z ) ( X + ∧ T h ( V Z )) , where coev Y : S → Σ ∞ T, + Y ∧ T h ( V Y ) ∧ S − n Y + d Y ) , − ( n Y + d Y ) is the coevaluation map of the Atiyah-Spanier-Whitehead duality on Y .Proof. Trivial. (cid:3)
Proposition 4.44.
Let E ∈ SH ( k ) be a motivic ring spectrum. Let X, Y, Z, W ∈ SmP roj ( k ) .Let α ∈ E n Y + d Y ) , ( n Y + d Y ) ( X + ∧ T h ( V Y )) , β ∈ E n Z + d Z ) , ( n Z + d Z ) ( Y + ∧ T h ( V Z )) and γ ∈ E n W + d W ) , ( n W + d W ) ( Z + ∧ T h ( V W )) . Let us denote by β ◦ † α for the composition of the aboveproposition and similarly for γ ◦ † β . Then ◦ † is associative and unital.Proof. We have that γ ◦ † ( β ◦ † α ) is the following composition by definition: γ ◦ † ( β ◦ † α ) : Σ ∞ T, + X ∧ T h ( V W ) coev Z −→ Σ ∞ T, + X ∧ Z ∧ T h ( V Z ) ∧ S − n Z + d Z ) , − ( n Z + d Z ) ∧ T h ( V W ) τ −→ Σ ∞ T, + X ∧ T h ( V Z ) ∧ Z ∧ T h ( V W ) ∧ S − n Z + d Z ) , − ( n Z + d Z ) ( β ◦ † α ) ∧ γ −→ E ∧ E ∧ S n W + d W ) , ( n W + d W ) µ E −→ E ∧ S n W + d W ) , ( n W + d W ) , hich can be rewritten as γ ◦ † ( β ◦ † α ) : Σ ∞ T, + X ∧ T h ( V W ) coev Z −→ Σ ∞ T, + X ∧ Z ∧ T h ( V Z ) ∧ S − n Z + d Z ) , − ( n Z + d Z ) ∧ T h ( V W ) τ −→ Σ ∞ T, + X ∧ T h ( V Z ) ∧ Z ∧ T h ( V W ) ∧ S − n Z + d Z ) , − ( n Z + d Z ) coev Y ∧ γ −→ Σ ∞ T, + X ∧ Y ∧ T h ( V Y ) ∧ S − n Y + d Y ) , − ( n Y + d Y ) ∧ T h ( V Z ) ∧ E ∧ S n W + d W ) − n Z + d Z ) , ( n W + d W ) − ( n Z + d Z ) τ −→ Σ ∞ T, + X ∧ T h ( V Y ) ∧ Y ∧ T h ( V Z ) ∧ E ∧ S n W + d W ) − n Y + d Y ) − n Z + d Z ) , ( n W + d W ) − ( n Z + d Z ) − ( n Y + d Y ) α ∧ β −→ E ∧ E ∧ E ∧ S n W + d W ) , ( n W + d W ) µ E ∧ id E −→ E ∧ E ∧ S n W + d W ) , ( n W + d W ) µ E −→ E ∧ S n W + d W ) , ( n W + d W ) . The composition ( γ ◦ † β ) ◦ † α is by definition: ( γ ◦ † β ) ◦ † α : Σ ∞ T, + X ∧ T h ( V W ) coev Y −→ Σ ∞ T, + X ∧ Y ∧ T h ( V Y ) ∧ S − n Y + d Y ) , − ( n Y + d Y ) ∧ T h ( V W ) τ → Σ ∞ T, + X ∧ T h ( V Y ) ∧ Y ∧ T h ( V W ) ∧ S − n Y + d Y ) , − ( n Y + d Y ) α ∧ ( γ ◦ † β ) −→ E ∧ E ∧ S n W + d W ) , ( n W + d W ) µ E −→ E ∧ S n W + d W ) , ( n W + d W ) , which can be rewritten as ( γ ◦ † β ) ◦ † α : Σ ∞ T, + X ∧ T h ( V W ) coev Y −→ Σ ∞ T, + X ∧ Y ∧ T h ( V Y ) ∧ S − n Y + d Y ) , − ( n Y + d Y ) ∧ T h ( V W ) τ → Σ ∞ T, + X ∧ T h ( V Y ) ∧ Y ∧ T h ( V W ) ∧ S − n Y + d Y ) , − ( n Y + d Y ) α ∧ coev Z −→ E ∧ Σ ∞ T, + Y ∧ Z ∧ T h ( V Z ) ∧ S − n Z + d Z ) , − ( n Z + d Z ) ∧ T h ( V W ) τ −→ Σ ∞ T, + Y ∧ T h ( V Z ) ∧ Z ∧ T h ( V W ) ∧ S − n Z + d Z ) , ( n Z + d Z ) ∧ E γ ∧ β −→ E ∧ E ∧ E ∧ S n W + d W ) , ( n W + d W ) µ E ∧ id E −→ E ∧ E ∧ S n W + d W ) , ( n W + d W ) µ E −→ E ∧ S n W + d W ) , ( n W + d W ) . Both γ ◦ † ( β ◦ † α ) and ( γ ◦ † β ) ◦ † α are equal to the following composition Σ ∞ T, + X ∧ T h ( V W ) coev Y ∧ coev Z −→ Σ ∞ T, + X ∧ Y ∧ T h ( V Y ) ∧ S − n Y + d Y ) , − ( n Y + d Y ) ∧ Z ∧ T h ( V Z ) ∧ S − n Z + d Z ) , − ( n Z + d Z ) ∧ T h ( V W ) τ −→ Σ ∞ T, + X ∧ T h ( V Y ) ∧ Y ∧ T h ( V Z ) ∧ Z ∧ T h ( V W ) ∧ S − n Y + d Y + n Z + d Z ) , − ( n Y + d Y + n Z + d Z ) α ∧ β ∧ γ −→ E ∧ E ∧ E ∧ S n W + d W ) , ( n W + d W ) µ E ∧ id E −→ E ∧ E ∧ S n W + d W ) , ( n W + d W ) µ E −→ E ∧ S n W + d W ) , ( n W + d W ) . (cid:3) Definition 4.45.
Let E ∈ SH ( k ) be a motivic ring spectrum. We define the category ofThom- E -correspondences Corr E ( k ) † to be the category, whose objects are Obj ( Corr E ( k ) † = Obj ( SmP roj ( k )) and morphisms are given by Corr E ( k ) † ( X, Y ) = E n Y + d Y ) ,n Y + d Y ( X + ∧ T h ( V Y )) , where V Y /Y is the duality vector bundle of rank n Y . Given α ∈ E n Y + d Y ) , ( n Y + d Y ) ( X + ∧ T h ( V Y )) nd β ∈ E n Z + d Z ) , ( n Z + d Z ) ( Y + ∧ T h ( V Z )) , we define their composition to be β ◦ † α : Σ ∞ T, + X ∧ T h ( V Z ) coev Y −→ Σ ∞ T, + X ∧ Y ∧ T h ( V Y ) ∧ S − n Y + d Y ) , − ( n Y + d Y ) ∧ T h ( V Z ) τ → Σ ∞ T, + X ∧ T h ( V Y ) ∧ Y ∧ T h ( V Z ) ∧ S − n Y + d Y ) , − ( n Y + d Y ) α ∧ β −→ E ∧ L S E ∧ S n Z + d Z ) , ( n Z + d Z ) µ E −→ E ∧ S n Z + d Z ) , ( n Z + d Z ) , where coev Y : S → Σ ∞ T, + Y ∧ T h ( V Y ) ∧ S − n Y + d Y ) , − ( n Y + d Y ) is the coevaluation map of the Atiyah-Spanier-Whitehead duality on Y .As we may write X + ∧ T h ( V Y ) = T h ( pr XY ∗ Y V Y ) , we have then the pullback map pr XY Z ∗ XY : E n Y + d Y ) , ( n Y + d Y ) ( T h ( pr XY ∗ Y V Y )) → E n Y + d Y ) , ( n Y + d Y ) ( T h ( pr XY Z ∗ XY V Y )) . Similarly pr XY Z ∗ Y Z : E n Z + d Z ) , ( n Z + d Z ) ( T h ( pr Y Z ∗ Z V Z )) → E n Z + d Z ) , ( n Z + d Z ) ( T h ( pr XY Z ∗ Y Z V Z )) . By taking cup product − ∪ E − : E n Y + d Y ) , ( n Y + d Y ) ( T h ( pr XY Z ∗ XY V Y )) ⊗ E n Z + d Z ) , ( n Z + d Z ) ( T h ( pr XY Z ∗ Y Z V Z )) → E n Y + d Y + n Z + d Z ) , ( n Y + d Y + n Z + d Z ) ( T h ( pr XY Z ∗ XY V Y ) ∧ T h ( pr XY Z ∗ Y Z V Z )) , and applying the pushforward pr XY Z ∗ XZ = ( coev Y ) ∗ we see that pr XY ZXZ ∗ ( pr XY Z ∗ XY α ∪ E pr XY Z ∗ Y Z β ) ∈ E n Z + d Z ) , ( n Z + d Z ) ( T h ( pr XZ ∗ Z V Z )) = E n Z + d Z ) , ( n Z + d Z ) ( X + ∧ T h ( V Z )) . Proposition 4.46.
Let E ∈ SH ( k ) be a motivic ring spectrum. Let X, Y, Z ∈ SmP roj ( k ) of dimension d X , d Y , d Z respectively. Let α ∈ E n Y + d Y ) , ( n Y + d Y ) ( X + ∧ T h ( V Y )) and β ∈ E n Z + d Z ) , ( n Z + d Z ) ( Y + ∧ T h ( V Z )) . Then the composition β ◦ † α in Corr E ( k ) † satisfies β ◦ † α = pr XY ZXZ ∗ ( pr XY Z ∗ XY α ∪ E pr XY Z ∗ Y Z β ) , where pr XY ZXZ ∗ = coev ∗ Y .Proof. Trivial. (cid:3)
Theorem 4.47. (Comparison) Let E ∈ SH ( k ) be a motivic ring spectrum. There is anequivalence of categories up to a natural -isomorphism ] Corr E ( k ) ≃ → Corr E ( k ) † ≃ → Corr E ( k ) . Proof.
We have the following association ] Corr E ( k ) → Corr E ( k ) † → Corr E ( k ) , X X X nd ] Corr E ( k )( X, Y ) defn = E d Y ,d Y ( X × Y, pr XY ∗ Y V Y ) th E ∼ = E d Y + n Y ) ,d Y + n Y ( X + ∧ T h ( V Y )) defn == SH ( k )[Σ ∞ T, + X ∧ T h ( V Y ) , E d Y + n Y ) ,d Y + n Y ] = Corr E ( k )( X, Y ) † D ∼ = SH ( k )[Σ ∞ T, + X, Σ ∞ T, + Y ∧ L S E ] defn = Corr E ( k )( X, Y ) , where th E denotes the twisted Thom isomorphism and D is the isomorphism induced byduality. It remains to check that the composition law in ] Corr E ( k ) is compatible with thecomposition law in Corr E ( k ) † and Corr E ( k ) via th E and D respectively. The compatibilityof composition laws via the Thom isomorphism th E follows from the Propositions 4.20, 4.33and 4.46. Now given a cohomology class α ∈ E d Y + n Y ) , ( d Y + n Y ) ( T h XY ( pr XY ∗ Y V Y )) we obtainits pullback by Σ ∞ T, + T h XY ( pr XY ∗ Y V Y ) α / / E d Y + n Y ) , ( d Y + n Y ) Σ ∞ T, + T h
XY Z ( pr XY Z ∗ Y V Y ) O O pr XY Z ∗ XY α ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ Similarly, given β ∈ E d Z + n Z ) , ( d Z + n Z ) ( T h
Y Z ( pr Y Z ∗ Z V Z )) we obtain its pullback by Σ ∞ T, + T h
Y Z ( pr Y Z ∗ Z V Z ) β / / E d Z + n Z ) , ( d Z + n Z ) Σ ∞ T, + T h
XY Z ( pr XY Z ∗ Z V Z ) O O pr XY Z ∗ Y Z β ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ So pr XY Z ∗ XY α ∪ E pr XY ZY Z β is the following composition pr XY Z ∗ XY α ∪ E pr XY ZY Z β : Σ ∞ T, + T h
XY Z ( pr XY Z ∗ Y V Y ) ∧ Σ ∞ T, + T h
XY Z ( pr XY Z ∗ Z V Z ) −∧− −→ E ∧ E ∧ S d Y + d Z + n Y + n Z ) ,d Y + d Z + n Y + n Z µ E → E ∧ S d Y + d Z + n Y + n Z ) ,d Y + d Z + n Y + n Z , which corresponds to the morphism Σ ∞ T, + T h
XY Z ( − pr XY Z ∗ Y T Y ) ∧ Σ ∞ T, + T h
XY Z ( − pr XY Z ∗ Z T Z ) → E . By definition the composition β ◦ α as composition of E -correspondences is given by thecomposition Σ ∞ T, + T h XZ ( − pr XZ ∗ Z T Z ) → Σ ∞ T, + T h
XY Z ( − pr XY Z ∗ Y T Y ) ∧ Σ ∞ T, + T h
XY Z ( − pr XY Z ∗ Z T Z ) → E , where the first map by construction is given as Σ ∞ T, + T h XZ ( − pr XZ ∗ Z T Z ) / / Σ ∞ T, + T h
XY Z ( − pr XY Z ∗ Y T Y ) ∧ Σ ∞ T, + T h
XY Z ( − pr XY Z ∗ Z T Z )Σ ∞ T, + X ∧ (Σ ∞ T, + Z ) ∨ id X ∧ coev Y ∧ id Z ∨ / / Σ ∞ T, + X ∧ Σ ∞ T, + Y ∧ (Σ ∞ T, + Y ) ∨ ∧ (Σ ∞ T, + Z ) ∨ his implies that the composition β ◦ α as E -correspondences is the same as X ∧ Y ∨ ∧ Y ∧ Z / / E ∧ E µ E / / E X ∧ Z ∨ coev Y O O ❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤ This shows that the composition laws of ] Corr E ( k ) and Corr E ( k ) are compatible. (cid:3) Proof of theorem 1.2
Homotopy t -structure. We recall in this section the notion homotopy t -structure interms of generators (cf. [Ay08]). Let k be a field. The subcategory SH ( k ) ≥ n is generatedunder homotopy colimits and extensions by { S p,q ∧ Σ ∞ P ( X + ) | X ∈ Sm/k, p − q ≥ n } , where S p,q = S p − qs ∧ S qt denotes the motivic spheres. We set SH ( k ) ≤ n = { E ∈ SH ( k ) | [ F, E ] = 0 , ∀ F ∈ SH ( k ) ≥ n +1 } The bigraded motivic homotopy sheaves are defined as π st A p,q ( E ) = a Nis ( U SH ( k )[ S p,q ∧ Σ ∞ P ( U + ) , E ] . We let π st A p ( E ) n defn = a Nis ( U SH ( k )[Σ ∞ P U + , S n − p,n ∧ E ]) For a fix p ∈ Z , π st A p ( E ) ∗ is considered as an abelian Z -graded sheaf. An abelian Nisnevichsheaf F ∈ Sh Nis ( Sm/k ) is called strictly A -invariant, if the map induced by the projection U × A → U : H iNis ( U, F ) → H iNis ( U × A , F ) is an isomorphism ∀ U ∈ Sm/k and ∀ i ≥ . For an abelian Nisnevich sheaf F ∈ Sh Nis ( Sm/k ) we will denote by F − ( U ) = Ker ( F ( U × k G m ) → F ( X )) , where the map is induced by the unit section of G m . Definition 5.1. (Morel).
A homotopy module is a pair ( F ∗ , ε ∗ ) , where F is a strictly A -invariant Z -graded abelian Nisnevich sheaf with ε n : F n ∼ = −→ ( F n +1 ) − . The following description of the homotopy t -structure is a consequence of F. Morel’s stable A -connectivity result (see for instance [Mor04a]): Theorem 5.2 (F. Morel) . Let k be field. (1) The triple ( SH ( k ) , SH ( k ) ≥ , SH ( k ) ≤ ) is a t -structure on SH ( k ) . (2) The heart of the homotopy t -structure π A ∗ ( k ) = SH ( k ) ≥ ∩ SH ( k ) ≤ is identified withthe category of homotopy modules. The homotopy t -structure is non-degenerated in the sense that for any U ∈ Sm/k and any E ∈ SH ( k ) , one has the morphism [Σ ∞ P ( U + ) , E ≥ n ] → [Σ ∞ P ( U + ) , E ] is an isomorphism for n ≤ and the morphism [Σ ∞ P ( U + ) , E ] → [Σ ∞ P ( U + ) , E ≤ n ] is an isomorphism for n > dim( U ) . By sending E E ≥ n and E E ≤ n − respectively, one has the following adjunctionsrespectively: i ≥ n : SH ( k ) ≥ n ⇆ SH ( k ) : τ ≥ n , τ ≤ n − : SH ( k ) ⇆ SH ( k ) ≤ n − : i ≤ n − , where we denote by i ≥ n and i ≤ n − the inclusion functors. We denote by H : π A ∗ ( k ) → SH ( k ) the inclusion functor. For a homotopy module F ∗ ∈ π A ∗ ( k ) we will call H ( F ∗ ) the Eilenberg-Maclane spectrum associated to F ∗ . Let S be now a Noetherian scheme of finite Krulldimension. We recall the rationally splitting of SH ( S ) Q constructed by F. Morel (see [CD10,§16.2]). The permutation isomorphism τ : Σ ∞ P , + G m, Q ∧ Σ ∞ P , + G m, Q → Σ ∞ P , + G m, Q ∧ Σ ∞ P , + G m, Q satisfies τ = 1 . This defines an element e ∈ End SH ( S ) Q ( Q ) , such that e = 1 . So we maydefine e + = e − , e − = e + 12 . Remark that e + and e − are idempotents. Hence we can define Q + = im ( e + ) and Q − =im ( e − ) . For any spectrum E ∈ SH ( S ) Q , one defines E + = Q + ∧ E and E − = Q − ∧ E . Thisleads to a splitting of stable homotopy category SH ( S ) Q + × SH ( S ) Q − ∼ = −→ SH ( S ) Q , ( E + , E − )
7→ E + ∧ E − Let us assume now S = Spec k . The algebraic Hopf fibration is the map A k − { } → P k , ( x, y ) [ x : y ] . This gives us the stable Hopf map in SH ( k ) η : Σ ∞ T, + G m → S k . Remark that from [Mor04a, 6.2.1] one has a homotopy fiber sequence in SH ( k ) : Σ ∞ T, + ( A k − { } ) S , ∧ η −→ Σ ∞ T, + P k Σ ∞ T, + ( i ) −→ Σ ∞ T, + P k , where i : P k ֒ → P k is the linear embedding. Following [Mor12] we define the Milnor-Witt K -theory of a field F without any assumption on char ( F ) : Definition 5.3.
Let F be a field. K MW ∗ ( F ) is the Z -graded associative unital ring freelygenerated by the symbols [ u ] , where u ∈ F × is of degree and a symbol η of degree − subject to the relation(1) [ u ] · [1 − u ] = 0 , ∀ u ∈ F × − { } . (2) [ uv ] = [ u ] + [ v ] + η · [ u ] · [ v ] , ∀ ( u, v ) ∈ ( F × ) . η · [ u ] = [ u ] · η, ∀ u ∈ F × .(4) Define h defn = η · [ −
1] + 2 . Then η · h = 0 .Let GW ( F ) be the Grothendieck-Witt ring of non-degenerate bilinear symmetric formsover F , where addition is given by orthogonal sum ⊕ and multiplication is given by tensorproduct ⊗ . There is a surjective ring homomorphism rk : GW ( F ) ։ Z , Q rk ( Q ) . The fundamental ideal is defined as I ( F ) defn = Ker ( rk : GW ( F ) ։ Z ) . Denote by I n ( F ) the n -th power of I ( F ) . If n ≤ one sets I n ( F ) = W ( F ) , where W ( F ) isthe Witt ring over F . Remark that W ( F ) = GW ( F ) / ( h ) , where ( h ) is the ideal generatedby hyperbolic spaces. By [Mor12, Lem. 3.10] there is a ring isomorphism GW ( F ) ∼ = −→ K MW ( F ) , h u i 7→ η · [ u ] . Let K M ∗ ( F ) be the Milnor K -theory K M ∗ ( F ) defn = T ens ∗ ( F × ) / h u ⊗ (1 − u ) i . There is a graded surjective homomorphism U : K MW ∗ ( F ) ։ K M ∗ ( F ) , [ u ]
7→ { u } , η . In fact, one can show that for each n there is a pullback diagram K MWn ( F ) U (cid:15) (cid:15) / / I n ( F ) (cid:15) (cid:15) K Mn ( F ) / / I n ( F ) /I n +1 ( F ) Following [Mor12, §3.2] we let K MWn be the n -th Milnor-Witt sheaf, which is a strictly A -invariant sheaf on ( Sm/k ) Nis . In [Mor04a, p. 437] Morel showed that one can define ahomotopy module K MW ∗ asscociated to the Milnor-Witt K -theory and in fact one has anisomorphism between homotopy modules π st A ( S ) ∗ ∼ = K MW ∗ . The homotopy module W ∗ is defined by setting every terms to be the unramified Witt sheaf W = a Nis ( U W ( U ) = W ( k ( U ))) and all the maps ε n are identity. Lemma 5.4.
Let k be a field. Let H K MW ∗ , Q be the Eilenberg-Maclane spectrum associatedto the Milnor-Witt K -theory homotopy module K MW ∗ , Q . There exists a strict motivic ringspectrum ˆ H K MW ∗ , Q ∈ Spect Σ T ( k ) Q , which is isomorphic to H K MW ∗ , Q in SH ( k ) Q .Proof. We have a splitting H K MW ∗ , Q = H ( K M ∗ , Q ) ∨ H ( W ∗ , Q ) . The result of Déglise [Deg13, Cor. 4.1.7] asserts that H ( K M ∗ ) is a strict H Z -module, where H Z denotes the motivic cohomology spectrum. The construction in [ALP15, §4] shows that he cofibrant replacement H ( W ∗ , Q ) cof is a commutative monoid object in Spect Σ T ( k ) Q , whichis isomorphic to H ( W ∗ , Q ) and S k [ η − ] . These imply that ˆ H K MW ∗ , Q = H K M ∗ , Q ∨ H ( W ∗ , Q ) cof is also a strict motivic ring spectrum in Spect Σ T ( k ) Q , which is isomorphic to H K MW ∗ , Q in SH ( k ) Q . (cid:3) Definition 5.5.
Let k be a field. We define the category of pure Chow-Witt motives to be CHW ( k ) Q = Mot ˆ H K MW ∗ , Q ( k ) Corollary 5.6.
Let k be a field. There is a functor CHW ( k ) Q → SH ( k ) Q . Proof.
This is a consequence of the Corollary 4.7 and Lemma 5.4. (cid:3)
Isomorphism between
Hom -groups.
Let k be a field. In this section we prove thatone has a fully faithful embedding CHW ( k ) Q → D A ,gm ( k ) Q . Remark that one has the equivalences of categories:
StHo A ,S ( k ) Q ∼ = D eff A ( k ) Q , StHo A , P ( k ) Q ∼ = D A ( k ) Q . For E ∈ SH ( k ) we define its stable A -cohomology as H p,qst A ( E, Z ) = SH ( k )( E, S p,q ) . We denote by SH ( k ) Q the localization of SH ( k ) . One has an adjunction L L Q : SH ( k ) ⇄ SH ( k ) Q : R U, which is induced by the Quillen adjunction L Q : Spect Σ T ( k ) ⇆ Spect Σ T ( k ) Q : U, where U : Spect Σ T ( k ) Q → Spect Σ T ( k ) is the forgetful functor by considering E Q = E ∧ Q = E ∧ hocolim ( S → S → S → · · · ) as a symmetric motivic T -spectrum in Spect Σ T ( k ) . For a motivic spectrum E ∈ SH ( k ) wedefine its rational stable A -cohomology as H p,qst A ( E, Q ) = SH ( k ) Q ( E Q , S p,q Q ) = SH ( k )( E Q , S p,q Q ) . We remark that by [Lev13, Lem. B2] if E is a compact object in SH ( k ) then one has anisomorphism H p,qst A ( E, Q ) = H p,qst A ( E, Z ) ⊗ Q . Similarly, we define the motivic cohomology of E as H p,qM ( E, Z ) = SH ( k )( E, H Z ∧ S p,q ) . If F ∗ ∈ π A ( k ) ∗ is a homotopy module, then the HF ∗ -cohomology of E is defined as H ( F ∗ ) p,q ( E ) = SH ( k )( E, S p,q ∧ HF ∗ ) and if E = Σ ∞ T, + X , where X ∈
Spc ( k ) + is a k -space (eg. Thom spaces), then the latercohomology is H p − qNis ( X , F q ) , where this cohomology is defined as H p − qNis ( X , F q ) = Ho A , + ( k )[ X , K ( F q )[ p − q ]] , here K ( − ) denotes the Eilenberg-Maclane functor. Theorem 5.7.
Let k be a field and E = Σ ∞ T, + T h ( V /X ) be the Thom spectrum of a vectorbundle V on a smooth k -scheme X . Let S be the motivic sphere spectrum. There exists acanonical isomorphism ϕ : H p,pst A ( T h ( V /X ) , Q ) ∼ = −→ H pNis ( T h ( V /X ) , K MWp ) Q , where ϕ is induced by the unit ϕ MW : S → H K MW ∗ .Proof. By stable A -connectivity theorem of Morel [Mor05] the motivic sphere spectrum S is − -connective. So we have a distinguished triangle ( S ) ≥ → S → Hπ ( S ) ∗ +1 → . By the computation of Morel we have π ( S ) ∗ = K MW ∗ . So after smashing with S p,p Q weobtain a distinguished triangle ( S ) ≥ ∧ S p,p Q → S p,p Q → H K MW ∗ ∧ S p,p Q +1 → . By taking [ T h ( V /X ) , − ] we have a long exact sequence · · · → [ T h ( V /X ) , ( S ) ≥ ∧ S p,p Q ] → [ T h ( V /X ) , S p,p Q ] ϕ → [ T h ( V /X ) , H K MW ∗ ∧ S p,p Q ] →→ [ T h ( V /X ) , ( S ) ≥ ∧ S p +1 ,p Q ] → · · · Now we have ( S ) ≥ ∧ S Q = ( S Q ) ≥ . By the work of C. D. Cisinski, F. Déglise ([CD10]) andthe work of A. Ananyevskiy, M. Levine, I. Panin ([ALP15]) we have S Q = H Q ∨ H W ∗ , Q . This implies ( S Q ) ≥ = ( H Q ) ≥ . The motivic cohomology spectrum H Q is also − -connective,so we have a distinguished triangle ( H Q ) ≥ → H Q → Hπ ( H Q ) ∗ +1 → . The homotopy module π ( H Q ) ∗ is K M ∗ , Q . We have (by [MVW06, Cor. 19.2] and by purity) H Q p,p ( T h ( V /X )) ∼ = −→ H K M, p,p ∗ ( T h ( V /X )) Q . Now from the splitting S Q = H Q ∨ H W ∗ , Q the map ϕ take the form: [ T h ( V /X ) , S p,p Q ] ∼ = H Q p,p ( T h ( V /X )) ⊕ H pNis ( T h ( V /X ) , W Q ) ϕ → H pNis ( T h ( V /X ) , K Mp ) Q ⊕ H pNis ( T h ( V /X ) , W Q ) ∼ = [ T h ( V /X ) , HK MW ∗ ∧ S p,p Q ] . This implies that ϕ is a canonical isomorphism. (cid:3) Corollary 5.8.
Let k be a field. The functor constructed in 5.6 CHW ( k ) Q → SH ( k ) Q is fully faithful roof. By definition
CHW ( k ) Q is the smallest pseudo-abelian full subcategory of the homo-topy category Ho k ( ˆ H K MW ∗ , Q − M od ) generated as an additive category by { Σ ∞ T, + X ∧ L S ˆ H K MW ∗ , Q | X ∈ SmP roj ( k ) } . The adjunction − ∧ L S ˆ H K MW ∗ , Q : SH ( k ) Q ⇄ Ho k ( ˆ H K MW ∗ , Q − M od ) : RU gives us a natural isomorphism Ho k ( ˆ H K MW ∗ , Q − M od )(Σ ∞ T, + X ∧ L S ˆ H K MW ∗ , Q , Σ ∞ T, + Y ∧ L S ˆ H K MW ∗ , Q ] ∼ = SH ( k ) Q [Σ ∞ T, + X, Σ ∞ T, + Y ∧ L S ˆ H K MW ∗ , Q ] . By duality 4.26 we have SH ( k ) Q [Σ ∞ T, + X, Σ ∞ T, + Y ∧ L S ˆ H K MW ∗ , Q ] ∼ = H n Y + d Y Nis ( T h ( V Y ) ∧ X + , K MWn Y + d Y ) Q , where d Y = dim( Y ) , V Y is the duality vector bundle given in the theorem 2.4 and n Y =rank( V Y ) . The corollary follows now from the Theorem 5.7. (cid:3) Appendix
In this appendix we simply recollect some facts and definitions in model categories. Allthe results are well-known and classical (see [Q67], [Hir03], [Hov99]).6.1.
Model Categories.Definition 6.1.
A model category M is a category with three classes of morphisms ( F ib ( M ) , Cof ( M ) , W ( M )) called fibrations, cofibrations and weak equivalences, such that:(1) M is closed under small limits and colimits.(2) If f, g ∈ M or ( M ) are composable and two out of f, g, g ◦ f are in W ( M ) , so is thethird one.(3) Given a commutative diagram A / / (cid:15) (cid:15) i (cid:15) (cid:15) X p (cid:15) (cid:15) (cid:15) (cid:15) B / / > > ⑦⑦⑦⑦ Y where i ∈ Cof ( M ) , p ∈ F ib ( M ) and either i or p is in W ( M ) , then there exists amorphism B → X making the diagram commutative.(4) W ( M ) , Cof ( M ) and F ib ( M ) are closed under retracts.(5) Given any morphism f : X → Y in M or ( M ) , there exist two functorial factorizations Z ❆❆❆❆❆❆❆❆ X > > ≃ > > ⑤⑤⑤⑤⑤⑤⑤⑤ f / / ❇❇❇❇❇❇❇❇ YW ≃ > > > > ⑥⑥⑥⑥⑥⑥⑥⑥ he first axiom implies that there exist an initial object ∅ and a final object ⋆ . We say M is pointed if ∅ ∼ = −→ ⋆ . Definition 6.2.
Let X ∈ Obj ( M ) be an object. X is called cofibrant if the natural mor-phism ∅ → X is in Cof ( M ) . X is called fibrant if the natural morphism X → ∗ is in F ib ( M ) .Let i : A → B and p : X → Y be two morphisms in M or ( M ) . We say i has left liftingproperty wrt. p or p has right lifting property wrt. i , if for every solide commutative diagram A / / (cid:15) (cid:15) i (cid:15) (cid:15) X p (cid:15) (cid:15) (cid:15) (cid:15) B / / > > ⑦⑦⑦⑦ Y the dotted morphism exists and makes the diagram commutative. Given two morphisms M or ( M ) ∋ f : A → B and M or ( M ) ∋ g : C → D , we say f is a retract of g , if there is acommutative diagram A / / f (cid:15) (cid:15) C / / g (cid:15) (cid:15) A f (cid:15) (cid:15) B / / D / / B where the horizontal composites are identities. Given an object X ∈ Obj ( M ) , the factoriza-tion axiom tells us that we can factor ∅ / / / / X cof ≃ / / / / X, where X cof is cofibrant. We call X cof a cofibrant replacement of X . Similarly, we can factor X / / ≃ / / X fib / / / / ⋆, where X fib is fibrant. We call X fib a fibrant replacement of X . Definition 6.3.
Let M , N be two model categories. A functor F : M → N is called a leftQuillen functor, if it has a right adjoint G : N → M and(1) If i ∈ Cof ( M ) , then F ( i ) ∈ Cof ( N ) .(2) If j ∈ Cof ( M ) ∩ W ( M ) , then F ( j ) ∈ Cof ( N ) ∩ W ( M ) .The right adjoint G : N → M is called a right Quillen functor and the adjunction F : M ⇆ N : G is called a Quillen adjunction. Definition 6.4.
Let F : M ⇆ N : G be a Quillen adjunction. F is called a left Quillen equivalence, if for every cofibrant object X ∈ Obj ( M ) and every fibrant object Y ∈ Obj ( N ) one has the following: A morphism f : X → GY is in W ( M ) iff its adjoint g = ε ( F,G ) ◦ F ( f ) : F X → Y is in W ( N ) . G is calledthen a right Quillen equivalence. The adjunction F : M ⇆ N : G is called a Quillen equivalence. efinition 6.5. Let X ∈ Obj ( M ) be an object in a model category M . The cylinder objectfor X is an object Cyl ( X ) , such that we have a factorization X ` X ∇ / / (cid:15) (cid:15) i (cid:15) (cid:15) XCyl ( X ) ≃ s ♦♦♦♦♦♦♦♦♦♦♦♦♦♦ where i ∈ Cof ( M ) and s ∈ W ( M ) . Definition 6.6.
Let X ∈ Obj ( M ) be an object in a model category M . A path object for X is an object P ( X ) , such that we have a factorization X ∆ / / r ≃ (cid:15) (cid:15) X × X P ( X ) p ♠♠♠♠♠♠♠♠♠♠♠♠♠ where r ∈ W ( M ) and p ∈ F ib ( M ) . Definition 6.7.
Let f, g : X ⇒ Y be two morphisms in M or ( M ) of a model category M . f is left homotopic to g if there is a cylinder object Cyl ( X ) for X , such that we have afactorization X ` X (cid:15) (cid:15) i (cid:15) (cid:15) ( f,g ) / / YCyl ( X ) LH ♦♦♦♦♦♦♦♦♦♦♦♦♦♦ The map LH is called a left homotopy from f to g . Definition 6.8.
Let f, g : X ⇒ Y be two morphisms in M or ( M ) of a model category M . f is right homotopic to g if there is a path object P ( Y ) for Y , such that we have a factorization X RH (cid:15) (cid:15) ( f,g ) / / Y × Y P ( Y ) p ♠♠♠♠♠♠♠♠♠♠♠♠♠ The map RH is called a right homotopy from f to g . Definition 6.9.
Let f, g : X ⇒ Y be two morphisms in M or ( M ) of a model category M . f is homotopic to g if f is left and right homotopic to g . Theorem 6.10. (Quillen [Q67, I.1 Thm. 1] ). Let M be a model category. There exists acategory Ho ( M ) = M [ W ( M ) − ] , which is called the homotopy category of M , where (1) Obj ( Ho ( M )) = Obj ( M ) . (2) Ho ( M )( X, Y ) = π (( X cof ) fib , ( Y cof ) fib ) , , where π denotes the set of homotopy classesand the composition law is induced by the composition law of M . heorem 6.11. (Quillen [Q67, I.4 Thm. 3] ). Let F : M ⇆ N : G be a Quillen adjunction. Then ( F, G ) induces an adjunction of homotopy categories L F : Ho ( M ) ⇆ Ho ( N ) : R G. Definition 6.12. (1) Let M be a model category. M is left proper, if in any pushoutdiagram A (cid:15) (cid:15) i (cid:15) (cid:15) h ≃ / / X (cid:15) (cid:15) B h / / Y where i ∈ Cof ( M ) and h ∈ W ( M ) , so h ∈ W ( M ) .(2) Let M be a model category. M is right proper, if in any pullback diagram A (cid:15) (cid:15) h ≃ / / X (cid:15) (cid:15) (cid:15) (cid:15) B h / / Y where p ∈ F ib ( M ) and h ∈ W ( M ) , so h ∈ W ( M ) .(3) M is proper, if it is left and right proper.Let ∆ denote the category, whose objects are ordered finite sets n = { < < · · · < n } , n ≥ and M or ( ∆ )( m, n ) = { f : m → n | i ≤ j = ⇒ f ( i ) ≤ f ( j ) } . There are cofaces δ i : n → n + 1 and codegeneracies σ i : n + 1 → n defined by δ i ( j ) = ( j, if j < ij + 1 , if j ≥ iσ i ( j ) = ( j, if j ≤ ij − , if j > i Cofaces and codegeneracies are generators for the maps in ∆ . They satisfy a list of relations(cf. [Weib94, §8]). Now one defines the category of simplicial sets as SSets defn = ∆ op ( Sets ) . So simplicial sets are just presheaves of sets on ∆ . For a general category A the categoryof simplicial objects and cosimplicial objects in A are defined to be ∆ op ( A ) and ∆ ( A ) respectively. Let Top be the category of compactly generated Hausdorff topological spaces.The geometric realization functor is defined by R : SSets → Top , X R ( X ) = Z n X ( n ) × ∆ n , here ∆ n is the presheaf M or ( ∆ )( − , n ) . There is an adjunction R : SSets ⇆ Top : S, where S is the singular functor S ( T ) : ∆ op → Sets, n Top ( R (∆ n ) , T ) . Here R (∆ n ) is R (∆ n ) = { ( x , · · · , x n ) ∈ R n +1 | x i ≥ , n X i =0 x i = 1 } . Theorem 6.13. (Quillen [Q67, II.3 Thm. 3] ). The category
SSets has a model categorystructure.
Definition 6.14.
Let M be a category. M is called simplicial if there is a functor M op × M → SSets , ( X, Y ) SSMap ( X, Y ) , such that(1) SSMap ( X, Y ) = M ( X, Y ) .(2) there exists a composition law ◦ : SSMap ( Y, Z ) × SSMap ( X, Y ) → SSMap ( X, Z ) , which is compatible with the composition law in M .(3) There is a simplicial sets map i X : ⋆ → SSMap ( X, X ) , ∀ X ∈ Obj ( M ) , where theassociativity of the composition law, right and left unit properties of i X follows fromthree commutative diagrams ([Hir03, Def. 9.1.2]). Definition 6.15.
Let M be a model category. M is called a simplicial model category if M is simplicial and(1) ∀ X ∈ Obj ( M ) there is an adjunction X ⊗ − : SSets ⇆ M : SSMap ( X, − ) , which is compatible with the simplicial structure on M .(2) ∀ Y ∈ Obj ( M ) there is an adjunction Y − : SSets ⇆ M op : SSMap ( − , Y ) , which is compatible with the simplicial structure on M .(3) For Cof ( M ) ∋ i : A → B and F ib ( M ) ∋ p : X → Y the map SSMap ( B, X ) ( i ∗ ,p ∗ ) / / / / SSMap ( A, X ) × SSMap ( A,Y ) SSMap ( B, Y ) is in F ib ( SSets , which is also in W ( SSets ) , if either i or p is in W ( M ) . Example 6.16.
SSets has a canonical simplicial model category structure.
SSMap ( X, Y ) is the simplicial set with SSMap ( X, Y ) n = SSets ( X × ∆ n , Y ) , with faces and degeneracies induced from the cosimplicial object ∆ • . roposition 6.17. Let M be a simplicial model category. If X is cofibrant and Y is fibrant,then Ho ( M )( X, Y ) = π SSMap ( X, Y ) . Consequently, for any objects
A, B ∈ Obj ( M ) one has Ho ( M )( A, B ) = π SSMap (( A cof ) fib , ( B cof ) fib ) . Localization.
All model categories in this subsection are being considered simplicial.
Definition 6.18.
Let M be a model category and V be a class of morphisms in M or ( M ) . Aleft localization of M wrt. V is a model category L V M together with a left Quillen functor F : M → L V M , such that:(1) The total left derived functor L F : Ho ( M ) → Ho ( L V M ) takes the images in Ho ( M ) of elements in V into isomorphisms in Ho ( L V M ) .(2) If N is a model category and T : M → N is a left Quillen functor such that L T : Ho ( M ) → Ho ( N ) take the images in Ho ( M ) of elements in V into isomorphisms in Ho ( N ) , then there is a unique left Quillen functor L V M → N , such that M F / / T (cid:15) (cid:15) L V M ∃ ! { { ✇ ✇ ✇ ✇ ✇ N Definition 6.19.
Let M be a model category and V be a class of morphisms in M or ( M ) .A right localization of M wrt. V is a model category R V M together with a right Quillenfunctor G : M → R V M , such that:(1) The total right derived functor R G : Ho ( M ) → Ho ( R V M ) takes the images in Ho ( M ) of elements in V into isomorphisms in Ho ( R V M ) (2) If N is a model category and T : M → N is a right Quillen functor such that R T takes the images in Ho ( M ) of elements in V into isomorphisms in Ho ( N ) , then thereis a unique right Quillen functor R V M → M , such that: M G / / T (cid:15) (cid:15) R V M ∃ ! { { ✇ ✇ ✇ ✇ ✇ N Definition 6.20.
Let M be a model category and V a class of morphisms in M or ( M ) .(1) An object X ∈ Obj ( M ) is called V -local if X is fibrant and for every f : A → B in V , SSMap ( B cof , X ) ≃ −→ SSMap ( A cof , X ) .(2) A morphism f : X → Y in M or ( M ) is a V -local equivalence if for every V -localobject T , SSMap ( Y cof , T ) ≃ −→ SSMap ( X cof , T ) (3) X ∈ Obj ( M ) is called V -colocal if X is cofibrant and for every f : A → B in V , SSMap ( X, A fib ) ≃ −→ SSMap ( X, B fib ) .(4) M or ( M ) ∋ f : X → is a V -colocal equivalence if for every V -colocal object T , SSMap ( T, X fib ) ≃ −→ SSMap ( T, Y fib ) . efinition 6.21. Let M be a model category and V be a class of morphisms in M or ( M ) .The left Bousfield localization (if it exists) of M wrt. V is a model category structure L V M on the underlying category M with:(1) W ( L V M ) is the class of V -local equivalences of M .(2) Cof ( L V M ) = Cof ( M ) .(3) F ib ( L V M ) = RLP ( Cof ( M ) ∩ W ( L V M )) . Definition 6.22.
Let M be a model category and V be a class of morphisms in M or ( M ) .The right Bousfield localization (if it exists) of M wrt. V is a model category structure R V M on the underlying category M with:(1) W ( R V M ) is the class of V -colocal equivalences of M .(2) F ib ( R V M ) = F ib ( M ) .(3) Cof ( R V M ) = LLP ( F ib ( M ) ∩ W ( R V M )) .6.3. Symmetric motivic T -Spectra. The reference for this subsection is [Jar00]. Let S be a Noetherian scheme of finite Krull dimension. Consider the category Sm/S of smoothof finite type S -schemes. A symmetric T -spectrum is a collection { X n } n ≥ , where X n ∈ ∆ op ( P rSh
Nis ( Sm/S )) + , together with the left actions Σ n × X n → X n , where Σ n is the n -th symmetric group. There are the bonding maps σ n : T ∧ X n → X n +1 , such that the interative composition T ∧ m ∧ X n → X n + m is Σ m × Σ n -equivariant. A morphism between symmetric T -spectra is a family { f n : X n → Y n } n ≥ , where the following diagram T ∧ X nσ n (cid:15) (cid:15) id ∧ f n / / T ∧ Y nσ n (cid:15) (cid:15) X n +1 f n +1 / / Y n +1 commutes and f n is Σ n -equivariant ∀ n ≥ . The category of symmetric T -spectra is denotedby Spect Σ T ( S ) . A symmetric sequence X is a family { X n | X n ∈ ∆ op ( P rSh
Nis ( Sm/S )) + } n ≥ with left actions Σ n × X n → X n . A morphism f : X → Y of symmetric sequences is a family { f n : X n → Y n } , where f n are Σ n -equivariant ∀ n ≥ . We denote the category of symmetric sequences of pointed simplicialpresheaves by ∆ op ( P rSh
Nis ( Sm/S )) Σ+ . Recall that there are families of functors F n : ∆ op ( P rSh
Nis ( Sm/S )) + → ∆ op ( P rSh
Nis ( Sm/S )) Σ+ , where ( F n ( X )) m = ( ⋆ if m = n W σ ∈ Σ n X if m = n nd Ev n : ∆ op ( P rSh
Nis ( Sm/S )) Σ+ → ∆ op ( P rSh
Nis ( Sm/S )) + , X X n . They are in fact adjoint to each other F n : ∆ op ( P rSh
Nis ( Sm/S )) + ⇆ ∆ op ( P rSh
Nis ( Sm/S )) Σ+ : Ev n . For two symmetric sequences X and Y , their product is defined as ( X ⊗ Y ) n = _ p + q = n Σ n ⊗ Σ p × Σ q X p ∧ Y q . The notation Σ n ⊗ Σ p × Σ q X p ∧ Y q means: there is an action γ of Σ p × Σ q on X p ∧ Y q via thecanonical embedding Σ p × Σ q ⊂ Σ n and also another action γ ′ : Σ p × Σ q × ( X p ∧ Y q ) → X p ∧ Y q .We let Σ n ⊗ Σ p × Σ q X p ∧ Y q = eq [ γ σ − γ ′ σ ] σ ∈ Σ p × Σ q . Now one can define F Σ n : ∆ op ( P rSh
Nis ( Sm/S )) + → Spect Σ T ( S ) , X 7→ S ⊗ F n ( X ) , where S denotes the motivic sphere spectrum S = ( S + , T ∧ S + , T ∧ ∧ S + , · · · )Σ n acts on S by permuting the T ∧ n factors and S + is pointed by S ` S . One has anadjunction F Σ n : ∆ op ( P rSh
Nis ( Sm/S )) + ⇆ Spect Σ T ( S ) : Ev n . In fact, one has F Σ0 ( S + ) = S . A symmetric T -spectrum X can be understood as a symmetricsequence with a module structure σ X : S ⊗ X → X over the motivic sphere spectrum S .Now we can define the smash product of symmetric T -spectra as X ∧ Y defn = coeq ( S ⊗ X ⊗ Y / / / / X ⊗ Y ) , where the top map is σ X ⊗ id Y and the bottom map is S ⊗ X ⊗ Y τ −→ X ⊗ S ⊗ Y id X ⊗ σ Y −→ X ⊗ Y. We just mention the following results of Jardine.
Theorem 6.23. (Jardine [Jar00, Thm. 4.2] ) The category
Spect Σ T ( S ) has a model categorystructure, which is proper and simplicial. Theorem 6.24. (Jardine [Jar00, Prop. 4.19] ). ( Spect Σ T ( S ) , S , ∧ ) is a symmetric monoidalmodel category. Now we discuss a little bit about the Quillen adjunction − ∧ E : Spect Σ T ( S ) ⇆ E − M od Σ : U, where E is a motivic strict ring spectrum (we always consider only commutative ring spec-trum). On the level of the underlying categories the unit and counit of the adjunction aredefined by η X : X ∼ = S ∧ X ϕ E ∧ id X −→ E ∧ X = U ( E ∧ X ) , and ε M : E ∧ U ( M ) = E ∧ M γ M −→ M. y [Jar00, Prop. 4.19] the category Spect Σ T ( S ) satisfies the axiom in [SS00, Def. 3.3]. By[SS00, Thm. 4.1] one can conclude that the adjunction − ∧ E : Spect Σ T ( S ) ⇆ E − M od Σ : U induces a model category structure on E − M od Σ . It is clear that the forgetful functor U : E − M od Σ → Spect Σ T ( S ) is a right Quillen functor, because F ib ( E − M od Σ ) and F ib ( E − M od Σ ) ∩ W ( E − M od Σ ) aredetected in Spect Σ T ( S ) . So we can claim that the adjunction above is a Quillen adjunction.Since E is a commutative ring spectrum, E − M od Σ has the closed symmetric monoidalcategory structure induced by the one on Spect Σ T ( S ) by declaring: − ∧ E − : E − M od Σ × E − M od Σ → E − M od Σ , ( M, N ) M ∧ E N and Hom E − Mod Σ : E − M od Σ × E − M od Σ → E − M od Σ , Hom E − Mod Σ ( M, N ) , where M ∧ E N defn = coeq ( E ∧ M ∧ N / / / / M ∧ N ) . The top map is γ M ∧ id and the bottom map is the composition E ∧ M ∧ N τ ∧ id −→ M ∧ E ∧ N id ∧ γ N −→ M ∧ N. The internal Hom is defined as
Hom E − Mod Σ ( M, N ) defn = eq (Hom Spect Σ T ( S ) ( M, N ) / / / / Hom
Spect Σ T ( S ) ( E ∧ M, N ) , where the top map is γ ∗ M = ◦ γ M and the bottom map is γ N ∗ : Hom Spect Σ T ( S ) ( M, N ) E ∧ −→ Hom
Spect Σ T ( S ) ( E ∧ M, E ∧ N ) γ N ◦ −→ Hom
Spect Σ T ( S ) ( E ∧ M, N ) . We should also mention the following theorem of Jardine:
Theorem 6.25. [Jar00, Thm. 4.31]
There is a Quillen equivalence V : Spect T ( S ) ⇆ Spect Σ T ( S ) : U, where Spect T ( S ) is the category of motivic T -spectra, V is the symmetrization functor and U is the forgetful functor. We remind the reader that throughout this work we take the motivic stable homotopycategory as SH ( k ) = Ho ( Spect Σ T ( k )) . The theorem of Jardine allows us to identify SH ( k ) equivalently to the A -stable homotopycategory SH P ( k ) ∼ = SH T ( k ) of Morel constructed in [Mor04a, Defn. 5.1, Rem. 5.1.10 andpp. 420], which is defined as the homotopy category of the motivic T -spectra Spect T ( k ) .Hence, we can use Morel computation of π st A ( S ) ∗ and his stable A -connectivity result. eferences [ABGHR14] M. Ando, A. J. Blumberg, D. Gepner, M. J. Hopkins, C. Rezk, An ∞ -categorical approach to R -line bundles, R -module Thom spectra, and twisted R -homology, J Topology, 7 ( ) , (2014).[ALP15] A. Ananyevskiy, M. Levine, I. Panin, Witt sheaves and the η -inverted sphere spectrum,arXiv:1504.04860v1 [math.AT], Preprint (2015).[AH11a] A. Asok, C. Haesemeyer, Stable A -homotopy and R -equivalence, J. Pur. Appl. Alg., 215 ,2469-2472, (2011).[AH11] A. Asok, C. Haesemeyer, The 0-th stable A -homotopy sheaf and quadratic zero cycles,arXiv:1108.3854v1 [math.AG], Preprint (2011).[Ay08] J. Ayoub, Les six opération de Grothendieck et le formalisme des cycles évanescents dans lemonde motivique I and II. Astérisque , , (2008).[CD10] C. D. Cisinski, F. Déglise, Triangulated category of mixed motives, arXiv:0912.2110v3[math.AG], Preprint (2012).[Deg13] F. Déglise, Orientable homotopy modules, American Journal of Math., (2), pp. 519-560,(2013).[Del87] P. Deligne, Le déterminant de la cohomologie, Current trends in arithmetic algebraic geometry,(Arata, Calif., 1985), Contemp. Math., vol. , Amer. Math. Soc., Providence, RI, 1987, pp.93-177.[EGA4] A. Grothendieck, J. Dieudonné, Éléments de géométrie algébrique. IV. Étude locale des schémaset des morphismes de schémas IV, Publ. Math. IHÉS , , , (1964-1967).[Fas07] J. Fasel, The Chow-Witt ring, Doc. Math., , 275-312, (2007).[Fas08] J. Fasel, Groupes de Chow-Witt, Mém. Soc. Math. Fr. (NS.), , (2008).[Hir03] P. S. Hirschhorn, Model categories and their localizations, Math. Surveys and Monographs, vol. , Amer. Math. Soc. (2003)[Hov99] M. Hovey, Model categories, Math. Surveys and Monographs, vol. , Amer. Math. Soc. (1999)[Hov01] M. Hovey, Spectra and symmetric spectra in general model categories, J. Pure Appl. Alg., 165 ( ) , 63-127, (2001).[Hu05] P. Hu, On the Picard group of the stable A -homotopy category, Topology, 44 ( ) , 609-640,(2005).[Jar00] J.F. Jardine, Motivic symmetric spectra, Doc. Math. , (2000), 445-553[Lev10] M. Levine, Slices and transfers, Doc. Math., 393-443, (2010).[Lev13] M. Levine, Convergence of Voevodsky’s slice tower, Doc. Math., , 907-941, (2013).[MVW06] C. Mazza, V. Voevodsky, C. Weibel, Lecture Notes on motivic cohomology, vol. 2, Clay Math.Mono., Amer. Math. Soc., Providence RI, (2006).[Mor04a] F. Morel, An introduction to A -homotopy theory, Contemporary developments in algebraic K -theory, 357-441, ICTP Lect. Notes, XV , Abdus Salam Int. Cent. Theoret. Phys., Trieste,(2004).[Mor04] F. Morel, On the motivic stable π of the sphere spectrum, in Axiomatic, Enriched and MotivicHomotopy Theory, 219-260, J.P.C. Greenlees (ed.), (2004), Kluwer Academic Publishers.[Mor05] F. Morel, The stable A -connectivity theorems, K -theory, , 1-68, (2005).[Mor12] F. Morel, A -algebraic topology over a field, vol. 2052, Lect. Notes in Math., Springer Heidelberg,(2012).[MV01] F. Morel, V. Voevodsky, A -homotopy theory of schemes, IHÉS Publ. Math., ( ) , 45-143,(1999).[NSO09] N. Naumann, M. Spitzweck, Paul Arne Østvær, Chern classes, K -theory and Landweber exact-ness over nonregular base schemes, in Motives and Algebraic Cycles: A Celebration in Honourof Spencer J. Bloch, Fields Inst. Comm., Vol. , (2009).[Q67] D. G. Quillen, Homotopical algebra, Lect. Notes in Math., No. , Springer, (1967).[Rio05] J. Riou, Dualité de Spanier-Whitehead en géométrie algébrique, C. R. Math. Acad. Sci. Paris , no. , (2005).[Rio10] J. Riou, Algebraic K -theory, A -homotopy and Riemann-Roch theorems, J. of Topology ,(2010), pp. 229-264. RO08] Oliver Röndigs, Paul Arne Østvær, Modules over motivic cohomology, Adv. in Math. vol. ,Issue , (2008).[SS00] S. Schwede, B. E. Shipley, Algebras and modules in monoidal model categories, Proc. LondonMath. Soc. , pp. 491-511, (2000).[Voe00] V. Voevodsky, Triangulated categories of motives over a field, in Cycles, transfers and motivichomology theories, Annals of Mathematics Studies, vol. 143, Princeton Univ. Press, (2000).[Voe03] V. Voevodsky, Motivic cohomology with Z / -coefficients, Publ. Math. IHÉS (2003).[Weib94] C. A. Weibel, An introduction to homological algebra, Cambridge stud. in adv. Math. ,(1994). E-mail address : [email protected]@gmail.com