E-motives and motivic stable homotopy

TL;DR

This paper develops the theory of pure E-motives within motivic homotopy theory, constructing twisted E-cohomology and embedding Chow-Witt motives into the geometric stable A^1-derived category, advancing the understanding of motivic categories.

Contribution

It introduces the category of pure E-motives for a motivic ring spectrum and embeds Chow-Witt motives into the geometric stable A^1-derived category.

Findings

Construction of pure E-motives using six functors formalism.

Definition of twisted E-cohomology in the motivic setting.

Fully faithful embedding of Chow-Witt motives into the geometric stable A^1-derived category.

Abstract

We introduce in this work the notion of the category of pure $\mathbf{E}$-Motives, where $\mathbf{E}$ is a motivic strict ring spectrum and construct twisted $\mathbf{E}$-cohomology by using six functors formalism of J. Ayoub. In particular, we construct the category of pure Chow-Witt motives $CHW(k)_{\mathbb{Q}}$ over a field $k$ and show that this category admits a fully faithful embedding into the geometric stable $\mathbb{A}^1$-derived category $D_{\mathbb{A}^1,gm}(k)_{\mathbb{Q}}$.

Equations

$$\underline{Chow}(k{)}^{op}\to {\mathbf{DM}}_{gm}(k),$$

$$[S^i,{\mathbb{G}}_m^{\wedge j}{]}_{{\mathbb{P}}^1}\otimes \mathbb{Q}=H_B^{j-i}(S,\mathbb{Q}(j))\oplus H_{Nis}^{-i}(S,\underline{\mathbb{W}}\otimes \mathbb{Q}),$$

$$CHW(k{)}_{\mathbb{Q}}\to D_{{\mathbb{A}}^1,gm}(k{)}_{\mathbb{Q}}.$$

$${\mathbf{E}}^{\ast }(X)\stackrel{\cong }{⟶}{\tilde{\mathbf{E}}}^{\ast +r}(Th(V)),$$

$$Th(V)\to Th(V)\wedge X_{+},$$

$$X_{+}\to X_{+}\wedge X_{+}$$

$$X_{+}\to Th(V).$$

$$\mathbf{E}\wedge Th(V)\to \mathbf{E}\wedge Th(V)\wedge X_{+}\to \mathbf{E}\wedge {\Sigma }^n\mathbf{E}\wedge X_{+}\stackrel{{\mu }_{\mathbf{E}}\wedge \mathrm{id}}{⟶}\mathbf{E}\wedge {\Sigma }^n{X}_{+}.$$

$$F_{\mathbf{E}}(\mathbf{E}\wedge {\Sigma }^n{X}_{+},\mathbf{E})\simeq F({\Sigma }^n{X}_{+},\mathbf{E})\stackrel{\simeq }{⟶}{F}_{\mathbf{E}}(\mathbf{E}\wedge Th(V),\mathbf{E})\simeq F(Th(V),\mathbf{E}),$$

$${\mathbf{E}}^{\ast ,\ast }(X)\stackrel{\cong }{⟶}{\mathbf{E}}^{\ast +2r,\ast +r}(Th(V)),$$

$$Th(-T_X)\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\mathbf{E}\to X^{\vee }{\wedge }_{\mathbb{S}}\mathbf{E}.$$

$$F_{\mathbf{E}}(Th(-T_X){\wedge }_{\mathbb{S}}\mathbf{E},\mathbf{E})\simeq F(Th(-T_X),\mathbf{E})\to \mathbf{E}.$$

$$F:\mathcal{A}\rightleftarrows \mathcal{B}:G,$$

$${\epsilon }_{(F,G)}:FG\to \mathrm{id},{\eta }_{(F,G)}:\mathrm{id}\to GF$$

$$FY\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$Y\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$coe{v}_A:1\to A\wedge A^{\vee },e{v}_A:A^{\vee }\wedge A\to 1,$$

$$A\cong 1\wedge A\stackrel{coe{v}_A\wedge \mathrm{id}}{⟶}A\wedge A^{\vee }\wedge A\stackrel{\mathrm{id}\wedge e{v}_A}{⟶}A\wedge 1\cong A$$

$$A^{\vee }\cong A^{\vee }\wedge 1\stackrel{\mathrm{id}\wedge coe{v}_A}{⟶}{A}^{\vee }\wedge A\wedge A^{\vee }\stackrel{e{v}_A\wedge \mathrm{id}}{⟶}1\wedge A^{\vee }\cong A^{\vee }$$

$$\alpha :{\mathrm{Hom}}_{\mathcal{C}}(-,A)\stackrel{\cong }{⟶}{\mathrm{Hom}}_{\mathcal{C}}(A^{\vee }\wedge -,1)$$

$$\alpha (\varphi )=e{v}_{A^{\vee }}\circ ({\mathrm{id}}_{A^{\vee }}\wedge \varphi ),$$

$${\alpha }^{-1}(\phi )=({\mathrm{id}}_A\wedge \phi )\circ (coe{v}_{A^{\vee }}\wedge {\mathrm{id}}_{-}).$$

$$Spc/k\rightleftarrows Sp{c}_{+}/k,$$

$${\mathrm{Hom}}_{{\mathbf{Ho}}_s^{Nis}(k)}(\mathcal{X},\mathcal{Z})\stackrel{\simeq }{\to }{\mathrm{Hom}}_{{\mathbf{Ho}}_s^{Nis}(k)}(\mathcal{X}\times {\mathbb{A}}^1,\mathcal{Z}).$$

$${\mathrm{Hom}}_{{\mathbf{Ho}}_s^{Nis}(k)}(\mathcal{Y},\mathcal{Z})\to {\mathrm{Hom}}_{{\mathbf{Ho}}_s^{Nis}(k)}(\mathcal{X},\mathcal{Z})$$

$$L_{{\mathbb{A}}^1}:{\mathbf{Ho}}_s^{Nis}(k)\rightleftarrows {\mathbf{Ho}}_{s,{\mathbb{A}}^1-loc}^{Nis}(k):i,$$

$${\mathrm{Hom}}_{{\mathbf{Ho}}_s^{Nis}(k)}(\mathcal{X},\mathcal{Y})\stackrel{\simeq }{\to }{\mathrm{Hom}}_{{\mathbf{Ho}}_{{\mathbb{A}}^1}(k)}(\mathcal{X},\mathcal{Y}).$$

$$Th(E/X)=E/E-s_0(X),$$

$$Th(E\times E^{\prime }/X\times Y)=Th(E/X)\wedge Th(E^{\prime }/Y).$$

Full text

1. 1 Introduction

2. 2 ${\mathbb{A}}^{1}$-homotopy category

3. 2.1 Unstable ${\mathbb{A}}^{1}$-homotopy category

4. 2.2 Stable ${\mathbb{A}}^{1}$-homotopy category

5. 3 ${\mathbb{A}}^{1}$-homological algebra

6. 3.1 Effective ${\mathbb{A}}^{1}$-derived category

7. 3.2 ${\mathbb{P}}^{1}$-stable ${\mathbb{A}}^{1}$-derived category

8. 4 $\mathbf{E}$-Motives

9. 4.1 $\mathbf{E}$-Correspondences

10. 4.2 Functoriality in motivic stable homotopy

11. 4.3 Cohomology with supports

12. 4.4 Relation to the category of twisted $E$-correspondences

13. 5 Proof of theorem 1.2

14. 5.1 Homotopy $t$-structure

15. 5.2 Isomorphism between ${\rm Hom}$-groups

16. 6 Appendix

17. 6.1 Model Categories

18. 6.2 Localization

19. 6.3 Symmetric motivic $T$-Spectra

1. 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 pure motives $\underline{Chow}(k)$ into the category of geometric motives $\mathbb{DM}_{gm}(k)$, hence also into the category of motives $\mathbf{DM}^{-}_{Nis}(k)$

[TABLE]

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 using Poincaré duality). In this note, we construct a category $CHW(k)_{{\mathbb{Q}}}$, which we call the category of pure Chow-Witt motives over a field $k$ and show that $CHW(k)_{{\mathbb{Q}}}$ admits a fully faithful embedding into the geometric ${\mathbb{P}}^{1}$-stable ${\mathbb{A}}^{1}$-derived category $D_{{\mathbb{A}}^{1},gm}(k)_{{\mathbb{Q}}}$ rationally. Our work can be viewed as an ${\mathbb{A}}^{1}$-version for Voevodsky’s embedding theorem. The advantage here is that by using duality formalism for ${\mathbb{P}}^{1}$-stable ${\mathbb{A}}^{1}$-derived category $D_{{\mathbb{A}}^{1}}(k)$ (see [Hu05, App. A] for stable ${\mathbb{A}}^{1}$-homotopy categories) and the six operations formalism of J. Ayoub [Ay08], we do not have to assume the resolution of singularities. However, unlike in motivic setting, one of the main problems here is that we don’t have cancellation theorem for the effective ${\mathbb{A}}^{1}$-derived category in general, see [AH11, Rem. 3.2.4], that is the reason why we can prove the embedding result only for ${\mathbb{Q}}$-coefficient. F. Morel conjectured in general that (see [Mor04]):

Conjecture 1.1**.**

[Mor04]** Let $S$ be a regular Noetherian scheme of finite Krull dimension. One has a direct decomposition in the rationally motivic stable homotopy category $\mathbf{StHo}_{{\mathbb{A}}^{1},{\mathbb{P}}^{1}}(S)$:

[TABLE]

where $H^{*}_{B}(-,{\mathbb{Q}}(*))$ denotes the Beilinson motivic cohomology, $\underline{\mathbb{W}}$ is the unramified Witt sheaf and $[-,-]_{{\mathbb{P}}^{1}}\otimes{\mathbb{Q}}$ denotes ${\rm Hom}_{\mathbf{StHo}_{{\mathbb{A}}^{1},{\mathbb{P}}^{1}}(S)}(-,-)_{{\mathbb{Q}}}$.

Over a general base scheme $S$ one can split the rational motivic sphere spectrum $\mathbf{1}_{{\mathbb{Q}}}=\mathbf{1}_{{\mathbb{Q}}+}\vee\mathbf{1}_{{\mathbb{Q}}-}$. The identifcation of the plus part $\mathbf{1}_{{\mathbb{Q}}+}=H_{B}$ has been done in [CD10, Thm 16.2.13] over any Noetherian scheme of finite Krull dimension $S$. The minus part $\mathbf{1}_{{\mathbb{Q}}-}=H\underline{\mathbf{W}}_{*{\mathbb{Q}}}$, where $H\underline{\mathbf{W}}_{*{\mathbb{Q}}}$ denotes the Eilenberg-Maclane spectrum associated to the rational Witt homotopy module $\underline{\mathbf{W}}_{*{\mathbb{Q}}}$, is given in the work of A. Ananyeskiy, M. Levine and I. Panin ([ALP15, §3, Thm. 5]) over fields $S={\rm Spec\,}k$. In general, the conjecture 1.1 over a regular Noetherian scheme $S$ of finite Krull dimension is still widely open, as far as I know. On the other hand, our interest started originally from the study of the existence of [math]-cycles of degree one on algebraic varieties. More precisely, Hélène Esnault asked (cf. [Lev10]): Given a 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 over an infinite perfect field of $char(k)\neq 2$ is equivalent to the assertion that the structure map $\mathbf{H}^{st{\mathbb{A}}^{1}}_{0}(X)\rightarrow\mathbf{H}^{st{\mathbb{A}}^{1}}_{0}({\rm Spec\,}k)$ is a split epimorphism, where $\mathbf{H}^{st{\mathbb{A}}^{1}}_{i}(X)$ denotes the ${\mathbb{P}}^{1}$-stable ${\mathbb{A}}^{1}$-homology sheaves, while in an earlier work [AH11a] they also showed that the existence of a $k$-rational point over an arbitrary field $k$ is equivalent to the condition that the structure map $\mathbb{H}_{0}^{{\mathbb{A}}^{1}}(X)\rightarrow\mathbb{H}_{0}^{{\mathbb{A}}^{1}}({\rm Spec\,}k)$ is split surjective. So roughly speaking, the obstruction to the lifting of a zero cycle of degree one to a rational point arises by passing from $S^{1}$-spectra to ${\mathbb{P}}^{1}$-spectra. As remarked by M. Levine, it is not to expect that the category of Chow-Witt motives $CHW(k)$ contains any information about the existence of rational points. Now we state our main theorem in this work:

Theorem 1.2**.**

Let $k$ be a field. There exists a category of pure Chow-Witt motives $CHW(k)_{{\mathbb{Q}}}$, which admits a fully faithful embedding

[TABLE]

In fact, one of the main steps in the work of [AH11] is to exhibit a natural isomorphism $H_{0}^{st{\mathbb{A}}^{1}}(X)(L)\rightarrow\widetilde{{\rm CH}}_{0}(X_{L})$ for any separable, finitely generated field extension $L/k$. So one may relate this step to our work as evaluating at a generic point, but much weaker than expected, since we can only prove the result for ${\mathbb{Q}}$-coefficient. Now our paper is organized as follows: we will review shortly ${\mathbb{A}}^{1}$-homotopy theory in section §2. Section §3 is devoted for ${\mathbb{A}}^{1}$-derived categories, in fact we will define the geometric ${\mathbb{P}}^{1}$-stable ${\mathbb{A}}^{1}$-derived category $D_{{\mathbb{A}}^{1},gm}(k)$ over a field $k$ in 3.9 at the end of §3. In fact, this is the subcategory of compact objects $D_{{\mathbb{A}}^{1},c}(k)$ of $D_{{\mathbb{A}}^{1}}(k)$ (see [CD10, Ex. 5.3.43]). In these §2 and §3 we simply steal everything which is needed from the presentation of [AH11]. For a complete treatment we strongly recommend the reader to [Ay08], [CD10] and [Mor12]. In section §4 we introduce the notion of pure $\mathbf{E}$-motives, where $\mathbf{E}$ is a motivic strict ring spectrum and relate several categories of $\mathbf{E}$-correspondences with each other via the twisted $\mathbf{E}$-cohomology. The twisted $\mathbf{E}$-cohomology appears since we will not assume the motivic ring spectrum $\mathbf{E}$ to be orientable. In topology, if $\mathbf{E}$ is a multiplicative cohomology theory and $V$ is an $\mathbf{E}$-orientable vector bundle of rank $r$, then one has a Thom-Dold isomorphism

[TABLE]

where $Th(V)$ is the Thom space of $V$ and the right hand side is the reduced cohomology. If $\mathbf{E}$ is a ring spectrum, then one can intepret this isomorphism as following: Via the Thom diagonal

[TABLE]

which is induced by the diagonal

[TABLE]

one can express $Th(V)$ as a comodule over $X_{+}$ and the comodule map is the natural map

[TABLE]

The geometric Thom isomorphism is the homotopy equivalence

[TABLE]

The composition is an $\mathbf{E}$-module map, hence one may take function spectrum

[TABLE]

which induces the Thom isomorphism on $\mathbf{E}$-cohomology. In algebraic geometry one has a similar result. For an oriented motivic ring spectrum $\mathbf{E}\in SH(S)$, where $S$ is a regular base, one has ([NSO09, Thm. 2.12])

[TABLE]

where $V$ is a vector bundle of rank $r$ on a smooth $S$-scheme $X$. The key point is that since $\mathbf{E}$ 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 $\mathbf{E}$ is not necessary oriented. One has to introduce twisted cohomology. Again in topology, by Atiyah duality one has a commutative diagram in the $(\infty,1)$-category $\mathbb{S}Mod$:

[TABLE]

where $PT:\mathbb{S}\rightarrow Th(-T_{X})$ is the Pontryagin-Thom collapse map. Let $\mathbf{E}$ be an $E_{\infty}$-ring spectrum. By taking $-\wedge_{\mathbb{S}}\mathbf{E}$ one obtains a map in the $(\infty,1)$-category $\mathbf{E}Mod$

[TABLE]

Taking function spectrum we have the (twisted) Umkehr map

[TABLE]

If $\mathbf{E}$ is non-oriented, there is no geometric Thom isomorphism. However, $F_{\mathbf{E}}(Th(-T_{X})\wedge_{\mathbb{S}}\mathbf{E},\mathbf{E})$ will give the twisted cohomology. This is the motivation from topology for us, since in algebraic geometry we also have the Atiyah-Spanier-Whitehead duality, but I do not know any $\infty$-categorical approach to twisted cohomology like the one in topology [ABGHR14]. So I introduce in section §4 the twisted $\mathbf{E}$-cohomology rather through the guide of the six functors formalism of J. Ayoub. The reader may recognize that the notion of $\mathbf{E}$-correspondences is similar to the construction of Jack Morava in topology. While it is very simple to define the category of $\mathbf{E}$-correspondences $Corr_{\mathbf{E}}(k)$, it is quite difficult to construct the cateogry $\widetilde{Corr}_{\mathbf{E}}(k)$ via twisted $\mathbf{E}$-cohomology. This category exists only up to a number of natural $2$-isomorphisms. This phenomenon reflexes the fact that we rely on six functors formalism, where Thom transformations are only $2$-isomorphic to each other. The composition in $\widetilde{Corr}_{\mathbf{E}}(k)$ is associative only up to a natural isomorphism induced by a natural $2$-isomorphism. In §5 we give the proof of the main theorem. In the appendix we give a minimal list of well-known facts and definitions of model categories. We fix now some notations throughout this work. For a pair of adjoint functors $F:{\mathcal{A}}\rightarrow{\mathcal{B}}$ and $G:{\mathcal{B}}\rightarrow{\mathcal{A}}$, we will adopt the notation in [CD10]

[TABLE]

where $F$ is left adjoint to $G$ and $G$ is right adjoint to $F$. Sometime we will write

[TABLE]

for the counit and unit of the adjunction repsectively. For every morphism $f:FY\rightarrow X$ in $Mor({\mathcal{B}})$, there is a unique morphism $g:Y\rightarrow GX$ in $Mor({\mathcal{A}})$ such that the following diagram commutes:

[TABLE]

For every morphism $g:Y\rightarrow GX$ in $Mor({\mathcal{A}})$, there is a unique morphism $f:FY\rightarrow X$ in $Mor({\mathcal{B}})$, such that the following diagram commutes:

[TABLE]

In a symmetric monoidal category $({\mathcal{C}},\wedge,\mathbf{1})$, an object $A$ is called strongly dualizable if there exists an object $A^{\vee}$ and morphisms

[TABLE]

such that the following compositions

[TABLE]

and

[TABLE]

are the identities ${\rm id}_{A}$ and ${\rm id}_{A^{\vee}}$. The natural isomorphism

[TABLE]

is given by

[TABLE]

and its inverse $\alpha^{-1}$ is given by

[TABLE]

Given two smooth $k$-schemes $X,Y\in Sm/k$ and two vector bundles ${\mathcal{E}},{\mathcal{E}}^{\prime}$ over $X$ resp. $Y$, we write ${\mathcal{E}}\times{\mathcal{E}}^{\prime}/X\times Y$ for the external sum over $X\times_{k}Y$. The ${\mathbb{P}}^{1}$- stable homotopy category over a base scheme $S$ will be denoted by $\mathbf{StHo}_{{\mathbb{A}}^{1},{\mathbb{P}}^{1}}(S)$ and we write $\mathbf{StHo}_{{\mathbb{A}}^{1},S^{1}}(S)$ for the $S^{1}$-stable homotopy category. Sometime when it is clear which category we are talking about, we just abbreviate our ${\mathbb{P}}^{1}$-stable homotopy category by $SH(S)$.

2. ${\mathbb{A}}^{1}$-homotopy category

2.1. Unstable ${\mathbb{A}}^{1}$-homotopy category

Let $Sm/k$ denote the category of separated smooth schemes of finite type over a field $k$. We write $Spc/k$ for the category $\Delta^{op}Sh_{Nis}(Sm/k)$ consisting of simplicial Nisnevich sheaves of sets on $Sm/k$. An object in $Spc/k$ is simply called a $k$-space, which is usually denoted by calligraphic letter ${\mathcal{X}}$. The Yoneda embedding $Sm/k\rightarrow Spc/k$ is given by sending a smooth scheme $X\in Sm/k$ to the corresponding representable sheaf ${\rm Hom}_{Sm/k}(-,X)$ then by taking the associated constant simplicial object, where all face and degeneracy maps are the identity. We will identify $Sm/k$ with its essential image in $Spc/k$. Denote by $Spc_{+}/k$ the category of pointed $k$-space, whose objects are $({\mathcal{X}},x)$, where ${\mathcal{X}}$ is a $k$-space and $x:{\rm Spec\,}k\rightarrow{\mathcal{X}}$ is a distinguished point. One has an adjoint pair

[TABLE]

which means that the functor $Spc/k\rightarrow Spc_{+}/k$ sending ${\mathcal{X}}\rightarrow{\mathcal{X}}_{+}={\mathcal{X}}\coprod{\rm Spec\,}k$ is left-adjoint to the forgetful functor $Spc_{+}/k\rightarrow Spc/k$. The category $Spc/k$ can be equipped with the injective local model structure $(C_{s},W_{s},F_{s})$, where cofibrations are monomorphisms, weak equivalences are stalkwise weak equivalences of simplicial sets and fibrations are morphisms with right lifting property wrt. morphisms in $C_{s}\cap W_{s}$. Denote by $\mathbb{Ho}_{s}^{Nis}(k)$ the resulting unpointed homotopy category as constructed by Joyal-Jardine (cf. [MV01, §2 Thm. 1.4]). We will write $\mathbb{Ho}_{s,+}^{Nis}(k)$ for the pointed homotopy category.

Definition 2.1**.**

[MV01]**

1. (1)

A $k$-space ${\mathcal{Z}}\in Spc/k$ is called ${\mathbb{A}}^{1}$-local if and only for any object ${\mathcal{X}}\in Spc/k$, the projection ${\mathcal{X}}\times{\mathbb{A}}^{1}\rightarrow{\mathcal{X}}$ induces a bijection

[TABLE] 2. (2)

Let ${\mathcal{X}}\rightarrow{\mathcal{Y}}\in Mor(Spc/k)$ be a morphism of $k$-spaces. It is an ${\mathbb{A}}^{1}$-weak equivalence if and only for any ${\mathbb{A}}^{1}$-local object ${\mathcal{Z}}$, the induced map

[TABLE]

is bijective.

In [MV01, §2 Thm. 3.2], F. Morel and V. Voevodsky proved that $Spc/k$ can be endowed with the ${\mathbb{A}}^{1}$-local injective model structure $(C,W_{{\mathbb{A}}^{1}},F_{{\mathbb{A}}^{1}})$, where cofibrations are monomorphisms, weak equivalences are ${\mathbb{A}}^{1}$-weak equivalences. The associated homotopy category obtained from $Spc/k$ by inverting ${\mathbb{A}}^{1}$-weak equivalences is denoted by $\mathbf{Ho}_{{\mathbb{A}}^{1}}(k)\stackrel{{\scriptstyle def}}{{=}}Spc/k[W_{{\mathbb{A}}^{1}}^{-1}]$. This category is called the unstable ${\mathbb{A}}^{1}$-homotopy category of smooth $k$-schemes. Let $\mathbf{Ho}_{s,{\mathbb{A}}^{1}-loc}^{Nis}(k)\subset\mathbf{Ho}_{s}^{Nis}(k)$ be the full subcategory consisting of ${\mathbb{A}}^{1}$-local objects. In fact, one has an adjoint pair (cf. [MV01])

[TABLE]

where $L_{{\mathbb{A}}^{1}}$ is the ${\mathbb{A}}^{1}$-localization functor sending ${\mathbb{A}}^{1}$-weak equivalences to isomorphisms. $L_{{\mathbb{A}}^{1}}$ induces thus an equivalence of categories $\mathbf{Ho}_{{\mathbb{A}}^{1}}(k)\rightarrow\mathbf{Ho}_{s,{\mathbb{A}}^{1}-loc}^{Nis}(k)$. This will imply that if ${\mathcal{X}}\in Spc/k$ is any object and ${\mathcal{Y}}$ is an ${\mathbb{A}}^{1}$-local object, then one has a canonical bijection

[TABLE]

We will write $\mathbf{Ho}_{{\mathbb{A}}^{1},+}(k)$ for the unstable pointed ${\mathbb{A}}^{1}$-homotopy category of smooth $k$-schemes. Recall

Definition 2.2**.**

Let $X\in Sm/k$ and $E$ be a vector bundle over $X$. The Thom space of $E$ is the pointed sheaf

[TABLE]

where $s_{0}:X\rightarrow E$ is the zero section of $E$.

Let $T\in Spc_{+}/k$ be the quotient sheaf ${\mathbb{A}}^{1}/({\mathbb{A}}^{1}-\{0\})$ pointed by the image of ${\mathbb{A}}^{1}-\{0\}$. Then $T\cong S^{1}_{t}\wedge S^{1}_{s}$ in $\mathbb{Ho}_{{\mathbb{A}}^{1},+}(k)$ ([MV01, Lem. 2. 15]). For a pointed space ${\mathcal{X}}\in Spc_{+}/k$, we denote by $\Sigma_{T}({\mathcal{X}},x)=T\wedge({\mathcal{X}},x)$. Remark that ${\mathbb{P}}^{n}/{\mathbb{P}}^{n-1}\cong T^{n}\stackrel{{\scriptstyle def}}{{=}}T^{\wedge n}$ is an ${\mathbb{A}}^{1}$-equivalence. In particular, we have $({\mathbb{P}}^{1},*)\cong T$ ([MV01, Cor. 2.18]). Recall

Proposition 2.3**.**

[MV01, §3 Prop. 2. 17]** Let $X,Y\in Sm/k$ and $E,E^{\prime}$ be vector bundles on $X$ and $Y$ respectively. One has

1. (1)

There is a canonical isomorphism of pointed sheaves

[TABLE] 2. (2)

There is a canonical isomorphism of pointed sheaves

[TABLE] 3. (3)

The canonical morphism of pointed sheaves

[TABLE]

is an ${\mathbb{A}}^{1}$-weak equivalence.

The following theorem due to Voevodsky will play an essential role for our purpose. However, as pointed out by M. Levine, the identities in $K_{0}(-)$ are not enough for us to construct maps between twisted $\mathbf{E}$-cohomology. Following a suggestion by M. Levine, we will refine this result of Voevodsky later (see 4.25).

Theorem 2.4**.**

[Voe03, Thm. 2.11]** Let $X\in SmProj/k$ a smooth projective variety of pure dimension $d_{X}$ over a field $k$. There exists an integer $n_{X}$ and a vector bundle $V_{X}$ over $X$ of rank $n_{X}$, such that

[TABLE]

where $T_{X}$ denotes the tangent bundle of $X$. Moreover, there exists a morphism $T^{\wedge n_{X}+d_{X}}\rightarrow Th(V_{X})$ in $\mathbb{Ho}_{{\mathbb{A}}^{1},+}(k)$, such that the induced map $H^{2d_{X}}_{{\mathcal{M}}}(X,{\mathbb{Z}}(d_{X}))\rightarrow{\mathbb{Z}}$ coincides with the degree map $\deg:{\rm CH}_{0}(X)\rightarrow{\mathbb{Z}}$, where $T=S^{1}_{s}\wedge{\mathbb{G}}_{m}$.

Remark 2.5**.**

One can always add a trivial bundle to $V_{X}$ in Voevodsky’s theorem 2.4 to increase $n_{X}$ appropriately.

2.2. Stable ${\mathbb{A}}^{1}$-homotopy category

Let $\mathbf{Spect}^{\Sigma}(k)$ be the category of symmetric spectra in $k$-spaces, which can be viewed as category of Nisnevich sheaves of symmetric spectra. By applying the construction in [Ay08, Def. 4.4.40, Cor. 4.4.42, Prop. 4.4.62], $\mathbf{Spect}^{\Sigma}(k)$ has the structure of a monoidal model category. Let $\mathbf{StHo}_{S^{1}}(k)$ be the resulting homotopy category. The stable ${\mathbb{A}}^{1}$-homotopy category of $S^{1}$-spectra $\mathbf{StHo}_{{\mathbb{A}}^{1},S^{1}}(k)$ is obtained from $\mathbf{StHo}_{S^{1}}(k)$ by Bousfield localization. Equivalently, the category $\mathbf{Spect}^{\Sigma}(k)$ can be equipped with an ${\mathbb{A}}^{1}$-local model structure (cf. [Ay08, Def. 4.5.12]). The homotopy category of this ${\mathbb{A}}^{1}$-local model structure is $\mathbf{StHo}_{{\mathbb{A}}^{1},S^{1}}(k)$, which is also known to be equivalent to the category $\mathbf{StHo}^{S^{1}}_{{\mathbb{A}}^{1}-loc}(Sm/k)$ constructed by F. Morel in [Mor05, Def. 4.1.1]. The ${\mathbb{A}}^{1}$-local symmetric sphere spectrum is defined by taking the functor

[TABLE]

with an action of symmetric groups, where $L_{{\mathbb{A}}^{1}}$ denotes the ${\mathbb{A}}^{1}$-localization functor. For a pointed space $({\mathcal{X}},x)$, its ${\mathbb{A}}^{1}$-local symmetric suspension spectrum is defined as the symmetric sequence

[TABLE]

together with symmetric groups actions. Let ${\mathcal{E}}$ be an ${\mathbb{A}}^{1}$-local symmetric spectrum in $Spc/k$. One defines ([AH11, Def. 2.1.11]) the $i$-th $S^{1}$-stable ${\mathbb{A}}^{1}$-homotopy sheaf $\pi_{i}^{st{\mathbb{A}}^{1},S^{1}}({\mathcal{E}})$ of ${\mathcal{E}}$ as the Nisnevich sheaf on $Sm/k$ associated to the presheaf

[TABLE]

Now we consider the symmetric $T$-spectra or ${\mathbb{P}}^{1}$-spectra ([Jar00]). ${\mathbb{P}}^{1}$ is pointed with $\infty$ and ${\mathbb{P}}^{1\wedge n}$ has a natural action of $\Sigma_{n}$ by permutation of the factors, so the association $\underline{n}\mapsto{\mathbb{P}}^{1\wedge n}$ is a symmetric sequence. A symmetric ${\mathbb{P}}^{1}$-spectrum is a symmetric sequence with a module structure over the sphere spectrum $\mathbb{S}^{0}$. Denote by $\mathbf{Spect}_{{\mathbb{P}}^{1}}^{\Sigma}(k)$ the full subcategory of the category of symmetric sequence in $k$-spaces $\mathbf{Fun}(\mathpzc{Sym},Spc_{+}/k)$ consisting of symmetric ${\mathbb{P}}^{1}$-spectra, which also has a model structure [Ay08, Def. 4.5.21]. Here we denote by $\mathpzc{Sym}$ the groupoid, whose objects are $\underline{n}$ and morphisms are given by bijections. Let $\mathbf{StHo}_{{\mathbb{A}}^{1},{\mathbb{P}}^{1}}(k)$ be the resulting homotopy category, which is called ${\mathbb{P}}^{1}$-stable ${\mathbb{A}}^{1}$-homotopy category. For a pointed space $({\mathcal{X}},x)$, we will write $\Sigma^{\infty}_{{\mathbb{P}}^{1}}({\mathcal{X}},x)$ for the suspension symmetric ${\mathbb{P}}^{1}$-spectrum, i.e., it is given by the functor $\underline{n}\mapsto{\mathbb{P}}^{1\wedge n}\wedge{\mathcal{X}}$ equipped with an action of symmetric group by permuting the first $n$-factors. Let $\mathbf{S}^{i}$ be the suspension symmetric ${\mathbb{P}}^{1}$-spectrum of $S^{i}_{s}$. If ${\mathcal{E}}$ is a symmetric ${\mathbb{P}}^{1}$-spectrum, then the $i$-th ${\mathbb{P}}^{1}$-stable ${\mathbb{A}}^{1}$-homotopy sheaf $\pi_{i}^{st{\mathbb{A}}^{1},{\mathbb{P}}^{1}}({\mathcal{E}})$ is defined as the Nisnevich sheaf on $Sm/k$ associated to the presheaf (cf. [AH11, Def. 2.1.14])

[TABLE]

Theorem 2.6**.**

[Mor05, Thm. 6.1.8 and Cor. 6.2.9]** Let ${\mathcal{E}}$ be an ${\mathbb{A}}^{1}$-local symmetric $S^{1}$-spectrum. The homotopy sheaves $\pi_{i}^{st{\mathbb{A}}^{1},S^{1}}({\mathcal{E}})$ are strictly ${\mathbb{A}}^{1}$-invariant.

One has a canonical isomorphism [AH11, Prop. 2.1.16]

[TABLE]

So one may view that $\mathbb{StHo}_{{\mathbb{A}}^{1},{\mathbb{P}}^{1}}(k)$ is obtained from $\mathbf{StHo}_{{\mathbb{A}}^{1},S^{1}}(k)$ by formally inverting the ${\mathbb{A}}^{1}$-localized suspension spectrum of ${\mathbb{G}}_{m}$. So from 2.6, we see that for a pointed $k$-space $({\mathcal{X}},x)$, the homotopy sheaves $\pi_{i}^{st{\mathbb{A}}^{1},{\mathbb{P}}^{1}}({\mathcal{X}})$ are also strictly ${\mathbb{A}}^{1}$-invariant. By the computation of F. Morel ([Mor04], [Mor12]), one can identify the Milnor-Witt $K$-theory sheaves with stable homotopy sheaves of spheres

[TABLE]

This identification allows us to conclude that $\mathbb{K}^{MW}_{n}$ are strictly ${\mathbb{A}}^{1}$-invariant sheaves.

3. ${\mathbb{A}}^{1}$-homological algebra

3.1. Effective ${\mathbb{A}}^{1}$-derived category

Let $Ch_{-}({\mathcal{A}}b_{k})$ be the category of chain complexes over the category ${\mathcal{A}}b_{k}$ of abelian Nisnevich sheaves. Denote by $Ch_{\geq 0}({\mathcal{A}}b_{k})$ the category of chain complexes of abelian Nisnevich sheaves on $Sm/k$, whose homoglocial degree $\geq 0$. The sheaf-theoretical Dold-Kan correspondence

[TABLE]

where $\Delta^{op}{\mathcal{A}}b_{k}$ is the cateogry of simplicial abelian Nisnevich sheaves, gives us via the inclusion functor $Ch_{\geq 0}({\mathcal{A}}b_{k})\hookrightarrow Ch_{-}({\mathcal{A}}b_{k})$, a functor

[TABLE]

By applying this functor on the Eilenberg-Maclane spectrum $H{\mathbb{Z}}$, we obtain a ring spectrum $\widetilde{H{\mathbb{Z}}}$ in $\mathbf{Fun}(\mathpzc{Sym},Ch_{-}({\mathcal{A}}b_{k}))$. Let $\mathbf{Spect}^{\Sigma}(Ch_{-}({\mathcal{A}}b_{k}))$ be the full subcategory of the category $\mathbf{Fun}(\mathpzc{Sym},Ch_{-}({\mathcal{A}}b_{k}))$ consisting of modules over $\widetilde{H{\mathbb{Z}}}$. On the other hand, by composing with the free abelian group functor

[TABLE]

one obtains a functor

[TABLE]

which sends the sphere symmetric sequence to $\widetilde{H{\mathbb{Z}}}$. This induces then a functor between categories of symmetric spectra

[TABLE]

In fact, by [Hov01, Thm. 9.3], this induces a Quillen functor, which one refers as Hurewicz functor

[TABLE]

Now the effective ${\mathbb{A}}^{1}$-derived category $D^{eff}_{{\mathbb{A}}^{1}}(k)$ is constructed by applying ${\mathbb{A}}^{1}$-localization on the category $\mathbf{Spect}^{\Sigma}(Ch_{-}({\mathcal{A}}b_{k}))$. By the work of Cisinski and Déglise (cf. [CD10, §5]), this category is equivalent to the ${\mathbb{A}}^{1}$-derived category constructed by F. Morel in [Mor12]. Let $({\mathcal{X}},x)\in Spc_{+}/k$ be a pointed space, and $\Sigma^{\infty}_{s}({\mathcal{X}},x)$ its suspension symmetric spectrum. We apply the Hurewicz functor on $\Sigma^{\infty}_{s}({\mathcal{X}},x)$ and then $L_{{\mathbb{A}}^{1}}^{ab}(-)$, so we may define a functor

[TABLE]

Here we write $L_{{\mathbb{A}}^{1}}^{ab}$ for the ${\mathbb{A}}^{1}$-localization functor on chain complexes to distinguish from the ${\mathbb{A}}^{1}$-localization $L_{{\mathbb{A}}^{1}}$ on spaces. If ${\mathcal{X}}\in Spc/k$ is not pointed, then we write $C^{{\mathbb{A}}^{1}}_{*}({\mathcal{X}})\stackrel{{\scriptstyle def}}{{=}}\widetilde{C}^{{\mathbb{A}}^{1}}_{*}({\mathcal{X}}_{+})$. Define ${\mathbb{Z}}[n]=\mathfrak{H}^{ab}(\Sigma^{\infty}_{s}S^{n}_{s})$.

Definition 3.1**.**

Let ${\mathcal{X}}\in Spc/k$ be a $k$-space. Its $i$-th ${\mathbb{A}}^{1}$-homology sheaf is the Nisnevich sheaf $\mathbf{H}^{{\mathbb{A}}^{1}}_{i}({\mathcal{X}})$ associated to the presheaf

[TABLE]

Consider $({\mathbb{P}}^{1},\infty)$ pointed by $\infty$. According to [MV01, Cor. 2.18], we have ${\mathbb{P}}^{1}=S^{1}_{s}\wedge{\mathbb{G}}_{m}$, so we have an identification $\widetilde{C}_{*}^{{\mathbb{A}}^{1}}({\mathbb{P}}^{1})=\widetilde{C}^{{\mathbb{A}}^{1}}_{*}(S^{1}_{s}\wedge{\mathbb{G}}_{m})$. We define the ${\mathbb{A}}^{1}$-Tate complex (called enhanced Tate (motivic) complex by A. Asok and C. Haesemeyer [AH11, Def. 2.1.25 and Def. 3.2.1 and Lem. 3.2.2]) as

[TABLE]

Definition 3.2**.**

Let ${\mathcal{X}}\in Spc/k$ be a $k$-space. The bigraded unstable ${\mathbb{A}}^{1}$-cohomology group $H^{p,q}_{{\mathbb{A}}^{1}}({\mathcal{X}},{\mathbb{Z}})$ is defined as

[TABLE]

The relationship between unstable ${\mathbb{A}}^{1}$-cohomology and Nisnevich hypercohomology with coefficient ${\mathbb{Z}}_{{\mathbb{A}}^{1}}(n)$ is given by the following

Proposition 3.3**.**

[AH11, Prop. 3.2.5]** Let $k$ be a field and ${\mathcal{X}}\in Spc/k$ be a $k$-space. One has

1. (1)

For any $p,q$, there is a canonical isomorphism

[TABLE] 2. (2)

The cohomology sheaves $\underline{H}^{p}({\mathbb{Z}}_{{\mathbb{A}}^{1}}(q))=0$, if $p>q$. 3. (3)

There is a canonical isomorphism $\underline{H}^{p}({\mathbb{Z}}_{{\mathbb{A}}^{1}}(p))\cong\mathbb{K}^{MW}_{p}$, for all $p>0$.

Remark 3.4**.**

By construction the complex ${\mathbb{Z}}_{{\mathbb{A}}^{1}}(n)$ is ${\mathbb{A}}^{1}$-local, hence by definition (cf. [Mor12, Def. 5.17]) one has immediately that the sheaves $\underline{H}^{p}({\mathbb{Z}}_{{\mathbb{A}}^{1}}(q))$ are strictly ${\mathbb{A}}^{1}$-invariant. **

3.2. ${\mathbb{P}}^{1}$-stable ${\mathbb{A}}^{1}$-derived category

Having defined an ${\mathbb{A}}^{1}$-Tate complex, the way that we stabilize the category $D^{eff}_{{\mathbb{A}}^{1}}(k)$ is to invert formally the ${\mathbb{A}}^{1}$-Tate complex to obtain the ${\mathbb{P}}^{1}$-stable ${\mathbb{A}}^{1}$-derived category $D_{{\mathbb{A}}^{1}}(k)$. This can be done by following the construction detailed in [CD10, §5]. As before, we take $D_{{\mathbb{A}}^{1}}(k)$ as the resulting homotopy category of the model category $\mathbb{Spect}^{\Sigma}_{{\mathbb{P}}^{1}}(Ch_{-}({\mathcal{A}}b_{k}))$ consisting of modules over the ${\mathbb{A}}^{1}$-localization of the normalized chain complex of the free abelian group on the sphere symmetric ${\mathbb{P}}^{1}$-spectrum. For a pointed space $({\mathcal{X}},x)\in Spc_{+}/k$, the stable ${\mathbb{A}}^{1}$-complex $\widetilde{C}_{*}^{st{\mathbb{A}}^{1}}({\mathcal{X}})$ of $({\mathcal{X}},x)$ is defined as $L_{{\mathbb{A}}^{1}}^{ab}(N{\mathbb{Z}}(\Sigma^{\infty}_{{\mathbb{P}}^{1}}({\mathcal{X}},x)))$ and if ${\mathcal{X}}\in Spc/k$ is an unpointed $k$-space, then we write $C_{*}^{st{\mathbb{A}}^{1}}({\mathcal{X}})$ for $\widetilde{C}_{*}^{st{\mathbb{A}}^{1}}({\mathcal{X}}_{+})$. The category $D_{{\mathbb{A}}^{1}}(k)$ has an unit object, denoted by $\mathbb{1}_{k}$, which is the complex $\widetilde{C}^{st{\mathbb{A}}^{1}}_{*}(\mathbb{S}^{0})$. Define $\mathbb{1}_{k}[n]=\mathbb{1}_{k}\otimes\widetilde{C}_{*}^{st{\mathbb{A}}^{1}}(S^{n}_{s})$ and $\widetilde{C}_{*}^{st{\mathbb{A}}^{1}}({\mathcal{X}})[n]=\widetilde{C}_{*}^{st{\mathbb{A}}^{1}}({\mathcal{X}})\otimes\mathbb{1}_{k}[n]$ for a $k$-space $({\mathcal{X}},x)\in Spc_{+}/k$.

Definition 3.5**.**

Let ${\mathcal{X}}\in Spc/k$ be a $k$-space. The $i$-th ${\mathbb{P}}^{1}$-stable ${\mathbb{A}}^{1}$-homology sheaf $\mathbb{H}^{st{\mathbb{A}}^{1}}_{i}({\mathcal{X}})$ is the Nisnevich sheaf associated to the presheaf

[TABLE]

Just like in case of stable ${\mathbb{A}}^{1}$-homotopy categories, one has the following result

Proposition 3.6**.**

[AH11, Prop. 2.1.29]** Let $U\in Sm/k$ and $({\mathcal{X}},x)\in Spc_{+}/k$. One has a canonical isomorphism

[TABLE]

The Hurewicz formalism induces the following functors, which one still calls Hurewicz functors (or abelianization functors)

[TABLE]

which give rise to morphisms of sheaves

[TABLE]

Definition 3.7**.**

Let ${\mathcal{X}}\in Spc/k$ be a $k$-space. The bigraded ${\mathbb{P}}^{1}$-stable ${\mathbb{A}}^{1}$-cohomology group $H^{p,q}_{st{\mathbb{A}}^{1}}({\mathcal{X}},{\mathbb{Z}})$ is defined as

[TABLE]

The advantage of ${\mathbb{P}}^{1}$-stable ${\mathbb{A}}^{1}$-derived category $D_{{\mathbb{A}}^{1}}(k)$ is that one has duality formalism. In the context of stable ${\mathbb{A}}^{1}$-homotopy theory, it was done in [Hu05, App. A] and also [Rio05]. Firstly we recall that Deligne introduced in [Del87, §4] virtual categories. If $f:X\rightarrow{\rm Spec\,}k$ is a smooth $k$-scheme, the category $\mathcal{V}(X)$ of virtual bundles on $X$ is identified to the fundamental groupoid of $\mathcal{K}(X)$ where $\mathcal{K}$ is some ${\mathbb{A}}^{1}$-fibrant genuine model of algebraic $K$-theory. An actual vector bundle $\xi$ defines an object $\xi$ in $\mathcal{V}(X)$ whose isomorphism class corresponds to $[\xi]\in K_{0}(X)$. A short exact sequence of vector bundles

[TABLE]

gives not just an equality $[\xi]=[\xi^{\prime}]+[\xi^{\prime\prime}]$ in $K_{0}(X)$ but also a specific isomorphism $\xi\cong\xi^{\prime}\oplus\xi^{\prime\prime}$ of objects in $\mathcal{V}(X)$. By using universal property of $\mathcal{V}(X)$ as a Picard category, one can define an isomorphism (see [Rio10, §4])

[TABLE]

We haven’t yet introduced in this section the $6$ operations formalism of J. Ayoub, however we 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])

[TABLE]

We discuss a little bit more about the Thom spectrum of virtual bundles. If $\xi$ is a virtual vector bundle on an affine variety $U$, then there exist an actual vector bundle $\xi^{\prime}$ on $U$ and an integer $n\geq 0$, such that

[TABLE]

So one may define

[TABLE]

If $X$ is a projective variety, one can define an affine variety, which is ${\mathbb{A}}^{1}$-weak equivalent to $X$ (see [Hu05, p. 10]): Consider first of all the projective space ${\mathbb{P}}^{N}$. One defines

[TABLE]

which is an ${\mathbb{A}}^{N}$-bundle $pr_{1}:U\rightarrow{\mathbb{P}}^{N}$. If $X$ is a projective variety, one has $i:X\hookrightarrow{\mathbb{P}}^{N}$ and the affine variety $\pi:i^{*}U\rightarrow X$ is an ${\mathbb{A}}^{1}$-weak equivalence, where $\pi$ is the pullback of $pr_{1}$ along the closed immersion $i$:

[TABLE]

If $-T_{X}$ is the virtual normal bundle on $X$ of the diagonal embedding $\Delta_{X}:X\hookrightarrow X\times_{k}X$, which is the virtual tangent bundle, then its Thom spectrum is defined to be the Thom spectrum $Th(\mu/i^{*}U)$, where $\mu$ is the complement of the pullback of the tangent bundle of $X$ along $\pi$. We state the following result in $D_{{\mathbb{A}}^{1}}(k)$, although the proof in case of $\mathbf{StHo}_{{\mathbb{A}}^{1},{\mathbb{P}}^{1}}(k)$ 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 $X\in SmProj/k$, then $C_{*}^{st{\mathbb{A}}^{1}}(X)$ is a strong dualizable object in $D_{{\mathbb{A}}^{1}}(k)$ and its dual is $C_{*}^{st{\mathbb{A}}^{1}}(X)^{\vee}=\widetilde{C}_{*}^{st{\mathbb{A}}^{1}}(Th(-T_{X}))$. Consequently, one has a canonical isomorphism

[TABLE]

Fortunately, we will use later duality via $6$ operations formalism of J. Ayoub, which is good enough for our main purpose. We end up this section by a definition:

Definition 3.9**.**

Let $k$ be a field. One defines the geometric stable ${\mathbb{A}}^{1}$-derived category $D_{{\mathbb{A}}^{1},gm}(k)$ over $k$ as the thick subcategory of $D_{{\mathbb{A}}^{1}}(k)$ generated by $C_{*}^{st{\mathbb{A}}^{1}}(X)$, where $X\in Sm/k$.

4. $\mathbf{E}$-Motives

4.1. $\mathbf{E}$-Correspondences

Let $k$ be a field and we denote by $SH(k)$ the motivic stable homotopy category. Throughout this section we fix a motivic spectrum $\mathbf{E}\in SH(k)$ together with a multiplication map

[TABLE]

and a unit map

[TABLE]

such that the fowlling diagrams commute

[TABLE]

[TABLE]

Such a triple $(\mathbf{E},\mu_{\mathbf{E}},\varphi_{\mathbf{E}})$ is called a motivic ring spectrum.

Proposition 4.1**.**

Let $X,Y,Z,W\in SmProj(k)$. Let $\alpha\in SH(k)[\Sigma^{\infty}_{T,+}X,\Sigma^{\infty}_{T,+}Y\wedge^{\mathbb{L}}\mathbf{E}]$, $\beta\in SH(k)[\Sigma^{\infty}_{T,+}Y,\Sigma^{\infty}_{T,+}Z\wedge^{\mathbb{L}}_{S}\mathbf{E}]$ and $\gamma\in SH(k)[\Sigma^{\infty}_{T,+}Z,\Sigma^{\infty}_{T,+}W\wedge^{\mathbb{L}}_{S}\mathbf{E}]$. Let’s denote

[TABLE]

and similarly for $\gamma\circ_{M}\beta$. Then $\circ_{M}$ is associative and unital.

Proof.

Both $\gamma\circ_{M}(\beta\circ_{M}\alpha)$ and $(\gamma\circ_{M}\beta)\circ_{M}\alpha$ are equal to the following composition

[TABLE]

∎

Definition 4.2**.**

The category of $\mathbf{E}$-correspondences $Corr_{\mathbf{E}}(k))$ is defined as:

[TABLE]

and

[TABLE]

where the composition

[TABLE]

is defined as

[TABLE]

Proposition 4.3**.**

There is a functor

[TABLE]

which sends a morphism $f:X\rightarrow Y$ of $k$-schemes to

[TABLE]

Proof.

The identity morphism in $Corr_{\mathbf{E}}(k)(X,X)$ is given by

[TABLE]

Let $\alpha\in Corr_{\mathbf{E}}(k)(X,X)$ be an arbitrary $\mathbf{E}$-correspondence. By definition we have

[TABLE]

Since $\mathbf{E}$ is a ring spectrum, we must have $\alpha\circ_{M}({\rm id}_{X}\wedge\varphi_{\mathbf{E}})=\alpha$. Similarly, $({\rm id}_{X}\wedge\varphi_{\mathbf{E}})\circ_{M}\alpha=\alpha$. We check the compatibility of the composition laws. Let $X\stackrel{{\scriptstyle f}}{{\rightarrow}}Y\stackrel{{\scriptstyle g}}{{\rightarrow}}Z$ be morphisms of $k$-schemes. By definition we have

[TABLE]

and

[TABLE]

The equality $\mathfrak{h}(g\circ f)=\mathfrak{h}(g)\circ_{M}\mathfrak{h}(f)$ follows from the fact that $\mathbf{E}$ is a ring spectrum. ∎

Let $\mathbf{Spect}_{T}^{\Sigma}(k)$ be the model category of symmetric motivic $T$-spectra ([Jar00]). Following [CD10], [Deg13, §2.2] we call $\mathbf{E}\in\mathbf{Spect}_{T}^{\Sigma}(k)$ a strict motivic ring spectrum, if $\mathbf{E}$ is a commutative monoid object in $\mathbf{Spect}_{T}^{\Sigma}(k)$. An $\mathbf{E}$-module spectrum is a pair $(M,\gamma_{M})$, where $M\in\mathbf{Spect}_{T}^{\Sigma}(k)$ and $\gamma_{M}:M\wedge\mathbf{E}\rightarrow M$, such that the following diagrams commute:

[TABLE]

[TABLE]

Given two $\mathbf{E}$-modules $(M,\gamma_{)}$ and $(N,\gamma_{N})$, an $\mathbf{E}$-module map is a map $f:M\rightarrow N$, such that the following diagram commutes:

[TABLE]

Given a strict motivic ring spectrum $\mathbf{E}$ one can form the model category $\mathbf{E}-Mod^{\Sigma}$ of $\mathbf{E}$-modules with respect to the symmetric monoidal model category $\mathbf{Spect}_{T}^{\Sigma}(k)$ (see e.g [SS00])and there is a Quillen adjunction of model categories (we will return to this point in the last discussion in the Appendix):

[TABLE]

where $U$ denotes the forgetful functor. This Quillen adjunction induces an adjunction between homotopy categories:

[TABLE]

where we denote by $Ho_{k}(\mathbf{E}-Mod)$ the homotopy category associated to the category of strict $\mathbf{E}$-modules.

Theorem 4.4**.**

Let $k$ be a field and $\mathbf{E}\in\mathbf{Spect}_{T}^{\Sigma}(k)$ be a strict motivic ring spectrum. There is a functor

[TABLE]

Proof.

Recall that we may regard $\Sigma^{\infty}_{T,+}X\wedge^{\mathbb{L}}_{S}\mathbf{E}$ as an $\mathbf{E}$-modules via the map

[TABLE]

Let us denote the assocation above by

[TABLE]

$F$ maps on morphisms as following: Given $\alpha:\Sigma^{\infty}_{T,+}X\rightarrow\Sigma^{\infty}_{T,+}Y\wedge^{\mathbb{L}}_{S}\mathbf{E}$, we associate

[TABLE]

We have to check firstly, that $({\rm id}_{Y}\wedge\mu_{\mathbf{E}})\circ(\alpha\wedge{\rm id}_{\mathbf{E}})$ is a morphism of $\mathbf{E}$-modules. Since $\mathbf{E}$ is a ring spectrum, there is a commutative diagram

[TABLE]

Now we have to check the compatibility of the composition laws. Given $\alpha:\Sigma^{\infty}_{T,+}X\rightarrow\Sigma^{\infty}_{T,+}Y\wedge^{\mathbb{L}}_{S}\mathbf{E}$ and $\beta:\Sigma^{\infty}_{T,+}Y\rightarrow\Sigma^{\infty}_{T,+}Z\wedge^{\mathbb{L}}_{S}\mathbf{E}$. Then $F(\beta)\circ F(\alpha)$ is the following composition

[TABLE]

The composition $F(\beta\circ_{M}\alpha)$ is

[TABLE]

where

[TABLE]

Hence, $F(\beta\circ_{M}\alpha)$ is the following composition

[TABLE]

Since $\mathbf{E}$ is a ring spectrum, we have a commutative diagram

[TABLE]

This implies that $F(\beta\circ_{M}\alpha)=F(\beta)\circ F(\alpha)$. ∎

Definition 4.5**.**

Let $k$ be a field and $\mathbf{E}\in\mathbf{Spect}_{T}^{\Sigma}(k)$ be a strict motivic ring spectrum. We define the category $\mathbf{Mot}_{\mathbf{E}}(k)$ of pure $\mathbf{E}$-motives over $k$ to be the smallest pseudo-abelian subcategory of $Ho_{k}(\mathbf{E}-Mod)$ generated as an additive category by $\{\Sigma^{\infty}_{T,+}X\wedge^{\mathbb{L}}_{S}\mathbf{E}|X\in SmProj(k)\}$. **

Remark 4.6**.**

We know that if $char(k)=0$ then there is an equivalence of categories

[TABLE]

where $\mathbf{DM}(k)$ denotes the category of big Voevodsky’s motives (cf. [RO08]). As the category of pure Grothendieck-Chow motives $\underline{Chow}(k)\hookrightarrow\mathbf{DM}(k)$ is embedded fully faithful into $DM(k)$, we raise a question: is $\mathbf{Mot}_{H{\mathbb{Z}}}(k)$ equivalent to $\underline{Chow}(k)$ via the equivalence above? We only know that

[TABLE]

where the first isomorphism comes from the adjunction

[TABLE]

The second isomorphism comes from duality, the third isomorphism is the Thom isomorphism for motivic cohomology and the last isomorphism is the comparison isomorphism of Voevodsky ([MVW06, Cor. 19.2]). The question is, if these isomorphisms are compatible with the equivalence $Ho_{k}(H{\mathbb{Z}}-Mod)\cong\mathbf{DM}(k)$? It seems the problem with the first three isomorphisms is not difficult, however it seems that the problem with the last isomorphism is hard. **

Corollary 4.7**.**

Let $k$ be a field and $\mathbf{E}$ be a strict motivic ring spectrum. There is a functor

[TABLE]

Proof.

This follows from the adjunction 4.1. ∎

4.2. Functoriality in motivic stable homotopy

Following [Ay08], we recall that the stable homotopy category of schemes defines a $2$-functor from category of quasi-projective smooth schemes over a field $QSProjSm/k$ to the category of symmetric monoidal closed triangulated categories. Remark that the six operations formalism works much more general than what we here require. However we restrict ourselves only to $QSProjSm/k$, since it is already enough for our aim. We will list now a minimal list of properties of the six operations formalism: for any morphism of schemes $f:T\rightarrow S$, there is a pullback functor

[TABLE]

such that $(f\circ g)^{*}=g^{*}\circ f^{*}$. Moreover,

1. (1)

One has an adjunction for any morphism of schemes $f:T\rightarrow S$

[TABLE]

If $f$ is smooth, then one has an adjunction

[TABLE] 2. (2)

Given a cartesian square

[TABLE]

and assume $f$ is smooth, then

[TABLE] 3. (3)

Let $f:Y\rightarrow X$ be a smooth morphism, ${\mathcal{E}}\in SH(Y)$ and ${\mathcal{F}}\in SH(X)$, the natural transformation

[TABLE]

is an isomorphism. 4. (4)

Let $i:Z\hookrightarrow X$ be a closed immersion with complement $j:U\hookrightarrow X$, then there is a distinguished triangle

[TABLE] 5. (5)

For any closed immersion $i:Z\hookrightarrow X$, one has an adjunction

[TABLE] 6. (6)

Given a cartesian square

[TABLE]

where $i:Z\hookrightarrow X$ is a closed immersion, then one has an isomorphism

[TABLE] 7. (7)

Let $i:Z\hookrightarrow X$ be a closed immersion, ${\mathcal{E}}\in SH(Z)$ and ${\mathcal{F}}\in SH(X)$, the natural transformation

[TABLE]

is an isomorphism. 8. (8)

For any separated morphism of finite type $f:Y\rightarrow X$, there is an adjunction

[TABLE] 9. (9)

For a smooth separated morphism of finite type $f:Y\rightarrow X$ with the relative tangent bundle $T_{f}$ there are canonical natural isomorphisms, which are dual to each other

[TABLE]

Moreover, for any separated morphism of finite type $f:Y\rightarrow X$, there exist natural isomorphisms

[TABLE]

[TABLE]

[TABLE] 10. (10)

If $f:Y\rightarrow X$ is a smooth projective morphism then $f_{\#}(\mathbf{1}_{Y})$ is strongly dualizable in $SH(X)$ with the dual

[TABLE]

Furthermore, one has $D_{X}(f_{*}K)\cong f_{*}D_{Y}(K\wedge_{Y}Th_{Y}(T_{f})),\forall K\in SH(Y)$.

We will need some facts about cohomology with supports in the next subsection.

4.3. Cohomology with supports

Let $S={\rm Spec\,}k$. We consider the category $SH(k)$. For a ring spectrum $\mathbf{E}\in SH(k)$ and a closed pair $(X,Z)$, where $\pi_{X}:X\rightarrow S$ is a smooth quasi-projective $k$-scheme and $i:Z\hookrightarrow X$ a smooth closed subscheme, one defines the cohomology with support as

[TABLE]

where we write $\mathbf{E}_{X}=\pi_{X}^{*}\mathbf{E}$. As $\Sigma^{\infty}_{T,+}X/X-Z:=\pi_{X\#}i_{*}(\mathbf{1}_{Z})$ in $SH(k)$, so the first isomorphism follows from the adjunction

[TABLE]

and the last isomorphism comes from the adjunction

[TABLE]

If $f:Y\rightarrow X$ is a smooth morphism of smooth quasi-projective $S$-schemes we have a canonical homomorphism

[TABLE]

where $T=Y\times_{X}Z$ defined as following: Consider the commutative diagram

[TABLE]

For a morphism

[TABLE]

we can associate to a morphism

[TABLE]

If $T\stackrel{{\scriptstyle j}}{{\hookrightarrow}}Z\stackrel{{\scriptstyle i}}{{\hookrightarrow}}X$ are closed immersions, we can define a pushforward on cohomology with supports

[TABLE]

as following: Given a morphism $\alpha:X/X-T\rightarrow\mathbf{E}_{X}\wedge S^{p,q}$ we associate $j_{!}(\alpha)=\alpha\circ\bar{j}$, where $\bar{j}:X/X-Z\rightarrow X/X-T$ is the canonical morphism in $SH(X)$ induced by the immersion $X-Z\hookrightarrow X-T$. If $\alpha\in\mathbf{E}^{p,q}_{Z}(X)$ and $\beta\in\mathbf{E}^{m,n}_{Z}(X)$ we define their product in $\mathbf{E}^{p+m,q+n}_{Z}(X)$ as a morphism

[TABLE]

If $\xi/X$ is a vector bundle over a smooth $k$-scheme $X$ with the zero section $s_{0}:X\rightarrow\xi$, then the $\mathbf{E}$-cohomology of the Thom spectrum $Th(\xi)$ is

[TABLE]

The pushforward defined as above works only for closed immersions. We will define later pushforward on $\mathbf{E}$-cohomology of Thom spectrum for projective smooth morphism using duality.

4.4. Relation to the category of twisted $E$-correspondences

Notation 4.8**.**

For a quasi-projective smooth $k$-scheme $\pi_{X}:X\rightarrow{\rm Spec\,}k$ and a vector bundle $p_{\xi}:\xi\rightarrow X$ with [math]-section $s_{X}:X\rightarrow\xi$ we will write $Th_{X}(\xi)=p_{\xi\#}s_{X!}(\mathbf{1}_{X})$ for the Thom transformation $Th(s_{X},p_{\xi})=p_{\xi\#}s_{X!}$ applying on $\mathbf{1}_{X}$. $Th_{X}(\xi)$ is an object in $SH(X)$ and $Th_{X}(-\xi)=s_{X}^{!}p_{\xi}^{*}(\mathbf{1}_{X})$ for its inverse as the inverse Thom transformation $Th^{-1}(s_{X},p_{\xi})$ applying on $\mathbf{1}_{X}$. The Thom spectrum will be denoted by $Th(\xi/X)$, which means

[TABLE]

Sometime we only write $Th(\xi)$ for the Thom spectrum, if it is clear which scheme $X$ we talk about. One can see easily that this definition coincides with the traditional definition of Thom spectrum as follow: Let $j:\xi-s_{X}(X)\hookrightarrow\xi$ be the open immersion with the complement $s_{X}:X\rightarrow\xi$. One has a localization sequence

[TABLE]

Applying $\pi_{\#}p_{\xi\#}$ and as $s_{X}{*}\cong s_{X!}$ is a natural $2$-isomorphism one has a natural isomorphism in $SH(k)$:

[TABLE]

Let $E\in SH(k)$ be a ring spectrum and $X/k$ a quasi-projective smooth $k$-scheme. Let $p_{\xi}:\xi\rightarrow X$ be a vector bundle of rank $r$ with the zero section $s:X\rightarrow\xi$. We define $E$-cohomology of $X$ twisted by a vector bundle as

[TABLE]

where we write $E_{X}^{2r,r}=E_{X}\wedge S^{2r,r}$. We denote by $E^{*,*}(X,\xi)$ the bigraded ring

[TABLE]

Remark that $E^{*,*}(X,\xi)$ is bigraded ring. Even if $E$ is a commutative ring spectrum, $E^{*,*}(X,\xi)$ is never bigraded commutative. If $\xi\in{\mathcal{V}}(X)$ is a virtual vector bundle of rank $r<0$ then $p_{-\xi}:-\xi\rightarrow X$ is an actual vector bundle, so we define

[TABLE]

This group has the following interpretation by Jouanolou trick: As $X$ is quasi-projective, so we have an immersion $i:X\hookrightarrow{\mathbb{P}}^{N}$. Via the Segre embedding ${\mathbb{P}}^{N}\times{\mathbb{P}}^{N}\hookrightarrow{\mathbb{P}}^{N^{2}+2N}$, $U$ is an affine variety. Let

[TABLE]

$pr_{1}:U\rightarrow{\mathbb{P}}^{N}$ is an ${\mathbb{A}}^{N}$-bundle. Consider the pullback diagram

[TABLE]

Then

[TABLE]

where $\zeta$ is an actual vector bundle on $U$, such that $\pi^{*}\xi\oplus{\mathcal{O}}^{n}=\zeta$.

Proposition 4.9**.**

Let $f:\xi\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\xi^{\prime}$ be an isomorphism of vector bundles on $X$

[TABLE]

There is a natural isomorphism

[TABLE]

Proof.

Consider the Cartesian squares

[TABLE]

One has two $2$-isomorphisms ([Ay08, §1.5.5])

[TABLE]

and

[TABLE]

which prove the Proposition. ∎

Proposition 4.10**.**

If $E$ is orientable in sense of [CD10, Def. 12.2.2], then there is a natural isomorphism

[TABLE]

Proof.

Since $E$ is orientable, one has by [CD10, Thm. 2.4.50 (3)] a canonical natural isomorphism

[TABLE]

This induces a natural isomorphism

[TABLE]

∎

Proposition 4.11**.**

(twisted Thom isomorphism) Let $X/k$ be a quasi-projective smooth $k$-scheme and $p_{\xi}:\xi\rightarrow X$ be a vector bundle of rank $r$ with the zero section $s:X\rightarrow\xi$. One has a natural isomorphism

[TABLE]

which we call the twisted Thom isomorphism.

Proof.

We have two adjunctions

[TABLE]

Hence, we have

[TABLE]

where the last natural isomorphism is induced by the adjunction $(\pi_{X\#},\pi_{X}^{*})$, where $\pi_{X}:X\rightarrow{\rm Spec\,}k$ is the structure morphism. So for a morphism

[TABLE]

the twisted Thom isomorphism is explicitly given by

[TABLE]

∎

Example 4.12**.**

The twisted Chow-Witt group $\widetilde{{\rm CH}}^{p}(X,\det\xi)$ defined by J. Fasel (cf. [Fas07] and [Fas08]) and also by F. Morel ([Mor12]) is an example of twisted cohomology. One has a natural isomorphism

[TABLE]

where $H(\underline{\mathbf{K}}^{MW}_{*})$ denotes the Eilenberg-Maclane spectrum associated to the homotopy module $\underline{\mathbf{K}}^{MW}_{*}$. We will discuss later about $H\underline{\mathbf{K}}^{MW}_{*}$ after introducing the homotopy t-structure. **

Before going further, we want to give a list of properties of Thom transformations that we will need for our constructions.

Proposition 4.13**.**

[Ay08, Prop. 2.3.19]** Let $X$ be a quasi-projective $k$-scheme and $\xi/X$ be a vector bundle. Let $f:Y\rightarrow X$ be a morphism. Then one has two natural $2$-isomorphisms

[TABLE]

which satisfy: For all $(K,L)\in Obj(SH(X)^{2})$, there are two commutative diagrams

[TABLE]

and

[TABLE]

Proposition 4.14**.**

[Ay08, Prop. 2.3.20]** Let $f:Y\rightarrow X$ be a $k$-morphism of quasi-projective schemes and $\xi/X$ be a vector bundle. There are two natural $2$-isomorphisms

[TABLE]

such that the following diagrams commute for all $(K,L)\in SH(X)\times SH(Y)$:

[TABLE]

[TABLE]

Let $f:Y\rightarrow X$ 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

[TABLE]

Consider the functor $f^{*}:SH(X)\rightarrow SH(Y)$. $f^{*}$ induces a map

[TABLE]

Let $s_{Y}$ be the [math]-section of the vector bundle $p_{f^{*}\xi}:f^{*}\xi\rightarrow Y$ and we write $f_{\xi}:f^{*}\xi\rightarrow\xi$. One has an exchange transformation (see [Ay08, Prop. 1.4.15])

[TABLE]

which is the following composition ($s_{*}\cong s_{!},s_{Y!}\cong s_{Y*}$ since $s$ and $s_{Y}$ are closed immersion):

[TABLE]

where $f_{\xi}:f^{*}\xi\rightarrow\xi$ is the induced map on vector bundles. Note that the exchange transformation $Ex^{*!}$ is an isomorphism, when $f$ is smooth ([Ay08, Cor. 1.4.17]). At this point we also notice that for an actual bundle $\xi$, the Thom transformation $Th_{X}(\xi)$ behaves well under pullback of a general morphism, since

[TABLE]

and we have a natural transformation

[TABLE]

which is an isomorphism, if $f$ is smooth (see [Ay08, Lem. 1.5.4]). However, he showed that $Th_{X}(\xi)$ and $Th_{X}(-\xi)$ are inverse to each other [Ay08, Thm. 1.5.7], hence $Th_{Y}(-f^{*}\xi)\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}f^{*}Th_{X}(-\xi)$ (cf. [Ay08, Rem. 1.5.10]) for all morphism not necessary smooth $f$. That is a very crucial point. Now consider the pullback diagram

[TABLE]

We have a natural isomorphism $f^{*}_{\xi}p^{*}_{\xi}\cong p^{*}_{f^{*}\xi}f^{*}$. Hence, we obtain a map

[TABLE]

which we define as pullback of twisted $E$-cohomology.

Remark 4.15**.**

The composition of pullback on twisted $E$-cohomology $g^{*}\circ f^{*}$ is only defined up to the natural isomorphism $(f\circ g)^{*}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}g^{*}f^{*}$. **

Remark 4.16**.**

Let $a:\xi\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\xi$ be an automorphism of a vector bundle $\xi$ of rank $r$ on $X$. Then one has the cartesian squares

[TABLE]

As in [Ay08, §1.5.5 p. 84] $a$ induces two $2$-isomorphisms between the Thom transformations

[TABLE]

and

[TABLE]

This induces an isomorphism, which is not necessary identity

[TABLE]

However, the two pullbacks induced on twisted $E$-cohomology along a morphism $f:Y\rightarrow X$ must not be on the same target $E^{p,q}(Y,f^{*}\xi)$, as there are two different pullback diagrams

[TABLE]

Consequently, there is no problem with maps between $E$-cohomology created by automorphisms of $\xi$. **

Remark 4.17**.**

Thanks to the Proposition 4.13. The pullback of cohomology of virtual vector bundles is defined in the same way. **

Proposition 4.18**.**

Let $Z\stackrel{{\scriptstyle g}}{{\rightarrow}}Y\stackrel{{\scriptstyle f}}{{\rightarrow}}X$ be morphisms of quasi-projective smooth $k$-schemes. Let $\xi/X$ be a vector bundle. Then one has up to a natural isomorphism induced by a natural $2$-isomorphism

[TABLE]

Proof.

Consider the chain of pullback bundles

[TABLE]

Let $s_{X},s_{Y}$ and $s_{Z}$ be the [math]-sections of $\xi,f^{*}\xi$ and $g^{*}f^{*}\xi$ respectively. The functoriality up to a natural isomorphism follows easily from the natural $2$-isomorphism

[TABLE]

∎

This motivates us to give the following definition:

Definition 4.19**.**

A twisted $E$-cohomology pre-theory is an association, which is contravariant in both variables:

[TABLE]

where $\mathbf{Ring}^{*}$ denotes the category of bigraded rings and ${\mathcal{V}}$ is the 2-category, where objects are categories of virtual vector bundles ${\mathcal{V}}(X)$ for $X\in QSProjSm(k)$ and

[TABLE]

[TABLE]

${\mathcal{A}}$ is the full subcategory of $QSProjSm(k)\times{\mathcal{V}}$ consisting of those pairs $(X,\xi)$, where $X\in QSProjSm(k)$ and $\xi\in{\mathcal{V}}(X)$. $Mor_{{\mathcal{A}}}((X,\xi),Y(,\eta))$ consists of pair $(f,g)$, where $f:X\rightarrow Y$ is a morphism of quasi-projective smooth $k$-schemes and $g:\xi\rightarrow\eta$ is a bundle map

[TABLE]

such that $\xi\rightarrow f^{*}\eta$ is a monomorphism in ${\mathcal{V}}(X)$. $E^{*,*}(-,-)$ sends such a pair $(X,\xi)$ to $E^{*,*}(X,\xi)$. Given an ${\mathcal{A}}$-morphism $(f,g):(X,\xi)\rightarrow(Y,\eta)$, $E^{*,*}(-,-)$ sends $(f,g)$ to the following composition

[TABLE]

where $f^{*}$ is the pullback map on twisted $E$-cohomology constructed as above and the last map is induced by

[TABLE]

as $\xi\rightarrow f^{*}\eta$ is a monomorphism in ${\mathcal{V}}(X)$. **

Proposition 4.20**.**

Let $f:Y\rightarrow X$ be a $k$-morphism of quasi-projective smooth $k$-schemes and $p_{\xi}:\xi\rightarrow X$ be a vector bundle of rank $r$ on $X$. There is a commutative diagram up to a natural isomorphism induced by a natural $2$-isomorphism

[TABLE]

where $f^{*}:E^{p+2r,q+r}(Th(\xi))\rightarrow E^{p+2r,q+r}(Th(f^{*}\xi))$ is the pullback given by

[TABLE]

Proof.

It is obvious by construction. Remark that for general morphism $f$ we always have the pullback

[TABLE]

Since $\pi_{X}:X\rightarrow{\rm Spec\,}k$ and $\pi_{Y}:Y\rightarrow{\rm Spec\,}k$ are smooth, then one has the natural isomorphisms via the adjunctions $(\pi_{X\#},\pi_{X}^{*})$ and $(\pi_{Y\#},\pi_{Y}^{*})$:

[TABLE]

and

[TABLE]

Explicitly, given a morphism

[TABLE]

we have

[TABLE]

and

[TABLE]

The two composition are natural isomorphism to each other, as we have the natural $2$-isomorphisms:

[TABLE]

∎

Let $f:Y\rightarrow X$ be a smooth projective morphism of projective smooth $k$-schemes of relative dimension $d=\dim(Y)-\dim(X)$ and $p_{\xi}:\xi\rightarrow X$ be a vector bundle of rank $r$ with the zero section $s:X\rightarrow\xi$. We define in the following a pushforward on twisted $E$-cohomology: Consider

[TABLE]

where $T_{f}$ is the normal bundle of the diagonal immersion $\delta:Y\rightarrow Y\times_{X}Y$.

[TABLE]

where $Th_{Y}(-T_{f})\in SH(Y)$ is the inverse of $Th_{Y}(T_{f})\in SH(Y)$. Since $E_{Y}=f^{*}E_{X}$, the adjunction

[TABLE]

gives us a natural isomorphism

[TABLE]

By the projection formula $Pr^{*}_{\#}$ and since $Th_{Y}(f^{*}\xi)\cong f^{*}Th_{X}(\xi)$ as $\xi$ is an actual bundle, we have then a natural isomorphism

[TABLE]

So we have then a natural isomorphism

[TABLE]

By [CD10, Prop. 2.4.31] we have

[TABLE]

where $D_{X}(f_{\#}\mathbf{1}_{Y})$ means the dual of $f_{\#}\mathbf{1}_{Y}$ in $SH(X)$. Hence there is a natural isomorphism

[TABLE]

From the counit of the adjunction $(f_{\#},f^{*})$

[TABLE]

we have an induced map

[TABLE]

By the twisted Thom isomorphism, the later group is

[TABLE]

Now we define formally:

Definition 4.21**.**

Let $f:Y\rightarrow X$ be a smooth projective morphism of projective smooth $k$-schemes of relative dimension $d=\dim(Y)-\dim(X)$ and $p_{\xi}:\xi\rightarrow X$ be a vector bundle. We define

[TABLE]

where $\pi_{f^{*}\xi}:f^{*}\xi\rightarrow Y$ is the pullback bundle and $s_{Y}$ is its [math]-section. The pushforward map is the induced map constructed as above

[TABLE]

Remark that our definition of projective pushforward $f_{*}$ doesn’t require $E$ to be an oriented cohomology theory, however we need the assumption on smoothness of $f$. The reason that we choose the notation $E^{p,q}(Y,f^{*}\xi-T_{f})$ is that this group should behave like the so-called cohomology twisted by formal difference of vector bundles. The shifting in the definition $(-2d,-d)$ reminds us that the inverse Thom transformation $Th_{Y}(-T_{f})$ should behave like the Thom spectrum of the virtual bundle $-T_{f}$ after taking $\pi_{Y\#}$, where $\pi_{Y}:Y\rightarrow{\rm Spec\,}k$ is the structure morphism of $Y$, as the rank of the virtual bundle $-T_{f}$ is $-d$. As already mentioned in $\S 3$ we refer the reader to [Rio10, §4] and [Del87] for the discussion on Picard category of virtual bundles. But we remind the reader again that we always work with an actual bundle $\xi$.

Remark 4.22**.**

Let $Z\stackrel{{\scriptstyle g}}{{\longrightarrow}}Y\stackrel{{\scriptstyle f}}{{\longrightarrow}}X$ be a sequence of composable morphisms. Then $f_{*}\circ g_{*}$ is not defined for a trivial reason: One has only a natural $2$-isomorphism

[TABLE]

which comes from the exact sequence

[TABLE]

The $2$-isomorphism $e_{\sigma}^{\vee}$ is not an identity. Consequently $f_{*}\circ g_{*}$ is only defined up to this specific natural $2$-isomorphism.

Remark 4.23**.**

Another variant to construct pushforward can be obtained as follows: One has a natural isomorphism via Thom transformation adjunctions Apply the functor $f_{*}:SH(Y)\rightarrow SH(X)$ we obtain a map

[TABLE]

By projection formula $Pr_{*}^{*}(f)$ we have a canonical isomorphism

[TABLE]

which induces a canonical isomorphism

[TABLE]

From the unit $\eta_{(f^{*},f_{*})}(X):\mathbf{1}_{X}\rightarrow f_{*}f^{*}\mathbf{1}_{X}\cong f_{*}\mathbf{1}_{Y}$ of the adjunction $(f^{*},f_{*})$ one obtains a map

[TABLE]

As $f$ is projective we have $f_{*}\cong f_{!}$ and since $f$ is smooth we have the canonical purity isomorphism

[TABLE]

which induces a canonical isomorphism

[TABLE]

The counit $\varepsilon_{(f_{\#},f^{*})}(X):f_{\#}\mathbf{1}_{Y}\cong f_{\#}f^{*}\mathbf{1}_{X}\rightarrow\mathbf{1}_{X}$ of the adjunction $(f_{\#},f^{*})$ induces then a map

[TABLE]

So we obtain a map defined as the composition of the maps above

[TABLE]

One can check the two constructions are equivalent. And again the Proposition 4.14 tells us that the composition of pushforward maps $f_{*}\circ g_{*}$ is only defined up to a specific natural $2$-isomorphism.

Proposition 4.24**.**

Let $Z\stackrel{{\scriptstyle g}}{{\rightarrow}}Y\stackrel{{\scriptstyle f}}{{\rightarrow}}X$ be smooth projective morphisms of projective smooth $k$-schemes of relative dimension $e$ resp. $d$. Let $\xi/X$ be a vector bundle of rank $r$. Then one has up to a natural $2$-isomorphism

[TABLE]

Proof.

We have an exact sequence of vector bundles on $Z$ [EGA4, 17.2.3]

[TABLE]

So we have an isomorphism (cf. [CD10, Rem. 2.4.52])

[TABLE]

where $-\wedge_{Z}-$ means relative wedge product over $Z$. Since $g^{*}$ is strong monoidal and since all $f$ and $g$ are smooth, which means that the $\wedge_{Z}$-inverse object of $Th_{Z}(T_{fg})$ is $Th_{Z}(-T_{fg})$ and $Th_{Z}(T_{g})^{-1}=Th_{Z}(-T_{g})$ and $(g^{*}Th_{Y}(T_{f}))^{-1}=g^{*}(Th_{Y}(-T_{f}))$ (cf. [CD10, Thm. 2.4.50 (3)]). Hence we have

[TABLE]

Functoriality of pushforward follows from this isomorphism as follow: We write $h=f\circ g$. Let $s_{Z}:Z\rightarrow g^{*}f^{*}\xi$ be the [math]-section of the vector bundle $p_{g^{*}f^{*}\xi}:g^{*}f^{*}\xi\rightarrow Z$. Let us recall the notation now: For an adjunction between categories

[TABLE]

we denote

[TABLE]

the counit and unit of the adjunction $(L,R)$ respectively. The composition $f_{*}\circ g_{*}$ is by construction the following composition:

[TABLE]

where $(1)$ is the natural isomorphism given by the adjunction of the Thom transformations $Th(s_{Z},p_{g^{*}f^{*}\xi})$ and $Th_{Z}(T_{fg})$, $(2)$ is the natural isomorphism given by the adjunction $(g_{\#},g^{*})$

[TABLE]

$(3)$ is the natural isomorphism given by the projection formula $Pr^{*}_{\#}(g)$

[TABLE]

$(4)$ is the natural isomorphism given by

[TABLE]

$(5)$ is the natural isomorphism given by the projection formula $Pr_{\#}^{*}(g)$, $(6)$ is the natural isomorphism given by duality in $SH(Y)$:

[TABLE]

$(7)$ is the natural isomorphism given by adjunction of duality in $SH(Y)$

[TABLE]

$(8)$ is the pushforward induced by the counit $g_{\#}\mathbf{1}_{Z}\cong g_{\#}g^{*}\mathbf{1}_{Y}\rightarrow\mathbf{1}_{Y}$

[TABLE]

$(9)$ is the natural isomorphism given by the adjunction $(f_{\#},f^{*})$

[TABLE]

$(10)$ is the natural isomorphism given by the projection formula $Pr_{\#}^{*}(f)$

[TABLE]

$(11)$ is the natural isomorphism given by duality in $SH(X)$:

[TABLE]

$(12)$ is the natural isomorphism given by the adjunction of duality in $SH(X)$

[TABLE]

and finally $(13)$ is the pushforward induced by the counit $f_{\#}\mathbf{1}_{Y}\cong f_{\#}f^{*}\mathbf{1}_{X}\rightarrow\mathbf{1}_{X}$:

[TABLE]

$h_{*}=(f\circ g)_{*}$ is the following composition:

[TABLE]

where $(1^{\prime})$ is the natural isomorphism given by the adjunction of the Thom transformations $Th(s_{Z},p_{g^{*}f^{*}\xi})$ and $Th_{Z}(T_{fg})$, $(2^{\prime})$ is the natural isomorphism given by the adjunction $(h_{\#},h^{*})$

[TABLE]

$(3^{\prime})$ is the natural isomorphism given by the projection formula $Pr_{\#}^{*}(h)$

[TABLE]

$(4^{\prime})$ is the natural isomorphism by duality in $SH(X)$:

[TABLE]

$(5^{\prime})$ is the natural isomorphism given by the adjunction of duality in $SH(X)$:

[TABLE]

and finally $(6^{\prime})$ is the pushforward induced by the counit $h_{\#}\mathbf{1}_{Z}\cong h_{\#}h^{*}\mathbf{1}_{X}\rightarrow\mathbf{1}_{X}$:

[TABLE]

The maps $(1)$ and $(1^{\prime})$ are identical. The diagram

[TABLE]

commutes, because $h_{\#}\cong f_{\#}\circ g_{\#}$, where

[TABLE]

is the natural isomorphism induced from the adjunction $(f_{\#},f^{*})$. Indeed, let

[TABLE]

be a morphism. Then one has

[TABLE]

and

[TABLE]

So we have $(2^{\prime})=(a_{1})\circ(2)$. Consider the pentagon

[TABLE]

where

[TABLE]

is the natural isomorphism induced by the adjunction $(f_{\#},f^{*})$ and

[TABLE]

is the natural isomorphism given by the projection formula $Pr_{\#}^{*}(f)$. We remind the reader that the isomorphism in the projection formula $Pr_{\#}^{*}(f)$ is given by the composition:

[TABLE]

The pentagon commutes since isomorphism induced by the projection formula $Pr_{\#}^{*}(h)$ is the composing of isomorphisms coming from projection formulas $Pr_{\#}^{*}(g)$ and $Pr_{\#}^{*}(f)$. Indeed, let

[TABLE]

be a morphism. We have

[TABLE]

and

[TABLE]

So we have

[TABLE]

For any $K\in SH(Z)$ one has commutative diagram (see [Ay08, §1.4.2, §1.5] and [CD10, Rem. 2.4.52])

[TABLE]

Now we take $K=\mathbf{1}_{Z}$ and dualize $D_{X}(-)$ the commutative diagram above. One has

[TABLE]

[TABLE]

and

[TABLE]

So we can conclude that

[TABLE]

Putting all together we have

[TABLE]

which means the pushforward on twisted $E$-cohomology satisfies $(f\circ g)_{*}=f_{*}\circ g_{*}$ up to a natural isomorphism induced by the natural $2$-isomorphism

[TABLE]

∎

Now we follow a suggestion by M. Levine to make a refinement to the result of Voevodsky in 2.4, since as pointed out by M. Levine it is not enough to use the identities in $K_{0}(-)$ to constructs maps between twisted $\mathbf{E}$-cohomology groups.

Proposition 4.25**.**

(A refinement of Voevodsky’s theorem) Let $X\in SmProj(k)$ of dimension $d_{X}$, where $k$ is a field. After fixing an embedding $X\hookrightarrow{\mathbb{P}}^{d}$ there exists a vector bundle $V_{X}$ on $X$ of rank $d^{2}+2d-d_{X}$, such that one has a specific isomorphism between objects in the Picard category of virtual bundles ${\mathcal{V}}(X)$ on $X$:

[TABLE]

Proof.

Case 1: $X={\mathbb{P}}^{d}$. One has an exact sequence

[TABLE]

By taking dual one also has

[TABLE]

There are two isomorphisms between objects in ${\mathcal{V}}(X)$:

[TABLE]

and

[TABLE]

Define

[TABLE]

As the Picard category ${\mathcal{V}}(X)=V(Vect(X))$ (the category of virtual objects associated to the exact category of vector bundles on $X$) has not just $\oplus$, but also a biexact functor

[TABLE]

which is distributive ([Del87]), one has an isomorphism in ${\mathcal{V}}(X)$:

[TABLE]

This implies that we have an isomorphism in ${\mathcal{V}}(X)$:

[TABLE]

Case 2: $X$ is smooth projective. Let $i:X\hookrightarrow{\mathbb{P}}^{d}$ be a closed embedding. One define

[TABLE]

where $N_{X/{\mathbb{P}}^{d}}$ denotes the normal bundle of $X$ in ${\mathbb{P}}^{d}$. In ${\mathcal{V}}(X)$ one has an isomorphism between objects

[TABLE]

From the exact sequence

[TABLE]

one has an isomorphism in ${\mathcal{V}}(X)$:

[TABLE]

This implies that we have a isomorphism in ${\mathcal{V}}(X)$:

[TABLE]

This implies that we have a specific isomorphism in ${\mathcal{V}}(X)$:

[TABLE]

∎

P. Hu in [Hu05] didn’t check if her construction is the same as the construction of Voevodsky. We notice that the map constructed by Voevodsky [Voe03, Thm. 2.11]

[TABLE]

is first of all only in $\mathbf{Ho}_{{\mathbb{A}}^{1},+}(k)$ and secondly very difficult to follow. We will take the refinement $V_{X}\oplus T_{X}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}{\mathcal{O}}_{X}^{d^{2}+2d}$ in ${\mathcal{V}}(X)$ and construct the Pontryagin-Thom collapse map

[TABLE]

by unpacking Hu’s construction. Firstly, for a projective smooth $k$-variety $i:X\hookrightarrow{\mathbb{P}}^{d}$, we have by definition

[TABLE]

If $PTV$ is already for ${\mathbb{P}}^{d}$ constructed, then $PTV$ for $X$ is defined by the composition

[TABLE]

where $q$ is the quotient map

[TABLE]

The isomorphism $V_{{\mathbb{P}}^{d}}/(V_{{\mathbb{P}}^{d}}-X)\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}Th(N_{X/V_{{\mathbb{P}}^{d}}})$ is the homotopy purity isomorphism ([MV01, §3 Thm.2.23]). For $X={\mathbb{P}}^{1}$ one has a commutative diagram in $\mathbf{Ho}_{{\mathbb{A}}^{1},+}(k)$ ([Hu05, pp. 9])

[TABLE]

because $pr_{1}:(X\times X)-\Delta_{X}\rightarrow X$ is an affine bundle. So one has a cofiber sequence in $\mathbf{Ho}_{{\mathbb{A}}^{1},+}(k)$

[TABLE]

For a vector bundle $\xi$ on $X$ one has

[TABLE]

One the other hand one has commutative diagram ([Hu05, (3.13)])

[TABLE]

So one obtains a cofiber sequence in $\mathbf{Ho}_{{\mathbb{A}}^{1},+}(k)$

[TABLE]

Now we take $\xi=V_{X}$ and by the refinement $V_{X}\oplus T_{X}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}{\mathcal{O}}_{X}^{d^{2}+2d}$ in ${\mathcal{V}}(X)$ we have then a cofiber sequence

[TABLE]

This gives rise to a map in $SH(k)$:

[TABLE]

To construct the Pontryagin-Thom collapse map

[TABLE]

such that the composition

[TABLE]

is the identity ${\rm id}_{\Sigma^{\infty}_{T,+}X}$ in $SH(k)$, it is enough to construct a map

[TABLE]

such that the composition $\varepsilon\circ(PTV\wedge{\rm id})$ is the collapse map $\Sigma^{\infty}_{T,+}X\rightarrow\mathbb{S}^{0}$, because $g^{V_{X}}$ is the composition

[TABLE]

By adjunction $\varepsilon$ gives us a map

[TABLE]

We remind the reader that P. Hu started with $X={\mathbb{P}}^{1}$ before [Hu05, Lem. 3.8], since she wanted to prove some particular results for projective quadric. For general $X$ and a vector bundle $\xi$ on $X$ one still has the map

[TABLE]

and hence a map

[TABLE]

and hence by applying $\xi=V_{X}$ one has a map

[TABLE]

Now we consider the linear embedding $i:{\mathbb{P}}^{d}\hookrightarrow{\mathbb{P}}^{d+1}$. By construction

[TABLE]

and so the diagram

[TABLE]

commutes, since it is adjoint to the commutativity of the diagram

[TABLE]

where we write $q^{{\mathbb{P}}^{d+1}}_{{\mathbb{P}}^{d}}$ for the quotient map and the last commutative diagram is obtained by $V_{{\mathbb{P}}^{d+1}}$-Thomification (i.e. we apply $g^{V_{{\mathbb{P}}^{d+1}}}$ on ${\mathbb{P}}^{d+1}\times{\mathbb{P}}^{d+1}\rightarrow Th(T_{{\mathbb{P}}^{d+1}})$) of the commutative diagram

[TABLE]

Consider the composition

[TABLE]

$({\mathbb{P}}^{d+1}-{\mathbb{P}}^{d})\times{\mathbb{P}}^{d}$ is mapped to $({\mathbb{P}}^{d+1}\times{\mathbb{P}}^{d+1})-\Delta_{{\mathbb{P}}^{d+1}}$. So the composition above induces a map

[TABLE]

where $j:({\mathbb{P}}^{d+1}-{\mathbb{P}}^{d})\hookrightarrow{\mathbb{P}}^{d+1}$ denotes the open immersion. After composing with the collapse map ${\mathbb{P}}^{d+1}_{+}\rightarrow\mathbb{S}^{0}$ and taking adjoint one obtains a map

[TABLE]

By construction there is a commutative diagram in $SH(k)$:

[TABLE]

Now the Claim $1$ in the proof of [Hu05, Lem. 3.8] implies that there is a morphism of distinguished triangles given by the commutative diagrams 4.2 and 4.3.

[TABLE]

So by induction on $d$ one can conclude that

[TABLE]

is an isomorphism in $SH(k)$ for all $d\geq 0$. Now we can construct the Pontryagin-Thom collapse map

[TABLE]

If $X\hookrightarrow{\mathbb{P}}^{d}$ is a smooth projective $k$-variety, we define $PTV$ for $X$ as the composition of $PTV$ for ${\mathbb{P}}^{d}$ with the quotient map

[TABLE]

So by construction we have:

Proposition 4.26**.**

Let $X$ be a smooth projective $k$-scheme. After fixing an embedding $i:X\hookrightarrow{\mathbb{P}}^{d}$, there is a commutative diagram in $SH(k)$

[TABLE]

where $PTH:\mathbb{S}^{0}\rightarrow\Sigma^{\infty}_{T,+}Th(-T_{X})$ is the map constructed in [Hu05, Lem. 3.18].

Proposition 4.27**.**

Let $f:Y\rightarrow X$ be a projective smooth morphism of projective smooth $k$-schemes of relative dimension $d=d_{Y}-d_{X}$. After fixing an embedding $X\hookrightarrow{\mathbb{P}}^{N}$, there is an isomorphism

[TABLE]

where $V_{X}$ and $V_{Y}$ are vector bundles on $X$ and $Y$ of rank $n_{X}$ and $n_{Y}$ as in theorem 2.4 respectively with the refinement in the Proposition 4.25. Moreover, the isomorphism $tt_{E}^{Y}$ is independent from the choice of the projective embeddings up to a unique canonical isomorphism.

Proof.

Let us denote by $s_{Y}$ the [math]-section of the vector bundle $p_{f^{*}V_{X}}:f^{*}V_{X}\rightarrow Y$ and by $s^{\prime}_{Y}:Y\rightarrow T_{f}$ the [math]-section of the relative tangent bundle. Let ${\mathcal{V}}(X)$ and ${\mathcal{V}}(Y)$ be the categories of virtual bundles on $X$ and $Y$ respectively. We have

[TABLE]

where we write $Th_{Y}(f^{*}V_{X}-T_{f})$ for the Thom transformation

[TABLE]

Let $\pi_{Y}:Y\rightarrow{\rm Spec\,}k$ be the structure morphism of $Y$. By the adjunction $(\pi_{Y\#},\pi_{Y}^{*})$ and since $E_{Y}=\pi_{Y}^{*}E$, we have then

[TABLE]

which comes from the fact that (cf. [Ay08, Thm. 1.5.9] and [Ay08, Rem. 1.5.10]):

[TABLE]

Now we apply the Voevodsky’s theorem 2.4 with a refinement as in Proposition 4.25. After fixing an embedding $X\hookrightarrow{\mathbb{P}}^{N}$ we have in ${\mathcal{V}}(X)$:

[TABLE]

Since $f$ is projective, there is a factorization

[TABLE]

where $Y\hookrightarrow{\mathbb{P}}^{M}\times_{k}X$ is a closed immersion. We take then the Segre embedding

[TABLE]

and apply the Proposition 4.25, so we have in ${\mathcal{V}}(Y)$ a specific isomorphism

[TABLE]

One has a functor [Del87, §4]

[TABLE]

Since $f$ is smooth we have an exact sequence

[TABLE]

which gives rise to an isomorphism in ${\mathcal{V}}(Y)$

[TABLE]

So we have then in ${\mathcal{V}}(Y)$ an isomorphsim

[TABLE]

where $-$ means $+$ the opposite object as explained in [Del87] and

[TABLE]

[TABLE]

Now since $Th$ defines a functor (cf. [Rio10, Def. 4.1.2])

[TABLE]

where ${\mathcal{V}}(Y)$ is the category of virtual bundles on $Y$, we can conclude that there is canonical isomorphism

[TABLE]

where the right hand side is by 4.26 canonical isomorphic to $D(Y_{+})\wedge S^{2(n_{X}+d_{X}),(n_{X}+d_{X})}$. So we can conclude that there is an isomorphism

[TABLE]

Now we have to show that this isomorphism is independent from the projective embeddings up to a unique canonical isomorphism. Let $Y\hookrightarrow{\mathbb{P}}^{N^{\prime}}$ be any closed embedding. Then we still have a canonical isomorphism

[TABLE]

As $D(Y_{+})$ is unique up to a canonical isomorphism we can conclude that $tt_{E}^{Y}$ is independent from the choice of the embeddings up to a unique canonical isomorphism. ∎

Remark 4.28**.**

As pointed out by M. Levine, one can simplify the arguments in the Proposition above by using the maps

[TABLE]

and

[TABLE]

which rigidify the situation considerably. **

Remark 4.29**.**

We will show later that with the refinement of $Th(V_{X})$ as in 4.26 the isomorphism $tt_{E}^{Y}$ is natural in sense that it is compatible with duality. **

Remark 4.30**.**

The isomorphism $tt_{E}^{Y}(V_{Y})$ in 4.27 is a natural candidate for a replacement of the twisted Thom isomorphism $th_{E}$ in 4.11 in case of $E$-cohomology twisted by formal difference of vector bundles. But we should remind the reader that we can only compute for a very particular case, namely $\xi=V_{X}$, where $V_{X}$ is the vector bundle as in Voevodsky’s theorem 2.4. **

Now we can compare:

Corollary 4.31**.**

Let $f:Y\rightarrow X$ be a smooth projective morphism of projective smooth $k$-schemes. There is an isomorphism up to a natural isomorphisms induced by the natural canonical isomorphism between duals

[TABLE]

Proof.

This is a consequence of 4.11 and 4.27. ∎

Remark 4.32**.**

The Corollary 4.31 is a surprising fact to us. At a first glance we have the impression that $E^{p,q}(Y,f^{*}V_{X}-T_{f})$ should depend relatively wrt. $f$ and $X$. At the end it turns out that $E^{p,q}(Y,f^{*}V_{X}-T_{f})$ is isomorphic to $E^{p,q}(Y,V_{Y})$, which depends absolutely only on $Y$. But it is clear that this is not the case for a general vector bundle $\xi$. **

Let $f:Y\rightarrow X$ be a smooth projective morphism of smooth projective $k$-schemes of dimension $d_{Y}$ and $d_{X}$ respectively. By Atiyah-Spanier-Whitehead duality and by the Proposition 4.26 we obtain its dual morphism in $SH(k):$

[TABLE]

where $V_{X}$ and $V_{Y}$ are vector bundles on $X$ and $Y$ of rank $n_{X}$ and $n_{Y}$ as in theorem 2.4 with a refinement in 4.25 respectively. By taking pullback of this map on $E$-cohomology and appyling Thom isomorphism we obtain a pushforward

[TABLE]

We show that the two pushforwards are the same and the isomorphism $tt^{Y}_{E}$ is natural in the sense that it is compatible with the duality in the following:

Proposition 4.33**.**

Let $f:Y\rightarrow X$ be a smooth projective $k$-morphism between smooth projective $k$-schemes. One has a commutative diagram up to natural isomorphisms induced by natural $2$-isomorphisms and the natural canonical isomorphism between duals

[TABLE]

Proof.

Let us denote by $p_{V_{X}}:V_{X}\rightarrow X$ the duality vector bundle on $X$ (cf. 2.4) with the zero-section $s_{X}:X\rightarrow V_{X}$ and $s_{Y}:Y\rightarrow f^{*}V_{X}$ the [math]-section of the pullback bundle $p_{f^{*}V_{X}}:f^{*}V_{X}\rightarrow Y$. By construction, the first pushforward map is the following composition:

[TABLE]

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 natural isomorphism given by the adjunction $(f_{\#},f^{*})$

[TABLE]

$(4)$ is the natural isomorphism given by projection formula $Pr_{\#}^{*}(f)$

[TABLE]

$(5)$ is the natural isomorphism induced by $f_{\#}Th_{Y}(-T_{f})\cong D_{X}(f_{\#}\mathbf{1}_{Y})$, $(6)$ is the natural isomorphism induced by adjunction of duality in $SH(X)$:

[TABLE]

and finally $(7)$ is the pushforward induced by the counit $\eta_{(f_{\#},f^{*})}:f_{\#}(\mathbf{1}_{Y})\cong f_{\#}f^{*}(\mathbf{1}_{X})\rightarrow\mathbf{1}_{X}$:

[TABLE]

The last isomorphism is the inverse of the twisted Thom isomorphism. So we have

[TABLE]

The map $tt_{E}^{Y}$ is the following composition

[TABLE]

where $(1^{\prime})=(1)$ , $(2^{\prime})=(2)$, $(3^{\prime})$ is the natural isomorphism induced by composing Thom transformations, $(4^{\prime})$ is induced by the isomorphism in ${\mathcal{V}}(Y)$:

[TABLE]

$(5^{\prime})$ is the cancellation in $SH(Y)$ and finally $(6^{\prime})$ is the natural isomorphism induced by the adjunction $(\pi_{Y\#},\pi_{Y}^{*})$ with $\pi_{Y}:Y\rightarrow{\rm Spec\,}k$ is the structure morphism of $Y$:

[TABLE]

Let us consider the diagram

[TABLE]

We have

[TABLE]

where the natural isomorphism is induced by the adjunction $(f_{\#},f^{*})$ as $f$ is smooth:

[TABLE]

By the projection formula $Pr_{\#}^{*}(f)$ we have a natural isomorphism

[TABLE]

which is explicitly written as

[TABLE]

But since $f$ is smooth and projective

[TABLE]

Thanks to Proposition 4.26 the composition $(f^{\vee})^{*}\circ tt^{Y}_{E}$ on $E$-cohomology is nothing but just the composition of natural isomorphisms above with the map induced by the unit $\mathbf{1}_{X}\rightarrow f_{*}f^{*}\mathbf{1}_{X}$ of the adjunction $(f^{*},f_{*})$:

[TABLE]

then followed by the natural isomorphism induced from the adjunction $(\pi_{X\#},\pi_{X}^{*})$:

[TABLE]

Indeed, by the very construction of the $6$ operations formalism [Ay08, Thm. 4.5.23], the stabilization functor $\Sigma^{\infty}_{T,+}:Sm/k\rightarrow SH(k)$ induces a morphism in $SH(k)$:

[TABLE]

which can be understood as a morphism $\pi_{Y\#}(\mathbf{1}_{Y})\rightarrow\pi_{X\#}(\mathbf{1}_{X})$, which in turn is the composition $\pi_{X\#}\circ\varepsilon_{(f_{\#},f^{*})}(\mathbf{1}_{X})$. In terms of six operations and by the Proposition 4.26, the dual objects in $SH(k)$ are:

[TABLE]

and

[TABLE]

Hence the pullback map $(f^{\vee})^{*}$ is just the pullback $SH(k)[-,E^{*,*}\wedge S^{*,*}]$ of the map

[TABLE]

We have to check that

[TABLE]

which means that we have to check

[TABLE]

But this is clear, since for a smooth projective morphism $\pi:T\rightarrow S$ one has a natural $2$-isomorphism

[TABLE]

which is the composition

[TABLE]

The equality, which we need to check above, follows simply from this fact and from the fact that, we have a natural $2$-isomorphism

[TABLE]

as $f$ is assumed to be smooth (cf. [Ay08, Thm. 1.5.9]). ∎

Let us construct the pullback for twisted $E$-cohomology of formal difference of vector bundles along a cartesian square. Let

[TABLE]

be a cartesian square of projective smooth $k$-schemes, where $f$ is smooth projective of relative dimension $d=\dim(Y)-\dim(X)$ and $u$ is any morphism. Let $pi_{\xi}:\xi\rightarrow X$ be a vector bundle of rank $r$ and denote by $s_{Y}:Y\rightarrow f^{*}\xi$ the [math]-section of the pullback bundle $p_{f^{*}\xi}:f^{*}\xi\rightarrow Y$. Consider

[TABLE]

By adjunction of Thom transformation we have

[TABLE]

where the isomorphism is

[TABLE]

By applying the functor $v^{*}:SH(Y)\rightarrow SH(Y^{\prime})$ we have an induced map

[TABLE]

We have $v^{*}Th_{Y}(f^{*}\xi)\cong Th_{Y^{\prime}}(v^{*}f^{*}\xi)=Th_{Y^{\prime}}(g^{*}u^{*}\xi)$ as $\xi$ is an actual bundle. Since $v^{*}$ is a monoidal functor, so we have

[TABLE]

But we have $v^{*}Th_{Y}(T_{f})\cong Th_{Y^{\prime}}(v^{*}T_{f})$, since $T_{f}$ is an actual bundle. By [EGA4, 16.5.12.2] one has $v^{*}T_{f}\cong T_{g}$, so $v^{*}Th_{Y}(T_{f})\cong Th_{Y^{\prime}}(T_{g})$. So we obtain the pullback map for twisted $\mathbf{E}$-cohomology of formal difference of vector bundle

[TABLE]

Proposition 4.34**.**

Consider the composition of cartesian squares of smooth projective $k$-schemes

[TABLE]

where $f$ is a smooth projective morphism, $u$ and $u^{\prime}$ are morphisms. Let $\xi$ be a vector bundle on $X$. Then up to natural isomorphisms induced by the natural $2$-isomorphism induced by $(-\circ-)^{*}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}(-)^{*}\circ(-)^{*}$ one has

[TABLE]

Proof.

Obvious. ∎

Proposition 4.35**.**

*(projective smooth base change)

Consider a cartesian square of projective smooth $k$-schemes*

[TABLE]

where $f$ is smooth projective of relative dimension $d=\dim(Y)-\dim(X)$ and $u$ is a morphism. Let $\xi/X$ be a vector bundle of rank $r$. One has a commutative diagram up to natural isomorphisms induced by natural $2$-isomorphisms

[TABLE]

Proof.

It is quite straightforward. We write $s_{Y}:Y\rightarrow f^{*}\xi$ and $s_{Y^{\prime}}:Y\rightarrow v^{*}f^{*}\xi$ for the [math]-sections of the vector bundles $p_{f^{*}\xi}:f^{*}\xi\rightarrow Y$ and $p_{v^{*}f^{*}\xi}:v^{*}f^{*}\xi\rightarrow Y^{\prime}$ respectively. $u^{*}f_{*}$ is the following composition

[TABLE]

where

[TABLE]

and

[TABLE]

$\alpha_{3}$ is the natural isomorphism

[TABLE]

$\Delta_{a}$ is the Cartesian square

[TABLE]

[TABLE]

$\Delta_{b}$ is the Cartesian square

[TABLE]

[TABLE]

$g_{*}v^{*}$ is the following composition

[TABLE]

where $\beta_{1}$ is the natural isomorphism

[TABLE]

and

[TABLE]

$\Delta_{1}$ is the Cartesian square

[TABLE]

[TABLE]

$\Delta_{2}$ is the Cartesian square

[TABLE]

[TABLE]

$\Delta_{3}$ is the Cartesian square

[TABLE]

[TABLE]

$\Delta_{4}$ is the Cartesian square

[TABLE]

[TABLE]

$\beta_{2}^{\prime}$ is the natural isomorphism

[TABLE]

$\beta_{3}$ is the natural isomorphism

[TABLE]

Gathering all together we have to check the following equality up to natural $2$-isomorphisms:

[TABLE]

This equality can be chased step by step by using the natural $2$-isomorphism

[TABLE]

which is the following composition

[TABLE]

and also the coherence of the exchange transformations. ∎

Now we construct the exceptional pullback for twisted $E$-cohomology. We keep the notation as above and let $i:T\hookrightarrow Y$ 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, $Bl_{T\times\{0\}}(Y\times{\mathbb{A}}^{1})$ is the blow-up of $Y\times{\mathbb{A}}^{1}$ with the center $T\times\{0\}$. The deformation space is the $k$-scheme

[TABLE]

Note that $D_{T}(T)=T\times{\mathbb{A}}^{1}$ is a closed subscheme of $D_{T}(Y)$. The scheme $D_{T}(Y)$ is fibred over ${\mathbb{A}}^{1}$. The flat morphism

[TABLE]

has $\pi^{-1}(1)=Y$ and $\pi^{-1}(0)=N_{T/Y}$. One has a deformation diagram of closed pairs

[TABLE]

The homotopy purity theorem of Morel-Voevodsky [MV01, §3 Thm. 2.23] states

[TABLE]

are isomorphism in $\mathbf{Ho}_{{\mathbb{A}}^{1},+}(k)$, which is generalized to motivic categories in [CD10, Thm. 2.4.35]. Consider now the adjunction

[TABLE]

Let

[TABLE]

be a cartesian square of smooth projective $k$-schemes, where $f$ is smooth projective of relative dimension $d=\dim(Y)-\dim(X)$, $k$ and $i$ are regular embeddings. Let $\xi$ be a vector bundle of rank $r$ on $X$. We define the exceptional pullback of twisted $E$-cohomology along a regular embedding $i:T\hookrightarrow Y$ as the following composition:

[TABLE]

Proposition 4.36**.**

Let

[TABLE]

be a chain of cartesian squares of smooth projective $k$-schemes, where $f$ is smooth projective, $i,i^{\prime},k,k^{\prime}$ are regular embeddings. Then we have $(i\circ i^{\prime})^{!}=i^{!}\circ i^{\prime!}$ up to natural isomorphisms induced by natural $2$-isomorphisms .

Proof.

Obvious. ∎

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

[TABLE]

where $f$ is smooth projective of relative dimension $d=\dim(Y)-\dim(X)$ and $k$ and $i$ are regular embeddings. Let $p_{\xi}:\xi\rightarrow X$ be a vector bundle of rank $r$. One has a commutative diagram up to a natural isomorphism

[TABLE]

Let $p_{\xi}:\xi\rightarrow X$ and $p_{\xi^{\prime}}:\xi^{\prime}\rightarrow X$ be two vector bundles of rank $r$ and $r^{\prime}$ resp. on $X$ with the zero sections $s:X\rightarrow\xi$ and $s^{\prime}:X\rightarrow\xi^{\prime}$ respectively. Let $s^{\prime\prime}:X\rightarrow\xi\oplus\xi^{\prime}$ to be the zero section of the bundle $\xi\oplus\xi^{\prime}$. We define the cup product

[TABLE]

as follow: Given morphisms in $SH(X)$

[TABLE]

and

[TABLE]

Then

[TABLE]

Remark 4.38**.**

If $f:T\rightarrow S$ is a morphism of finite type between schemes, then we have

[TABLE]

Proposition 4.39**.**

(projection formula) Let $f:Y\rightarrow X$ be a smooth projective morphism of smooth projective $k$-schemes of relative dimension $d=\dim(Y)-\dim(X)$. Let $\xi$ and $\xi^{\prime}$ be two vector bundles on $X$. Let $a\in E^{p,q}(X,\xi)$ and $b\in E^{p^{\prime},q^{\prime}}(Y,f^{*}\xi^{\prime}-T_{f})$. Then one has up to natural isomorphisms

[TABLE]

in $E^{p+p^{\prime}-2d,q+q^{\prime}-d}(X,\xi\oplus\xi^{\prime})$.

Proof.

This follows from the projective smooth base change 4.35 by standard argument. Consider the commutative diagram

[TABLE]

We have

[TABLE]

where $\Gamma_{f}=(f\times{\rm id})\Delta_{X}$. ∎

Proposition 4.40**.**

Let $\mathbf{E}\in SH(k)$ be a motivic ring spectrum. Let $X,Y,Z\in SmProj(k)$. Let $\alpha\in\mathbf{E}^{2d_{Y},d_{Y}}(X\times Y,pr_{Y}^{XY*}V_{Y})$ and $\beta\in\mathbf{E}^{2d_{Z},d_{Z}}(Y\times Z,pr_{Z}^{YZ*}V_{Z})$, where $V_{Y}$ and $V_{Z}$ are the vector bundles given in theorem 2.4 with a refinement in 4.25. Then we have up to natural isomorphisms

[TABLE]

Proof.

This follows from our construction of pullback, pushforward and cup product and the projections fit to the following commutative diagram

[TABLE]

∎

Proposition 4.41**.**

Let $\mathbf{E}\in SH(k)$ be a motivic ring spectrum. Let $X,Y,Z,W\in SmProj(k)$. Let $\alpha\in\mathbf{E}^{2d_{Y},d_{Y}}(X\times Y,pr_{Y}^{XY*}V_{Y})$, $\beta\in\mathbf{E}^{2d_{Z},d_{Z}}(Y\times Z,pr_{Z}^{YZ*}V_{Z})$ and $\gamma\in\mathbf{E}^{2d_{W},d_{W}}(Z\times W,pr_{W}^{ZW*}V_{W})$. Let’s denote

[TABLE]

and similarly for $\gamma\circ\beta$. Then $\circ$ is associative up to natural isomorphisms induced by $2$-isomorphisms.

Proof.

We have

[TABLE]

where $(1)$ is the definition, $(2)$ follows from smooth projective base change:

[TABLE]

$(3)$ follows from the compatibility of pullback and $\cup_{\mathbf{E}}$ (4.38), functoriality of pullback (4.18) and the projection formula (4.39), $(4)$ follows from functoriality of pullback (4.18) and pushforward (4.24), $(5)$ follows from the associativity of $\cup_{\mathbf{E}}$, which is a consequence of our requirement that $\mathbf{E}$ is a motivic ring spectrum (see the beginning of §4.1). Symmetrically, the last expression is exactly $(\gamma\circ\beta)\circ\alpha$. ∎

Definition 4.42**.**

Let $\mathbf{E}\in SH(k)$ be a motivic ring spectrum. We define the category of twisted $\mathbf{E}$-correspondences $\widetilde{Corr}_{\mathbf{E}}(k)$ to be the category, whose objects are

[TABLE]

and morphisms are given by

[TABLE]

where $V_{Y}/Y$ is the vector bundle given in the theorem 2.4. Given $\alpha\in\mathbf{E}^{2d_{Y},d_{Y}}(X\times Y,pr_{Y}^{XY*}V_{Y})$ and $\beta\in\mathbf{E}^{2d_{Z},d_{Z}}(Y\times Z,pr_{Z}^{YZ*}V_{Z})$ we define their composition to be

[TABLE]

which is associative up to natural isomorphisms. **

Proposition 4.43**.**

Let $\mathbf{E}\in SH(k)$ be a motivic ring spectrum. Let $X,Y,Z\in SmProj(k)$. Let $\alpha\in\mathbf{E}^{2(n_{Y}+d_{Y}),(n_{Y}+d_{Y})}(X_{+}\wedge Th(V_{Y}))$ and $\beta\in\mathbf{E}^{2(n_{Z}+d_{Z}),n_{Z}+d_{Z}}(Y_{+}\wedge Th(V_{Z}))$. Then the following composition

[TABLE]

lies in $\mathbf{E}^{2(n_{Z}+d_{Z}),(n_{Z}+d_{Z})}(X_{+}\wedge Th(V_{Z}))$, where

[TABLE]

is the coevaluation map of the Atiyah-Spanier-Whitehead duality on $Y$.

Proof.

Trivial. ∎

Proposition 4.44**.**

Let $\mathbf{E}\in SH(k)$ be a motivic ring spectrum. Let $X,Y,Z,W\in SmProj(k)$. Let $\alpha\in\mathbf{E}^{2(n_{Y}+d_{Y}),(n_{Y}+d_{Y})}(X_{+}\wedge Th(V_{Y}))$, $\beta\in\mathbf{E}^{2(n_{Z}+d_{Z}),(n_{Z}+d_{Z})}(Y_{+}\wedge Th(V_{Z}))$ and $\gamma\in\mathbf{E}^{2(n_{W}+d_{W}),(n_{W}+d_{W})}(Z_{+}\wedge Th(V_{W}))$. Let us denote by $\beta\circ^{\dagger}\alpha$ for the composition of the above proposition and similarly for $\gamma\circ^{\dagger}\beta$. Then $\circ^{\dagger}$ is associative and unital.

Proof.

We have that $\gamma\circ^{\dagger}(\beta\circ^{\dagger}\alpha)$ is the following composition by definition:

[TABLE]

which can be rewritten as

[TABLE]

The composition $(\gamma\circ^{\dagger}\beta)\circ^{\dagger}\alpha$ is by definition:

[TABLE]

which can be rewritten as

[TABLE]

Both $\gamma\circ^{\dagger}(\beta\circ^{\dagger}\alpha)$ and $(\gamma\circ^{\dagger}\beta)\circ^{\dagger}\alpha$ are equal to the following composition

[TABLE]

∎

Definition 4.45**.**

Let $\mathbf{E}\in SH(k)$ be a motivic ring spectrum. We define the category of Thom-$\mathbf{E}$-correspondences $Corr_{\mathbf{E}}(k)^{\dagger}$ to be the category, whose objects are

[TABLE]

and morphisms are given by

[TABLE]

where $V_{Y}/Y$ is the duality vector bundle of rank $n_{Y}$. Given

[TABLE]

and

[TABLE]

we define their composition to be

[TABLE]

where

[TABLE]

is the coevaluation map of the Atiyah-Spanier-Whitehead duality on $Y$. **

As we may write $X_{+}\wedge Th(V_{Y})=Th(pr_{Y}^{XY*}V_{Y})$, we have then the pullback map

[TABLE]

Similarly

[TABLE]

By taking cup product

[TABLE]

and applying the pushforward $pr_{XZ}^{XYZ*}=(coev_{Y})^{*}$ we see that

[TABLE]

Proposition 4.46**.**

Let $\mathbf{E}\in SH(k)$ be a motivic ring spectrum. Let $X,Y,Z\in SmProj(k)$ of dimension $d_{X},d_{Y},d_{Z}$ respectively. Let $\alpha\in\mathbf{E}^{2(n_{Y}+d_{Y}),(n_{Y}+d_{Y})}(X_{+}\wedge Th(V_{Y}))$ and $\beta\in E^{2(n_{Z}+d_{Z}),(n_{Z}+d_{Z})}(Y_{+}\wedge Th(V_{Z}))$. Then the composition $\beta\circ^{\dagger}\alpha$ in $Corr_{\mathbf{E}}(k)^{\dagger}$ satisfies

[TABLE]

where $pr_{XZ*}^{XYZ}=coev_{Y}^{*}$.

Proof.

Trivial. ∎

Theorem 4.47**.**

(Comparison) Let $\mathbf{E}\in SH(k)$ be a motivic ring spectrum. There is an equivalence of categories up to a natural $2$-isomorphism

[TABLE]

Proof.

We have the following association

[TABLE]

and

[TABLE]

where $th_{\mathbf{E}}$ denotes the twisted Thom isomorphism and $D$ is the isomorphism induced by duality. It remains to check that the composition law in $\widetilde{Corr}_{\mathbf{E}}(k)$ is compatible with the composition law in $Corr_{\mathbf{E}}(k)^{\dagger}$ and $Corr_{\mathbf{E}}(k)$ via $th_{\mathbf{E}}$ and $D$ respectively. The compatibility of composition laws via the Thom isomorphism $th_{\mathbf{E}}$ follows from the Propositions 4.20, 4.33 and 4.46. Now given a cohomology class $\alpha\in\mathbf{E}^{2(d_{Y}+n_{Y}),(d_{Y}+n_{Y})}(Th_{XY}(pr_{Y}^{XY*}V_{Y}))$ we obtain its pullback by

[TABLE]

Similarly, given $\beta\in\mathbf{E}^{2(d_{Z}+n_{Z}),(d_{Z}+n_{Z})}(Th_{YZ}(pr_{Z}^{YZ*}V_{Z}))$ we obtain its pullback by

[TABLE]

So $pr_{XY}^{XYZ*}\alpha\cup_{\mathbf{E}}pr_{YZ}^{XYZ}\beta$ is the following composition

[TABLE]

which corresponds to the morphism

[TABLE]

By definition the composition $\beta\circ\alpha$ as composition of $\mathbf{E}$-correspondences is given by the composition

[TABLE]

where the first map by construction is given as

[TABLE]

This implies that the composition $\beta\circ\alpha$ as $\mathbf{E}$-correspondences is the same as

[TABLE]

This shows that the composition laws of $\widetilde{Corr}_{\mathbf{E}}(k)$ and $Corr_{\mathbf{E}}(k)$ are compatible. ∎

5. Proof of theorem 1.2

5.1. Homotopy $t$-structure

We recall in this section the notion homotopy $t$-structure in terms of generators (cf. [Ay08]). Let $k$ be a field. The subcategory $SH(k)_{\geq n}$ is generated under homotopy colimits and extensions by

[TABLE]

where $S^{p,q}=S^{p-q}_{s}\wedge S_{t}^{q}$ denotes the motivic spheres. We set

[TABLE]

The bigraded motivic homotopy sheaves are defined as

[TABLE]

We let

[TABLE]

For a fix $p\in{\mathbb{Z}}$, $\pi^{st{\mathbb{A}}^{1}}_{p}(E)_{*}$ is considered as an abelian ${\mathbb{Z}}$-graded sheaf. An abelian Nisnevich sheaf $F\in Sh_{Nis}(Sm/k)$ is called strictly ${\mathbb{A}}^{1}$-invariant, if the map induced by the projection $U\times{\mathbb{A}}^{1}\rightarrow U$:

[TABLE]

is an isomorphism $\forall U\in Sm/k$ and $\forall i\geq 0$. For an abelian Nisnevich sheaf $F\in Sh_{Nis}(Sm/k)$ we will denote by

[TABLE]

where the map is induced by the unit section of ${\mathbb{G}}_{m}$.

Definition 5.1**.**

*(Morel). *A homotopy module is a pair $(F_{*},\varepsilon_{*})$, where $F$ is a strictly ${\mathbb{A}}^{1}$-invariant ${\mathbb{Z}}$-graded abelian Nisnevich sheaf with

[TABLE]

The following description of the homotopy $t$-structure is a consequence of F. Morel’s stable ${\mathbb{A}}^{1}$-connectivity result (see for instance [Mor04a]):

Theorem 5.2** (F. Morel).**

Let $k$ be field.

1. (1)

The triple $(SH(k),SH(k)_{\geq 0},SH(k)_{\leq 0})$ is a $t$-structure on $SH(k)$. 2. (2)

The heart of the homotopy $t$-structure $\pi_{*}^{{\mathbb{A}}^{1}}(k)=SH(k)_{\geq 0}\cap SH(k)_{\leq 0}$ is identified with the category of homotopy modules. 3. (3)

The homotopy $t$-structure is non-degenerated in the sense that for any $U\in Sm/k$ and any $E\in SH(k)$, one has the morphism

[TABLE]

is an isomorphism for $n\leq 0$ and the morphism

[TABLE]

is an isomorphism for $n>\dim(U)$.

By sending $E\mapsto E_{\geq n}$ and $E\mapsto E_{\leq n-1}$ respectively, one has the following adjunctions respectively:

[TABLE]

where we denote by $i_{\geq n}$ and $i_{\leq n-1}$ the inclusion functors. We denote by

[TABLE]

the inclusion functor. For a homotopy module $F_{*}\in\pi_{*}^{{\mathbb{A}}^{1}}(k)$ we will call $H(F_{*})$ the Eilenberg-Maclane spectrum associated to $F_{*}$. Let $S$ be now a Noetherian scheme of finite Krull dimension. We recall the rationally splitting of $SH(S)_{{\mathbb{Q}}}$ constructed by F. Morel (see [CD10, §16.2]). The permutation isomorphism

[TABLE]

satisfies $\tau^{2}=1$. This defines an element $e\in End_{SH(S)_{{\mathbb{Q}}}}(\mathbf{1}_{{\mathbb{Q}}})$, such that $e^{2}=1$. So we may define

[TABLE]

Remark that $e_{+}$ and $e_{-}$ are idempotents. Hence we can define $\mathbf{1}_{{\mathbb{Q}}+}={\rm im\,}(e_{+})$ and $\mathbf{1}_{{\mathbb{Q}}-}={\rm im\,}(e_{-})$. For any spectrum ${\mathcal{E}}\in SH(S)_{{\mathbb{Q}}}$, one defines ${\mathcal{E}}_{+}=\mathbf{1}_{{\mathbb{Q}}+}\wedge{\mathcal{E}}$ and ${\mathcal{E}}_{-}=\mathbf{1}_{{\mathbb{Q}}-}\wedge{\mathcal{E}}$. This leads to a splitting of stable homotopy category

[TABLE]

Let us assume now $S={\rm Spec\,}k$. The algebraic Hopf fibration is the map

[TABLE]

This gives us the stable Hopf map in $SH(k)$

[TABLE]

Remark that from [Mor04a, 6.2.1] one has a homotopy fiber sequence in $SH(k)$:

[TABLE]

where $i:{\mathbb{P}}^{1}_{k}\hookrightarrow{\mathbb{P}}^{2}_{k}$ 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 ${\mathbb{Z}}$-graded associative unital ring freely generated by the symbols $[u]$, where $u\in F^{\times}$ is of degree $1$ and a symbol $\eta$ of degree $-1$ subject to the relation

1. (1)

$[u]\cdot[1-u]=0,\forall u\in F^{\times}-\{1\}.$ 2. (2)

$[uv]=[u]+[v]+\eta\cdot[u]\cdot[v],\forall(u,v)\in(F^{\times})^{2}.$ 3. (3)

$\eta\cdot[u]=[u]\cdot\eta,\forall u\in F^{\times}$. 4. (4)

Define $h\stackrel{{\scriptstyle defn}}{{=}}\eta\cdot[-1]+2$. Then $\eta\cdot h=0$.

Let $GW(F)$ be the Grothendieck-Witt ring of non-degenerate bilinear symmetric forms over $F$, where addition is given by orthogonal sum $\oplus$ and multiplication is given by tensor product $\otimes$. There is a surjective ring homomorphism

[TABLE]

The fundamental ideal is defined as

[TABLE]

Denote by $I^{n}(F)$ the $n$-th power of $I(F)$. If $n\leq 0$ one sets $I^{n}(F)=W(F)$, where $W(F)$ is the Witt ring over $F$. Remark that $W(F)=GW(F)/(h)$, where $(h)$ is the ideal generated by hyperbolic spaces. By [Mor12, Lem. 3.10] there is a ring isomorphism

[TABLE]

Let $K^{M}_{*}(F)$ be the Milnor $K$-theory

[TABLE]

There is a graded surjective homomorphism

[TABLE]

In fact, one can show that for each $n$ there is a pullback diagram

[TABLE]

Following [Mor12, §3.2] we let $\underline{\mathbf{K}}^{MW}_{n}$ be the $n$-th Milnor-Witt sheaf, which is a strictly ${\mathbb{A}}^{1}$-invariant sheaf on $(Sm/k)_{Nis}$. In [Mor04a, p. 437] Morel showed that one can define a homotopy module $\underline{\mathbf{K}}^{MW}_{*}$ asscociated to the Milnor-Witt $K$-theory and in fact one has an isomorphism between homotopy modules

[TABLE]

The homotopy module $\underline{\mathbf{W}}_{*}$ is defined by setting every terms to be the unramified Witt sheaf $\mathbf{W}=a_{Nis}(U\mapsto W(U)=W(k(U)))$ and all the maps $\varepsilon_{n}$ are identity.

Lemma 5.4**.**

Let $k$ be a field. Let $H\underline{\mathbf{K}}^{MW}_{*,{\mathbb{Q}}}$ be the Eilenberg-Maclane spectrum associated to the Milnor-Witt $K$-theory homotopy module $\underline{\mathbf{K}}^{MW}_{*,{\mathbb{Q}}}$. There exists a strict motivic ring spectrum $\hat{H}\underline{\mathbf{K}}^{MW}_{*,{\mathbb{Q}}}\in\mathbf{Spect}^{\Sigma}_{T}(k)_{{\mathbb{Q}}}$, which is isomorphic to $H\underline{\mathbf{K}}^{MW}_{*,{\mathbb{Q}}}$ in $SH(k)_{{\mathbb{Q}}}$.

Proof.

We have a splitting

[TABLE]

The result of Déglise [Deg13, Cor. 4.1.7] asserts that $H(\underline{\mathbf{K}}^{M}_{*})$ is a strict $H{\mathbb{Z}}$-module, where $H{\mathbb{Z}}$ denotes the motivic cohomology spectrum. The construction in [ALP15, §4] shows that the cofibrant replacement $H(\underline{\mathbf{W}}_{*,{\mathbb{Q}}})^{cof}$ is a commutative monoid object in $\mathbf{Spect}^{\Sigma}_{T}(k)_{{\mathbb{Q}}}$, which is isomorphic to $H(\underline{\mathbf{W}}_{*,{\mathbb{Q}}})$ and $\mathbb{S}^{0}_{k}[\eta^{-1}]$. These imply that

[TABLE]

is also a strict motivic ring spectrum in $\mathbf{Spect}^{\Sigma}_{T}(k)_{{\mathbb{Q}}}$, which is isomorphic to $H\underline{\mathbf{K}}^{MW}_{*,{\mathbb{Q}}}$ in $SH(k)_{{\mathbb{Q}}}$. ∎

Definition 5.5**.**

Let $k$ be a field. We define the category of pure Chow-Witt motives to be

[TABLE]

Corollary 5.6**.**

Let $k$ be a field. There is a functor

[TABLE]

Proof.

This is a consequence of the Corollary 4.7 and Lemma 5.4. ∎

5.2. Isomorphism between ${\rm Hom}$-groups

Let $k$ be a field. In this section we prove that one has a fully faithful embedding

[TABLE]

Remark that one has the equivalences of categories:

[TABLE]

For $E\in SH(k)$ we define its stable ${\mathbb{A}}^{1}$-cohomology as

[TABLE]

We denote by $SH(k)_{{\mathbb{Q}}}$ the localization of $SH(k)$. One has an adjunction

[TABLE]

which is induced by the Quillen adjunction

[TABLE]

where $U:\mathbf{Spect}_{T}^{\Sigma}(k)_{{\mathbb{Q}}}\rightarrow\mathbf{Spect}_{T}^{\Sigma}(k)$ is the forgetful functor by considering

[TABLE]

as a symmetric motivic $T$-spectrum in $\mathbf{Spect}_{T}^{\Sigma}(k)$. For a motivic spectrum $E\in SH(k)$ we define its rational stable ${\mathbb{A}}^{1}$-cohomology as

[TABLE]

We remark that by [Lev13, Lem. B2] if $E$ is a compact object in $SH(k)$ then one has an isomorphism

[TABLE]

Similarly, we define the motivic cohomology of $E$ as $H^{p,q}_{M}(E,{\mathbb{Z}})=SH(k)(E,H{\mathbb{Z}}\wedge S^{p,q})$. If $F_{*}\in\pi_{0}^{{\mathbb{A}}^{1}}(k)_{*}$ is a homotopy module, then the $HF_{*}$-cohomology of $E$ is defined as

[TABLE]

and if $E=\Sigma^{\infty}_{T,+}{\mathcal{X}}$, where ${\mathcal{X}}\in Spc(k)_{+}$ is a $k$-space (eg. Thom spaces), then the later cohomology is $H^{p-q}_{Nis}({\mathcal{X}},F_{q})$, where this cohomology is defined as

[TABLE]

where $K(-)$ denotes the Eilenberg-Maclane functor.

Theorem 5.7**.**

Let $k$ be a field and $E=\Sigma^{\infty}_{T,+}Th(V/X)$ be the Thom spectrum of a vector bundle $V$ on a smooth $k$-scheme $X$. Let $\mathbb{S}^{0}$ be the motivic sphere spectrum. There exists a canonical isomorphism

[TABLE]

where $\varphi$ is induced by the unit $\varphi_{MW}:\mathbb{S}^{0}\rightarrow H\underline{\mathbf{K}}^{MW}_{*}$.

Proof.

By stable ${\mathbb{A}}^{1}$-connectivity theorem of Morel [Mor05] the motivic sphere spectrum $\mathbb{S}^{0}$ is $-1$-connective. So we have a distinguished triangle

[TABLE]

By the computation of Morel we have $\pi_{0}(\mathbb{S}^{0})_{*}=\underline{\mathbf{K}}^{MW}_{*}$. So after smashing with $S^{2p,p}_{{\mathbb{Q}}}$ we obtain a distinguished triangle

[TABLE]

By taking $[Th(V/X),-]$ we have a long exact sequence

[TABLE]

Now we have $(\mathbb{S}^{0})_{\geq 1}\wedge\mathbb{S}^{0}_{{\mathbb{Q}}}=(\mathbb{S}^{0}_{{\mathbb{Q}}})_{\geq 1}$. By the work of C. D. Cisinski, F. Déglise ([CD10]) and the work of A. Ananyevskiy, M. Levine, I. Panin ([ALP15]) we have

[TABLE]

This implies $(\mathbb{S}^{0}_{{\mathbb{Q}}})_{\geq 1}=(H{\mathbb{Q}})_{\geq 1}$. The motivic cohomology spectrum $H{\mathbb{Q}}$ is also $-1$-connective, so we have a distinguished triangle

[TABLE]

The homotopy module $\pi_{0}(H{\mathbb{Q}})_{*}$ is $\underline{\mathbf{K}}^{M}_{*,{\mathbb{Q}}}$. We have (by [MVW06, Cor. 19.2] and by purity)

[TABLE]

Now from the splitting $\mathbb{S}^{0}_{{\mathbb{Q}}}=H{\mathbb{Q}}\vee H\underline{\mathbf{W}}_{*,{\mathbb{Q}}}$ the map $\varphi$ take the form:

[TABLE]

This implies that $\varphi$ is a canonical isomorphism. ∎

Corollary 5.8**.**

Let $k$ be a field. The functor constructed in 5.6

[TABLE]

is fully faithful

Proof.

By definition $CHW(k)_{{\mathbb{Q}}}$ is the smallest pseudo-abelian full subcategory of the homotopy category $Ho_{k}(\hat{H}\underline{\mathbf{K}}^{MW}_{*,{\mathbb{Q}}}-Mod)$ generated as an additive category by

[TABLE]

The adjunction

[TABLE]

gives us a natural isomorphism

[TABLE]

By duality 4.26 we have

[TABLE]

where $d_{Y}=\dim(Y)$, $V_{Y}$ is the duality vector bundle given in the theorem 2.4 and $n_{Y}={\rm rank}(V_{Y})$. The corollary follows now from the Theorem 5.7. ∎

6. Appendix

In this appendix we simply recollect some facts and definitions in model categories. All the results are well-known and classical (see [Q67], [Hir03], [Hov99]).

6.1. Model Categories

Definition 6.1**.**

A model category ${\mathcal{M}}$ is a category with three classes of morphisms

[TABLE]

called fibrations, cofibrations and weak equivalences, such that:

1. (1)

${\mathcal{M}}$ is closed under small limits and colimits. 2. (2)

If $f,g\in Mor({\mathcal{M}})$ are composable and two out of $f,g,g\circ f$ are in $W({\mathcal{M}})$, so is the third one. 3. (3)

Given a commutative diagram

[TABLE]

where $i\in Cof({\mathcal{M}})$, $p\in Fib({\mathcal{M}})$ and either $i$ or $p$ is in $W({\mathcal{M}})$, then there exists a morphism $B\rightarrow X$ making the diagram commutative. 4. (4)

$W({\mathcal{M}}),Cof({\mathcal{M}})$ and $Fib({\mathcal{M}})$ are closed under retracts. 5. (5)

Given any morphism $f:X\rightarrow Y$ in $Mor({\mathcal{M}})$, there exist two functorial factorizations

[TABLE]

The first axiom implies that there exist an initial object $\varnothing$ and a final object $\star$. We say ${\mathcal{M}}$ is pointed if $\varnothing\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\star$.

Definition 6.2**.**

Let $X\in Obj({\mathcal{M}})$ be an object. $X$ is called cofibrant if the natural morphism $\varnothing\rightarrow X$ is in $Cof({\mathcal{M}})$. $X$ is called fibrant if the natural morphism $X\rightarrow*$ is in $Fib({\mathcal{M}})$. **

Let $i:A\rightarrow B$ and $p:X\rightarrow Y$ be two morphisms in $Mor({\mathcal{M}})$. We say $i$ has left lifting property wrt. $p$ or $p$ has right lifting property wrt. $i$, if for every solide commutative diagram

[TABLE]

the dotted morphism exists and makes the diagram commutative. Given two morphisms $Mor({\mathcal{M}})\ni f:A\rightarrow B$ and $Mor({\mathcal{M}})\ni g:C\rightarrow D$, we say $f$ is a retract of $g$, if there is a commutative diagram

[TABLE]

where the horizontal composites are identities. Given an object $X\in Obj({\mathcal{M}})$, the factorization axiom tells us that we can factor

[TABLE]

where $X^{cof}$ is cofibrant. We call $X^{cof}$ a cofibrant replacement of $X$. Similarly, we can factor

[TABLE]

where $X^{fib}$ is fibrant. We call $X^{fib}$ a fibrant replacement of $X$.

Definition 6.3**.**

Let ${\mathcal{M}},{\mathcal{N}}$ be two model categories. A functor $F:{\mathcal{M}}\rightarrow{\mathcal{N}}$ is called a left Quillen functor, if it has a right adjoint $G:{\mathcal{N}}\rightarrow{\mathcal{M}}$ and

1. (1)

If $i\in Cof({\mathcal{M}})$, then $F(i)\in Cof({\mathcal{N}})$. 2. (2)

If $j\in Cof({\mathcal{M}})\cap W({\mathcal{M}})$, then $F(j)\in Cof({\mathcal{N}})\cap W({\mathcal{M}})$.

The right adjoint $G:{\mathcal{N}}\rightarrow{\mathcal{M}}$ is called a right Quillen functor and the adjunction

[TABLE]

is called a Quillen adjunction. **

Definition 6.4**.**

Let

[TABLE]

be a Quillen adjunction. $F$ is called a left Quillen equivalence, if for every cofibrant object $X\in Obj({\mathcal{M}})$ and every fibrant object $Y\in Obj({\mathcal{N}})$ one has the following: A morphism $f:X\rightarrow GY$ is in $W({\mathcal{M}})$ iff its adjoint $g=\varepsilon_{(F,G)}\circ F(f):FX\rightarrow Y$ is in $W({\mathcal{N}})$. $G$ is called then a right Quillen equivalence. The adjunction

[TABLE]

is called a Quillen equivalence. **

Definition 6.5**.**

Let $X\in Obj({\mathcal{M}})$ be an object in a model category ${\mathcal{M}}$. The cylinder object for $X$ is an object $Cyl(X)$, such that we have a factorization

[TABLE]

where $i\in Cof({\mathcal{M}})$ and $s\in W({\mathcal{M}})$. **

Definition 6.6**.**

Let $X\in Obj({\mathcal{M}})$ be an object in a model category ${\mathcal{M}}$. A path object for $X$ is an object ${\mathcal{P}}(X)$, such that we have a factorization

[TABLE]

where $r\in W({\mathcal{M}})$ and $p\in Fib({\mathcal{M}})$. **

Definition 6.7**.**

Let $f,g:X\rightrightarrows Y$ be two morphisms in $Mor({\mathcal{M}})$ of a model category ${\mathcal{M}}$. $f$ is left homotopic to $g$ if there is a cylinder object $Cyl(X)$ for $X$, such that we have a factorization

[TABLE]

The map $LH$ is called a left homotopy from $f$ to $g$. **

Definition 6.8**.**

Let $f,g:X\rightrightarrows Y$ be two morphisms in $Mor({\mathcal{M}})$ of a model category ${\mathcal{M}}$. $f$ is right homotopic to $g$ if there is a path object ${\mathcal{P}}(Y)$ for $Y$, such that we have a factorization

[TABLE]

The map $RH$ is called a right homotopy from $f$ to $g$. **

Definition 6.9**.**

Let $f,g:X\rightrightarrows Y$ be two morphisms in $Mor({\mathcal{M}})$ of a model category ${\mathcal{M}}$. $f$ is homotopic to $g$ if $f$ is left and right homotopic to $g$.

Theorem 6.10**.**

(Quillen [Q67, I.1 Thm. 1]). Let ${\mathcal{M}}$ be a model category. There exists a category $Ho({\mathcal{M}})={\mathcal{M}}[W({\mathcal{M}})^{-1}]$, which is called the homotopy category of ${\mathcal{M}}$, where

1. (1)

$Obj(Ho({\mathcal{M}}))=Obj({\mathcal{M}})$. 2. (2)

$Ho({\mathcal{M}})(X,Y)=\pi((X^{cof})^{fib},(Y^{cof})^{fib}),$, where $\pi$ denotes the set of homotopy classes and the composition law is induced by the composition law of ${\mathcal{M}}$.

Theorem 6.11**.**

(Quillen [Q67, I.4 Thm. 3]). Let

[TABLE]

be a Quillen adjunction. Then $(F,G)$ induces an adjunction of homotopy categories

[TABLE]

Definition 6.12**.**

1. (1)

Let ${\mathcal{M}}$ be a model category. ${\mathcal{M}}$ is left proper, if in any pushout diagram

[TABLE]

where $i\in Cof({\mathcal{M}})$ and $h_{1}\in W({\mathcal{M}})$, so $h_{2}\in W({\mathcal{M}})$. 2. (2)

Let ${\mathcal{M}}$ be a model category. ${\mathcal{M}}$ is right proper, if in any pullback diagram

[TABLE]

where $p\in Fib({\mathcal{M}})$ and $h_{2}\in W({\mathcal{M}})$, so $h_{1}\in W({\mathcal{M}})$. 3. (3)

${\mathcal{M}}$ is proper, if it is left and right proper.

Let $\mathbf{\Delta}$ denote the category, whose objects are ordered finite sets

[TABLE]

and

[TABLE]

There are cofaces $\delta^{i}:\underline{n}\rightarrow\underline{n+1}$ and codegeneracies $\sigma^{i}:\underline{n+1}\rightarrow\underline{n}$ defined by

[TABLE]

[TABLE]

Cofaces and codegeneracies are generators for the maps in $\mathbf{\Delta}$. They satisfy a list of relations (cf. [Weib94, §8]). Now one defines the category of simplicial sets as

[TABLE]

So simplicial sets are just presheaves of sets on $\mathbf{\Delta}$. For a general category ${\mathcal{A}}$ the category of simplicial objects and cosimplicial objects in ${\mathcal{A}}$ are defined to be $\mathbf{\Delta}^{op}({\mathcal{A}})$ and $\mathbf{\Delta}({\mathcal{A}})$ respectively. Let $\mathbf{Top}$ be the category of compactly generated Hausdorff topological spaces. The geometric realization functor is defined by

[TABLE]

where $\Delta^{n}$ is the presheaf $Mor(\mathbf{\Delta})(-,\underline{n})$. There is an adjunction

[TABLE]

where $S$ is the singular functor

[TABLE]

Here $R(\Delta^{n})$ is

[TABLE]

Theorem 6.13**.**

(Quillen [Q67, II.3 Thm. 3]). The category $\mathbf{SSets}$ has a model category structure.

Definition 6.14**.**

Let ${\mathcal{M}}$ be a category. ${\mathcal{M}}$ is called simplicial if there is a functor

[TABLE]

such that

1. (1)

$\mathbf{SSMap}(X,Y)_{0}={\mathcal{M}}(X,Y)$. 2. (2)

there exists a composition law

[TABLE]

which is compatible with the composition law in ${\mathcal{M}}$. 3. (3)

There is a simplicial sets map $i_{X}:\star\rightarrow\mathbf{SSMap}(X,X),\forall X\in Obj({\mathcal{M}})$, where the associativity of the composition law, right and left unit properties of $i_{X}$ follows from three commutative diagrams ([Hir03, Def. 9.1.2]).

Definition 6.15**.**

Let ${\mathcal{M}}$ be a model category. ${\mathcal{M}}$ is called a simplicial model category if ${\mathcal{M}}$ is simplicial and

1. (1)

$\forall X\in Obj({\mathcal{M}})$ there is an adjunction

[TABLE]

which is compatible with the simplicial structure on ${\mathcal{M}}$. 2. (2)

$\forall Y\in Obj({\mathcal{M}})$ there is an adjunction

[TABLE]

which is compatible with the simplicial structure on ${\mathcal{M}}$. 3. (3)

For $Cof({\mathcal{M}})\ni i:A\rightarrow B$ and $Fib({\mathcal{M}})\ni p:X\rightarrow Y$ the map

[TABLE]

is in $Fib(\mathbf{SSets}$, which is also in $W(\mathbf{SSets})$, if either $i$ or $p$ is in $W({\mathcal{M}})$.

Example 6.16**.**

$\mathbf{SSets}$* has a canonical simplicial model category structure. $\mathbf{SSMap}(X,Y)$ is the simplicial set with*

[TABLE]

with faces and degeneracies induced from the cosimplicial object $\Delta^{\bullet}$.

Proposition 6.17**.**

Let ${\mathcal{M}}$ be a simplicial model category. If $X$ is cofibrant and $Y$ is fibrant, then

[TABLE]

Consequently, for any objects $A,B\in Obj({\mathcal{M}})$ one has

[TABLE]

6.2. Localization

All model categories in this subsection are being considered simplicial.

Definition 6.18**.**

Let ${\mathcal{M}}$ be a model category and ${\mathcal{V}}$ be a class of morphisms in $Mor({\mathcal{M}})$. A left localization of ${\mathcal{M}}$ wrt. ${\mathcal{V}}$ is a model category $L_{{\mathcal{V}}}{\mathcal{M}}$ together with a left Quillen functor $F:{\mathcal{M}}\rightarrow L_{{\mathcal{V}}}{\mathcal{M}}$, such that:

1. (1)

The total left derived functor $\mathbb{L}F:Ho({\mathcal{M}})\rightarrow Ho(L_{{\mathcal{V}}}{\mathcal{M}})$ takes the images in $Ho({\mathcal{M}})$ of elements in ${\mathcal{V}}$ into isomorphisms in $Ho(L_{{\mathcal{V}}}{\mathcal{M}})$. 2. (2)

If ${\mathcal{N}}$ is a model category and $T:{\mathcal{M}}\rightarrow{\mathcal{N}}$ is a left Quillen functor such that $\mathbb{L}T:Ho({\mathcal{M}})\rightarrow Ho({\mathcal{N}})$ take the images in $Ho({\mathcal{M}})$ of elements in ${\mathcal{V}}$ into isomorphisms in $Ho({\mathcal{N}})$, then there is a unique left Quillen functor $L_{{\mathcal{V}}}{\mathcal{M}}\rightarrow{\mathcal{N}}$, such that

[TABLE]

Definition 6.19**.**

Let ${\mathcal{M}}$ be a model category and ${\mathcal{V}}$ be a class of morphisms in $Mor({\mathcal{M}})$. A right localization of ${\mathcal{M}}$ wrt. ${\mathcal{V}}$ is a model category $R_{{\mathcal{V}}}{\mathcal{M}}$ together with a right Quillen functor $G:{\mathcal{M}}\rightarrow R_{{\mathcal{V}}}{\mathcal{M}}$, such that:

1. (1)

The total right derived functor ${\mathbb{R}}G:Ho({\mathcal{M}})\rightarrow Ho(R_{{\mathcal{V}}}{\mathcal{M}})$ takes the images in $Ho({\mathcal{M}})$ of elements in ${\mathcal{V}}$ into isomorphisms in $Ho(R_{{\mathcal{V}}}{\mathcal{M}})$ 2. (2)

If ${\mathcal{N}}$ is a model category and $T:{\mathcal{M}}\rightarrow{\mathcal{N}}$ is a right Quillen functor such that ${\mathbb{R}}T$ takes the images in $Ho({\mathcal{M}})$ of elements in ${\mathcal{V}}$ into isomorphisms in $Ho({\mathcal{N}})$, then there is a unique right Quillen functor $R_{{\mathcal{V}}}{\mathcal{M}}\rightarrow{\mathcal{M}}$, such that:

[TABLE]

Definition 6.20**.**

Let ${\mathcal{M}}$ be a model category and ${\mathcal{V}}$ a class of morphisms in $Mor({\mathcal{M}})$.

1. (1)

An object $X\in Obj({\mathcal{M}})$ is called ${\mathcal{V}}$-local if $X$ is fibrant and for every $f:A\rightarrow B$ in ${\mathcal{V}}$, $\mathbf{SSMap}(B^{cof},X)\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\mathbf{SSMap}(A^{cof},X)$. 2. (2)

A morphism $f:X\rightarrow Y$ in $Mor({\mathcal{M}})$ is a ${\mathcal{V}}$-local equivalence if for every ${\mathcal{V}}$-local object $T$, $\mathbf{SSMap}(Y^{cof},T)\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\mathbf{SSMap}(X^{cof},T)$ 3. (3)

$X\in Obj({\mathcal{M}})$ is called ${\mathcal{V}}$-colocal if $X$ is cofibrant and for every $f:A\rightarrow B$ in ${\mathcal{V}}$, $\mathbf{SSMap}(X,A^{fib})\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\mathbf{SSMap}(X,B^{fib})$. 4. (4)

$Mor({\mathcal{M}})\ni f:X\rightarrow$ is a ${\mathcal{V}}$-colocal equivalence if for every ${\mathcal{V}}$-colocal object $T$, $\mathbf{SSMap}(T,X^{fib})\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\mathbf{SSMap}(T,Y^{fib})$.

Definition 6.21**.**

Let ${\mathcal{M}}$ be a model category and ${\mathcal{V}}$ be a class of morphisms in $Mor({\mathcal{M}})$. The left Bousfield localization (if it exists) of ${\mathcal{M}}$ wrt. ${\mathcal{V}}$ is a model category structure $L_{{\mathcal{V}}}{\mathcal{M}}$ on the underlying category ${\mathcal{M}}$ with:

1. (1)

$W(L_{{\mathcal{V}}}{\mathcal{M}})$ is the class of ${\mathcal{V}}$-local equivalences of ${\mathcal{M}}$. 2. (2)

$Cof(L_{{\mathcal{V}}}{\mathcal{M}})=Cof({\mathcal{M}})$. 3. (3)

$Fib(L_{{\mathcal{V}}}{\mathcal{M}})=RLP(Cof({\mathcal{M}})\cap W(L_{{\mathcal{V}}}{\mathcal{M}}))$.

Definition 6.22**.**

Let ${\mathcal{M}}$ be a model category and ${\mathcal{V}}$ be a class of morphisms in $Mor({\mathcal{M}})$. The right Bousfield localization (if it exists) of ${\mathcal{M}}$ wrt. ${\mathcal{V}}$ is a model category structure $R_{{\mathcal{V}}}{\mathcal{M}}$ on the underlying category ${\mathcal{M}}$ with:

1. (1)

$W(R_{{\mathcal{V}}}{\mathcal{M}})$ is the class of ${\mathcal{V}}$-colocal equivalences of ${\mathcal{M}}$. 2. (2)

$Fib(R_{{\mathcal{V}}}{\mathcal{M}})=Fib({\mathcal{M}})$. 3. (3)

$Cof(R_{{\mathcal{V}}}{\mathcal{M}})=LLP(Fib({\mathcal{M}})\cap W(R_{{\mathcal{V}}}{\mathcal{M}}))$.

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 smooth of finite type $S$-schemes. A symmetric $T$-spectrum is a collection $\{X_{n}\}_{n\geq 0}$, where $X_{n}\in\mathbf{\Delta}^{op}(PrSh_{Nis}(Sm/S))_{+}$, together with the left actions

[TABLE]

where $\Sigma_{n}$ is the $n$-th symmetric group. There are the bonding maps

[TABLE]

such that the interative composition

[TABLE]

is $\Sigma_{m}\times\Sigma_{n}$-equivariant. A morphism between symmetric $T$-spectra is a family $\{f_{n}:X_{n}\rightarrow Y_{n}\}_{n\geq 0}$, where the following diagram

[TABLE]

commutes and $f_{n}$ is $\Sigma_{n}$-equivariant $\forall n\geq 0$. The category of symmetric $T$-spectra is denoted by $\mathbf{Spect}_{T}^{\Sigma}(S)$. A symmetric sequence $X$ is a family $\{X_{n}|X_{n}\in\mathbf{\Delta}^{op}(PrSh_{Nis}(Sm/S))_{+}\}_{n\geq 0}$ with left actions

[TABLE]

A morphism $f:X\rightarrow Y$ of symmetric sequences is a family $\{f_{n}:X_{n}\rightarrow Y_{n}\}$, where $f_{n}$ are $\Sigma_{n}$-equivariant $\forall n\geq 0$. We denote the category of symmetric sequences of pointed simplicial presheaves by $\mathbf{\Delta}^{op}(PrSh_{Nis}(Sm/S))_{+}^{\Sigma}$. Recall that there are families of functors

[TABLE]

where

[TABLE]

and

[TABLE]

They are in fact adjoint to each other

[TABLE]

For two symmetric sequences $X$ and $Y$, their product is defined as

[TABLE]

The notation $\Sigma_{n}\otimes_{\Sigma_{p}\times\Sigma_{q}}X_{p}\wedge Y_{q}$ means: there is an action $\gamma$ of $\Sigma_{p}\times\Sigma_{q}$ on $X_{p}\wedge Y_{q}$ via the canonical embedding $\Sigma_{p}\times\Sigma_{q}\subset\Sigma_{n}$ and also another action $\gamma^{\prime}:\Sigma_{p}\times\Sigma_{q}\times(X_{p}\wedge Y_{q})\rightarrow X_{p}\wedge Y_{q}$. We let

[TABLE]

Now one can define

[TABLE]

where $\mathbb{S}^{0}$ denotes the motivic sphere spectrum

[TABLE]

$\Sigma_{n}$ acts on $\mathbb{S}^{0}$ by permuting the $T^{\wedge n}$ factors and $S_{+}$ is pointed by $S\coprod S$. One has an adjunction

[TABLE]

In fact, one has $F^{\Sigma}_{0}(S_{+})=\mathbb{S}^{0}$. A symmetric $T$-spectrum $X$ can be understood as a symmetric sequence with a module structure $\sigma_{X}:\mathbb{S}^{0}\otimes X\rightarrow X$ over the motivic sphere spectrum $\mathbb{S}^{0}$. Now we can define the smash product of symmetric $T$-spectra as

[TABLE]

where the top map is $\sigma_{X}\otimes{\rm id}_{Y}$ and the bottom map is

[TABLE]

We just mention the following results of Jardine.

Theorem 6.23**.**

(Jardine [Jar00, Thm. 4.2]) The category $\mathbf{Spect}_{T}^{\Sigma}(S)$ has a model category structure, which is proper and simplicial.

Theorem 6.24**.**

(Jardine [Jar00, Prop. 4.19]). $(\mathbf{Spect}_{T}^{\Sigma}(S),\mathbb{S}^{0},\wedge)$ is a symmetric monoidal model category.

Now we discuss a little bit about the Quillen adjunction

[TABLE]

where $\mathbf{E}$ is a motivic strict ring spectrum (we always consider only commutative ring spectrum). On the level of the underlying categories the unit and counit of the adjunction are defined by

[TABLE]

and

[TABLE]

By [Jar00, Prop. 4.19] the category $\mathbf{Spect}_{T}^{\Sigma}(S)$ satisfies the axiom in [SS00, Def. 3.3]. By [SS00, Thm. 4.1] one can conclude that the adjunction

[TABLE]

induces a model category structure on $\mathbf{E}-Mod^{\Sigma}$. It is clear that the forgetful functor

[TABLE]

is a right Quillen functor, because $Fib(\mathbf{E}-Mod^{\Sigma})$ and $Fib(\mathbf{E}-Mod^{\Sigma})\cap W(\mathbf{E}-Mod^{\Sigma})$ are detected in $\mathbf{Spect}^{\Sigma}_{T}(S)$. So we can claim that the adjunction above is a Quillen adjunction. Since $\mathbf{E}$ is a commutative ring spectrum, $\mathbf{E}-Mod^{\Sigma}$ has the closed symmetric monoidal category structure induced by the one on $\mathbf{Spect}_{T}^{\Sigma}(S)$ by declaring:

[TABLE]

and

[TABLE]

where

[TABLE]

The top map is $\gamma_{M}\wedge{\rm id}$ and the bottom map is the composition

[TABLE]

The internal Hom is defined as

[TABLE]

where the top map is $\gamma_{M}^{*}=\circ\gamma_{M}$ and the bottom map is

[TABLE]

We should also mention the following theorem of Jardine:

Theorem 6.25**.**

[Jar00, Thm. 4.31]** There is a Quillen equivalence

[TABLE]

where $\mathbf{Spect}_{T}(S)$ 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 homotopy category as

[TABLE]

The theorem of Jardine allows us to identify $SH(k)$ equivalently to the ${\mathbb{A}}^{1}$-stable homotopy category $SH^{{\mathbb{P}}^{1}}(k)\cong SH^{T}(k)$ of Morel constructed in [Mor04a, Defn. 5.1, Rem. 5.1.10 and pp. 420], which is defined as the homotopy category of the motivic $T$-spectra $\mathbf{Spect}_{T}(k)$. Hence, we can use Morel computation of $\pi_{0}^{st{\mathbb{A}}^{1}}(\mathbb{S}^{0})_{*}$ and his stable ${\mathbb{A}}^{1}$-connectivity result.