Trace formalism for motivic cohomology

We will use the language of $\infty$-categories, but except for §6, this is used just to facilitate the presentation. See the remark at the end of this paragraph for some explanation.

Let $\mathcal{P}\mathrm{r}^{\mathrm{L}}_{\mathrm{st}}$ be the full subcategory of $\mathcal{P}\mathrm{r}^{\mathrm{L}}$ (cf. [Lur09, Definition 5.5.3.1]) spanned by stable $\infty$-categories. We have the functor $\mathcal{SH}\colon\mathrm{Sch}_{/k}^{\mathrm{op}}\rightarrow\mathcal{P}\mathrm{r}^{\mathrm{L}}_{\mathrm{st}}$ sending $T$ to Voevodsky-Morel’s stable homotopy $\infty$-category $\mathcal{SH}(T)$ (cf. [CD15, §2.1] or [CD19, Example 1.4.3] for model categorical treatment and [Abe22a, §6.7] and references therein for $\infty$-categorical treatment). Let be a commutative ring. Then Voevodsky defined the motivic Eilenberg-MacLane spectrum $\mathbb{H}{}_{k}$, which is an $\mathbb{E}_{\infty}$-algebra of $\mathcal{SH}(\mathrm{Spec}(k))$. By pulling back, this spectrum yields a spectrum $\mathbb{H}{}_{T/k}$ on $\mathcal{SH}(T)$, and defines an “absolute ring $\mathcal{SH}$-spectrum” in the sense of [Deg18, Definition 1.1.1]. The absolute ring $\mathcal{SH}$-spectrum $\mathbb{H}{}_{T/k}$ is equipped with an “orientation” in the sense of [Deg18, Definition 2.2.2] by [Deg18, Example 2.2.4]. Under this situation, all the results of [Deg18, Introduction, Theorem 1] can be applied. We do not try to recall the definitions of each terminology, but instead, we sketch what we can get by fixing these data.

We put $\mathcal{D}_{T}:=\mathrm{Mod}_{\mathbb{H}{}_{T}}(\mathcal{SH}(T))$, the symmetric monoidal $\infty$-category of $\mathbb{H}{}_{T}$-module objects in $\mathcal{SH}(T)$. Then the assignment $\mathcal{D}_{T}$ to $T$ can be promoted to a functor $\mathcal{D}\colon\mathrm{Sch}_{/S}^{\mathrm{op}}\rightarrow\mathcal{P}\mathrm{r}^{\mathrm{L}}_{\mathrm{st}}$ which yields “motivic categories” in the sense of [CD19]. This can be checked from [CD19, Proposition 5.3.1 and Proposition 7.2.18]. We may find a summary of the axioms of what this means in [Abe22a, §6.1], and also references. Among other things, we may use “six functors”. In this $\infty$-categorical context, we can find a construction of six functor formalism in [Abe22a, §6.8], which follows the idea of [Kha16]. Let $X\in\mathrm{Sch}_{/S}$. Then $\mathcal{D}_{X}$ is a symmetric monoidal stable $\infty$-category. Given a morphism $f\colon X\rightarrow Y$ in $\mathrm{Sch}_{/S}$, the functor $\mathcal{D}$ induces the functor $\mathcal{D}_{Y}\rightarrow\mathcal{D}_{X}$, which we denote by $f^{*}$ in accordance with the six functor formalism of Grothendieck. The functor $f^{*}$ admits a right adjoint, which we denote by $f_{*}$. We also have the “extraordinary pushforward functor” $f_{!}\colon\mathcal{D}_{X}\rightarrow\mathcal{D}_{Y}$ as well as its right adjoint $f^{!}$. We have the natural transform $f_{!}\rightarrow f_{*}$ which is an isomorphism when $f$ is proper.

The orientation on $\mathbb{H}\Lambda$ yields an orientation on $\mathcal{D}$ in the sense of [CD19, Definition 2.4.12] by [CD19, Example 2.4.40] and [Deg18, §2.2.5]. For $n\in\mathbb{Z}$, we denote the $n$-th Tate twist by $(n)$, the $n$-th shift by $[n]$, and $(n)[2n]$ by $\left<n\right>$. We often denote the unit object of $\mathcal{D}_{T}$ by _T. By fixing an orientation, we have a canonical isomorphism $f^{*}(d)[2d]\cong f^{!}$ for any smooth morphism $f$ in $\mathrm{Sch}_{/S}$ (cf. [CD19, Theorem 2.4.50]). In fact, the fundamental class constructed in [Deg18, Introduction, Theorem 1] can be seen as a generalization of this isomorphism.

Let $f\colon X\rightarrow S$. For $\mathcal{F}\in\mathcal{D}_{S}$, we set

Here, we view $\mathrm{H}^{\mathrm{BM}}$ as a spectrum. When the coefficient ring is obvious, we abbreviate $\mathrm{H}^{\mathrm{BM}}(X/S,\Lambda(n))$ by $\mathrm{H}^{\mathrm{BM}}(X/S,n)$. We write $\mathrm{H}^{\mathrm{BM}}_{m}(X/S,\mathcal{F})$ for $\pi_{m}\mathrm{H}^{\mathrm{BM}}(X/S,\mathcal{F})$, and call it the Borel-Moore homology. Note that $\pi_{m}\mathrm{H}^{\mathrm{BM}}(X/S,n)$ coincides with $(\mathbb{H}\Lambda)^{\mathrm{BM}}_{m,n}(X/S)$ in [Deg18]. Assume we are given a closed subscheme $Z\subset X$ and denote the complement by $U$. By localization sequence of 6-functor formalism, we have the long exact sequence

We introduce the $p$dh-topology as follows.

We call $\ell^{\prime}$dh-topology what is called $\ell$dh-topology in [CD15, §5.2]. Obviously, cdh-topology is coarser than $p$dh-topology, and $p$dh-topology is coarser than $\ell^{\prime}$dh-topology for any $\ell\neq p$.

Let $S$ be an object of $\mathrm{Sch}_{/k}$. Recall that the theorem of Temkin [T], which is a refinement of Gabber’s prime-to-$\ell$ alteration theorem, states as follows: there exists an alteration $S^{\prime}\rightarrow S$ whose generic degree is some power of $p$ and $S^{\prime}$ is smooth. Without Temkin’s theorem, $p$dh-topology might have been useless, but armed with the theorem, we can show the following statement as usual.

For any $S\in\mathrm{Sch}_{/k}$, we may find a $p$dh-hypercovering $S_{\bullet}\rightarrow S$ such that $S_{i}$ is $k$-smooth by standard use of the lemma above and [SGA4, Exposé $\mathrm{V}^{\mathrm{bis}}$, Proposition 5.1.3].

We have the following $p$dh-descent, which is a straightforward corollary of a $\ell^{\prime}$dh-descent result by S. Kelly.

Now, let $\mathcal{G}\in\mathcal{D}_{S}$. Then we have

We write $\mathrm{Hom}^{k}:=\pi_{-k}\mathrm{Hom}$. Assume that $\mathrm{Hom}^{k}\bigl{(}p_{i}^{*}\mathcal{G}_{i},p_{i}^{*}\mathcal{F}_{i}\bigr{)}\cong 0$ for any $i\in\mathbf{\Delta}$. Then the complex of -indexed diagrams $\bigl{\{}\mathrm{Hom}\bigl{(}p_{i}^{*}\mathcal{G}_{i},p_{i}^{*}\mathcal{F}_{i}\bigr{)}\bigr{\}}_{i\in\mathbf{\Delta}}$ belongs to $D^{+}(\mathrm{Ab}^{\mathbf{\Delta}})$, and induces a spectral sequence

Let us recall the definition of bivariant theory after Fulton and MacPherson very briefly.

Our Borel-Moore homology $\mathrm{H}^{\mathrm{BM}}_{a}(X/S,\Lambda(b))$ defines a bivariant theory (in an extended sense because it is bigraded), cf. [Deg18, §1.2.8]. By associating the graded group $\bigoplusop\displaylimits_{k}\mathrm{H}^{\mathrm{BM}}_{2k}(X/S,\Lambda(k))$ to $X\rightarrow S$, we define the bivariant theory denoted by $\mathrm{H}^{\mathrm{BM}}_{2*}(X/S,\Lambda(*))$. This bivariant theory has a canonical orientation as follows. Let $q\colon\mathbb{A}^{1}_{S}\rightarrow S$ be the projection. Then we have a morphism

where the isomorphism is defined using [CD19, Theorem 2.4.50.3] and the canonical identification of $\mathrm{MTh}_{\mathbb{A}^{1}}(T_{q})$ with ${}_{\mathbb{A}^{1}}(1)[2]$, where $\mathrm{MTh}$ is the motivic Thom spectrum defined in [CD19, Definition 2.4.12]. The class of the above morphism in $\mathrm{H}^{\mathrm{BM}}_{2}(X/S,\Lambda(1))\cong\pi_{0}\mathrm{Hom}_{D(S)}(q_{!}q^{*}{}_{S}(1)[2],{}_{S})$ is the canonical orientation of $\mathrm{H}^{\mathrm{BM}}_{2*}(X/S,\Lambda(*))$.

Let us introduce another main player of this paper, $z(-,-)$, from [SV]. Let $f\colon X\rightarrow S$ be a morphism, and $d\geq 0$ be an integer. Recall that Suslin and Voevodsky⁽²⁾⁽²⁾(2)In fact, Suslin and Voevodsky used the notation $z(X/S,d)$ as a presheaf on $\mathrm{Sch}_{/S}$. Our $z(X/S,d)$ is the global sections of it. introduced Abelian groups $z_{\mathrm{equi}}(f,d)$ and $z(f,d)$, or $z_{\mathrm{equi}}(X/S,d)$ and $z(X/S,d)$ if no confusion may arise. We do not recall the precise definition of these groups, but content ourselves with giving ideas of how these groups are defined. Both groups are certain subgroups of the free Abelian group $Z(X)$ generated by integral subscheme of $X$. If we are given an element $w\in Z(X)$ we may consider the “support” denoted by $\mathrm{Supp}(w)$ in an obvious manner. Naively thinking, we wish to define $z(X/S,d)$ as a subgroup of $Z(X)$ consisting of $w$ such that $\mathrm{Supp}(w)\rightarrow S$ is equidimensional of dimension $d$ over generic points of $S$. However, if we defined $z(X/S,d)$ in this way, the association $z(X_{T}/T,d)$ to $T$ would not be functorial. In order to achieve this functoriality, Suslin and Voevodsky introduces an ingenious compatibility conditions. We do not recall these compatibility conditions, but here is an illuminating example: Let $Z\subset X$ be a closed immersion such that the morphism $Z\rightarrow S$ is flat. Then the associated cycle $[Z]$, called a flat cycle, belongs to $z(X/S,d)$. Now, the group $z_{\mathrm{equi}}(X/S,d)$ is a subgroup of $z(X/S,d)$. The element $w$ belongs to $z_{\mathrm{equi}}(X/S,d)$ if and only if the morphism $\mathrm{Supp}(w)\rightarrow S$ is equidimensional (of relative dimension $d$). By the compatibility conditions we mentioned above, if we are given a morphism $S^{\prime}\rightarrow S$, we have the pullback homomorphism $z_{(\mathrm{equi})}(X/S,d)\rightarrow z_{(\mathrm{equi})}(X\times_{S}S^{\prime}/S^{\prime},d)$. This enables us to define presheaves $\underline{z}_{(\mathrm{equi})}(X/S,d)$ on $\mathrm{Sch}_{/S}$. Then $\underline{z}(X/S,d)$ is a cdh-sheaf, and the cdh-sheafification of $\underline{z}_{\mathrm{equi}}(X/S,d)$ coincides with $\underline{z}(X/S,d)$. Furthermore, flat cycles generate $\underline{z}(X/S,d)$ cdh-locally, and can be thought of as a building pieces (cf. [SV, Theorem 4.2.11]). The following theorem compactly summarizes some aspects of [SV].

Our main theorem is as follows.

A proof of this theorem is given at the end of Section 5. Let us introduce a notation. Let $f\colon X\rightarrow S$ be a flat morphism of relative dimension $d$. Then $[X]$ is an element of $z_{\mathrm{equi}}(f,d)$. If we are given $\tau$ as above, we have $\tau([X])\in\mathrm{H}^{\mathrm{BM}}_{2d}(f,\Lambda(d))$. This element is denoted by $\mathrm{Tr}^{\tau}_{f}$.

Before going to the next section, let us show the most important property to construct the trace map, namely the vanishing of suitable higher homotopies. For a morphism $f\colon X\rightarrow S$, we put $\dim(f):=\max\bigl{\{}\dim(f^{-1}(s))\mid s\in S\bigr{\}}$.

For a scheme $Z$, we often denote $\dim(Z)$ by $d_{Z}$. Let $f\colon X\rightarrow S$ be (any) separated morphism of finite type such that $S$ is smooth equidimensional, and put $d_{f}:=d_{X}-d_{S}$. In this case, let us construct a morphism $t_{f}\colon f_{!}{}_{X}\!\left<d_{f}\right>\rightarrow{}_{S}$, which we will show to be equal to $\mathrm{Tr}_{f}$ when $f$ is flat.

Let us start to construct $t_{f}$. Considering componentwise, it suffices to construct the morphism when $S$ is connected. For any separated scheme $X$ of finite type over $k$, we have the canonical isomorphism

by [Jin16, Corollary 3.9]. We have

where the first isomorphism follows since $g^{*}\!\left<d\right>\xrightarrow{\sim}g^{!}$ for any equidimensional smooth morphism $g$ of relative dimension $d$. Let $X=\bigcupop\displaylimits_{i\in I}X_{i}$ be the irreducible components, and let $I^{\prime}\subset I$ be the subset of $i$ such that $\dim(X_{i})=d_{X}$. Let $\xi_{i}$ be the generic point of $X_{i}$. The element in $\mathrm{H}^{\mathrm{BM}}_{2d}(X/S,\Lambda(d))$ corresponding via the isomorphism above to the element $\sumop\displaylimits_{i\in I^{\prime}}\mathrm{lg}(\mathcal{O}_{X,\xi_{i}})\cdot[X_{i,\mathrm{red}}]\in\mathrm{CH}_{d_{X}}(X;\Lambda)$ on the right hand side is defined to be $t_{f}$.

Let us end this paragraph with a simple observation. Let $U\subset X$ be an open dense subscheme. Then the restriction map $\mathrm{H}^{\mathrm{BM}}_{2d_{f}}(X/S,\Lambda(d_{f}))\rightarrow\mathrm{H}^{\mathrm{BM}}_{2d_{f}}(U/S,\Lambda(d_{f}))$ is an isomorphism. Indeed, we have $\mathrm{H}^{\mathrm{BM}}_{2d_{h}}(X/Z,\Lambda(d_{h}))\cong\mathrm{CH}_{d_{X}}(X;\Lambda)\cong{}^{\oplus r_{X}}$, where $r_{X}$ is the set of irreducible components of $X$ of dimension $d_{X}$ by the computation above. Since $r_{X}$ and $r_{U}$ are the same, we get the claim.

By the setup 2.1, we may apply [Deg18, Introduction, Theorem 1]. In particular, for a morphism between smooth schemes $f\colon X\rightarrow Y$ we have the fundamental class $\overline{\eta}_{f}\in\mathrm{H}^{\mathrm{BM}}_{2d_{f}}(X/Y,\Lambda(d_{f}))$. When $Y=\mathrm{Spec}(k)$, we sometimes denote $\overline{\eta}_{f}$ by $\overline{\eta}_{X}$. As we expect, we have the following comparison.