Motivic Cohomology II: categories of étale motives

by eoinmackall

Where we’re at, and where we’re going

In this blog post we’ll give the first formal construction of a category of motives. In particular, we’ll construct the category of (geometric) étale motives. These aren’t necessarily the motives that we’ll want to work with. For example, the final step of the construction will give us a category which we can’t currently prove has a full or faithful embedding from the previous step — except in the case our coefficients contain \mathbb{Q}. In this case we are only helped along knowing more precise results from the theory of Voevodsky’s geometric (Nisnevich) motives — in particular Voevodsky’s cancellation theorem.

This means we won’t stay long with this category. However, we can describe properties of the categories that make up the construction. It’s possible to point out relations when the coefficient system is particularly well-behaved. In this setting well-behaved means torsion; it is analogous to the fact that étale cohomology is better behaved with torsion coefficients.

However, I haven’t taken up the task of doing this here. I may come back to this section at a later date to add more of what is known about the category of geometric étale motives as it’s constructed below. (If you’re reading this paragraph I haven’t done that yet).

Preliminaries

A1 Weak Equivalence

Our construction of categories of motives will require this section and the next. For the set up: let C_{\tau} be a Grothendieck topology where it makes sense to consider sheaves with transfers (i.e. a topology satisfying T1). Let F,G be two complexes of C_{\tau}-sheaves with transfers with coefficients in a ring R (i.e. compelxes of elements of \mathsf{Sh}_{C_{\tau}}(\mathsf{Cor}_k(R))).

(5.1) Definition: A morphism f:F\rightarrow G in the derived category of bounded above chain complexes D^-(\mathsf{Sh}_{C_{\tau}}(\mathsf{Cor}_k(R))) is called an \mathbb{A}^1-weak equivalence if the cone of f is in the smallest thick subcategory \mathscr{E}_{\mathbb{A}} satisfying:

i) for every smooth scheme X the cone of R_{tr}(X\times \mathbb{A}^1)\rightarrow R_{tr}X, which is the two term complex

[\cdots \rightarrow 0\rightarrow R_{tr}(X\times \mathbb{A}^1)\rightarrow R_{tr}X\rightarrow 0\rightarrow \cdots ]

is contained in \mathscr{E}_{\mathbb{A}}

ii) \mathscr{E}_{\mathbb{A}} is closed under direct sums which exist in D^-(\mathsf{Sh}_{C_{\tau}}(\mathsf{Cor}_k(R))).

Remark: The last part, regarding existence, is necessary since arbitrary direct sums of sheaves of bounded above chain complexes need not be bounded above.

By the last two paragraphs of the thick subcategories section here, the class of \mathbb{A}^1-weak equivalences form a saturated multiplicative system of morphisms.

Tate Twists

We’ll consider the scheme \mathbb{A}^1\setminus \{0\} pointed by x:\text{Spec}(k)\rightarrow \mathbb{A}^1\setminus \{0\}, defined by \{pt\}\mapsto 1. Let \pi:\mathbb{A}^1\setminus\{0\}\rightarrow \text{Spec}(k) be the structure map.

These two morphisms induce morphisms on presheaves with transfers

x_{tr}:R_{tr}\rightarrow R_{tr}(\mathbb{A}^1\setminus \{0\})

\pi_{tr}:R_{tr}(\mathbb{A}^1\setminus\{0\})\rightarrow R_{tr}.

Further, since the equality \pi\circ x=\text{id}_{\text{Spec}(k)} holds as morphisms of schemes, and the inclusion to presheaves with transfers is fully faithful, we have the equality as presheaves with transfers \pi_{tr}\circ x_{tr} =\text{id}_{R_{tr}}. If we define R_{tr}(\mathbb{A}^1\setminus\{0\},1) to be the cokernel of the morphism x_{tr} (a posteriori, equivalently the kernel of \pi_{tr}) then this shows the short exact sequence

0\rightarrow R_{tr}\xrightarrow{x_{tr}} R_{tr}(\mathbb{A}^1\setminus \{0\})\rightarrow R_{tr}(\mathbb{A}^1\setminus\{0\},1)\rightarrow 0

is split. In other words, we have an isomorphism R_{tr}(\mathbb{A}^1\setminus\{0\})\cong R_{tr}(\mathbb{A}^1\setminus\{0\},1)\oplus R_{tr}.

(5.2) Definition: the complex R(1):=C_\ast R_{tr}(\mathbb{A}^1\setminus\{0\},1)[-1] will be called the Tate complex in this blog (at least until I give it another name in a later post).

Using the Tate complex we can define Tate twists in derived categories of complexes of sheaves in suitable Grothendieck topologies C_{\tau}. That is,

(5.3) Definition: the functor a_{C_{\tau}}\circ-\otimes R(1):D^-(\mathsf{Sh}_{C_{\tau}}(Cor_k(R))), which is the sheafification of the tensor product of these complexes, M_\bullet\mapsto M_\bullet\otimes^{tr}_L R(1)\mapsto a_{C_\tau}(M_\bullet \otimes_L^{tr} R(1)), is called the Tate twist operation.

For brevity, we’ll write M(n):= a_{C_{\tau}}(M\otimes_L^{tr} R(n)):= a_{C_{\tau}}(M\otimes_L^{tr} R(1)^{\otimes^{tr}_L n}) for the nth iteration of the Tate twist for any postive n.

This is related to the Tate twist defined in Grothendieck’s categories of pure Chow motives in the following way: as above we define R_{tr}(\mathbb{P}^1,\infty) to be the cokernel of the morphism \infty: R_{tr}\rightarrow R_{tr}(\mathbb{P}^1), induced from the map of schemes \{pt\}\mapsto \infty. This cokernel will be called the Lefschetz motive. The following diagram shows the situation we are in, with the middle row exact and obtained from applying the  Čech complex from Proposition (2.8) to the cover _0\mathbb{A}^1:=\text{Spec}(k[t]) and _\infty\mathbb{A}^1:=\text{Spec}(k[1/t]) of \mathbb{P}^1; the objects on the bottom rows are the cokernels of the vertical morphisms; the diagram is commutative.

\begin{matrix} & & R_{tr} & \xrightarrow{\sim} & R_{tr} & \xrightarrow{\sim} & R_{tr} & &\\ & & \downarrow &  & \downarrow & & \downarrow & &\\0 & \rightarrow & R_{tr}(\mathbb{A}^1\setminus\{0\}) & \rightarrow & R_{tr}(\mathbb{A}^1) & \rightarrow & R_{tr}(\mathbb{P}^1) &\rightarrow  & 0 \\ & & \downarrow &  & \downarrow & & \downarrow & &\\ & & R_{tr}(\mathbb{A}^1\setminus\{0\},1) & \rightarrow & R_{tr}(\mathbb{A}^1, 1) & \rightarrow & R_{tr}(\mathbb{P}^1,1) & &\end{matrix}

It follows the bottom row is exact (from a diagram chase and an application of the snake lemma). Applying C_\ast results in the distinguished triangle

C_\ast R_{tr}(\mathbb{A}^1\setminus \{0\},1)\rightarrow C_\ast R_{tr}(\mathbb{A}^1,1)\rightarrow C_\ast R_{tr}(\mathbb{P}^1,1)\rightarrow C_\ast R_{tr}(\mathbb{A}^1\setminus \{0\},1)[1].

Our calculations of the algebraic singular cohomology (search for “example”) of \mathbb{A}^n for n\geq 0 show C_\ast R_{tr}(\mathbb{A}^1,1)\simeq 0. So we have an isomorphism of complexes C_\ast R_{tr}(\mathbb{P}^1,1)\rightarrow C_\ast R_{tr}(\mathbb{A}^1\setminus \{0\},1)[1]=Z(1)[2]. The last thing to do is to assure ourselves this is independent of our choice of point. In other words, C_\ast R_{tr}(\mathbb{P}^1,1)\cong C_\ast R_{tr}(\mathbb{P}^1,\infty). But this is clear since even on the level of schemes there is an automorphism of \mathbb{P}^1 with 1 \mapsto \infty.

Later, we will show the inclusion of Chow motives into the motives defined by Voevodsky sends the Lefschetz motive (in the category of pure Chow motives) to the Lefschetz motive (in the category of geometric Nisnevich motives) which the above shows is isomorphic with the object Z(1)[2].

Etale Motives

In defining Etale motives, we’ll go through various notations. All of these I will explain.

(5.4) Definition: The category \text{DM}_{\acute{e}t}^{\text{eff},-}(k,R) will be our new notation for the category D^-(\mathsf{Sh}_{\acute{e}t}(Cor_k(R))).

Our new notation stands for \text{DM}^-, the derived category of bounded above motives, \acute{e}t, in the étale topology, \text{eff}, which are effective, (k,R), over the field k with coefficients in R. All of these are relatively self explanatory except, possibly, the adjective effective. Typically, effective is tacked on to an algebro-geometric term when there will later be an algebraic construction generalizing one which had geometric origin. For example, in Chow groups there are effective cycle classes, those which can be interpreted as equivalence classes represented by an explicit geometrically obtained cycle (like the class of a point [pt]), and noneffective cycle classes, those which are algebraically constructed (like the negative of the class of a point -[pt]). We will similarly extend our category to a larger one but, not yet.

But first, we don’t want to consider all bounded above complexes. There are bound to be quite a few complexes which could reasonably be discarded (those which are not generated by the smooth schemes for instance). For this reason we take only a smaller category in our construction.

(5.5) Definition: \text{DM}_{\acute{e}t}^{\text{eff}}(k,R) is defined to be the smallest thick subcategory of \text{DM}_{\acute{e}t}^{\text{eff},-}(k,R) which contains all of the objects, and subobjects of, R_{tr}X for all smooth schemes X and is closed under quasi-isomorphisms, direct sums, shifts, and cones.

(5.6) Definition: The category of geometric étale motives over k with coefficients in R, \text{DM}_{\acute{e}t}(k,R), is constructed as the \mathbb{P}^1-stabilization of the \mathbb{A}^1-localization of \text{DM}_{\acute{e}t}^{\text{eff}}(k,R).

These two notions I haven’t defined yet, the \mathbb{P}^1-stabilization and the \mathbb{A}^1-localization, are exactly where the preliminaries for this blog post are used.

(5.7) Definition: The \mathbb{A}^1-localization of \text{DM}^{\text{eff}}_{\acute{e}t}(k,R) is the localization, in the sense of Propositions (3.3) and (3.4) here, at the multiplicative system formed by \mathbb{A}^1 weak equivalences.

(5.8) Definition: the \mathbb{P}^1-stabilization of the \mathbb{A}^1-localization of \text{DM}^{\text{eff}}_{\acute{e}t}(k,R) is the localization, in the sense of Proposition (3.14) ii), of the Tate twist operation.

There is a more accurate notion of what one wants from \mathbb{P}^1-stabilization, which is described in this ICM article by Ayoub. It is related to the homotopical aspect of these constructions so, for the time being, I will avoid them.

In this article above, Ayoub also mentions how the category \text{DM}_{\acute{e}t}(k,R) is defective — that it is not a triangulated category for instance. If I could take a guess, I think this is a result of not knowing whether the \mathbb{P}^1-stabilization gives a full and faithful embedding \text{DM}_{\acute{e}t}^{\text{eff}}(k,R)\rightarrow \text{DM}_{\acute{e}t}(k,R). This discrepancy disappears when it is known that the latter category agrees with the category of geometric Nisnevich motives (which is known if, for example, \mathbb{Q}\subset R), which is where I will head in the next post.

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