ScalingStacks

Let $k$ be a perfect field and $\operatorname{pt}=\operatorname{Spec}(k).$ All varieties are considered to be over $k.$

For a variety $X$ over $k$ we consider the triangulated category $\operatorname{DM}(X,\mathbb{Q})$ of rational motivic sheaves on $X.$¹¹ 1 There are various definitions of $\operatorname{DM}(X,\mathbb{Q})$ that are equivalent, see [CD19, Section C.3]. We leave the choice of definition to the reader. In the literature, objects in $\operatorname{DM}(X,\mathbb{Q})$ are often referred to as relative motives or simply motives. However, we prefer the term motivic sheaf and reserve the term motive for objects in $\operatorname{DM}(k,\mathbb{Q})$. In the special case $X=\operatorname{Spec}(k),$ the category $\operatorname{DM}(k,\mathbb{Q})$ agrees with Voevodsky’s triangulated category of mixed motives over $k.$

The system of categories $\operatorname{DM}(X,\mathbb{Q})$ has remarkable properties, some of which we will recall now.

First, the work of Ayoub [Ayo07a, Ayo07b] and Cisinski–Déglise [CD19] shows that $\operatorname{DM}(X,\mathbb{Q})$ can be equipped with a six-functor-formalism which works very similarly as in the setting of $\ell$-adic sheaves. Important objects are the motive $\operatorname{M}(X)$ and motive with compact support $\operatorname{M}^{c}(X)$ of a variety $f:X\to\operatorname{pt}$ which can be expressed in terms of the six functors as

There is an autoequivalence $(1)=-\otimes\mathbb{Q}(1)$ called Tate twist on $\operatorname{DM}$ which commutes with the six functors and can be defined by splitting the motive of the projective line as

For each prime $\ell$ invertible in $k$, there is an $\ell$-adic realisation functor to the category $\operatorname{D}(X,\mathbb{Q}_{\ell})$ of $\ell$-adic sheaves on $X$

which is compatible with the six functors and the Tate twist, see [Ayo14].

Morphisms in $\operatorname{DM}(X,\mathbb{Q})$ are best understood in terms of higher Chow groups, as defined by Bloch [Blo86]. For $X$ smooth, there is a natural isomorphism

where $\mathbb{Q}_{X}$ denotes the tensor unit and $\operatorname{CH}$ a higher Chow group. In particular, for $n=2m$ one obtains the usual Chow group of codimension-$m$ cycles $\operatorname{CH}^{m}(X,0)_{\mathbb{Q}}=\operatorname{CH}^{m}(X)_{\mathbb{Q}}.$ In this case the realisation functor $\operatorname{Real}_{\ell}$ yields the cycle class map to $\ell$-adic cohomology

For $X=\operatorname{pt}$ and $k=\mathbb{F}_{q}$ or $k=\overline{\mathbb{F}}_{p}$ these hom-groups are particularly simple:

In the following the purity property in (3.1) will be crucial. For this reason, we will from now on restrict to the case $k=\overline{\mathbb{F}}_{p}.$

We will now recall the definition of the category of pure/mixed Tate motives, see [Lev05], which will play for us the role of a graded version of the derived category of sheaves on the point.

For example, the decompostion of a projective space into affine spaces

induces a decomposition of its motive into a direct sum of Tate motives

which shows that $\operatorname{M}(\mathbb{P}^{n})$ is pure Tate.

We will use the following generalization.

As the notation suggests, the category of pure Tate motives is the heart of a weight structure:

Using that $k=\overline{\mathbb{F}}_{p}$ we obtain a simple decription of the category of Tate motives:

We note that the weight structure on mixed Tate motives considered here is just the shadow of the Chow weight structure on $\operatorname{DM}^{gm}(k,\mathbb{Q}),$ see Example 2.2.

Soergel–Virk–Wendt [SVW18] introduced an equivariant version of the above formalism. For a variety $X$ with an action of a linear algebraic group $G$ they define the category $\operatorname{DM}_{G}(X,\mathbb{Q})$ of $G$-equivariant motivic sheaves on $X.$ This system of categories still carries a six-functor-formalism³³ 3 For technical reasons, Soergel–Virk–Wendt construct a six-functor-formalism for the full subcategory $\operatorname{DM}^{+}_{G}(X,\mathbb{Q})\subset\operatorname{DM}_{G}(X,\mathbb{Q})$ of objects that are bounded below with respect to the homotopy $t$-structure, see [SVW18, Section I.6]. All objects that we consider here are automatically in $\operatorname{DM}^{+}_{G}(X,\mathbb{Q})$ and we simply ignore this technicality. and induction/restriction functors changing the group.

Similarly to the non-equivariant case, there is a realization functor to the equivariant derived category of $\ell$-adic sheaves $\operatorname{D}_{G}(X,\mathbb{Q}_{\ell})$ of Bernstein–Lunts [BL94]

Moreover, for $X$ smooth there is a natural isomorphism

to the equivariant higher Chow groups as defined by Totaro [Tot99] and Edidin–Graham [EG98].

There is a forgetful functor from equivariant to non-equivariant motivic sheaves

commuting with the six functors.

An important property of equivariant (motivic) sheaves is the induction equivalence, which allows to describe $G$-equivariant motivic sheaves on a $G$-orbit $G/H$ in terms of $H$-equivariant motives on a point:

There is also an equivariant version of the category of mixed Tate motives defined in Section 3.2. For the remainder of the section let $k=\overline{\mathbb{F}}_{p}.$

We now explain how the explicit description of this category in [SVW18, Theorem II.3.1] can be obtained using the weight complex functor.

Denote by $G^{0}\subset G$ the connected component of the identity. Then one can consider the object $\operatorname{Ind}^{G}_{G^{0}}(\mathbb{Q})\in\operatorname{DM}_{G}(k)$ which plays the role of the local system on $BG$ with free action of the component group $\pi_{0}(G)=G/G^{0}.$ The heart of $w$ admits the following explicit description by [SVW18, Proposition II.4.5].

For the moment, assume that $G=G_{0}$ is connected. By [SVW18, Proposition I.7.6] the categories $\operatorname{DM}_{G}$ just depend on the quotient $G/R_{u}(G)$ of $G$ by its unipotent radical. Hence, we can assume that $G$ is reductive. Denote by $T$ a maximal torus in $G$ and by $W=N_{G}(T)/T$ the Weyl group. Let $S=\operatorname{Sym}(X(T)_{\mathbb{Q}})$ be the symmetric algebra of the rationalized character lattice of $T.$ The algebra $S$ is isomorphic to a polynomial ring in $\operatorname{rank}(T)$ many variables and graded where we put $X(T)$ in degree one.

Since we are only considering rational coefficients, the $T$- and $G$-equivariant Chow rings agree with equivariant cohomology rings. In particular, the Chern class map induces isomorphisms of graded algebras

see [Tot99] or [EG98, Section 3.2]. Similarly, for higher Chow groups

using [Kri17, Theorem 1.5] and [Kri13, Theorem 5.7].

If $G$ is not connected, we consider the extension algebra

which can be thought of as the Chow ring of $BG$ with coefficients in the local system given by the regular representation of $\pi_{0}(G).$ For an explicit description denote by $S^{W}\ltimes\mathbb{Q}[\pi_{0}(G)]$ the twisted group algebra, see [SVW18, A.2.3], which as a graded vector space is just the tensor product $S^{W}\otimes\mathbb{Q}[\pi_{0}(G)]$ with $\mathbb{Q}[\pi_{0}(G)]$ concentrated in degree zero.

With Corollary 2.16 we obtain the following explicit description of $\operatorname{DTM}_{G}(k,\mathbb{Q}).$

Since $S^{W}\ltimes\mathbb{Q}[\pi_{0}(G)]$ has finite cohomological dimension, $\operatorname{D}^{\mathbb{Z}}_{\operatorname{perf}}(S^{W}\ltimes\mathbb{Q}[\pi_{0}(G)])$ is just the bounded derived category of finitely generated graded modules. A similar result was shown in [SVW18, Theorem II.3.1] using slightly different arguments.

Let $\ell\neq p$ be a prime. The categories of equivariant mixed Tate motives $\operatorname{DTM}_{G}(k)$ can be regarded as graded versions of the categories of equivariant sheaves $\operatorname{D}_{G}(\operatorname{pt},\mathbb{Q}_{\ell})$ defined by Bernstein–Lunts [BL94]. Under the realisation functor $\operatorname{Real}_{\ell},$ see (3.1), the Tate motive $\mathbb{Q}(1)$ gets mapped to the Tate module

which can be identified with $\mathbb{Q}_{\ell}$ by choosing a compatible system of $\ell^{n}$-th roots of unity in $k=\overline{\mathbb{F}}_{p}.$ This induces a natural equivalence of functors

Hence, intuitively, the Tate twist $(1)$ can be regarded as a shift of grading and $\operatorname{Real}_{\ell}$ as a functor forgetting the grading. Restricted to mixed Tate motives, the functor $\operatorname{Real}_{\ell}$ becomes a degrading functor in the sense of [BGS96, Section 4.3].

We need a notion of local purity for motivic sheaves on a variety $X.$ That is, we want to consider motivic sheaves whose restriction to every point is pure or mixed Tate:

It is sometimes convenient to work with the following equivalent orbitwise definition.

Objects that are pointwise pure Tate have remarkable properties. They behave very similarly to pure Tate objects on a point, particulary if there are only finitely many $G$-orbits. They satisfy the following extension vanishing:

Restricted to pointwise mixed Tate objects $\operatorname{Real}_{\ell}$ is a degrading functor after passing to $\mathbb{Q}_{\ell}$-coefficients.

Pointwise pure Tate objects can be obtained from pushforwards along proper maps whose fibers have pure Tate motives.