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 $k=\overline{\mathbb{F}}_{p}.$