Subsection 3.6. (0N02) · ScalingStacks

3.6. Pointwise Tate

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:

0MYF

Definition 3.16. Let $?\in\{*,!\}.$ An object $M\in\operatorname{DM}_{G}(X)$ is called $?$ -pointwise mixed Tate or $?$ -pointwise pure Tate, respectively, if for each point $i_{x}:\operatorname{pt}\to X$

\displaystyle i_{x}^{?}\operatorname{For}M\in\operatorname{DTM}(k,\mathbb{Q})\text{ or }i_{x}^{?}\operatorname{For}M\in\operatorname{DTM}(k,\mathbb{Q})^{w=0}\text{, respectively.}

Here the functor $i_{x}^{?}\operatorname{For}$ is the composition

\operatorname{DM}_{G}(X,\mathbb{Q})\stackrel{{\scriptstyle\operatorname{For}}}{{\to}}\operatorname{DM}(X,\mathbb{Q})\stackrel{{\scriptstyle i_{x}^{?}}}{{\to}}\operatorname{DM}(k,\mathbb{Q}).

The object $M$ is called pointwise mixed (pure) Tate if it is $*$ - and $!$ -pointwise mixed (pure) Tate.

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

0MYG

Proposition 3.17. Let $?\in\{*,!\}$ and $M\in\operatorname{DM}_{G}(X).$ Then $M$ is $?$ -pointwise mixed or pure Tate if and only if for each orbit $\mathcal{O}\hookrightarrow X$

\displaystyle(i,s)^{*}j^{?}M\in\operatorname{DTM}_{H}(k,\mathbb{Q})\text{ or }(i,s)^{*}j^{?}M\in\operatorname{DTM}_{H}(k,\mathbb{Q})^{w=0}\text{, respectively.}

Here $j:G/H\cong\mathcal{O}\hookrightarrow X$ and the functor $(i,s)^{*}j^{?}$ is the composition

\operatorname{DM}_{G}(X,\mathbb{Q})\stackrel{{\scriptstyle j^{?}}}{{\to}}\operatorname{DM}_{G}(G/H,\mathbb{Q})\stackrel{{\scriptstyle(i,s)^{*}}}{{\to}}\operatorname{DM}_{H}(\operatorname{pt},\mathbb{Q})

of pullback to the orbit and the induction equivalence (3.6).

0MYH

Remark 3.18. In [SVW18] this equivalent orbitwise definition is used.

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:

0MYI

Proposition 3.19. Assume that the $G$ -action on $X$ has finitely many orbits. Let $M,N\in\operatorname{DM}_{G}(X)$ be $*$ - and $!$ -pointwise pure Tate. Then $\operatorname{Hom}_{\operatorname{DM}_{G}(X)}(M,N[n])=0$ for all $n\neq 0.$

0MYJ

Proof. The statement can be shown by an induction on the number of orbits. Denote by $j:G/H\cong\mathcal{O}\hookrightarrow X$ and $i:Z=X\backslash\mathcal{O}\to X$ the inclusion of an open orbit $\mathcal{O}$ and its closed complement $Z.$ Then the localisation triangle induces an exact sequence

Since $i^{*}M$ and $i^{!}N$ are $*$ - and $!$ -pointwise pure Tate, respectively, the first term of the sequence vanishes by induction. The last term vanishes since by assumption $j^{*}M$ and $j^{!}N$ correspond to objects in $\operatorname{DTM}_{H}(k,\mathbb{Q})^{w=0}$ via the induction equivalence (3.6) and thus have no non-trivial extension by Proposition 3.12. See [SVW18, Corollary II.4.19] for a similar proof. ∎

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

0MYK

Proposition 3.20. Assume that the $G$ -action on $X$ has finitely many orbits. Let $\ell\neq p$ be a prime and $M,N\in\operatorname{DM}_{G}(X).$ Then the natural isomorphisms $\operatorname{Real}_{\ell}\circ(1)\to\operatorname{Real}_{\ell},$ see (3.9), induces isomorphisms

\bigoplus_{n\in\mathbb{Z}}\operatorname{Hom}_{\operatorname{DM}_{G}(k,\mathbb{Q}_{\ell})}(M,N(n))\stackrel{{\scriptstyle\sim}}{{\to}}\operatorname{Hom}_{\operatorname{D}_{G}(k,\mathbb{Q}_{\ell})}(\operatorname{Real}_{\ell}(M),\operatorname{Real}_{\ell}(N))

if $M,N$ are $*$ - and $!$ -pointwise mixed Tate, respectively, and

\operatorname{Hom}_{\operatorname{DM}_{G}(X,\mathbb{Q}_{\ell})}(M,N)\stackrel{{\scriptstyle\sim}}{{\to}}\operatorname{Hom}_{\operatorname{D}_{G}(X,\mathbb{Q}_{\ell})}(\operatorname{Real}_{\ell}(M),\operatorname{Real}_{\ell}(N))

if $M,N$ are $*$ - and $!$ -pointwise pure Tate, respectively.

0MYL

Proof. As in the proof of Proposition 3.19 the statement can be reduced to the case of a point where it is the same as Proposition 3.15. ∎

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

0MYM

Proposition 3.21. Let $\mu:M\to N$ be a $G$ -equivariant proper map. Assume that $M$ is smooth and that the motives of the fibers of $\mu$ are pure Tate,

\operatorname{M}(\mu^{-1}(\{x\})\in\operatorname{DTM}(k,\mathbb{Q})^{w=0}.

Then the object $\mu_{!}(\mathbb{Q}_{M})\in\operatorname{DM}_{G}(N,\mathbb{Q})$ is pointwise pure Tate.

0MYN

Proof. We first show that $\mu_{!}(\mathbb{Q}_{M})$ is $*$ -pointwise pure Tate. Let $i_{x}:\operatorname{pt}\to N$ be the inclusion of a point $x\in X.$ We have to show that

i_{x}^{*}\operatorname{For}\mu_{!}(\mathbb{Q}_{M})\in\operatorname{DTM}(k,\mathbb{Q})^{w=0}.

By applying base change with respect to the Cartesian diagram

and the fact that $\operatorname{For}$ commutes with the six operations, we have

i_{x}^{*}\operatorname{For}\mu_{!}(\mathbb{Q}_{M})=\mu^{\prime}_{!}l^{*}\mathbb{Q}_{M}=\mu^{\prime}_{!}\mathbb{Q}_{\mu^{-1}(x)}\in\operatorname{DM}(k,\mathbb{Q}).

Now $\mu^{\prime}_{!}\mathbb{Q}_{\mu^{-1}(x)}$ is pure Tate since it is Verdier dual to the motive $\operatorname{M}^{c}(\mu^{-1}(\{x\})=\operatorname{M}(\mu^{-1}(\{x\})$ and Verdier duality preserves pure Tate motives.

That $\mu_{!}(\mathbb{Q}_{M})=\mu_{*}(\mathbb{Q}_{M})$ is $!$ -pointwise pure Tate follows by using Verdier dual arguments. ∎

Jens Niklas Eberhardt, Catharina Stroppel

Original source: arXiv:2109.00305v2