Companions on Artin stacks

Abstract

Deligne’s conjecture that $\ell$-adic sheaves on normal schemes over a finite field admit $\ell '$-companions was proved by L. Lafforgue in the case of curves and by Drinfeld in the case of smooth schemes. In this paper, we extend Drinfeld’s theorem to smooth Artin stacks and deduce Deligne’s conjecture for coarse moduli spaces of smooth Artin stacks. We also extend related theorems on Frobenius eigenvalues and traces to Artin stacks.

Appendix: Structure of pure perverse sheaves

The goal of this appendix is to prove the geometric semisimplicity of pure perverse sheaves (Theorem 6.1).

Let $\iota :\overline{\mathbb {Q}_\ell }\rightarrow \mathbb {C}$ be an embedding. Let X be a stack. Let $w\in \mathbb {R}$ and let $K\in D(X,\overline{\mathbb {Q}_\ell })$. We say that K has $\iota$-weights$\le w$ if the ith cohomology sheaf $\mathcal {H}^i K$ of K has punctual $\iota$-weights $\le w+i$ for all i, and K has $\iota$-weights $\ge w$ if DK has $\iota$-weights $\le -w$. We say that K is $\iota$-pure of weight w if it has $\iota$-weights $\le w$ and $\ge w$.

Theorem 6.1

Let X be a stack and let $\mathcal {P}$ be an $\iota$-pure perverse Weil $\overline{\mathbb {Q}_\ell }$-sheaf on X. Then the pullback of $\mathcal {P}$ to $X\otimes _{\mathbb {F}_q} \overline{\mathbb {F}_q}$ is semisimple.

The case of affine stabilizers is a theorem of Sun [32, Theorem 3.11], extending the case of schemes [3, Théorème 5.3.8]. Note that the decomposition theorem of pure complexes [32, Theorem 3.12] does not extend to general stacks, as shown in [32, Section 1].

As in the case of schemes [3, Proposition 5.3.9], Theorem 6.1 has the following consequence on the structure of pure perverse sheaves. As before we let $\mathcal {E}_n$ denote the $\overline{\mathbb {Q}_\ell }$-sheaf on $\mathrm {Spec}(\mathbb {F}_q)$ of stalk $\overline{\mathbb {Q}_\ell }^n$ on which $\mathrm {Frob}_q$ acts unipotently with one Jordan block.

Corollary 6.2

Let X be a stack. The indecomposable $\iota$-pure perverse Weil $\overline{\mathbb {Q}_\ell }$-sheaves on X are of the form $\mathcal {P}\otimes \pi _X^*\mathcal {E}_n$ with $\mathcal {P}$ simple, where $\pi _X:X\rightarrow \mathrm {Spec}(\mathbb {F}_q)$. Moreover, for every simple perverse Weil $\overline{\mathbb {Q}_\ell }$-sheaf $\mathcal {P}$, there exists a unique $m\ge 1$ such that $\mathcal {P}\simeq p_* \mathcal {Q}$, where $p:X\otimes _{\mathbb {F}_q} \mathbb {F}_{q^m}\rightarrow X$ is the projection, $\mathcal {Q}$ is geometrically simple (i.e. the pullback of $\mathcal {Q}$ to $X\otimes _{\mathbb {F}_{q^m}}\overline{\mathbb {F}_q}$ is simple) and not isomorphic to any of its conjugates under $\mathrm {Gal}(\mathbb {F}_{q^m}/\mathbb {F}_q)$.

The first assertion of the corollary still holds with $\overline{\mathbb {Q}_\ell }$ replaced by a finite (or algebraic) extension of $\mathbb {Q}_\ell$.

The key to the proof of Theorem 6.1 is a weight estimate.

Proposition 6.3

Let X be a stack and let $\pi :X\rightarrow \mathrm {Spec}(\mathbb {F}_q)$ be the projection. Let $K\in D^{\ge 0}(X,\overline{\mathbb {Q}_\ell })$ be a complex of $\iota$-weights $\ge w$ and vanishing i-th cohomology for $i<0$. Then for all $i\ge 0$, $R^i \pi _* K$ has $\iota$-weights $\ge w+\lceil \frac{i}{2}\rceil$. Moreover $H^i(X\otimes _{\mathbb {F}_q} \overline{\mathbb {F}_q},K)^{\mathrm {Gal}(\overline{\mathbb {F}_q}/\mathbb {F}_q)}=0$ for $i>0$ if $w\ge 0$, and $R\Gamma (X,K)=0$ if $w>0$.

The estimate is optimal. Indeed, for $X=BA$, where A is an Abelian variety, and a of weight 1, $R^i\pi _*(\overline{\mathbb {Q}_\ell }\oplus \overline{\mathbb {Q}_\ell }^{(a)}[-1])$ is pure of weight $\lceil \frac{i}{2}\rceil$. Unlike the case of schemes or stacks with affine stabilizers, $R^i\pi _* K$ is not of $\iota$-weights $\ge w+i$ in general.

Proof

The second assertion follows from the first one and the short exact sequence

$$\begin{aligned} 0\rightarrow H^{i-1}(X\otimes _{\mathbb {F}_q} \overline{\mathbb {F}_q},K)_{\mathrm {Gal}(\overline{\mathbb {F}_q}/\mathbb {F}_q)} \rightarrow H^i(X,K)\rightarrow H^{i}(X\otimes _{\mathbb {F}_q} \overline{\mathbb {F}_q},K)^{\mathrm {Gal}(\overline{\mathbb {F}_q}/\mathbb {F}_q)}\rightarrow 0. \end{aligned}$$

Note that for any stratification of X into locally closed substacks $(j_\alpha :X_\alpha \rightarrow X)_\alpha$ such that the closure of every stratum is a union of strata, K is a successive extension of $Rj_{\alpha *}Rj_\alpha ^! K$, with $Rj_\alpha ^! K\in D^{\ge 0}$ of $\iota$-weights $\ge w$. Thus we may assume that X is smooth of dimension d and K has lisse cohomology sheaves. We may further assume $K=\mathcal {F}[-n]$, with $\mathcal {F}$ lisse of $\iota$-weights $\ge w+n$ and $n\ge 0$. Then the $\iota$-weights of are at most

$$\begin{aligned} d+\frac{2d+n-i}{2}-(w+n)-2d=-w-\frac{i+n}{2}\le -w-\frac{i}{2} \end{aligned}$$

by [31, Theorem 1.4]. We conclude by the fact that the $\iota$-weights are in $w+\mathbb {Z}$.

Corollary 6.4

Let X be a stack and let $\mathcal {P}$ and $\mathcal {Q}$ be perverse $\overline{\mathbb {Q}_\ell }$-sheaves on X, with $\mathcal {P}$ of $\iota$-weights $\le w$, and $\mathcal {Q}$ of $\iota$-weights $\ge w$. Then for $i>0$, $\mathrm {Hom}^i(\mathcal {P}_{\overline{\mathbb {F}_q}},\mathcal {Q}_{\overline{\mathbb {F}_q}})^{\mathrm {Gal}(\overline{\mathbb {F}_q}/\mathbb {F}_q)}=0$, so that the canonical map $\mathrm {Hom}^i(\mathcal {P},\mathcal {Q})\rightarrow \mathrm {Hom}^i(\mathcal {P}_{\overline{\mathbb {F}_q}},\mathcal {Q}_{\overline{\mathbb {F}_q}})$ is zero. Moreover, if $\mathcal {Q}$ has $\iota$-weights $>w$, then $R\mathrm {Hom}(\mathcal {P},\mathcal {Q})=0$.

For perverse Weil $\overline{\mathbb {Q}_\ell }$-sheaves and $i=1$, the first assertion holds with $\mathrm {Hom}^1$ replaced by $\mathrm {Ext}^1$ and $\mathrm {Gal}(\overline{\mathbb {F}_q}/\mathbb {F}_q)$ replaced by $W(\overline{\mathbb {F}_q}/\mathbb {F}_q)$.

Proof

We apply the proposition to $K=R\mathcal {H} om (\mathcal {P},\mathcal {Q}) \in D^{\ge 0}(X,\overline{\mathbb {Q}_\ell })$, which has $\iota$-weights $\ge 0$. If $\mathcal {Q}$ has $\iota$-weights $>w$, then K has $\iota$-weights $>0$. $\square$

The proof of Theorem 6.1 is then identical to the proof of [3, Théorème 5.3.8], with [3, Proposition 5.1.15] replaced by Corollary 6.4.