TITLE: When is the etale cohomology of $\mathrm{Sym}^n(X)$ isomorphic to the $\Sigma_n$-invariants in the étale cohomology of $X^n$?
QUESTION [15 upvotes]: Suppose $X$ is a smooth projective variety defined over an arbitrary algebraically closed field $k$, and consider the action of $\Sigma_n$ on the $n$-fold product $X^n$. Is it true that $H_{\acute{e}t}^i(\mathrm{Sym}^n(X),\mathbb{Q}_\ell)\cong H_{\acute{e}t}^i(X^n,\mathbb{Q}_\ell)^{\Sigma_n}$? In particular, what happens in the case where $\operatorname{char} k=p>0$?
In Grothendieck's Toh\^oku paper Sec. 5.2, he determines sufficient conditions to ensure that, for a topological space $X$ with a finite group $G$ acting on it (not necessarily faithfully), $H^i(X/G,\mathcal{A})\cong H^i(X,\mathcal{A})^G$ for a sheaf $\mathcal{A}$ (Cor. to Prop. 5.2.3). In characteristic zero, comparison theorems allow me to appeal to this result. In positive characteristic, if the variety lifts to characteristic zero, then I can make the same argument, but it seems like there ought to be a direct proof of this fact.
I am particularly interested in when $X$ is a surface, but would be happy to know of any general results (with references) similar to Grothendieck's result above.

REPLY [3 votes]: The following is a particular case of (SGA 4.3, XVII Th. 5.5.21) : Let $X$ be a quasi-projective scheme over an algebraically closed field $k$. Then for any $n \geq 0$ and any $r \geq 1$ we have
$$
R \Gamma_c(\mathrm{Sym}_{k}^n(X), \mathbb{Z}/r\mathbb{Z}) = L \Gamma^n ( R\Gamma_c(X, \mathbb{Z}/r\mathbb{Z})).
$$
The functor $L \Gamma^n$ is the left derived functor of the non-additive functor $\Gamma^n$, which coincides on flat modules with the ``symmetric tensor'' functor. The $R \Gamma_c$ denotes higher direct image with compact supports.
(SGA 4.3, XVII Th. 5.5.21) gives a more general statement, in a relative situation, with more general coefficients.