Question

Let $F=(F_n)_n$ be an $\ell$-adic sheaf on $X_{et}$, for a variety $X$ over an algebraically closed field $k$ of characteristic not equal to $\ell$. Does the presheaf sending $U$ to $H^i(U,F):=\lim_n H^i(U,F_n)$ sheafify to zero?

Answer 1

CORRECTED ANSWER: I believe that the answer is no, at least in some contexts.

For example, suppose that $X = $Spec $k$, with $k$ a field, and $F = {\mathbb Z}_{\ell}(1)$. Then $U = $Spec $l$ for some finite separable extension $l$ of $k$, and $H^1(U,F) = \ell$-adic completion of $l^{\times}$, which I will denote by $\widehat{l^{\times}}$.

Thus the stalk of the presheaf $U \mapsto H^1(U,F)$ (and hence of the associated sheaf) at the (unique) geometric point of $X$ is the direct limit over $l$ of $\widehat{l^{\times}}.$

This direct limit need not vanish. For example, if $k$ is finite, then so is $l$, and $\widehat{l^{\times}}$ is just the $\ell$-Sylow subgroup of $l$. Thus the stalk in this case is just $\bar{k}^{\times}[\ell^{\infty}],$ the group of $\ell$-power roots of unity in $\bar{k}$.

This fits with a certain intuition, namely that one has to go to smaller and small etale neighbourhoods to trivialize $F_n$ as $n$ increases, and hence one can't kill of cohomology classes in $H^i(U,F)$ just by restricting to some $V$.

I think that the answer is yes. Here is a proof (hopefully blunder-free):

It is true for the presheaf $U \mapsto H^i(U,F_1).$ In other words, if we fix $U$, then for each element $h \in H^i(U,F_1)$ and each geometric point $x$ of $U$, there is an etale n.h. $V$ of $x$ such that $h_{| V} = 0.$ Since $H^i(U,F_1)$ is finite dimensional, there is a $V$ that works for the whole of $H^i(U,F_1)$ at once.

I claim that then $H^i(U,F_n)$ restricts to $0$ on $V$ as well.

To see this, consider the exact sequence $0 \to F_n \to F_{n+1} \to F_1 \to 0.$ Applying $H^i(U,\text{--})$ to this yields a middle exact sequence $H^i(U,F_n) \to H^i(U,F_{n+1}) \to H^i(U,F_1).$ Applying $H^i(V,\text{--})$ yields a middle exact sequence $H^i(V,F_n)\to H^i(V,F_{n+1}) \to H^i(V,F_1).$ Restriction gives a map from the first of these sequences to the second. It is zero on the two outer terms, by induction together with the case $n = 1$ proved above, and so is zero on the inner term.

This shows that restricting from $U$ to $V$ kills $H^i(U,F_n)$ for all $n$, and hence $H^i(U,F)$, as required.

EDIT: As was noted in the comment below, this proof assumes that $F$ is ${\mathbb Z}_{\ell}$ -flat. Let me sketch an argument that hopefully handles the general case:

Put $F$ in a short exact sequence $0 \to F_{tors} \to F \to F_{fl} \to 0.$ The same kind of argument as above reduces us to checking $F_{fl}$ and $F_{tors}$ separately. The above proof handles the case of $F_{fl}$, while $F_{tors} = F_{tors,n}$ for some large enough $n$, and so the projective limit collapses in this case and there is nothing to check.

(Note: I am assuming some basic kind of finiteness assumption on $F$ here, so that the above makes sense. Constructibility should be enough.)