# Testing the vanishing of cohomology fiberwise for a proper morphism from an Artin stack

Let $S$ be a Noetherian scheme, let $f\colon\mathscr{X} \rightarrow S$ be a proper morphism with $\mathscr{X}$ an algebraic stack, and let $\mathscr{F}$ be an $S$-flat coherent sheaf on $\mathscr{X}$. If for some point $s \in S$ and some $n \ge 0$ we have $H^n(\mathscr{X}_s, \mathscr{F}_s) = 0$, does it then follow that $R^nf_*(\mathscr{F}) = 0$ in a neighborhood of $s$?

For schemes, the claim is part of the cohomology and base change formalism. I have looked at some references on similar material for algebraic stacks (Grothendieck existence, semicontinuity of cohomology...), but could not find (or overlooked) the above statement. A pointer to the literature would be very welcome.

• Coherence of higher direct images is proved in Olsson's paper "On proper coverings of Artin stacks", from which the theorem on formal functions is deduced similarly to the scheme case (so completion of the $s$-stalk is the inverse limit of $n$th cohomologoes on infinitesimal fibers, all of which vanish when $n$th cohomology on the fiber vanishes due to the $S$-flatness of $\mathscr{F}$). See Theorem 11.1(ii) in Olsson's paper "Sheaves on artin stacks" for the theorem on formal functions for a proper Artin stack over an adic noetherian ring (doesn't require the earlier parts of that paper). – nfdc23 Dec 28 '15 at 16:47
• In addition to the references given by nfdc23, please also confer Theorem A of the following: Jack Hall, "Cohomology and Base Change for Algebraic Stacks", Math. Zeitschrift, vol 278, pp. 401-429, (2014) maths-people.anu.edu.au/~hallj/data/publications/Coho_BC.pdf – Jason Starr Dec 28 '15 at 17:41
• @nfdc23: I was trying to argue it like this but I did not see why the vanishing of the cohomology of the fiber suffices for the vanishing of the cohomologies of the infinitesimal fibers. Could you explain this (i.e. could you explain the "due to the $S$-flatness of $\mathscr{F}$" part of your comment; in particular, I overlook the way in which flatness is relevant for this)? – O-Ren Ishii Dec 28 '15 at 20:28
• @JasonStarr: That looks like a very useful reference for these matters (and, I suspect, answers my question). I will look it over carefully. – O-Ren Ishii Dec 28 '15 at 20:30
• @O-RenIshii: By compatibility of quasi-coherent cohomology with flat base change, WLOG $S={\rm{Spec}}(A)$ for complete local noetherian $(A, \mathfrak{m})$. Let $X_m$ be the $m$th infinitesimal special fiber, $\mathscr{F}_m=\mathscr{F}|_{X_m}$, and $k=A/\mathfrak{m}$. It suffices to show ${\rm{H}}^n(X_m, \mathscr{F}_m)=0$ for all $m > 0$ given the case $m=0$. By $A$-flatness, $\mathscr{F}_{m+1} \twoheadrightarrow \mathscr{F}_m$ has kernel $\mathscr{F}_0\otimes_k (\mathfrak{m}^{m+1}/\mathfrak{m}^{m+2})=\mathscr{F}_0^{\oplus d_m}$ where $d_m:=\dim_k \mathfrak{m}^{m+1}/\mathfrak{m}^{m+2}$. – nfdc23 Dec 28 '15 at 22:06