Vanishing of cohomology on the fiber and base change

Question

Let $S$ be a Noetherian scheme and $\mathcal F$ a coherent sheaf on $\mathbb P^n\times S$, flat over $S$. Let $s_0\in S$ and suppose that $\mathcal F$ induces a coherent sheaf $\mathcal F_{s_0}$ on $\mathbb P^n_{s_0}=p_2^{-1}(s_0)$. If $H^{i+1}(\mathbb P^n, \mathcal F_{s_0})=0$, is it true that there is an open neighbourhood $U$ of $s_0$ such that for any $s\in U$ the natural map $R^i p_{2,*}(\mathcal F)\otimes k(s)\to H^i(\mathbb P^n_s, \mathcal F_s)$ is an isomorphism? Equivalently, is $R^i p_{2,*}(\mathcal F)$ locally free in $U$? It seems that Mumford uses it in his Lectures on curves on an algebraic surface, but I don't know why this should be true. I'm especially interested in the case $i=0$.

Answer 1

Here is the statement you are looking for.

Theorem. Let $f:X \to S$ be a proper morphism of Noetherian schemes and $\mathcal{F}$ be a coherent sheaf over $X$, flat over $S$. If $s_0\in S$ is such that $H^1(f^{-1}(s_0),\mathcal{F}_{s_0})=0$, then there exists an open neighborhood $U$ of $s_0$ such that $f_*\mathcal{F}|_U$ is a locally free $\mathcal{O}_U$-module.

This is an application of Grothendieck's results on cohomology and base change in EGA III. See, for instance, Theorem 12.11 in Hartshorne's Algebraic Geometry.

Assuming this "Cohomology and Base Change Theorem", the proof goes roughly as follows. The condition $H^1(f^{-1}(s_0),\mathcal{F}_{s_0})=0$ implies that $R^1f_*\mathcal{F} \otimes k(s_0) = 0$. Using that $R^1f_*\mathcal{F}$ is coherent and Nakayama's lemma, we deduce that there exists an open neighborhood $U$ of $s_0$ such that $R^1f_*\mathcal{F}$ vanishes over $U$. In particular, $R^1f_*\mathcal{F}$ is locally free over $U$. By the Cohomology and Base Change Theorem applied to $i=1$, this implies that the natural map $$ f_*\mathcal{F} \otimes k(s) \to H^0(f^{-1}(s),\mathcal{F}_{s}) $$ is an isomorphism for every $s\in U$. This last fact implies that $f_*\mathcal{F}$ is locally free over $U$ (by the same theorem, "applied to $i=0$")

Another nice reference for all of these is the following preprint https://arxiv.org/pdf/1312.7320.pdf.