Let $X$ be a complete nonsingular curve and $S$ a scheme over $k$ algebraically closed, and $\cal{F}$ a coherent sheaf on $X \times S$, generated by finitely many global sections and flat over $S$ (via the projection). So my question is the following: is the locus of points $s\in S$ such that $\cal{F}_s$ is locally free on $X$ open? And if so, is there an easy way to see this?
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3$\begingroup$ Welcome new contributor. That is false without a hypothesis that the curve is proper. When the curve is proper, you can prove that with “Fitting ideals”. $\endgroup$– Jason StarrCommented Mar 13, 2022 at 1:38
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$\begingroup$ Thank you! Yes, the curves I'm working with are in fact proper (I edited the question). Thank you so much! $\endgroup$– nolatosCommented Mar 13, 2022 at 1:42
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$\begingroup$ @JasonStarr Is it important that $X$ is a curve? And maybe it could be generalized to a relatively proper map $X\to S$ instead of a trivial family? $\endgroup$– Z. MCommented Mar 13, 2022 at 11:42
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$\begingroup$ @Z.M You are correct. The same result holds for a proper, flat morphism. $\endgroup$– Jason StarrCommented Mar 13, 2022 at 12:24
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To elaborate on Jason's comment: consider a morphism of schemes $f:\mathrm{X}\rightarrow\mathrm{S}$, and an $f$-flat coherent sheaf $\mathscr{F}$ on $\mathrm{X}$. Then
(1) the singular locus $\mathrm{Sing}(\mathscr{F})$ of $\mathscr{F}$ (given by the points $x\in\mathrm{X}$ such that $\mathscr{F}$ is not locally free at $x$) is closed - if $r$ is the rank of $\mathscr{F}$, it is cut out by the $r$-th Fitting ideal sheaf of $\mathscr{F}$;
(2) $\mathscr{F}$ is locally free at $x\in\mathrm{X}$ if and only if $\mathscr{F}_{f(x)}$ is locally free at $x$, where $\mathscr{F}_{f(x)}$ denotes the restriction of $\mathscr{F}$ to $f^{-1}(f(x))$ - this follows from flatness of $\mathscr{F}$ and the Nakayama lemma.
By (1) and (2), $f(\mathrm{Sing}(\mathscr{F}))$ consists of all $s\in\mathrm{S}$ such that $\mathscr{F}_{s}$ is not locally free. If $f$ is proper, then of course the latter image is closed in $\mathrm{S}$.
Lemma 2.1.8 in The Geometry of Moduli Spaces of Sheaves by Huybrechts & Lehn is a good reference.
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$\begingroup$ Thank you so much, that was very helpful! $\endgroup$– nolatosCommented Mar 14, 2022 at 21:19