The Stacks project

Lemma 69.22.8. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\overline{y}$ be a geometric point of $Y$. Assume

  1. $Y$ locally Noetherian,

  2. $f$ is proper, and

  3. $X_{\overline{y}}$ has discrete underlying topological space.

Then for any coherent sheaf $\mathcal{F}$ on $X$ we have $(R^ pf_*\mathcal{F})_{\overline{y}} = 0$ for all $p > 0$.

Proof. Let $\kappa (\overline{y})$ be the residue field of the local ring of $\mathcal{O}_{Y, \overline{y}}$. As in Lemma 69.22.7 we set $X_{\overline{y}} = X_1 = \mathop{\mathrm{Spec}}(\kappa (\overline{y})) \times _ Y X$. By Morphisms of Spaces, Lemma 67.34.8 the morphism $f : X \to Y$ is quasi-finite at each of the points of the fibre of $X \to Y$ over $\overline{y}$. It follows that $X_{\overline{y}} \to \overline{y}$ is separated and quasi-finite. Hence $X_{\overline{y}}$ is a scheme by Morphisms of Spaces, Proposition 67.50.2. Since it is quasi-compact its underlying topological space is a finite discrete space. Then it is an affine scheme by Schemes, Lemma 26.11.8. By Lemma 69.17.3 it follows that the algebraic spaces $X_ n$ are affine schemes as well. Moreover, the underlying topological of each $X_ n$ is the same as that of $X_1$. Hence it follows that $H^ p(X_ n, \mathcal{F}_ n) = 0$ for all $p > 0$. Hence we see that $(R^ pf_*\mathcal{F})_{\overline{y}}^\wedge = 0$ by Lemma 69.22.7. Note that $R^ pf_*\mathcal{F}$ is coherent by Lemma 69.20.2 and hence $R^ pf_*\mathcal{F}_{\overline{y}}$ is a finite $\mathcal{O}_{Y, \overline{y}}$-module. By Algebra, Lemma 10.97.1 this implies that $(R^ pf_*\mathcal{F})_{\overline{y}} = 0$. $\square$

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