Lemma 30.20.8 (02OE)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 30: Cohomology of Schemes
Section 30.20: The theorem on formal functions
Lemma 30.20.8 (cite)

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Lemma 30.20.8. Let $f : X \to Y$ be a morphism of schemes. Let $y \in Y$ . Assume

$Y$ locally Noetherian,
$f$ is proper, and
$f^{-1}(\{ y\} )$ is finite.

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

Proof. The fibre $X_ y$ is finite, and by Morphisms, Lemma 29.20.7 it is a finite discrete space. Moreover, the underlying topological space of each infinitesimal neighbourhood $X_ n$ is the same. Hence each of the schemes $X_ n$ is affine according to Schemes, Lemma 26.11.8. Hence it follows that $H^ p(X_ n, \mathcal{F}_ n) = 0$ for all $p > 0$ . Hence we see that $(R^ pf_*\mathcal{F})_ y^\wedge = 0$ by Lemma 30.20.7. Note that $R^ pf_*\mathcal{F}$ is coherent by Proposition 30.19.1 and hence $R^ pf_*\mathcal{F}_ y$ is a finite $\mathcal{O}_{Y, y}$ -module. By Nakayama's lemma (Algebra, Lemma 10.20.1) if the completion of a finite module over a local ring is zero, then the module is zero. Whence $(R^ pf_*\mathcal{F})_ y = 0$ . $\square$

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