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

1. $Y$ locally Noetherian,

2. $f$ is proper, and

3. $f^{-1}(\{ y\} )$ 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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