Lemma 30.20.9. Let $f : X \to Y$ be a morphism of schemes. Let $y \in Y$. Assume

$Y$ locally Noetherian,

$f$ is proper, and

$\dim (X_ y) = d$.

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

**Proof.**
The fibre $X_ y$ is of finite type over $\mathop{\mathrm{Spec}}(\kappa (y))$. Hence $X_ y$ is a Noetherian scheme by Morphisms, Lemma 29.15.6. Hence the underlying topological space of $X_ y$ is Noetherian, see Properties, Lemma 28.5.5. Moreover, the underlying topological space of each infinitesimal neighbourhood $X_ n$ is the same as that of $X_ y$. Hence $H^ p(X_ n, \mathcal{F}_ n) = 0$ for all $p > d$ by Cohomology, Proposition 20.20.7. Hence we see that $(R^ pf_*\mathcal{F})_ y^\wedge = 0$ by Lemma 30.20.7 for $p > d$. 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 $(R^ pf_*\mathcal{F})_ y = 0$.
$\square$

## Comments (4)

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