The Stacks project

Lemma 69.22.6. Let $A$ be a ring. Let $I \subset A$ be an ideal. Assume $A$ is Noetherian and complete with respect to $I$. Let $f : X \to \mathop{\mathrm{Spec}}(A)$ be a proper morphism of algebraic spaces. Let $\mathcal{F}$ be a coherent sheaf on $X$. Then

\[ H^ p(X, \mathcal{F}) = \mathop{\mathrm{lim}}\nolimits _ n H^ p(X, \mathcal{F}/I^ n\mathcal{F}) \]

for all $p \geq 0$.

Proof. This is a reformulation of the theorem on formal functions (Theorem 69.22.5) in the case of a complete Noetherian base ring. Namely, in this case the $A$-module $H^ p(X, \mathcal{F})$ is finite (Lemma 69.20.3) hence $I$-adically complete (Algebra, Lemma 10.97.1) and we see that completion on the left hand side is not necessary. $\square$

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