Lemma 69.22.6 (08B0)—The Stacks project

Processing math: 100%

Table of contents
Part 4: Algebraic Spaces
Chapter 69: Cohomology of Algebraic Spaces
Section 69.22: The theorem on formal functions
Lemma 69.22.6 (cite)

numberstags

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. $\square$

Comments (0)

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$ ). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Comments (0)

Post a comment