The Stacks project

Theorem 30.20.5 (Theorem on formal functions). Let $A$ be a Noetherian ring. Let $I \subset A$ be an ideal. Let $f : X \to \mathop{\mathrm{Spec}}(A)$ be a proper morphism. Let $\mathcal{F}$ be a coherent sheaf on $X$. Fix $p \geq 0$. The system of maps

\[ H^ p(X, \mathcal{F})/I^ nH^ p(X, \mathcal{F}) \longrightarrow H^ p(X, \mathcal{F}/I^ n\mathcal{F}) \]

define an isomorphism of limits

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

where the left hand side is the completion of the $A$-module $H^ p(X, \mathcal{F})$ with respect to the ideal $I$, see Algebra, Section 10.96. Moreover, this is in fact a homeomorphism for the limit topologies.

Proof. This follows from Lemma 30.20.4 as follows. Set $M = H^ p(X, \mathcal{F})$, $M_ n = H^ p(X, \mathcal{F}/I^ n\mathcal{F})$, and denote $N_ n = \mathop{\mathrm{Im}}(M \to M_ n)$. By Lemma 30.20.4 parts (2) and (3) we see that $(M_ n)$ is a Mittag-Leffler system with $N_ n \subset M_ n$ equal to the image of $M_ k$ for all $k \gg n$. It follows that $\mathop{\mathrm{lim}}\nolimits M_ n = \mathop{\mathrm{lim}}\nolimits N_ n$ as topological modules (with limit topologies). On the other hand, the $N_ n$ form an inverse system of quotients of the module $M$ and hence $\mathop{\mathrm{lim}}\nolimits N_ n$ is the completion of $M$ with respect to the topology given by the kernels $K_ n = \mathop{\mathrm{Ker}}(M \to N_ n)$. By Lemma 30.20.4 part (1) we have $K_ n \subset I^{n - c}M$ and since $N_ n \subset M_ n$ is annihilated by $I^ n$ we have $I^ n M \subset K_ n$. Thus the topology defined using the submodules $K_ n$ as a fundamental system of open neighbourhoods of $0$ is the same as the $I$-adic topology and we find that the induced map $M^\wedge = \mathop{\mathrm{lim}}\nolimits M/I^ nM \to \mathop{\mathrm{lim}}\nolimits N_ n = \mathop{\mathrm{lim}}\nolimits M_ n$ is an isomorphism of topological modules1. $\square$

[1] To be sure, the limit topology on $M^\wedge $ is the same as its $I$-adic topology as follows from Algebra, Lemma 10.96.3. See More on Algebra, Section 15.36.

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