The Stacks project

Theorem 69.22.5 (Theorem on formal functions). In Situation 69.22.1. 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. In fact, this follows immediately from Lemma 69.22.4. We spell out the details. Set $M = H^ p(X, \mathcal{F})$ and $M_ n = H^ p(X, \mathcal{F}/I^ n\mathcal{F})$. Denote $N_ n = \mathop{\mathrm{Im}}(M \to M_ n)$. By the description of the limit in Homology, Section 12.31 we have

\[ \mathop{\mathrm{lim}}\nolimits _ n M_ n = \{ (x_ n) \in \prod M_ n \mid \varphi _ i(x_ n) = x_{n - 1}, \ n = 2, 3, \ldots \} \]

Pick an element $x = (x_ n) \in \mathop{\mathrm{lim}}\nolimits _ n M_ n$. By Lemma 69.22.4 part (3) we have $x_ n \in N_ n$ for all $n$ since by definition $x_ n$ is the image of some $x_{n + m} \in M_{n + m}$ for all $m$. By Lemma 69.22.4 part (1) we see that there exists a factorization

\[ M \to N_ n \to M/I^{n - c_1}M \]

of the reduction map. Denote $y_ n \in M/I^{n - c_1}M$ the image of $x_ n$ for $n \geq c_1$. Since for $n' \geq n$ the composition $M \to M_{n'} \to M_ n$ is the given map $M \to M_ n$ we see that $y_{n'}$ maps to $y_ n$ under the canonical map $M/I^{n' - c_1}M \to M/I^{n - c_1}M$. Hence $y = (y_{n + c_1})$ defines an element of $\mathop{\mathrm{lim}}\nolimits _ n M/I^ nM$. We omit the verification that $y$ maps to $x$ under the map

\[ M^\wedge = \mathop{\mathrm{lim}}\nolimits _ n M/I^ nM \longrightarrow \mathop{\mathrm{lim}}\nolimits _ n M_ n \]

of the lemma. We also omit the verification on topologies. $\square$

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