# The Stacks Project

## Tag 02OC

Theorem 29.20.5 (Theorem on formal functions). Let $A$ be a Noetherian ring. Let $I \subset A$ be an ideal. Let $f : X \to \mathop{\rm Spec}(A)$ 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 $$H^p(X, \mathcal{F})^\wedge \longrightarrow \mathop{\rm lim}\nolimits_n H^p(X, \mathcal{F}/I^n\mathcal{F})$$ 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 10.95. Moreover, this is in fact a homeomorphism for the limit topologies.

Proof. In fact, this follows immediately from Lemma 29.20.4. 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 $N_n = \mathop{\rm Im}(M \to M_n)$. By the description of the limit in Homology, Section 12.28 we have $$\mathop{\rm lim}\nolimits_n M_n = \{(x_n) \in \prod M_n \mid \varphi_i(x_n) = x_{n - 1}, ~n = 2, 3, \ldots\}$$ Pick an element $x = (x_n) \in \mathop{\rm lim}\nolimits_n M_n$. By Lemma 29.20.4 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 29.20.4 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 $\mathop{\rm lim}\nolimits_n M/I^nM$. We omit the verification that $y$ maps to $x$ under the map $$M^\wedge = \mathop{\rm lim}\nolimits_n M/I^nM \longrightarrow \mathop{\rm lim}\nolimits_n M_n$$ of the lemma. We also omit the verification on topologies. $\square$

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\begin{theorem}[Theorem on formal functions]
\label{theorem-formal-functions}
Let $A$ be a Noetherian ring.
Let $I \subset A$ be an ideal.
Let $f : X \to \Spec(A)$ 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
$$H^p(X, \mathcal{F})^\wedge \longrightarrow \lim_n H^p(X, \mathcal{F}/I^n\mathcal{F})$$
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 \ref{algebra-section-completion}.
Moreover, this is in fact a homeomorphism for the limit topologies.
\end{theorem}

\begin{proof}
In fact, this follows immediately from
Lemma \ref{lemma-ML-cohomology-powers-ideal}. 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 $N_n = \Im(M \to M_n)$.
By the description of the limit in Homology, Section
\ref{homology-section-inverse-systems} we have
$$\lim_n M_n = \{(x_n) \in \prod M_n \mid \varphi_i(x_n) = x_{n - 1}, \ n = 2, 3, \ldots\}$$
Pick an element $x = (x_n) \in \lim_n M_n$.
By Lemma \ref{lemma-ML-cohomology-powers-ideal} 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 \ref{lemma-ML-cohomology-powers-ideal} 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 $\lim_n M/I^nM$.
We omit the verification that $y$ maps to $x$ under the
map
$$M^\wedge = \lim_n M/I^nM \longrightarrow \lim_n M_n$$
of the lemma. We also omit the verification on topologies.
\end{proof}

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