Lemma 115.8.3. Let $I$ be a finitely generated ideal of a ring $A$. Let $X$ be a scheme over $\mathop{\mathrm{Spec}}(A)$. Let
\[ \ldots \to \mathcal{F}_3 \to \mathcal{F}_2 \to \mathcal{F}_1 \]
be an inverse system of $\mathcal{O}_ X$-modules such that $\mathcal{F}_ n = \mathcal{F}_{n + 1}/I^ n\mathcal{F}_{n + 1}$. Assume
\[ \bigoplus \nolimits _{n \geq 0} H^0(X, I^ n\mathcal{F}_{n + 1}) \]
satisfies the ascending chain condition as a graded $\bigoplus _{n \geq 0} I^ n/I^{n + 1}$-module. Then the limit topology on $M = \mathop{\mathrm{lim}}\nolimits \Gamma (X, \mathcal{F}_ n)$ is the $I$-adic topology.
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