The Stacks project

Lemma 52.3.3. Let $A$ be a Noetherian ring complete with respect to a principal ideal $(f)$. 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. Assume

  1. $\Gamma (X, \mathcal{F}_1)$ is a finite $A$-module,

  2. the equivalent conditions of Lemma 52.3.1 hold.

Then

\[ M = \mathop{\mathrm{lim}}\nolimits \Gamma (X, \mathcal{F}_ n) \]

is a finite $A$-module, $f$ is a nonzerodivisor on $M$, and $M/fM$ is the image of $M$ in $\Gamma (X, \mathcal{F}_1)$.

Proof. By Lemma 52.3.2 and its proof we have $M/fM \subset H^0(X, \mathcal{F}_1)$. From (1) and the Noetherian property of $A$ we get that $M/fM$ is a finite $A$-module. Observe that $\bigcap f^ nM = 0$ as $f^ nM$ maps to zero in $H^0(X, \mathcal{F}_ n)$. By Algebra, Lemma 10.96.12 we conclude that $M$ is finite over $A$. $\square$

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