Lemma 52.3.1 (0EH9)—The Stacks project

Lemma 52.3.1. Let $X$ be a scheme. Let $f \in \Gamma (X, \mathcal{O}_ X)$. Let

\[ \ldots \to \mathcal{F}_3 \to \mathcal{F}_2 \to \mathcal{F}_1 \]

be an inverse system of quasi-coherent $\mathcal{O}_ X$-modules. The following are equivalent

for all $n \geq 1$ the map $f : \mathcal{F}_{n + 1} \to \mathcal{F}_{n + 1}$ factors through $\mathcal{F}_{n + 1} \to \mathcal{F}_ n$ to give a short exact sequence $0 \to \mathcal{F}_ n \to \mathcal{F}_{n + 1} \to \mathcal{F}_1 \to 0$,
for all $n \geq 1$ the map $f^ n : \mathcal{F}_{n + 1} \to \mathcal{F}_{n + 1}$ factors through $\mathcal{F}_{n + 1} \to \mathcal{F}_1$ to give a short exact sequence $0 \to \mathcal{F}_1 \to \mathcal{F}_{n + 1} \to \mathcal{F}_ n \to 0$
there exists an $\mathcal{O}_ X$-module $\mathcal{G}$ which is $f$-divisible such that $\mathcal{F}_ n = \mathcal{G}[f^ n]$.
there exists an $\mathcal{O}_ X$-module $\mathcal{F}$ which is $f$-torsion free such that $\mathcal{F}_ n = \mathcal{F}/f^ n\mathcal{F}$.

Proof. The equivalence of (1), (2), (3) and the implication (4) $\Rightarrow $ (1) are proven in Cohomology, Lemma 20.36.1. Assume (1) holds. Set $\mathcal{F} = \mathop{\mathrm{lim}}\nolimits \mathcal{F}_ n$. By Lemma 52.2.1 part (4) we have $\mathcal{F}_ n = \mathcal{F}/f^ n\mathcal{F}$. Let $U \subset X$ be open and $s = (s_ n) \in \mathcal{F}(U) = \mathop{\mathrm{lim}}\nolimits \mathcal{F}_ n(U)$. Choose $n \geq 1$. If $fs = 0$, then $s_{n + 1}$ is in the kernel of $\mathcal{F}_{n + 1} \to \mathcal{F}_ n$ by condition (1). Hence $s_ n = 0$. Since $n$ was arbitrary, we see $s = 0$. Thus $\mathcal{F}$ is $f$-torsion free. $\square$

The Stacks project

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