The Stacks project

Lemma 52.25.1. For any coherent triple $(\mathcal{F}, \mathcal{F}_0, \alpha )$ there exists a coherent $\mathcal{O}_ X$-module $\mathcal{F}'$ such that $f : \mathcal{F}' \to \mathcal{F}'$ is injective, an isomorphism $\alpha ' : \mathcal{F}'|_ U \to \mathcal{F}$, and a map $\alpha '_0 : \mathcal{F}'/f\mathcal{F}' \to \mathcal{F}_0$ such that $\alpha \circ (\alpha ' \bmod f) = \alpha '_0|_{U_0}$.

Proof. Choose a finite $A$-module $M$ such that $\mathcal{F}$ is the restriction to $U$ of the coherent $\mathcal{O}_ X$-module associated to $M$, see Local Cohomology, Lemma 51.8.2. Since $\mathcal{F}$ is $f$-torsion free, we may replace $M$ by its quotient by $f$-power torsion. On the other hand, let $M_0 = \Gamma (X_0, \mathcal{F}_0)$ so that $\mathcal{F}_0$ is the coherent $\mathcal{O}_{X_0}$-module associated to the finite $A/fA$-module $M_0$. By Cohomology of Schemes, Lemma 30.10.5 there exists an $n$ such that the isomorphism $\alpha _0$ corresponds to an $A/fA$-module homomorphism $\mathfrak m^ n M/fM \to M_0$ (whose kernel and cokernel are annihilated by a power of $\mathfrak m$, but we don't need this). Thus if we take $M' = \mathfrak m^ n M$ and we let $\mathcal{F}'$ be the coherent $\mathcal{O}_ X$-module associated to $M'$, then the lemma is clear. $\square$

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