The Stacks project

Lemma 91.11.3. Let $(f, f')$ be a morphism of first order thickenings of ringed topoi as in Situation 91.9.1. Let $\mathcal{F}'$ be an $\mathcal{O}'$-module and set $\mathcal{F} = i^*\mathcal{F}'$. Assume that $\mathcal{F}'$ is flat over $\mathcal{O}_{\mathcal{B}'}$ and that $(f, f')$ is a strict morphism of thickenings. Then the following are equivalent

  1. $\mathcal{F}'$ is an $\mathcal{O}'$-module of finite presentation, and

  2. $\mathcal{F}$ is an $\mathcal{O}$-module of finite presentation.

Proof. The implication (1) $\Rightarrow $ (2) follows from Modules on Sites, Lemma 18.23.4. For the converse, assume $\mathcal{F}$ of finite presentation. We may and do assume that $\mathcal{C} = \mathcal{C}'$. By Lemma 91.11.2 we have a short exact sequence

\[ 0 \to \mathcal{I} \otimes _{\mathcal{O}_ X} \mathcal{F} \to \mathcal{F}' \to \mathcal{F} \to 0 \]

Let $U$ be an object of $\mathcal{C}$ such that $\mathcal{F}|_ U$ has a presentation

\[ \mathcal{O}_ U^{\oplus m} \to \mathcal{O}_ U^{\oplus n} \to \mathcal{F}|_ U \to 0 \]

After replacing $U$ by the members of a covering we may assume the map $\mathcal{O}_ U^{\oplus n} \to \mathcal{F}|_ U$ lifts to a map $(\mathcal{O}'_ U)^{\oplus n} \to \mathcal{F}'|_ U$. The induced map $\mathcal{I}^{\oplus n} \to \mathcal{I} \otimes \mathcal{F}$ is surjective by right exactness of $\otimes $. Thus after replacing $U$ by the members of a covering we can find a lift $(\mathcal{O}'|_ U)^{\oplus m} \to (\mathcal{O}'|_ U)^{\oplus n}$ of the given map $\mathcal{O}_ U^{\oplus m} \to \mathcal{O}_ U^{\oplus n}$ such that

\[ (\mathcal{O}'_ U)^{\oplus m} \to (\mathcal{O}'_ U)^{\oplus n} \to \mathcal{F}'|_ U \to 0 \]

is a complex. Using right exactness of $\otimes $ once more it is seen that this complex is exact. $\square$

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