# stacks/stacks-project

Use earlier lemma in local-cohomology

This is a rare instance where a commit removes more lines than it adds.
 @@ -3104,24 +3104,8 @@ \section{Improving coherent modules} \end{lemma} \begin{proof} Let us show that $\text{depth}_{\mathcal{O}_{X, x}}(\mathcal{F}'_x) \geq 2$ for $x \in X$, $x \not \in U$. Namely, let $V$ be the punctured spectrum of $\mathcal{O}_{X, x}$. Then $V$ contains the inverse image of $U$ along $\Spec(\mathcal{O}_{X, x}) \to X$. Since $\mathcal{F}' = j_*(\mathcal{F}|_U) = j_*(\mathcal{F}'|_U)$, the same is true after base change by the flat morphism $\Spec(\mathcal{O}_{X, z}) \to X$ (Cohomology of Schemes, Lemma \ref{coherent-lemma-flat-base-change-cohomology}). A fortiori, the canonical map $\mathcal{F}'_x \to H^0(V, \mathcal{F}'|_V)$ is an isomorphism. This means that $H^i_{\mathfrak m_x}(\mathcal{F}'_x)$ is zero for $i = 0, 1$, see Lemma \ref{lemma-finiteness-pushforwards-and-H1-local}. Thus the depth is at least $2$ by Dualizing Complexes, Lemma \ref{dualizing-lemma-depth}. \medskip\noindent We have $\text{depth}_{\mathcal{O}_{X, x}}(\mathcal{F}'_x) \geq 2$ by Divisors, Lemma \ref{divisors-lemma-depth-pushforward} part (3). The uniqueness of $\mathcal{F} \to \mathcal{F}'$ follows from Divisors, Lemma \ref{divisors-lemma-depth-2-hartog}. The compatibility with flat pullbacks follows from