# stacks / stacks-project

Shorten a proof

Thanks to B. Shih
https://stacks.math.columbia.edu/tag/01BU#comment-4738
aisejohan committed Dec 10, 2019
1 parent c0eb7a0 commit 4ebdfea1f03bfc41c7d3c711c8c78a21c0f053a5
Showing with 3 additions and 8 deletions.
1. +3 −8 modules.tex
 @@ -1843,14 +1843,9 @@ \section{Coherent modules} $0 \to \mathcal{K} \to \mathcal{K}_3 \to \mathcal{F}_1$ where $\mathcal{K}_3$ is the module of relations among the images of the sections $s_i$ in $\mathcal{F}_3$. Since $\mathcal{F}_3$ is coherent we see that $\mathcal{K}_3$ is finite type. Since $\mathcal{F}_1$ is coherent we see that the image $\mathcal{I}$ of $\mathcal{K}_3 \to \mathcal{F}_1$ is coherent. Hence $\mathcal{K}$ is the kernel of the map $\mathcal{K}_3 \to \mathcal{I}$ between a finite type sheaf and a coherent sheaves and hence finite type by (2). Since $\mathcal{F}_1$ is coherent we see that $\mathcal{K}$ is the kernel of a map from a finite type module to a coherent module and hence finite type by (2). \medskip\noindent Proof of (5). This follows because (3) and (4) show that