Skip to content
Permalink
Browse files

Shorten a proof

  • Loading branch information
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

0 comments on commit 4ebdfea

Please sign in to comment.
You can’t perform that action at this time.