Lemma 9.2.64AA3D030C896

Proof. A locally finitely generated submodule of a coherent sheaf is coherent, because its sections have the same relations when they are regarded as sections of the larger sheaf. The image of a morphism from a coherent sheaf is locally generated by the images of local generators of the source, so the image of a morphism between coherent sheaves is coherent. For the kernel of a morphism $\phi \colon \mathcal F\to \mathcal G$ between coherent sheaves, choose local generators $F_1,\ldots ,F_ p$ of $\mathcal F$ and local generators $R_ j=(R_{j1},\ldots ,R_{jp})$ of the relations of $\phi (F_1),\ldots ,\phi (F_ p)$. If $\sum _ ia_ iF_ i$ lies in the kernel, then $(a_ i)$ is a relation of the $\phi (F_ i)$, hence a combination of the $R_ j$; so the sections $\sum _ iR_{ji}F_ i$ generate the kernel, which is therefore coherent.

Let $\mathcal F\subset \mathcal S$ be coherent sheaves and consider $\mathcal G=\mathcal S/\mathcal F$; this covers cokernels, because images are coherent. Local generators of $\mathcal S$ map to local generators of $\mathcal G$. Given sections $G_1,\ldots ,G_ p$ of $\mathcal G$, lift them locally to sections $S_1,\ldots ,S_ p$ of $\mathcal S$, and choose local generators $F_1,\ldots ,F_ k$ of $\mathcal F$. A tuple $(a_ i)$ is a relation of the $G_ i$ exactly when $\sum _ ia_ iS_ i\in \mathcal F$, that is, when $(a,b)$ is a relation of $(S_1,\ldots ,S_ p,F_1,\ldots ,F_ k)$ in $\mathcal S$ for some $b$. Hence the projections to the first $p$ coordinates of local generators of the latter relations generate the relations of the $G_ i$, and $\mathcal G$ is coherent.

Now let $\mathcal F$ and $\mathcal G$ be coherent. Local lifts of generators of $\mathcal G$, together with local generators of $\mathcal F$, generate $\mathcal S$. Given sections $S_1,\ldots ,S_ p$ of $\mathcal S$, let $R_1,\ldots ,R_ l$ generate the relations of their images in $\mathcal G$. The combinations $H_ j=\sum _ iR_{ji}S_ i$ lie in $\mathcal F$; let $Q_1,\ldots ,Q_ k$ generate their relations. Every relation $(a_ i)$ of the $S_ i$ is first a relation of their images, so $a=\sum _ jc_ jR_ j$, and then $\sum _ jc_ jH_ j=\sum _ ia_ iS_ i=0$, so $c$ is a combination of $Q_1,\ldots ,Q_ k$. Hence the tuples $\sum _ jQ_{\nu j}R_ j$, $1\leq \nu \leq k$, which are relations of the $S_ i$, generate all of them, and $\mathcal S$ is coherent. The remaining case of the exact sequence, with $\mathcal S$ and $\mathcal G$ coherent, is the kernel case. A finite direct sum is an extension $0\to \mathcal F\to \mathcal F\oplus \mathcal G\to \mathcal G\to 0$. Finally, if $\mathcal A$ is coherent, so is $\mathcal A^ r$, and the last assertion follows from the first sentence of the proof. $\square$