Kernels, cokernels, and finite extensions of coherent module sheaves

Let $\mathcal{F}$ be coherent and fix $x\in X$. Local finite type gives, after choosing a neighborhood $U$ of $x$, a finite-rank free module sheaf map $\pi :{\mathcal{O}}_U^{\oplus k}\to \mathcal{F}{\mathrm{\mid }}_U$ that is surjective on every stalk. By [F1], its kernel is locally of finite type; after shrinking to $V\subseteq U$, choose finitely many sections generating that kernel on $V$ and let $\rho :{\mathcal{O}}_V^{\oplus m}\to {\mathcal{O}}_V^{\oplus k}$ be the corresponding map. The sequence ${\mathcal{O}}_V^{\oplus m}\stackrel{\rho }{\to }{\mathcal{O}}_V^{\oplus k}\stackrel{\pi }{\to }\mathcal{F}{\mathrm{\mid }}_V\to 0$ is exact by [F2], [F3], and [F4]. Thus every coherent module has a local finite presentation.

Let $f:\mathcal{F}\to \mathcal{G}$ be a morphism of coherent modules and set $\mathcal{K}=\ker f$. Choose locally a finite-rank free epimorphism $\pi :{\mathcal{O}}^{\oplus k}\to \mathcal{F}$. The composite $f\pi :{\mathcal{O}}^{\oplus k}\to \mathcal{G}$ has a locally finite-type kernel by [F1]; its image under $\pi$ is exactly $\mathcal{K}$ on each stalk, since $\pi$ is stalkwise surjective. Hence $\mathcal{K}$ is locally of finite type by [F4]. For every finite-rank map $\eta :{\mathcal{O}}^{\oplus p}\to \mathcal{K}$, its composite with the inclusion $\mathcal{K}\hookrightarrow \mathcal{F}$ has the same kernel; [F1] for $\mathcal{F}$ makes that kernel locally of finite type. Thus $\mathcal{K}$ is coherent.

Let $\mathcal{C}=\mathrm{coker}f$. It is locally of finite type because it is a quotient of the locally finite-type module $\mathcal{G}$. To check its relation condition, fix a finite-rank map $\gamma :{\mathcal{O}}^{\oplus p}\to \mathcal{C}$ and a point $x$. Since $\mathcal{G}\to \mathcal{C}$ is a sheaf epimorphism, choose local lifts of the finitely many images of the standard basis near $x$; they define $\tilde{\gamma }:{\mathcal{O}}^{\oplus p}\to \mathcal{G}$ lifting $\gamma$. Also choose a local finite-rank epimorphism $\delta :{\mathcal{O}}^{\oplus m}\to \mathcal{F}$. Define $ϵ:{\mathcal{O}}^{\oplus (m+p)}\to \mathcal{G}$ by $ϵ(a,b)=f(\delta (a))+\tilde{\gamma }(b)$. By [F1], $\ker ϵ$ is locally of finite type. Projection of $\ker ϵ$ to the second factor has image exactly $\ker \gamma$ on every stalk: an element of $\ker ϵ$ maps to zero in $\mathcal{C}$, and any stalk element $b\in \ker \gamma$ has $\tilde{\gamma }(b)\in \mathrm{im}f$, so a stalk preimage under $\delta$ completes it to an element of $\ker ϵ$. The image of a locally finite-type module is locally finite type, so $\ker \gamma$ is locally of finite type. This holds for every $\gamma$, hence $\mathcal{C}$ is coherent.

Consider a short exact sequence $0\to {\mathcal{F}}^{\mathrm{\prime }}\stackrel{i}{\to }\mathcal{E}\stackrel{p}{\to }{\mathcal{F}}^{\mathrm{\prime }\mathrm{\prime }}\to 0$. If $\mathcal{E}$ and ${\mathcal{F}}^{\mathrm{\prime }\mathrm{\prime }}$ are coherent, then ${\mathcal{F}}^{\mathrm{\prime }}=\ker p$ is coherent by step 2.1. If ${\mathcal{F}}^{\mathrm{\prime }}$ and $\mathcal{E}$ are coherent, then ${\mathcal{F}}^{\mathrm{\prime }\mathrm{\prime }}=\mathrm{coker}i$ is coherent by step 2.2.

It remains to assume ${\mathcal{F}}^{\mathrm{\prime }}$ and ${\mathcal{F}}^{\mathrm{\prime }\mathrm{\prime }}$ coherent. Local finite generators of ${\mathcal{F}}^{\mathrm{\prime }}$ together with local lifts to $\mathcal{E}$ of finite generators of ${\mathcal{F}}^{\mathrm{\prime }\mathrm{\prime }}$ generate $\mathcal{E}$, so $\mathcal{E}$ is locally of finite type. For any finite-rank map $g:{\mathcal{O}}^{\oplus k}\to \mathcal{E}$, the kernel of $pg$ is locally of finite type by [F1] for ${\mathcal{F}}^{\mathrm{\prime }\mathrm{\prime }}$; choose a local finite-rank epimorphism $\gamma :{\mathcal{O}}^{\oplus m}\to \ker (pg)$. Exactness gives a map $\delta :{\mathcal{O}}^{\oplus m}\to {\mathcal{F}}^{\mathrm{\prime }}$ with $i\delta =g\gamma$. By [F1] for ${\mathcal{F}}^{\mathrm{\prime }}$, $\ker \delta$ is locally of finite type. On each stalk, $\gamma$ maps $\ker \delta$ onto $\ker g$: inclusion follows from $i\delta =g\gamma$, and if $s\in \ker g$, then $pg(s)=0$, so $s=\gamma (t)$ for some $t$ and $i\delta (t)=g(s)=0$; injectivity of $i$ gives $\delta (t)=0$. Therefore $\ker g$ is locally of finite type. Since this holds for every $g$, $\mathcal{E}$ is coherent.