Coherent sheaves on a locally Noetherian scheme

Noetherian affine charts inside any open: let $x\in X$ and let $U\subseteq X$ be an open neighbourhood of $x$. By [F6] there is an affine open $W=\mathrm{Spec}A\ni x$ with $A$ Noetherian; the set $W\cap U$ is an open neighbourhood of $x$ in $W$, so by the basis property of distinguished opens [F7] there is $f\in A$ with $x\in D(f)\subseteq W\cap U$. Then $D(f)=\mathrm{Spec}{A}_f\subseteq U$ is affine and $A_f$ is Noetherian by [F5].

Finite type is detected on every affine chart: let $V=\mathrm{Spec}B\subseteq X$ be affine and let $\mathcal{F}$ be of finite type. Since $\mathcal{F}$ is quasi-coherent, [F9] gives $\mathcal{F}{\mathrm{\mid }}_V\cong \tilde{N}$ with $N=\mathrm{\Gamma }(V,\mathcal{F}{\mathrm{\mid }}_V)$. For $x\in V$ choose by [F2] an affine open $U_x\subseteq X$ with $\mathcal{F}{\mathrm{\mid }}_{U_x}\cong \widetilde{M_x}$ and $M_x$ finitely generated, and by [F7] choose $g_x\in B$ with $x\in D_V(g_x)\subseteq U_x\cap V$; then $D_V(g_x)$ is an affine open subscheme of $U_x$ with coordinate ring $B_{g_x}$, so [F10] gives $\mathcal{F}{\mathrm{\mid }}_{D_V(g_x)}\cong \widetilde{(B_{g_x}{\otimes }_{O(U_x)}{M}_x)}$ with a finitely generated module, while $(\tilde{N}){\mathrm{\mid }}_{D_V(g_x)}\cong \widetilde{(N_{g_x})}$ by [F3]; hence $N_{g_x}$ is finitely generated over $B_{g_x}$. Since $V=D_V(g_1)\cup \cdots \cup D_V(g_k)$ for finitely many $g_i\in B$ with $(g_1,\dots ,g_k)=B$ by quasi-compactness [F7], and each $N_{g_i}$ is finitely generated, choose finitely many elements of $N$ whose images generate each $N_{g_i}$ and let $N^{\mathrm{\prime }}\subseteq N$ be the submodule they generate; then $(N\mathrm{/}{N}^{\mathrm{\prime }}{)}_{g_i}=0$ for all $i$, so $N\mathrm{/}{N}^{\mathrm{\prime }}=0$ by the localisation criterion of [F3], and $N=N^{\mathrm{\prime }}$ is finitely generated.

Claim 1, forward direction: by definition a coherent module is quasi-coherent of finite type, so coherent implies finite type.

Claim 1, converse: let $\mathcal{F}$ be quasi-coherent of finite type, let $U\subseteq X$ be open, $n\ge 0$ and $\tau :{\mathcal{O}}_U^n\to \mathcal{F}{\mathrm{\mid }}_U$; we show that $\ker \tau$ is of finite type. By step 1.1 the open $U$ is covered by affine charts $V=\mathrm{Spec}B\subseteq U$ with $B$ Noetherian; on such a chart $\mathcal{F}{\mathrm{\mid }}_V=\tilde{N}$ with $N$ finitely generated by step 1.2, and $\tau {\mathrm{\mid }}_V$ is induced by a $B$-linear map $\psi :B^n\to N$ with $\ker (\tau {\mathrm{\mid }}_V)=\widetilde{(\ker \psi )}$ [F1, F9]. As $B$ is Noetherian, the finite free module $B^n$ is Noetherian, so its submodule $\ker \psi$ is finitely generated [F4], so $\ker (\tau {\mathrm{\mid }}_V)$ is of finite type [F3]. Kernel sheaves restrict to open subschemes and finite type is local on $X$ [F3, F2], so $\ker \tau$ is of finite type over the cover of $U$ by these charts; as this holds for every $(U,n,\tau )$, $\mathcal{F}$ is coherent [F1].

Claim 2, kernels, images and cokernels: let $\phi :\mathcal{F}\to \mathcal{G}$ be a morphism of coherent modules. By [F8] the sheaves $\ker \phi$, $\mathrm{im}\phi$ and $\mathrm{coker}\phi$ are quasi-coherent. Cover $X$ by Noetherian affine charts $V=\mathrm{Spec}B$ (step 1.1); on each of them $\mathcal{F}{\mathrm{\mid }}_V=\tilde{M}$ and $\mathcal{G}{\mathrm{\mid }}_V=\tilde{N}$ with $M,N$ finitely generated (step 1.2), and $\phi {\mathrm{\mid }}_V=\tilde{\psi }$ for a unique $B$-linear map $\psi :M\to N$ [F9]; by exactness of $\widetilde{(-)}$ the sheaves $\ker (\phi {\mathrm{\mid }}_V)$, $\mathrm{im}(\phi {\mathrm{\mid }}_V)$ and $\mathrm{coker}(\phi {\mathrm{\mid }}_V)$ are the associated sheaves of $\ker \psi$, $\mathrm{im}\psi$ and $\mathrm{coker}\psi$ [F3]. Since $B$ is Noetherian and $M$ finitely generated, $\ker \psi$ is finitely generated, and $\mathrm{im}\psi$, $\mathrm{coker}\psi$ are quotients of finitely generated modules, hence finitely generated [F4]; therefore the three sheaves are of finite type on the covering charts and hence of finite type [F2], and being quasi-coherent they are coherent by step 2.1.

Claim 3, extensions: let $0\to \mathcal{F}\to \mathcal{E}\to \mathcal{G}\to 0$ be exact with $\mathcal{F},\mathcal{G}$ coherent and $\mathcal{E}$ quasi-coherent. Cover $X$ by Noetherian affine charts $V=\mathrm{Spec}B$ (step 1.1); restriction to the open subscheme $V$ is exact, so $0\to \tilde{M}\to \mathcal{E}{\mathrm{\mid }}_V\to \tilde{N}\to 0$ is exact with $\mathcal{F}{\mathrm{\mid }}_V=\tilde{M}$, $\mathcal{G}{\mathrm{\mid }}_V=\tilde{N}$ and $M,N$ finitely generated (steps 1.2, [F3]). Since $\mathcal{E}{\mathrm{\mid }}_V$ is quasi-coherent [F8] and $V$ is affine, [F9] gives $\mathcal{E}{\mathrm{\mid }}_V\cong \tilde{P}$ with $P=\mathrm{\Gamma }(V,\mathcal{E}{\mathrm{\mid }}_V)$; the functor $\mathrm{\Gamma }(V,-)$ is an equivalence between quasi-coherent ${\mathcal{O}}_V$-modules and $B$-modules, hence exact, so applying it to $0\to \tilde{M}\to \tilde{P}\to \tilde{N}\to 0$ yields the exact sequence $0\to M\to P\to N\to 0$ [F9]. With $M$ and $N$ finitely generated, $P$ is finitely generated: lift finitely many generators of $N$ to $P$ and add generators of $M$ [F4]. Therefore $\mathcal{E}{\mathrm{\mid }}_V$ is of finite type on every chart of the cover, so $\mathcal{E}$ is of finite type [F2]; being quasi-coherent by hypothesis, $\mathcal{E}$ is coherent by step 2.1.

Claim 2, finite direct sums: for coherent $\mathcal{F},\mathcal{G}$ the biproduct $\mathcal{F}\oplus \mathcal{G}$ is quasi-coherent [F8], and on a Noetherian affine chart as in step 3.1 it is $\widetilde{(M\oplus N)}$ with $M\oplus N$ finitely generated [F3, F9], hence is of finite type by locality [F2]; by step 2.1 it is coherent.

Choice accounting: the arguments use the family of all Noetherian affine charts, which is determined by the data, together with finite subcovers and finite generating sets extracted from the given modules and morphisms; no chart, generator family or isomorphism is selected by an infinite simultaneous choice, so the only Axiom of Choice is the inherited one recorded in [F11].