Kernels and cokernels of quasi-coherent modules

The affine chart: let $U=\mathrm{Spec}A\subseteq X$ be affine with $\mathcal{F}{\mathrm{\mid }}_U\cong \tilde{M}$, $\mathcal{G}{\mathrm{\mid }}_U\cong \tilde{N}$, and let $u:M\to N$ be the $A$-linear map with $\phi {\mathrm{\mid }}_U$ corresponding to $\tilde{u}$ under the full faithfulness of [F3]. The two sequences of $A$-modules $0⟶\ker u⟶M⟶\mathrm{im}u⟶0,0⟶\mathrm{im}u⟶N⟶\mathrm{coker}u⟶0$ are exact, and localising them at a prime $\mathfrak{p}$ remains exact by [F4]; under the stalk identifications $(\tilde{M}{)}_{\mathfrak{p}}=M_{\mathfrak{p}}$, $(\tilde{N}{)}_{\mathfrak{p}}=N_{\mathfrak{p}}$ and $(\tilde{K}{)}_{\mathfrak{p}}=K_{\mathfrak{p}}$ of [F5], and the naturality of these identifications, the stalk sequences of the sheaf maps $\widetilde{(\ker u)}\to \tilde{M}\to \widetilde{(\mathrm{im}u)}$ and $\widetilde{(\mathrm{im}u)}\to \tilde{N}\to \widetilde{(\mathrm{coker}u)}$ are exact at every prime, so by [F2] the sequences of ${\mathcal{O}}_U$-modules $0⟶\widetilde{(\ker u)}⟶\tilde{M}\stackrel{\tilde{u}}{\to }\tilde{N}⟶\widetilde{(\mathrm{coker}u)}⟶0$ and the intermediate image sequence are exact. Consequently $\ker (\phi {\mathrm{\mid }}_U)\cong \widetilde{(\ker u)}$, $\mathrm{im}(\phi {\mathrm{\mid }}_U)\cong \widetilde{(\mathrm{im}u)}$ and $\mathrm{coker}(\phi {\mathrm{\mid }}_U)\cong \widetilde{(\mathrm{coker}u)}$ canonically.

The affine isomorphisms are restrictions of the global sheaves: by the locality of [F1], restricting the global kernel, image and cokernel gives the kernel, image and cokernel of $\phi {\mathrm{\mid }}_U$; combined with step 1.1 this yields the three displayed isomorphisms of claim (1) on every affine open $U$ on which $\mathcal{F}$ and $\mathcal{G}$ are both associated.

Global quasi-coherence: for $x\in X$ choose affine opens $U_{\mathcal{F}}\ni x$ and $U_{\mathcal{G}}\ni x$ on which $\mathcal{F}$ and $\mathcal{G}$ are associated, respectively; then $U_{\mathcal{F}}\cap U_{\mathcal{G}}$ is an open neighbourhood of $x$ in the affine scheme $U_{\mathcal{F}}$ and hence contains a distinguished open $D(f)\ni x$ by [F6], on which both are associated. Therefore the family of all affine opens on which both are associated covers $X$, and on each such member step 2.1 exhibits $\ker \phi$, $\mathrm{im}\phi$ and $\mathrm{coker}\phi$ as associated sheaves; by the affine cover criterion [F6] all three are quasi-coherent, which is claim (1).

The abelian subcategory: the zero sheaf is quasi-coherent, since it is $\tilde{0}$ on every affine chart, and the binary biproduct of quasi-coherent modules is quasi-coherent: on an affine chart with $\mathcal{F}{\mathrm{\mid }}_U\cong \tilde{M}$, $\mathcal{G}{\mathrm{\mid }}_U\cong \tilde{N}$ the objectwise direct sum has sections $\tilde{M}(D(f))\oplus \tilde{N}(D(f))=M_f\oplus N_f\cong (M\oplus N{)}_f$ on distinguished opens, compatibly with restriction by [F9], so it is $\widetilde{(M\oplus N)}$ on the covering family of common affine charts and hence quasi-coherent by [F6]. Since $\mathrm{M}\mathrm{o}\mathrm{d}({\mathcal{O}}_X)$ is abelian by [F7] and $\mathrm{QCoh}(X)$ is a full subcategory closed under kernels, cokernels and finite biproducts computed in $\mathrm{M}\mathrm{o}\mathrm{d}({\mathcal{O}}_X)$, the definition [F8] makes $\mathrm{QCoh}(X)$ an abelian subcategory and the inclusion a full additive exact embedding; images are kernels of cokernels, so they are covered as well. This is claim (2).

No separation hypothesis: steps 1.1 to 4.1 use only affine charts, localisations, and the local definition of quasi-coherence; intersecting two affine charts and passing to a distinguished open is always possible in an affine scheme, and no statement about intersections of affine opens being affine, quasi-compactness or quasi-separatedness of $X$ is invoked. This is claim (3).

Choice accounting: the covering family used in steps 3.1 and 4.1 is the family of all affine opens on which the relevant sheaves are associated, which is determined by the data, so no chart or module is selected; the map $u$ of step 1.1 is the unique map corresponding to $\phi {\mathrm{\mid }}_U$ under the affine equivalence, and all identifications are the canonical ones of [F3] and [F5]. Hence the only use of the Axiom of Choice is the inherited one recorded in the Statement through [F10].