Affine quasi-coherent sheaves are modules

The zero ring: if $A=0$ then $X=\mathrm{Spec}A=\mathrm{\varnothing }$ (The underlying space of an affine spectrum). The only $A$-module is $0$, whose associated sheaf is the zero sheaf on the empty space, and the only sheaf on $\mathrm{\varnothing }$ is the zero sheaf, so $M\mapsto \tilde{M}$ and $\mathrm{\Gamma }$ are the identity functors between the one-object, one-morphism categories and each claim holds. Hence assume $A\mathrm{\ne }0$, so that $X\mathrm{\ne }\mathrm{\varnothing }$, for the rest of the proof; then $\mathrm{\Gamma }(X,\tilde{M})=M$ in particular is [F6].

Each point has a basic neighbourhood on which $\mathcal{F}$ is associated to its own sections: let $x\in X$; by [F1] there are an affine open $V=\mathrm{Spec}B\subseteq X$ containing $x$ and a $B$-module $N$ with $\mathcal{F}{\mathrm{\mid }}_V\cong \tilde{N}$; by [F2] choose $f\in A$ with $x\in D(f)\subseteq V$. In $V=\mathrm{Spec}B$ the open $D(f)$ is the distinguished open cut out by the image of $f$ in $B$, with ring of sections $A_f$; applying [F5] to the affine open subscheme $D(f)=\mathrm{Spec}{A}_f$ of $V$ gives $(\tilde{N}){\mathrm{\mid }}_{D(f)}\cong \widetilde{(A_f{\otimes }_{B}N)}$. Restricting the isomorphism $\mathcal{F}{\mathrm{\mid }}_V\cong \tilde{N}$ therefore yields $\mathcal{F}{\mathrm{\mid }}_{D(f)}\cong \widetilde{(A_f{\otimes }_{B}N)}$, and by [F6] applied to the ring $A_f$ and the $A_f$-module $A_f{\otimes }_{B}N$ this module is $\mathrm{\Gamma }(D(f),\mathcal{F})$. Hence $\mathcal{F}{\mathrm{\mid }}_{D(f)}\cong \widetilde{\mathrm{\Gamma }(D(f),\mathcal{F})}$ for such an $f$.

The unit and naturality: claim 1 is the case $f=1$ of the canonical identification $\mathrm{\Gamma }(D(f),\tilde{M})\cong M_f$ of [F6], which is natural in $M$: an $A$-linear map $u:M\to N$ makes the square with the unit maps commute because ${\tilde{u}}_X=u$ under these identifications. Hence $\mathrm{\Gamma }(X,-)\circ (-{)}^{\sim }\cong {\mathrm{i}\mathrm{d}}_{{\mathrm{Mod}}_A}$.

Full faithfulness: let $M,N$ be $A$-modules. If $\psi :\tilde{M}\to \tilde{N}$ is a morphism of ${\mathcal{O}}_X$-modules, put $u={\psi }_X:M\to N$ under the identifications $\mathrm{\Gamma }(X,\tilde{M})=M$, $\mathrm{\Gamma }(X,\tilde{N})=N$ of [F6]. For $m\mathrm{/}{f}^k\in M_f=\tilde{M}(D(f))$ one has $m\mathrm{/}{f}^k=f^{-k}\cdot (m{\mathrm{\mid }}_{D(f)})$ in the module structure, so ${\psi }_{D(f)}(m\mathrm{/}{f}^k)=f^{-k}{\psi }_{D(f)}(m{\mathrm{\mid }}_{D(f)})=f^{-k}u(m){\mathrm{\mid }}_{D(f)}={\tilde{u}}_{D(f)}(m\mathrm{/}{f}^k)\mathrm{.}$ Thus $\psi$ and $\tilde{u}$ have the same components on every distinguished open, so $\psi =\tilde{u}$ by [F7]. Conversely $\widetilde{({\psi }_X)}=\psi$ just shown, and $(\tilde{u}{)}_X=u$, so $u\mapsto \tilde{u}$ is a bijection with inverse $\psi \mapsto {\psi }_X$; this proves claim 3.

Finite principal refinement: the distinguished opens $D(f)$ with $\mathcal{F}{\mathrm{\mid }}_{D(f)}\cong \widetilde{\mathrm{\Gamma }(D(f),\mathcal{F})}$ cover $X$ by step 1.2, and $X$ is compact by [F3]; fix a finite subcover $X=D(f_1)\cup \cdots \cup D(f_r)$, with $r\ge 1$ because $X\mathrm{\ne }\mathrm{\varnothing }$. Then $f_1,\dots ,f_r$ generate the unit ideal by [F4].

Descent datum: for $i=1,\dots ,r$ put $M_i=\mathrm{\Gamma }(D(f_i),\mathcal{F})$ and fix an isomorphism ${\theta }_i:\mathcal{F}{\mathrm{\mid }}_{D(f_i)}\to \widetilde{M_i}$ as provided by step 1.2. For each pair $(i,j)$ the open $D(f_i{f}_j)=D(f_i)\cap D(f_j)$ carries two identifications, so ${\phi }_{ij}:={\theta }_j\circ {\theta }_i^{-1}:(M_i{)}_{f_j}⟶(M_j{)}_{f_i}$ is an isomorphism of $A_{f_i{f}_j}$-modules, using $\widetilde{M_i}(D(f_i{f}_j))=(M_i{)}_{f_j}$ and $\widetilde{M_j}(D(f_i{f}_j))=(M_j{)}_{f_i}$ from [F6]. These isomorphisms satisfy ${\phi }_{ii}=\mathrm{i}\mathrm{d}$ and, on the triple overlap $D(f_i{f}_j{f}_k)$, ${\phi }_{jk}\circ {\phi }_{ij}={\phi }_{ik}$, because the $\theta$'s are isomorphisms of the same restrictions of $\mathcal{F}$. Thus $f_1,\dots ,f_r$ and the datum $(M_i,{\phi }_{ij})$ satisfy the hypotheses of [F9].

The descent module: by [F9] applied to the datum of step 3.1 there is an $A$-module $M=\ker (u-v)\subseteq {\prod }_i{M}_i$, with $u,v$ as in [F9], and canonical isomorphisms $M_{f_i}\cong M_i$ compatible with the ${\phi }_{ij}$; the construction uses no choice principle.

The descent module is $\mathrm{\Gamma }(X,\mathcal{F})$: the finite family $X=D(f_1)\cup \cdots \cup D(f_r)$ is a cover, so by [F8] the restriction map $e:\mathrm{\Gamma }(X,\mathcal{F})\to {\prod }_i\mathrm{\Gamma }(D(f_i),\mathcal{F})={\prod }_i{M}_i$ identifies $\mathrm{\Gamma }(X,\mathcal{F})$ with $\ker (d_0-d_1)$. Under the identifications $\mathrm{\Gamma }(D(f_i{f}_j),\mathcal{F})=(M_i{)}_{f_j}$ induced by ${\theta }_i$ and $=(M_j{)}_{f_i}$ induced by ${\theta }_j$, the component $(i,j)$ of $d_0$ becomes $(m_i){\mathrm{\mid }}_{f_j}$ and that of $d_1$ becomes ${\theta }_i{\theta }_j^{-1}((m_j){\mathrm{\mid }}_{f_i})={\phi }_{ij}^{-1}((m_j){\mathrm{\mid }}_{f_i})$; hence $d_0-d_1$ corresponds exactly to $u-v$ of step 4.1, and $e$ identifies $\mathrm{\Gamma }(X,\mathcal{F})\cong M$ canonically.

The same computation on every basic open: fix $g\in A$; the open $D(g)$ is covered by $D(g{f}_1),\dots ,D(g{f}_r)$, and ${\theta }_i$ identifies $\mathcal{F}(D(g{f}_i))$ with $(M_i{)}_g$ and $\mathcal{F}(D(g{f}_i{f}_j))$ with $(M_i{)}_{g{f}_j}=(M_i{)}_{f_{j}g}$ by [F6] applied to $\widetilde{M_i}$. Transporting $d_0,d_1$ as in step 5.1 exhibits these identifications and the isomorphisms ${\phi }_{ij}$ localised at $g$ as the localisation at $g$ of the maps $u,v$. By [F8] the restriction map $\mathrm{\Gamma }(D(g),\mathcal{F})\to {\prod }_i(M_i{)}_g$ identifies $\mathrm{\Gamma }(D(g),\mathcal{F})$ with $\ker ((u-v{)}_g)$, and by [F10] this kernel is $(\ker (u-v){)}_g=M_g$. Hence for every $g\in A$ there is a canonical isomorphism ${\beta }_g:\tilde{M}(D(g))=M_g⟶\mathrm{\Gamma }(D(g),\mathcal{F}),$ natural in $g$ in the sense that the ${\beta }_g$ commute with the restriction maps $M_g\to M_h$ and $\mathrm{\Gamma }(D(g),\mathcal{F})\to \mathrm{\Gamma }(D(h),\mathcal{F})$ for $D(h)\subseteq D(g)$; both sides are induced by the same restriction maps out of $\mathrm{\Gamma }(X,\mathcal{F})$ and out of ${\prod }_i{M}_i$.

The counit is an isomorphism: by [F7] the compatible family ${\beta }_g$, $g\in A$, is the family of distinguished-open components of a unique morphism $\tilde{M}\to \mathcal{F}$. On $D(f_i)=\mathrm{Spec}{A}_{f_i}$ its components on the distinguished opens $D(f_{i}h)$ are the maps ${\beta }_{f_{i}h}$ of step 6.1, all isomorphisms; applying [F7] on the affine scheme $D(f_i)$ shows that the restriction of the morphism to $D(f_i)$ is an isomorphism, and since the $D(f_i)$ cover $X$, [F12] shows that the morphism itself is an isomorphism. Since $\mathrm{\Gamma }(X,\mathcal{F})\cong M$ by step 5.1, this is the canonical comparison morphism ${\kappa }_{\mathcal{F}}$ of the Statement: for each $g$ the map ${\beta }_g$ and the localisation at $g$ of the restriction $\mathrm{\Gamma }(X,\mathcal{F})\to \mathrm{\Gamma }(D(g),\mathcal{F})$ both factor the same restriction $\mathrm{\Gamma }(X,\mathcal{F})\to {\prod }_i(M_i{)}_g$ through the localisation $M\to M_g$, and the restriction $\mathrm{\Gamma }(D(g),\mathcal{F})\to {\prod }_i(M_i{)}_g$ is injective by [F8], so by the universal property of localisation the two maps $M_g\to \mathrm{\Gamma }(D(g),\mathcal{F})$ coincide. This proves claim 2.

Conclusion: by step 7.1 every quasi-coherent $\mathcal{F}$ satisfies $\mathcal{F}\cong \widetilde{\mathrm{\Gamma }(X,\mathcal{F})}$, so $(-{)}^{\sim }$ is essentially surjective; by step 1.4 it is full and faithful, hence an equivalence of categories, and by steps 1.3 and 7.1 the unit and counit relating $(-{)}^{\sim }$ and $\mathrm{\Gamma }(X,-)$ are natural isomorphisms, so the two functors are quasi-inverse; this proves claim 4. For a prime $\mathfrak{p}\in X$, [F11] together with the isomorphism $\mathcal{F}\cong \widetilde{\mathrm{\Gamma }(X,\mathcal{F})}$ identifies the stalk ${\mathcal{F}}_{\mathfrak{p}}$ with the localisation $\mathrm{\Gamma }(X,\mathcal{F}{)}_{\mathfrak{p}}$ of the global module. As to the Axiom of Choice: the only uses are the compactness of $\mathrm{Spec}A$ in step 2.1 (through [F3], whose proof uses AC) and the associated-sheaf machinery inherited through [F5], [F6], [F7] and [F11]; step 2.1 selects witnesses only for the finitely many members of a finite subcover, the cover in step 1.2 is the set of all distinguished opens with the stated property and not a per-point choice, and the principal module descent [F9] and its use in steps 3.1, 4.1, 5.1 and 6.1 are choice-free. All constructions are canonical.