Structure sheaf of a quasi-compact quasi-separated morphism is affine-local

Fix an affine open $U=\mathrm{Spec}R\subseteq S$ and write $X_U=f^{-1}(U)$, so $\mathrm{\Gamma }(X_U,{\mathcal{O}}_X)=(f_{\ast }{\mathcal{O}}_X)(U)$ by [F9]. The open $U$ is quasi-compact by [F2], so $X_U$ is quasi-compact by [F1]; since affine opens form a basis of $X_U$, there are finitely many affine opens $U_1,\dots ,U_n\subseteq X_U$ with $X_U=U_1\cup \cdots \cup U_n$.

For every pair $i,j$ the intersection $U_i\cap U_j$ is quasi-compact by [F1], since $U_i,U_j$ are affine and both lie over the affine open $U$. Choose finitely many affine opens $U_{ijk}\subseteq U_i\cap U_j$ covering $U_i\cap U_j$; one may take $U_{iii}=U_i$. The restrictions of ${\mathcal{O}}_X$ give maps $\mathrm{\Gamma }(X_U,{\mathcal{O}}_X)⟶{\prod }_i\mathrm{\Gamma }(U_i,{\mathcal{O}}_X)\stackrel{\mathrm{\partial }}{\to }{\prod }_{i,j,k}\mathrm{\Gamma }(U_{ijk},{\mathcal{O}}_X),$ where $\mathrm{\partial }((s_i{)}_i{)}_{i,j,k}=s_i{\mathrm{\mid }}_{U_{ijk}}-s_j{\mathrm{\mid }}_{U_{ijk}}$; the composite is zero.

For the second assertion let $R\to R^{\mathrm{\prime }}$ be flat and put $U^{\mathrm{\prime }}=\mathrm{Spec}{R}^{\mathrm{\prime }}$, so the fibre product $X_U{\times }_U{U}^{\mathrm{\prime }}$ has the affine open cover $\{U_i{\times }_U{U}^{\mathrm{\prime }}\}$ with $U_i{\times }_U{U}^{\mathrm{\prime }}\cong \mathrm{Spec}(C_i{\otimes }_R{R}^{\mathrm{\prime }})$ where $C_i=\mathrm{\Gamma }(U_i,{\mathcal{O}}_X)$, and pairwise intersections covered by the affine opens $U_{ijk}{\times }_U{U}^{\mathrm{\prime }}$; here $\mathrm{\Gamma }(U_i{\times }_U{U}^{\mathrm{\prime }},{\mathcal{O}}_{X_U{\times }_U{U}^{\mathrm{\prime }}})=C_i{\otimes }_R{R}^{\mathrm{\prime }}=\mathrm{\Gamma }(U_i,{\mathcal{O}}_X){\otimes }_R{R}^{\mathrm{\prime }}$ and likewise for the triple intersections.

This sequence is exact at the middle term. If a section over $X_U$ restricts to zero on every $U_i$ it is zero because the $U_i$ cover $X_U$; conversely if $(s_i{)}_i$ satisfies $\mathrm{\partial }(s_i)=0$, then $s_i$ and $s_j$ agree on each member $U_{ijk}$ of a cover of $U_i\cap U_j$, so by [F5] they agree on $U_i\cap U_j$; the sheaf axiom [F5] then glues the family to a unique section of ${\mathcal{O}}_X$ over $X_U=X_U$. Hence $0\to \mathrm{\Gamma }(X_U,{\mathcal{O}}_X)\to {\prod }_i\mathrm{\Gamma }(U_i,{\mathcal{O}}_X)\to {\prod }_{i,j,k}\mathrm{\Gamma }(U_{ijk},{\mathcal{O}}_X)$ is exact.

Fix $r\in R$ and apply the exact functor $-{\otimes }_R{R}_r$ of [F7] to the sequence of step 2.1, using [F8] to move the tensor product inside the finite products. Since each $U_i$, and each $U_{ijk}$, is affine and its structure map to $U=\mathrm{Spec}R$ makes its ring of sections an $R$-algebra, [F6] identifies $\mathrm{\Gamma }(U_i,{\mathcal{O}}_X){\otimes }_R{R}_r=\mathrm{\Gamma }(U_i,{\mathcal{O}}_X{)}_r=\mathrm{\Gamma }(U_i\cap f^{-1}(D(r)),{\mathcal{O}}_X)$ and likewise for the $U_{ijk}$, where $U_i\cap f^{-1}(D(r))$ and $U_{ijk}\cap f^{-1}(D(r))$ are principal opens of the affine schemes $U_i$, $U_{ijk}$ and hence affine. So the localised sequence is exact.

Tensoring the exact sequence of step 2.1 with $R^{\mathrm{\prime }}$ is exact because $R^{\mathrm{\prime }}$ is flat over $R$, and by [F8] the tensored sequence has the terms computed in step 1.3, so it is the sheaf equaliser sequence of the pulled-back cover of $X_U{\times }_U{U}^{\mathrm{\prime }}$; its kernel is therefore $\mathrm{\Gamma }(X_U{\times }_U{U}^{\mathrm{\prime }},{\mathcal{O}}_X)$ by the argument of step 2.1, while the kernel of the original sequence is $M=\mathrm{\Gamma }(X_U,{\mathcal{O}}_X)$ by step 2.1. Since tensor product of the exact sequence preserves the kernel, $M{\otimes }_R{R}^{\mathrm{\prime }}\cong \mathrm{\Gamma }(X_U{\times }_U{U}^{\mathrm{\prime }},{\mathcal{O}}_X)$.

The opens $U_i\cap f^{-1}(D(r))$ form a finite affine open cover of $X_U\cap f^{-1}(D(r))=f^{-1}(D(r))$, and their intersections are covered by the affine opens $U_{ijk}\cap f^{-1}(D(r))$; the canonical map on restrictions gives exactly the localised sequence of step 3.1. By the same sheaf argument as in step 2.1, its kernel is $\mathrm{\Gamma }(f^{-1}(D(r)),{\mathcal{O}}_X)$, so step 3.1 yields a canonical isomorphism $M{\otimes }_R{R}_r\cong \mathrm{\Gamma }(f^{-1}(D(r)),{\mathcal{O}}_X)$ with $M=\mathrm{\Gamma }(X_U,{\mathcal{O}}_X)$; it is multiplicative because all maps are restriction maps of the structure sheaf and localisation maps of rings.

If $D(s)\subseteq D(r)$, the localisation $M{\otimes }_R{R}_r\to M{\otimes }_R{R}_s$ corresponds under step 4.1 to the restriction $\mathrm{\Gamma }(f^{-1}(D(r)),{\mathcal{O}}_X)\to \mathrm{\Gamma }(f^{-1}(D(s)),{\mathcal{O}}_X)$: both are induced by restricting sections along $f^{-1}(D(s))\subseteq f^{-1}(D(r))$, and the identification with localisation in step 3.1 is natural in the localised ring. Since $U$ was arbitrary and these identifications are compatible with restriction and multiplication, $U\mapsto \mathrm{\Gamma }(f^{-1}U,{\mathcal{O}}_X)$ together with them is precisely the affine-local module-associated structure of [F10] for $f_{\ast }{\mathcal{O}}_X$.

All selections in the proof are of finitely many members of a fixed cover of a quasi-compact space or of a basis of an affine scheme, which are finitely many existential instantiations and not applications of a choice principle; the gluing in [F5] is unique, and the AC-free statements are used only. Hence the lemma is choice-free. [F5, step 1.1, step 2.1] $\mathrm{\square }$