Affine-local quasi-coherent algebras before general sheaf theory

Let $S$ be a scheme. A sheaf $\mathcal{A}$ of commutative unital ${\mathcal{O}}_S$-algebras is affine-locally module-associated if, for every affine open $U=\mathrm{Spec}R\subseteq S$, there is an $R$-algebra $B_U$ and an isomorphism of ${\mathcal{O}}_U$-algebras $\mathcal{A}{\mathrm{\mid }}_U\cong \widetilde{B_U}\mathrm{.}$ The isomorphism is required to identify restriction to every principal open $D(r)\subseteq U$ with localization $B_U\to B_U[\phi (r{)}^{-1}]$ induced by the structure map $\phi :R\to B_U$; equivalently, $\mathrm{\Gamma }(D(r),\mathcal{A})=B_U[\phi (r{)}^{-1}]$ under this identification. The notation $\widetilde{B_U}$ is the standard module-associated sheaf on $\mathrm{Spec}R$: on the principal-open basis it has sections $B_U[\phi (r{)}^{-1}]$, with the usual localization maps as restrictions. This is the construction in Stacks Schemes §26.5, Definition 26.5.3 and Lemma 26.5.4; module localization and the spectrum conventions are those of Multiplicative subsets and the localisation $S^{-1}R$ as equivalence classes of fractions and The underlying space of an affine spectrum. Here $S$ and its structure sheaf are as in Schemes and morphisms over a base, and the affine ring--scheme correspondence is the one in Affine schemes are contravariantly equivalent to commutative rings.

This is the local algebra condition used on this page. It makes no appeal to a later general theorem about quasi-coherent sheaves.