Affineness is local on the target

Given: The objects, hypotheses and conventions in the statement above, including the Axiom of Choice (The Axiom of Choice).

A morphism of schemes $f:X\to S$ is affine when $f^{-1}(U)$ is affine for every affine open subscheme $U\subseteq S$. Here the inverse image carries the restricted structure sheaf, as in def-affine-open-subscheme, and $f$ is a morphism of locally ringed spaces as in def-morphism-of-schemes. The empty scheme is affine, being $\mathrm{Spec}0$. Affineness of a morphism does not require its total source or target to be affine. (Affine morphisms)

Let $X$ be a scheme, $R=\mathrm{\Gamma }(X,{\mathcal{O}}_X)$, and $f_1,\dots ,f_n\in R$. Write $X_f$ for the open locus where the germ of $f$ is a unit. If $(f_1,\dots ,f_n)=R$ and each $U_i=X_{f_i}$ is affine, then $X$ is affine. In fact the canonical morphism $X\to \mathrm{Spec}R$ is an isomorphism. Empty $X$ and $n=0$ are allowed. (Affineness from a finite principal cover)

For $f\in A$, the morphism induced by $A\to A_f$ identifies $\mathrm{Spec}(A_f)$ with the open locally ringed subspace $D(f)$ of $\mathrm{Spec}A$. (A principal localization identifies its spectrum with a distinguished open)

Every affine scheme is quasi-compact. (Every affine scheme is quasi-compact)

Assume the Axiom of Choice (def-axiom-of-choice). In a nonzero commutative ring, every proper ideal is contained in a maximal ideal. (In a nonzero commutative ring, every proper ideal is contained in a maximal ideal)