Affine acyclicity of quasi-coherent sheaves

Take $\mathcal{B}=\{D(f):f\in A\}$, which contains $X=D(1)$ and is closed under finite intersections because $D(f)\cap D(g)=D(fg)$ [F3]. For $U=D(g)\in \mathcal{B}$ let $\mathrm{C}\mathrm{o}\mathrm{v}(U)$ be the set of finite covers of $U$ by distinguished opens $D(h_1),\dots ,D(h_r)\subseteq U$ with ${\sum }_j{A}_g{h}_j=A_g$; equivalently, after writing $h_j$ for its image in $A_g$, these are the finite covers of $U$ by basic opens. Every open cover of $U$ has a refinement in $\mathrm{C}\mathrm{o}\mathrm{v}(U)$ because $U$ is affine hence quasi-compact and distinguished opens form a basis [F3, F4], and finite intersections of members of a cover in $\mathrm{C}\mathrm{o}\mathrm{v}(U)$ are again distinguished opens, hence lie in $\mathcal{B}$.

Fix $U=D(g)\in \mathcal{B}$ and a cover in $\mathrm{C}\mathrm{o}\mathrm{v}(U)$ given by $h_1,\dots ,h_r\in A_g$ generating the unit ideal of the ring $A_g$. The Čech complex of this cover with values in $\mathcal{F}$ has terms ${⨁}_{i_0<\cdots <i_p}\mathcal{F}(D(h_{i_0}\cdots h_{i_p}))$ [F6]. By the affine quasi-coherent equivalence of [F6] and the associated-sheaf section formula [F5], the quasi-coherent restriction $\mathcal{F}{\mathrm{\mid }}_U$ is isomorphic to $\tilde{N}$ for $N=\mathcal{F}(U)$, so $\mathcal{F}(D(h_{i_0}\cdots h_{i_p}))\cong N_{h_{i_0}\cdots h_{i_p}}$; the Čech complex is therefore the augmented alternating complex of the ring $A_g$ and the module $N$ with respect to the generating elements $h_1,\dots ,h_r$, and [F2] shows that it is exact in every positive degree, that is, ${\check{H}}^p(\mathcal{U},\mathcal{F})=0$ for every $p>0$.

The basis $\mathcal{B}$, the assignment $\mathrm{C}\mathrm{o}\mathrm{v}$ and the sheaf $\mathcal{F}$ satisfy all hypotheses of [F1]: $\mathcal{B}$ contains $X$ and is closed under finite intersections, each $\mathrm{C}\mathrm{o}\mathrm{v}(U)$ is a nonempty cofinal family of finite covers with intersections in $\mathcal{B}$ by step 1.1, and the required positive Čech vanishing is step 2.1. Hence $H^q(U,\mathcal{F}{\mathrm{\mid }}_U)=0$ for every $U\in \mathcal{B}$ and every $q>0$; taking $U=X=D(1)$ gives $H^q(X,\mathcal{F})=0$ for every $q>0$, which is the statement.

Boundary cases: if $A=0$ then $X=\mathrm{\varnothing }$, the basis is $\mathcal{B}=\{\mathrm{\varnothing }\}$, $\mathrm{C}\mathrm{o}\mathrm{v}(\mathrm{\varnothing })$ contains the empty cover, whose Čech complex is the zero complex, so step 2.1 holds vacuously and [F1] gives the vanishing; if $\mathcal{F}=0$ the same steps apply with $N=0$. The Axiom of Choice is used through [F1], [F2] and the affine quasi-coherent equivalence in [F6], and no other selection is made.