Higher direct images localize over an affine base

Let $V=\mathrm{Spec}A\subseteq S$ be affine. Since $f$ is quasi-compact, $X_V:=f^{-1}V$ is quasi-compact, so choose a finite affine open cover $U_0,\dots ,U_r$ of $X_V$. Every $U_i$ lies over the affine open $V$ of $S$, and $X_V$ is separated over $V$; consequently, by the separation criterion [F6], for all tuples $i_0,\dots ,i_p$ the intersection $U_{i_0\dots i_p}=U_{i_0}\cap \cdots \cap U_{i_p}$ is affine (the empty intersection only arises for the empty tuple and is $X_V$ itself, which we do not need).

Fix $a\in A$ and put $X_a:=f^{-1}(D(a))\subseteq X_V$ and $U_i^a:=U_i\cap X_a$. Since $U_i\subseteq X_V$, the morphism $U_i\to V$ is a morphism of affine schemes, so $U_i^a$ is the distinguished open $D(a)\subseteq U_i=\mathrm{Spec}{\mathcal{O}}_X(U_i)$ and hence is affine; more generally $U_{i_0\dots i_p}^a=D(a)\subseteq U_{i_0\dots i_p}$ is affine by step 1.1. The finitely many $U_i^a$ cover $X_a$, so $X_a$ is quasi-compact, and $X_a$ is separated as an open subscheme of the separated $V$-scheme $X_V$ composed with the separated morphism $V\to \mathrm{Spec}\mathbb{Z}$ [F6]. Hence the Čech comparison [F4] applies to the finite affine cover $U_0^a,\dots ,U_r^a$ of $X_a$ and gives isomorphisms ${\check{H}}^q({\mathcal{U}}^a,\mathcal{F})⟶H^q(X_a,\mathcal{F}),q\ge 0.$

For every tuple $i_0,\dots ,i_p$ the ring ${\mathcal{O}}_X(U_{i_0\dots i_p}^a)={\mathcal{O}}_X(U_{i_0\dots i_p}{)}_a$ is the localisation at $a$, and by the affine equivalence [F3] the quasi-coherent module $\mathcal{F}{\mathrm{\mid }}_{U_{i_0\dots i_p}}=\widetilde{\mathcal{F}(U_{i_0\dots i_p})}$ has sections $\mathcal{F}(U_{i_0\dots i_p}^a)=\mathcal{F}(U_{i_0\dots i_p}{)}_a=\mathcal{F}(U_{i_0\dots i_p}){\otimes }_A{A}_a,$ and the Čech differentials, which are assembled from the restriction maps of $\mathcal{F}$, are the localisations of the corresponding maps of the complex $C^{\bullet }(\mathcal{U},\mathcal{F})$ for the cover $\mathcal{U}$ of $X_V$. Hence there are canonical isomorphisms of complexes of $A_a$-modules $C^p({\mathcal{U}}^a,\mathcal{F})\cong C^p(\mathcal{U},\mathcal{F}){\otimes }_A{A}_a,p\ge 0,$ and for $D(b)\subseteq D(a)$ the restriction of Čech cochains $C^p({\mathcal{U}}^a,\mathcal{F})\to C^p({\mathcal{U}}^b,\mathcal{F})$ corresponds to the canonical map $C^p(\mathcal{U},\mathcal{F}){\otimes }_A{A}_a\to C^p(\mathcal{U},\mathcal{F}){\otimes }_A{A}_b$.

Combining steps 2.1, 2.2 with the exactness of localisation [F7], for every $a\in A$ there are canonical isomorphisms of $A_a$-modules $H^q(X_a,\mathcal{F})\cong {\check{H}}^q({\mathcal{U}}^a,\mathcal{F})\cong H^q(C^{\bullet }(\mathcal{U},\mathcal{F}){\otimes }_A{A}_a)\cong H^q(C^{\bullet }(\mathcal{U},\mathcal{F})){\otimes }_A{A}_a,$ where the last identification is the canonical isomorphism of [F7]; for $a=1$ this gives $M:=H^q(X_V,\mathcal{F})\cong H^q(C^{\bullet }(\mathcal{U},\mathcal{F}))$ and hence, for every $a$, a canonical isomorphism ${\psi }_a:H^q(X_a,\mathcal{F})⟶M_a=H^q(X_V,\mathcal{F}{)}_a,$ $A_a$-linear and natural in $a$.

The isomorphisms ${\psi }_a$ are compatible with restriction: for $D(b)\subseteq D(a)$ the square $H^q(X_a,\mathcal{F})\stackrel{{\psi }_a}{\to }{M}_a,\text{restriction}\downarrow \downarrow \text{localisation},H^q(X_b,\mathcal{F})\stackrel{{\psi }_b}{\to }{M}_b$ commutes. Indeed the right column of step 3.1 is functorial in the coefficient ring by [F7], the middle terms are the cochain complexes of step 2.2 whose transition maps are localisation, and the left vertical map corresponds to the middle one under the comparison isomorphisms of step 2.1 by the compatibility of [F5] applied to the open subspace $X_b\subseteq X_a$.

Write $Q:=R^q{f}_{\ast }\mathcal{F}{\mathrm{\mid }}_V$, the restriction of the sheafification of the presheaf $P$ of [F1] to the affine open $V$, and let $\pi :P{\mathrm{\mid }}_V\to Q$ be the sheafification map. For each distinguished open $D(a)\subseteq V$ define ${\varphi }_a:M_a\stackrel{{\psi }_a^{-1}}{\to }{H}^q(X_a,\mathcal{F})=P(D(a))\stackrel{{\pi }_{D(a)}}{\to }Q(D(a))\mathrm{.}$ By step 4.1 and the fact that $\pi$ is a morphism of presheaves, the maps ${\varphi }_a$ are compatible with restrictions: for $D(b)\subseteq D(a)\subseteq V$ one has ${\varphi }_a(s){\mathrm{\mid }}_{D(b)}={\varphi }_b(s{\mathrm{\mid }}_{D(b)})$ for all $s\in M_a=M(D(a))$ by [F2]. Since the distinguished opens cover $V$ and $Q$ is a sheaf, gluing over covers by distinguished opens [F8] defines a unique morphism of ${\mathcal{O}}_V$-modules $\varphi :\tilde{M}⟶Q=R^q{f}_{\ast }\mathcal{F}{\mathrm{\mid }}_V$ whose component on $D(a)$ is ${\varphi }_a$.

The morphism $\varphi$ is an isomorphism. For $x\in V$ the distinguished opens containing $x$ are cofinal among its open neighbourhoods, so by [F1] and [F2] $Q_x={\mathrm{colim}}_{D(a)\ni x}P(D(a))\cong {\mathrm{colim}}_{D(a)\ni x}{M}_a={\tilde{M}}_x,$ where the middle isomorphism is induced by the natural isomorphisms ${\psi }_a$ of step 3.1, which are compatible with restriction by step 4.1. Under this identification the stalk map ${\varphi }_x$ is the identity, hence an isomorphism; as this holds for every $x\in V$, [F8] shows that $\varphi$ is an isomorphism of ${\mathcal{O}}_V$-modules.

Conclusion. For every affine open $V\subseteq S$ step 5.2 exhibits $(R^q{f}_{\ast }\mathcal{F}){\mathrm{\mid }}_V$ as the associated sheaf of the $A$-module $H^q(f^{-1}V,\mathcal{F})$, proving claim 2. Since the affine opens cover $S$, claim 1 follows from locality of quasi-coherence. Claim 3 is the description of sections of an associated sheaf on distinguished opens [F2] applied to the isomorphism of step 5.2. If $V=\mathrm{\varnothing }$ then $A=0$, $X_V=\mathrm{\varnothing }$, $P(D(a))=0=M_a$ and the identifications are trivial; the zero sheaf and degree $q=0$ are included since ${\psi }_a$ and $\varphi$ exist for every $q\ge 0$ and $M=H^0(X_V,\mathcal{F})=\mathcal{F}(X_V)$. The Axiom of Choice and the Axiom of Dependent Choice are inherited from [F1] and [F4] and from the choice of the finite cover in step 1.1; the remaining steps (localisation, gluing over the distinguished-open basis) make no further choices.