Finite coherent cohomology for proper schemes

The cover. Since $f$ is proper it is separated and of finite type by [F1], hence quasi-compact, and $\mathrm{Spec}A$ is quasi-compact, so $X=f^{-1}(\mathrm{Spec}A)$ is quasi-compact. If $X=\mathrm{\varnothing }$, take the empty cover and put $r=-1$, $n=0$. Otherwise take a finite affine open cover, discard any empty members, and reindex it as $U_0,\dots ,U_r$; it still covers $X$, has $r\ge 0$, and has $n=r+1\ge 1$ members.

The intersections. For every nonempty finite subset $I\subseteq \{0,\dots ,r\}$ the intersection $U_I:={\bigcap }_{i\in I}{U}_i$ is affine, say $U_I=\mathrm{Spec}{B}_I$ with $B_I={\mathcal{O}}_X(U_I)$, and if such an intersection is empty, it is the affine scheme $\mathrm{Spec}0$; in particular every $U_I$ maps into the affine base $\mathrm{Spec}A$.

The bounded Čech complex. The scheme $X$ is quasi-compact and separated by 1.1 and [F1], the cover is finite affine with affine intersections by 1.2, and the coherent $\mathcal{F}$ is quasi-coherent, so [F3] applies: the ordered Čech complex $K^{\bullet }:=C^{\bullet }(\mathcal{U},\mathcal{F})$ of $\mathcal{U}=(U_0,\dots ,U_r)$ satisfies $K^p=0$ for $p<0$ and for $p>r$, and its cohomology computes the sheaf cohomology, $H^p(K^{\bullet })\cong {\check{H}}^p(\mathcal{U},\mathcal{F})\cong H^p(X,\mathcal{F})$ for every $p\ge 0$.

Vanishing above the length of the cover. The complex $K^{\bullet }$ is concentrated in degrees $0,\dots ,r$ by 1.3, so $H^p(K^{\bullet })=0$ for every $p>r$, and therefore $H^p(X,\mathcal{F})=0$ for every $p>r$; with $n=r+1$ when $X\mathrm{\ne }\mathrm{\varnothing }$ this is the assertion $H^p(X,\mathcal{F})=0$ for $p\ge n$, and when $X=\mathrm{\varnothing }$ one has $K^{\bullet }=0$ and all cohomology vanishes, so $H^p(X,\mathcal{F})=0$ for every $p\ge 0=n$.

Finiteness in every degree. Since $A$ is Noetherian, $f$ is proper and $\mathcal{F}$ is coherent, [F4] shows that each $R^q{f}_{\ast }\mathcal{F}$ is a coherent ${\mathcal{O}}_{\mathrm{Spec}A}$-module; by [F4] again, applied to the affine open $\mathrm{Spec}A$, there is a canonical isomorphism $R^q{f}_{\ast }\mathcal{F}\cong \widetilde{H^q(X,\mathcal{F})}$. Hence the associated sheaf $\widetilde{H^q(X,\mathcal{F})}$ is coherent, hence of finite type by [F5], and since $A$ is Noetherian [F5] gives that $H^q(X,\mathcal{F})$ is a finitely generated $A$-module. This holds for every $q\ge 0$, in particular for $q=0$.

Boundaries and choice accounting. If $X=\mathrm{\varnothing }$ the cover is empty and 1.4 gives $H^q(X,\mathcal{F})=0$ for all $q\ge 0$, which is finitely generated and has the asserted vanishing with $n=0$. If $\mathcal{F}=0$ then all cohomology vanishes and both claims hold for every cover. The zero ring $A=0$ is Noetherian and $\mathrm{Spec}0=\mathrm{\varnothing }$, so $X=\mathrm{\varnothing }$ and the empty case applies. The one-member case $n=1$ is a proper affine source: then $r=0$, and 1.4 gives $H^q(X,\mathcal{F})=0$ for $q\ge 1$, the classical affine vanishing consistent with 1.3. Degree $q=0$ is treated in 1.5 together with all other degrees; the vanishing bound $n$ is the number of members of a chosen cover, depends only on $X$, and is unaffected by $\mathcal{F}$. The Axiom of Choice [F6] is used for the finite affine cover of 1.1, the associated-sheaf and localisation machinery of [F5] and the higher-direct-image and Čech-comparison suppliers of [F3] and [F4]; the Axiom of Dependent Choice [F6] is inherited from those same suppliers and from the localisation and distinguished-open arguments of [F5].