Projective n-space has quasi-coherent cohomological dimension at most n

Set $X={\mathbb{P}}_A^n$ and let $\mathcal{U}=(U_0,\dots ,U_n)$ be the ordered family of standard charts, with $U_i=\mathrm{Spec}{B}_i$ and $B_i=A[x_{\mathrm{\ell }}^{(i)}:\mathrm{\ell }\mathrm{\ne }i]$. By [F1] each $U_i$ is an open affine subscheme and the $U_i$ cover $X$; in particular $X$ is a scheme over $\mathrm{Spec}\mathbb{Z}$.

Every finite intersection of distinct members $U_{i_0},\dots ,U_{i_m}$ of $\mathcal{U}$ is affine: inside the affine chart $U_{i_0}$ it is ${\bigcap }_{\mathrm{\ell }=1}^{m}D(x_{i_{\mathrm{\ell }}}^{(i_0)})=D(x_{i_1}^{(i_0)}\cdots x_{i_m}^{(i_0)})$ by [F1] and [F2], and $D(f)$ is affine with ring $(B_{i_0}{)}_f$ by [F2]. For $m=0$ this is the affine chart $U_{i_0}$ itself.

$X$ is quasi-compact: $\pi :X\to \mathrm{Spec}A$ is of finite type, hence quasi-compact, so $X={\pi }^{-1}(\mathrm{Spec}A)$ is quasi-compact by [F3].

$X$ is separated as a scheme. Apply the criterion [F4] to the structure morphism $X\to \mathrm{Spec}\mathbb{Z}$, the single affine open $W=\mathrm{Spec}\mathbb{Z}$ of the base and the affine cover $\{U_i\}$ of $X$: for $j=k$ the ring $U_j$ is affine and $B_j{\otimes }_{\mathbb{Z}}{B}_j\to \mathrm{\Gamma }(U_j,{\mathcal{O}}_X)=B_j$ is the multiplication map, which is surjective; for $j\mathrm{\ne }k$ the intersection $U_j\cap U_k$ is affine by 1.2, $\mathrm{\Gamma }(U_j\cap U_k,{\mathcal{O}}_X)=(B_j{)}_{x_k^{(j)}}$ by [F1] and [F2], and the natural map $B_j{\otimes }_{\mathbb{Z}}{B}_k\to (B_j{)}_{x_k^{(j)}}$ sends $b\otimes 1\mapsto b$ and, by the reciprocal transition formula of [F1], $1\otimes x_m^{(k)}\mapsto x_m^{(j)}\mathrm{/}{x}_k^{(j)}$ for $m\mathrm{\ne }j,k$ and $1\otimes x_j^{(k)}\mapsto 1\mathrm{/}{x}_k^{(j)}$; its image therefore contains $B_j$ and the inverse $1\mathrm{/}{x}_k^{(j)}$, hence is the whole localisation $(B_j{)}_{x_k^{(j)}}=B_j[1\mathrm{/}{x}_k^{(j)}]$. The criterion gives that $X\to \mathrm{Spec}\mathbb{Z}$ is separated.

By [F5], applied to the quasi-compact separated scheme $X$, the finite affine open cover $U_0,\dots ,U_n$ and the quasi-coherent module $\mathcal{F}$, the canonical comparison map ${\check{H}}^q(\mathcal{U},\mathcal{F})\to H^q(X,\mathcal{F})$ is an isomorphism for every $q\ge 0$.

For $q>n$ the index set $\{0,\dots ,n\}$ of the ordered cover $\mathcal{U}$ has no increasing $(q+1)$-tuple, so $C^q(\mathcal{U},\mathcal{F})=0$ by [F6] and hence ${\check{H}}^q(\mathcal{U},\mathcal{F})=0$. By the isomorphism of 1.5, $H^q(X,\mathcal{F})=0$ for every $q>n$.

Boundary and choice accounting. For $n=0$ the cover is the single chart $U_0=\mathrm{Spec}A$, only $C^0$ is nonzero, and the conclusion $H^q=0$ for $q>0$ follows from 1.6 (it also follows from the affinity of $X=\mathrm{Spec}A$). For $A=0$ the ring $B_i$ is the zero ring, $U_i=\mathrm{Spec}0=\mathrm{\varnothing }$, so $X=\mathrm{\varnothing }$ by [F1] and $\{U_0,\dots ,U_n\}$ is still an affine open cover of $X$; the same argument applies verbatim, and $C^q=0$ for all $q$ by [F6] since $\mathcal{F}$ vanishes on the empty scheme. The case $\mathcal{F}=0$ is trivially included. Nothing is asserted for $q\le n$. The Axiom of Choice is a hypothesis, consumed exactly through the Čech comparison theorem [F5] (and the sheaf-cohomology framework it refers to); the remaining steps 1.1-1.4 and 1.6 make no choice, the indexes $0,\dots ,n$ being finite and the gluing data being fixed by the definition.