Finite-support coherent sheaves have finite-dimensional global sections

By [F1], the cohomology groups in the Statement are finite-dimensional under the stated compactness and coherence assumptions. If $S=\mathrm{\varnothing }$, [F4] makes $\mathcal{F}=0$, so all claims hold with $\mathrm{\ell }=0$. Suppose $S\mathrm{\ne }\mathrm{\varnothing }$. When $r=0$, $S$ consists of the unique point of $P^0$; take its neighborhood to be all of $P^0$. When $r\ge 1$, [F5] and finite Hausdorff separation give pairwise disjoint open neighborhoods $V_x$ of the points of $S$. In either case, on each punctured neighborhood $V_x\setminus \{x\}$ every stalk of $\mathcal{F}$ vanishes by the definition of $S$, so the restriction there is the zero sheaf by [F4].

For each $x\in S$, the germ map from $\mathcal{F}{\mathrm{\mid }}_{V_x}$ to the skyscraper sheaf $i_{x,\ast }{\mathcal{F}}_x$ is an isomorphism. It is injective on sections because a section with zero germ at $x$ and zero stalk at every other point has zero germ everywhere, hence is zero by the uniqueness clause [F6]. It is surjective on sections over an open $W\subseteq V_x$ containing $x$: represent a desired germ by a section on a smaller neighborhood of $x$, restrict to the punctured overlap where $\mathcal{F}$ is zero, and glue that section to zero on $W\setminus \{x\}$ using [F6]. For $W$ not containing $x$, both section groups are zero.

The open complement $P^r\setminus S$ has zero restriction of $\mathcal{F}$, and the neighborhoods $V_x$ are pairwise disjoint. The local isomorphisms of step 2.1 therefore glue to a sheaf isomorphism $\mathcal{F}\cong {⨁}_{x\in S}{i}_{x,\ast }{\mathcal{F}}_x$, where the sum is finite. Each skyscraper is flasque by [F2], since every restriction is either the identity or the map onto $0$; hence their finite direct sum is flasque. By [F3] and the ambient AC [A1], $H^q(P^r,\mathcal{F})=0$ for $q>0$.

Taking global sections of the finite direct-sum decomposition in step 3.1 and using [F7] gives $H^0(P^r,\mathcal{F})=\mathrm{\Gamma }(P^r,\mathcal{F})\cong {⨁}_{x\in S}{\mathcal{F}}_x$. Its dimension is the finite sum $\mathrm{\ell }$ from the Statement.