Finite-dimensional coherent cohomology on compact projective space

If $r=0$, $P^0(\mathbb{C})$ is a point and ${\mathcal{O}}_{P^0}=\mathbb{C}$. Coherence makes its module of global sections finitely generated over $\mathbb{C}$, hence finite-dimensional. Sheaves on a point are abelian groups and global sections is the identity functor, so positive right-derived global sections vanish. Every finite-presentation quotient topology on this finite-dimensional space is Hausdorff; finite-dimensional complex vector spaces have a unique Hausdorff locally convex topology, so the cover/presentation topology is independent.

Suppose $1\le r\le 3$ and fix $x\in P^r$. Apply [A1] to $\mathcal{F}$ at $x$ with length $2r$ to obtain a chart neighborhood $W_0$ with a partial free resolution. Its first relation kernel $K_0$ is coherent by [A1]. Apply [A1] again to $K_0$ at $x$ with length $2r$, and choose a neighborhood $W_1\subseteq W_0$ on which that kernel resolution is defined. Apply [A2] to $\mathcal{F}$ at $x$ to obtain a polydisc neighborhood $W_2$. Choose concentric coordinate polydiscs $D_x^{\mathrm{\prime }}⋐D_x$ with $\overline{D_x}\subseteq W_1\cap W_2$. Then $D_x^{\mathrm{\prime }}$ is a neighborhood of $x$.

For any finite family of centers $x_1,\dots ,x_s$ with nested pairs $D_{x_i}^{\mathrm{\prime }}⋐D_{x_i}$ as in the local construction above and with the $D_{x_i}^{\mathrm{\prime }}$ covering $P^r$, put $U_i=D_{x_i}$ and $U_i^{\mathrm{\prime }}=D_{x_i}^{\mathrm{\prime }}$. Conditional on such a choice, the refinement from $\mathcal{U}=(U_i)$ to ${\mathcal{U}}^{\mathrm{\prime }}=(U_i^{\mathrm{\prime }})$ uses the same index set and sends each index $i$ to itself. In every degree its cochain map is componentwise restriction $\mathrm{\Gamma }(U_I,\mathcal{F})\to \mathrm{\Gamma }(U_I^{\mathrm{\prime }},\mathcal{F})$. It commutes with the Čech differential because restriction of sections commutes with each term of the alternating sum [F8].

The sets $D_x^{\mathrm{\prime }}$ cover $P^r$. Compactness [F1] gives finitely many centers $x_1,\dots ,x_s$ whose smaller polydiscs still cover; set $U_i=D_{x_i}$ and $U_i^{\mathrm{\prime }}=D_{x_i}^{\mathrm{\prime }}$. Each singleton $U_i$ and $U_i^{\mathrm{\prime }}$ is $\mathcal{F}$-acyclic by [A2]. For every nonempty finite intersection of at least two of the $U_i$, the intersection lemma [F4] applies using the length-$2r$ resolution of $\mathcal{F}$ on the closure of any participating $U_i$. It also applies to the corresponding intersections of the $U_i^{\mathrm{\prime }}$, since their closures lie inside those same resolution neighborhoods. Thus both finite covers are acyclic for $\mathcal{F}$.

On any nonempty intersection $V=U_I$ or $V=U_I^{\mathrm{\prime }}$, use the first presentation of $\mathcal{F}$ from a chosen participating $U_{i_0}$. Its relation kernel is $K_0{\mathrm{\mid }}_V$, and the same local Oka construction in step 1.2 gives a length-$2r$ resolution of $K_0$ on a neighborhood of $\overline{U_{i_0}}$. Applying [F4] to $K_0$ on $V$ gives $H^1(V,K_0)=0$. The long exact sequence of $0\to K_0\to {\mathcal{O}}_V^{p_0}\to \mathcal{F}{\mathrm{\mid }}_V\to 0$ then makes the section map $\mathrm{\Gamma }(V,{\mathcal{O}}_V^{p_0})\to \mathrm{\Gamma }(V,\mathcal{F})$ surjective. Therefore [F3] supplies the required Fréchet section and Čech topologies on both covers.

For every index set $I$, $\overline{U_I^{\mathrm{\prime }}}\subseteq {\bigcap }_{i\in I}\overline{U_i^{\mathrm{\prime }}}\subseteq U_I$, and this closure is compact because it is a closed subset of the finite intersection of the compact sets $\overline{U_i^{\mathrm{\prime }}}$. Every nonzero component of the cochain map in step 1.3 is therefore a restriction with relatively compact target, hence compact by [F5]. For the finitely many components in a fixed degree, choose in each source a zero-neighborhood whose restriction has compact closure; their finite product is a zero-neighborhood and the product of those compact closures is compact. Thus the full cochain map in that degree is compact.

Both covers in step 2.1 are acyclic for $\mathcal{F}$. By [F7], the induced map on Čech cohomology is the identity comparison through $H^q(P^r,\mathcal{F})$, hence an isomorphism in every degree. In particular it is surjective.

Each Čech cochain space is a finite product of Fréchet section spaces by [F3], and restrictions are continuous, so the two Čech cochain complexes are Fréchet complexes. By steps 3.2–3.3, their cochain map is compact in degree $q$ and surjective on degree-$q$ cohomology. Apply [F6] to conclude that $H^q(C^{\bullet }({\mathcal{U}}^{\mathrm{\prime }},\mathcal{F}))$ is Hausdorff and finite-dimensional. The Leray comparison [F7] transfers both conclusions to $H^q(P^r,\mathcal{F})$. If $q$ is at least the number of cover members, the Čech cochain space is zero and the conclusion is immediate as well.

To compare two finite admissible covers, retain the given presentation ${\pi }_I$ on every nonempty intersection $U_I$ of either parent cover, and write $K_I=\ker {\pi }_I$; there are only finitely many such coherent kernels by [A1]. First shrink each parent cover to open sets $B_i$ with $\overline{B_i}\subset U_i$, still covering $P^r$: at every point choose a coordinate polydisc with compact closure in some parent member, take a finite subcover by compactness [F1], and for each parent member take the union of its finitely many assigned polydiscs. Do this for both parent covers. At a point $x$, repeat step 1.2 for $\mathcal{F}$ and its new presentation kernel. In addition, for every parent intersection $U_I$ containing $x$, use [A1] to choose a length-$2r$ resolution neighborhood of its particular kernel $K_I$. Intersect these finitely many neighborhoods. Choose nested polydiscs $D_x^{\mathrm{\prime }}⋐D_x$ whose closures lie in this intersection and in one selected $B_i$ from each parent cover. Also choose $\overline{D_x}$ disjoint from every $\overline{B_j}$ whose parent $U_j$ does not contain $x$; those are finitely many closed sets missing $x$. Compactness gives a finite cover $\mathcal{W}$ by some of the $D_x^{\mathrm{\prime }}$, with the selected parent assignments. As in steps 1.2, 2.1 and 3.1, its intersections are $\mathcal{F}$-acyclic and have surjective finite-presentation section maps. Now take a nonempty refined intersection $T=D_{x_0}^{\mathrm{\prime }}\cap \cdots \cap D_{x_k}^{\mathrm{\prime }}$ and one parent cover. Let $I$ be its distinct assigned parent indices. Since $D_{x_0}^{\mathrm{\prime }}$ meets each selected $B_i$, the avoidance condition implies $x_0\in U_i$ for every $i\in I$. Hence the closure of $D_{x_0}^{\mathrm{\prime }}$ lies inside a resolution neighborhood for this very $K_I$. Applying [F4] to $K_I$ on $T$ gives $H^1(T,K_I)=0$, so [F9] proves surjectivity of the section map of the SAME parent presentation ${\pi }_I{\mathrm{\mid }}_T$. Its section map on $U_I$ is surjective by parent admissibility. Moreover $\bar{T}$ is compact in $U_I$, since each refined polydisc has compact closure in its assigned parent. Thus [F5] applies to $\mathrm{\Gamma }(U_I,\mathcal{F})\to \mathrm{\Gamma }(T,\mathcal{F})$ using that common presentation. By [F3], the resulting target quotient topology agrees with the chosen one on $T$. This works for both parent covers and all tuples; no newly chosen presentation is substituted for a parent presentation without proving its section surjectivity.

For either parent cover, the standard refinement cochain map to $\mathcal{W}$ sends a component to its restriction with the sign of the permutation that sorts the parent indices, and sends it to zero when two parent indices repeat. The alternating formula [F8] shows it is a cochain map: with distinct parents, deletion commutes with restriction and permutation signs; with a repeated parent, the two faces deleting that repeated pair cancel. Each nonzero component restriction is continuous by [F3] and compact by [F5]; its refined intersection has compact closure in the corresponding parent intersection by step 4.2. The covers are finite, so the finite-product argument of step 3.2 makes both cochain maps compact. Their induced cohomology maps are isomorphisms by [F7], hence surjective and open by [F6]; the maps themselves are continuous as cochain maps, so these bijections are topological isomorphisms. Therefore the two original Čech quotient topologies agree.