Finiteness of cohomology of coherent sheaf for proper morphisms

So, I gave you a reference, but I think (it would need a verification) that the argument can be shortened as follows (some nontrivial prerequisites of homological algebra and algebraic geometry are still needed – but it might be easier than the complete Stacks argument).

The cohomology LES gives you a “two implies three” statement for exact sequences of coherent sheaves $0 \rightarrow F \rightarrow G \rightarrow H \rightarrow 0$. Moreover, we can assume by induction that it’s true for coherent sheaves supported on a proper closed subset of $X$.

Lemma 1: let $F$ be a coherent sheaf on a Noetherian scheme $X$, $U$ be an open subset, $G \subset F_{|U}$ be a coherent subsheaf. Then $G=H_{|U}$ for some coherent subsheaf $H$ of $F$.

Proof (after stacks, Lemma 28.22.2, among others): assume first $X$ affine, $X=\mathrm{Spec}\,A$. Then write $M=F(X)$, $U=\bigcup_i{D(f_i)}$ a finite reunion, $G(D(f_i))=M_i$. Let $N$ be the $A$-submodule made of the $a \in M$ such that $a_{|D(f_i)} \in G(D(f_i))$ for each $i$. Then $\tilde{N}_{|U} \subset G$. To prove the equality, it is enough to show that $\tilde{N}(D(f_i))$ surjects onto $G(D(f_i))$ for all $i$. But, if $a \in G(D(f_i))$, as $U$ is quasi-compact, it means that $af_i^n$ extends to a global section of $G$ for all large enough $n$. But $af_i^n$ (as $a \in F(D(f_i))$) also extends for all large enough $n$ to a global section of $F$. So said section lies in $N$ and thus $a \in N(D(f_i))$.

Assume next $X=U \cap V$, $V$ affine. Then we can find (by the preceding case) a subsheaf $H \subset F_{|V}$ such that $H_{|U\cap V}=G_{|U \cap V}$ and then glue $G$ and $H$.

In general, write $X=U \cup V_1\cup V_2\cup \ldots \cup V_p$ where the $V_i$ are affine. Write $U_i=U \cup V_1 \cup \ldots \cup V_i$. By the case above, we can extend $G$ from $U_i$ to $U_{i+1}$.

Lemma 2: in this setting (we come back to a proper scheme $X$ over an affine Noetherian scheme $S=\mathrm{Spec}\,A$ and we assume by Noetherian induction that the cohomology groups of all coherent sheaves supported on proper closed subsets are finitely generated), let $F,F’$ be two coherent sheaves on $X$, $U$ a nonempty open subset and $i: F_{|U} \rightarrow F’_{|U}$ an isomorphism. Then, for all $p \geq 0$, $H^p(X,F)$ is finitely generated iff $H^p(X,F’)$ is.

Proof: let $G_1=F\oplus F’$, there exists a coherent subsheaf $H$ whose restriction to $U$ is the graph of $i$. Let $G=G_1/H$. Now, both injections $F \rightarrow G,F’\rightarrow G$ are isomorphisms above $U$, so their kernels and cokernels are supported on a proper closed subset of $X$. In particular, studying the cohomology LES, it follows $H^p(X,F)$ finitely generated iff $H^p(X,G)$ finitely generated iff $H^p(G,F’)$ finitely generated.

Now, let $\pi: X’ \rightarrow X$ be a “Chow map”, ie projective, surjective, with $X’/S$ projective and an isomorphism above some everywhere dense open subset $U$ of $X$. Let $L$ be a large tensor power of an ample line bundle on $X’$, so that for every affine $V \subset X$, $L_{|\pi^{-1}(V)}$ is acyclic and $L$ is acyclic. We’ll show that $\pi_*L$ is acyclic.

Consider an injective resolution $L \rightarrow I_{\cdot}$. In the category of abelian sheaves, $\pi_*$ is the right adjoint of a left exact functor ($\pi^{-1}$) so formally $\pi_*$ maps injectives to injectives. Moreover, the cohomology of $\pi_*I_{\cdot}$ is, above every affine open subset $V \subset X$, the Cech cohomology of $L_{|\pi^{-1}(V)}$, so is concentrated in degree $0$, and thus the complex $\pi_*I_{\cdot}$ is still exact and thus an injective resolution of $L’=\pi_*L$. The cohomology of its image under $f_*$ computes $H^{\cdot}(X’,L)$ (which is in degree $0$) and also the cohomology of $\pi_*L$, and thus $\pi_*L$ is acyclic.

Up to restricting $U$, we may assume that $U$ is affine (take small open subsets in every irreducible component that are pairwise disjoint) and $L$ is free on $U$. In particular, it follows that for every coherent sheaf $F$ on $X$, we have a surjective map above $U$ denoted as $\psi: (\pi_*L)^{\oplus r}_{|U} \rightarrow F_{|U}$. There is a coherent subsheaf $K \subset (\pi_*L)^{\oplus r}$ such that $K_{|U}$ is the kernel of $\psi$. Then let $F’= (\pi_*L)^{\oplus r}/K$. As $F_{|U}$ and $F’_{|U}$ are isomorphic, it follows that $H^p(X,F)$ is finitely generated iff $H^p(X,F’)$ is finitely generated (by Lemma 2) iff $H^{p+1}(X,K)$ is finitely generated.

The conclusion follows by “descending induction” and remembering that Cech cohomology of quasi-coherent sheaves vanishes in large enough degree (independently of the quasi-coherent sheaf of interest).