Serre's vanishing theorem on affine schemes

Remarks

The idea of the proof for $X = \operatorname{Spec} A$ with $A$ Noetherian: if $I$ is an injective $A$-module, the associated sheaf $\tilde{I}$ is flasque. Resolve $\mathcal{F} = \tilde{M}$ by $\tilde{I}^\bullet$ for an injective resolution $M \to I^\bullet$; this is a flasque resolution, and taking global sections gives back the exact complex $I^\bullet$. In general one uses that the Čech complex of a quasi-coherent sheaf for a cover by standard open sets $D(f_i)$ is exact in positive degrees.

Example: on a separated scheme $X$ with an affine open cover, the theorem shows that the cover is acyclic for quasi-coherent sheaves, so Čech cohomology of this cover computes $H^i(X, \mathcal{F})$.

The affine hypothesis is necessary: on $X = \mathbb{P}^1_k$, which is not affine, $H^1(X, \mathcal{O}(-2)) \cong k \ne 0$. Conversely, Serre's criterion says that a Noetherian scheme $X$ is affine if $H^1(X, \mathcal{F}) = 0$ for every quasi-coherent sheaf $\mathcal{F}$, and it suffices to ask this for coherent ideal sheaves.

Going deeper

For a student

The theorem is the base of every computation of coherent cohomology: one covers a separated scheme by affine open sets and computes the Čech complex. It gives the long exact sequence argument that a surjection of quasi-coherent sheaves on an affine scheme is surjective on global sections. With Serre's converse criterion, it characterizes affine schemes cohomologically, in the same way as Cartan's Theorem B for Stein manifolds.