nLab Serre's criterion of affineness

Idea

Serre’s criterion of affineness characterizes affine morphisms of schemes in terms of exactness properties of the corresponding functors among the categories of quasicoherent sheaves.

If $f\colon X\to Y$ is a quasicompact morphism of algebraic schemes and $X$ is separated, then $f$ is affine iff it is cohomologically affine, that is, the direct image functor $f_*$ is exact.

Statement

EGA II Theorem 5.2.1 Let $X$ be a reduced quasicompact scheme or a noetherian scheme. The following conditions are equivalent

a) X is an affine scheme.

b) There is a family of elements $f_\alpha\in\Gamma(X,\mathcal{O}_X)$ such that (the corresponding principal localizations) $X_{f_\alpha}$ are affine and the ideal generated by $f_\alpha$-s in $A$ is $A$.

c) The functor $\Gamma(X,\mathcal{F})$ is exact in $\mathcal{F}$ in the category of quasicoherent $\mathcal{O}_X$-modules, in other words, if

$$0\rightarrow \mathcal{F}'\rightarrow\mathcal{F}\rightarrow\mathcal{F}''\rightarrow 0 \,\,\,\,\,\,\, (*)$$

is an exact sequence of quasicoherent $\mathcal{O}_X$-modules, the sequence

$$0\rightarrow \Gamma(X,\mathcal{F}')\rightarrow\Gamma(X,\mathcal{F})\rightarrow\Gamma(X,\mathcal{F}'')\rightarrow 0$$

is exact.

c)‘ The property c) holds for all exact sequences () of quasicoherent $\mathcal{O}_X$-modules such that $\mathcal{F}$ is a sub-$\mathcal{O}_X$-module of a finite product $\mathcal{O}_X^n$.

d) $H^1(X,\mathcal{F})=0$ for each quasi-coherent sheaf of $\mathcal{O}_X$-modules $\mathcal{F}$

d)‘ $H^1(X,\mathcal{I})=0$ for each quasi-coherent sheaf $\mathcal{I}$ of ideals in $\mathcal{O}_X$

EGA II Corollary 5.2.2 Let $f:X\to Y$ be a quasicompact separated morphism of schemes. The following conditions are equivalent:

a) $f$ is an affine homomorphism.

b) The functor $f_*$ is exact on the category of quasicoherent $\mathcal{O}_X$-modules.

c) For each quasi-coherent $\mathcal{O}_X$-module $\mathcal{F}$, we have $\mathbf{R}^1 f_*(\mathcal{F})=0$.

c’) For every quasi-coherent sheaf $\mathcal{I}$ of ideals in $\mathcal{O}_X$, we have $\mathbf{R}^1 f_*(\mathcal{I})=0$.

Literature

• See also EGA EGA IV 1.7.17

• Stacks project tag/0656

• Akhil Mathew, Serre’s criterion for affineness as Morita theory, a blog post