Affine scheme/Cohomological criterion/Introduction/Section

A scheme ${}U$ is called affine if it is isomorphic to the spectrum of some commutative ring ${}R$. If the scheme is of finite type over a field (or a ring) ${}K$ (if we have a variety), then this is equivalent to saying that there exist global functions

${}g_{1},\ldots ,g_{m}\in \Gamma (U,{\mathcal {O}}_{U})\,$

such that the mapping

$U\longrightarrow {{\mathbb {A} }_{K}^{m}},x\longmapsto {\left(g_{1}(x),\ldots ,g_{m}(x)\right)},$

is a closed embedding. The relation to cohomology is given by the following well-known theorem of Serre.

Theorem

Let ${}X$ denote a noetherian scheme. Then the following properties are equivalent.

1. ${}X$ is an affine scheme.
2. For every quasicoherent sheaf ${}{\mathcal {F}}$ on ${}X$ and all ${}i\geq 1$ we have ${}H^{i}(X,{\mathcal {F}})=0$.
3. For every coherent ideal sheaf ${}{\mathcal {I}}$ on ${}X$ we have ${}H^{1}(X,{\mathcal {I}})=0$.

It is in general a difficult question whether a given scheme ${}U$ is affine. For example, suppose that ${}X=\operatorname {Spec} {\left(R\right)}$ is an affine scheme and

${}U=D({\mathfrak {a}})\subseteq X\,$

is an open subset (such schemes are called quasiaffine) defined by an ideal ${}{\mathfrak {a}}\subseteq R$. When is ${}U$ itself affine? The cohomological criterion above simplifies to the condition that ${}H^{i}(U,{\mathcal {O}}_{X})=0$ for ${}i\geq 1$.

Of course, if ${}{\mathfrak {a}}=(f)$ is a principal ideal (or up to radical a principal ideal), then

${}U=D(f)\cong \operatorname {Spec} {\left(R_{f}\right)}\,$

is affine. On the other hand, if ${}(R,{\mathfrak {m}})$ is a local ring of dimension ${}\geq 2$, then

${}D({\mathfrak {m}})\subset \operatorname {Spec} {\left(R\right)}\,$

is not affine, since

${}H^{d-1}(U,{\mathcal {O}}_{X})=H_{\mathfrak {m}}^{d}(R)\neq 0\,$

by the relation between sheaf cohomology and local cohomology and a Theorem of Grothendieck.