# Affine scheme/Cohomological criterion/Introduction/Section

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

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

such that the mapping

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

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

## Theorem

Let ${\displaystyle {}X}$ denote a noetherian scheme. Then the following properties are equivalent.
1. ${\displaystyle {}X}$ is an affine scheme.
2. For every quasicoherent sheaf ${\displaystyle {}{\mathcal {F}}}$ on ${\displaystyle {}X}$ and all ${\displaystyle {}i\geq 1}$ we have ${\displaystyle {}H^{i}(X,{\mathcal {F}})=0}$.
3. For every coherent ideal sheaf ${\displaystyle {}{\mathcal {I}}}$ on ${\displaystyle {}X}$ we have ${\displaystyle {}H^{1}(X,{\mathcal {I}})=0}$.

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

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

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

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

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

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

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

is not affine, since

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

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