nLab affine variety

Idea

Affine $k$-variety is a locus of zeros of a set of polynomials in the affine $n$-dimensional space $\mathbf{A}^n_k$. Usually $k$ is taken to be a field.

Definition

Given a field $k$, an affine $k$-variety is a maximal spectrum (= set of maximal ideals) of a finitely generated noetherian (commutative unital) $k$-commutative algebra without nilpotents, equipped with the Zariski topology; the algebra can be recovered as the coordinate ring of the variety; this correspondence is an equivalence of categories, if the morphisms are properly defined.

Affine varieties can be embedded as closed subvarieties into an affine space (in the sense of algebraic geometry). As topological spaces affine varieties are noetherian.

Properties

Cohomology

For $X$ an affine variety then its abelian sheaf cohomology with coefficients in the structure sheaf satisfies

$$H^{\bullet \geq 1}(X,\mathcal{O}_X) = 0 \,.$$

The converse requires in addition some finiteness condition. (Ballico 08).

Related concepts

• affine scheme

• spectrum of a commutative ring

• Stein space, Cartesian space

References

• E. Ballico, A characterization of affine varieties 2008