nLab
affine variety

Contents

Idea

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

Definition

Given a field kk, an affine kk-variety is a maximal spectrum (= set of maximal ideals) of a finitely generated noetherian (commutative unital) kk-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 varietes 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 XX an affine variety then its abelian sheaf cohomology with coefficients in the structure sheaf satisfies

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

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

References

  • E. Ballico, A characzerization of affine varieties 2008

Revised on May 28, 2014 08:23:44 by Urs Schreiber (89.204.154.192)