What is "affine variety"

n. (context algebraic geometry English) A set of points (in ''n''-dimensional space) which satisfy a set of equations which have a polynomial of ''n'' variables on one side and a zero on the other side.

In algebraic geometry, an affine variety over an algebraically closed field k is the zero-locus in the affine n-space k of some finite family of polynomials of n variables with coefficients in k that generate a prime ideal. If the condition of generating a prime ideal is removed, such a set is called an (affine) algebraic set. A Zariski open subvariety of an affine variety is called a quasi-affine variety.

If X is an affine variety defined by a prime ideal I, then the quotient ring

k[x, …, x]/I
is called the coordinate ring of X. This ring is precisely the set of all regular functions on X; in other words, it is the space of global sections of the structure sheaf of X. A theorem of Serre gives a cohomological characterization of an affine variety; it says an algebraic variety is affine if and only if

H(X, F) = 0
for any i > 0 and any quasi-coherent sheaf F on X. (cf. Cartan's theorem B.) This makes the cohomological study of an affine variety non-existent, in a sharp contrast to the projective case in which cohomology groups of line bundles are of central interest.

An affine variety plays a role of a local chart for algebraic varieties; that is to say, general algebraic varieties such as projective varieties are obtained by gluing affine varieties. Linear structures that are attached to varieties are also (trivially) affine varieties; e.g., tangent spaces, fibers of algebraic vector bundles.

An affine variety is, up to an equivalence of categories a special case of an affine scheme, which is precisely the spectrum of a ring. In complex geometry, an affine variety is an analog of a Stein manifold.