Use this tag for questions related to an affine variety over an algebraically-closed field.
In algebraic-geometry, an affine variety over an algebraically-closed field $k$ is the zero locus in the affine $n$-space $k^n$ 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 sub-variety 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 $k[x_1, \ldots, x_n]/I$ is called the coordinate ring of $X$. That ring is precisely the set of all regular functions on $X,$ i.e., the space of global sections of the structure sheaf of $X.$ A theorem of Serre gives a cohomological characterization of an affine variety: the theorem states that an algebraic variety is affine if and only if $H^i(X,F) = 0$ for any $i > 0$ and any quasi-coherent sheaf $F$ on $X.$ That makes the cohomological study of an affine variety non-existent in 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; i.e, 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.
Affine varieties are, up to an equivalence of categories, special cases of affine-schemes, which are precisely spectrums of a ring. In complex-geometry, affine varieties are analogs of Stein manifolds.
