Coordinate ring

In algebraic geometry, an affine variety over an algebraically closed field k is the zero-locus in the affine n-space ${\displaystyle k^{n}}$ 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 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

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

for any ${\displaystyle i>0}$ 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.

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