homomorphism sends to a polynomial in . This corresponds to the morphism of varieties defined by Structure sheaf Equipped with the structure sheaf described below, an affine variety is a locally ringed space. Given an affine variety X with coordinate ring A, the sheaf of k-algebras is defined by letting be the ring of regular functions on U. Let D(f) = { x | f(x) ≠ 0 } for each f in A. They form a base for the topology of X and so is determined by its values on the open sets D(f). (See also: sheaf of modules#Sheaf associated to a module.) The key fact, which relies on Hilbert nullstellensatz in the essential way, is the following: Proof: The inclusion ⊃ is clear. For the opposite, let g be in the left-hand side and , which is an ideal. If x is in D(f), then, since g is regular near x, there is some open affine neighborhood D(h) of x such that ; that is, hm g is in A and thus x is not in V(J). In other words, and thus the Hilbert nullstellensatz implies f is in the radical of J; i.e., . The claim, first of all, implies that X is a "locally ringed" space since where . Secondly, the claim implies that is a sheaf; indeed, it says if a function is regular (pointwise) on D(f), then it must be in the coordinate ring of D(f); that is, "regular-ness" can be patched together. Hence, is a locally ringed space. Serre's theorem on affineness 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 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. Affine algebraic groups An affine variety over an algebraically closed field is called an affine algebraic group if it has: A multiplication , which is a regular morphism that follows the associativity axiom—that is, such that for all points , and in An identity element such that for every in An inverse morphism, a regular bijection such that for every in Together, these define a group structure on the variety. The above morphisms are often written using