Affine Scheme

An affine scheme is the ring spectrum of a commutative ring with identity, regarded as a locally ringed space with a structure sheaf. Its underlying set consists of the prime ideals of . A locally ringed space that is locally isomorphic to an affine scheme is called a scheme (Itô 1986, p. 69). An affine scheme is a generalization of the notion of affine variety, where the coordinate ring is replaced by any commutative unit ring. The ring spectrum carries the Zariski topology and its structure sheaf; an arbitrary topological space is not sufficient.

For an affine scheme , the ring of global sections of its structure sheaf recovers , so (Stacks Project, Tag 01HV). The structure sheaf carries information that is not determined by the underlying topological space. For example, if is a field, then and the affine scheme each have one point, but they are not isomorphic. The latter has a nonzero nilpotent element in its coordinate ring, which is the ring of dual numbers over .