An affine scheme is a scheme that as a sheaf on the opposite category CRing of commutative rings (or equivalently as a sheaf on the subcategory of finitely presented rings) is representable. In a ringed space picture an affine scheme is a locally ringed space which is locally isomorphic to the prime spectrum of a commutative ring. Affine schemes form a full subcategory of the category of schemes.
The correspondence extends to a functor . The fundamental theorem on morphisms of schemes says that there is a bijection
In other words, for fixed , and for varying there is a restricted functor
and the functor from schemes to presheaves on is fully faithful. Thus the general schemes if defined as ringed spaces, indeed form a full subcategory of the category of presheaves on .
There is an analogue of this theorem for relative noncommutative schemes in the sense of Rosenberg.
There is no similar equation the other way round, that is “”. As a mnemonic, note that with ordinary Galois connections between power sets, one is always homming into (not out of) the functorial construction. More geometrically, consider the example and . Then the left hand side consists of all the -valued points of (of which there are many). On the other hand, the right hand side only contains the unique ring homomorphism , since .
A relative affine scheme over a scheme is a relative scheme isomorphic to the spectrum of a (commutative unital) algebra in the category of quasicoherent -modules; such a “relative” spectrum has been introduced by Grothendieck. It is characterized by the property that for every open the inverse image is an open affine subscheme of isomorphic to and such open affines glue in such a way that corresponds to the restriction morphism of algebras.
Relative affine scheme is a concrete way to represent an affine morphism of schemes.
Given a commutative unital ring there is an equivalence of categories between the category of -modules and the category of quasicoherent sheaves of -modules given on objects by where is the unique sheaf such that the restriction on the principal Zariski open subsets is given by the localization where is the principal Zariski open set underlying , and the restrictions are given by the canonical maps among the localizations. The action of is defined using a similar description of . Its right adjoint (quasi)inverse functor is given by the global sections functor .