Sheaf of algebras


In algebraic geometry, a sheaf of algebras on a ringed space X is a sheaf of commutative rings on X that is also a sheaf of -modules. It is quasi-coherent if it is so as a module.
When X is a scheme, just like a ring, one can take the global Spec of a quasi-coherent sheaf of algebras: this results in the contravariant functor from the category of quasi-coherent -algebras on X to the category of schemes that are affine over X. Moreover, it is an equivalence: the quasi-inverse is given by sending an affine morphism to

Affine morphism

A morphism of schemes is called affine if has an open affine cover 's such that are affine. For example, a finite morphism is affine. An affine morphism is quasi-compact and separated; in particular, the direct image of a quasi-coherent sheaf along an affine morphism is quasi-coherent.
The base change of an affine morphism is affine.
Let be an affine morphism between schemes and a locally ringed space together with a map. Then the natural map between the sets:
is bijective.

Examples

Given a ringed space S, there is the category of pairs consisting of a ringed space morphism and an -module. Then the formation of direct images determines the contravariant functor from to the category of pairs consisting of an -algebra A and an A-module M that sends each pair to the pair.
Now assume S is a scheme and then let be the subcategory consisting of pairs such that is an affine morphism between schemes and a quasi-coherent sheaf on. Then the above functor determines the equivalence between and the category of pairs consisting of an -algebra A and a quasi-coherent -module.
The above equivalence can be used to do the following construction. As before, given a scheme S, let A be a quasi-coherent -algebra and then take its global Spec:. Then, for each quasi-coherent A-module M, there is a corresponding quasi-coherent -module such that called the sheaf associated to M. Put in another way, determines an equivalence between the category of quasi-coherent -modules and the quasi-coherent -modules.