Ringed space

A topological space with a sheaf of rings . The sheaf is called the structure sheaf of the ringed space . It is usually understood that is a sheaf of associative and commutative rings with a unit element. A pair is called a morphism from a ringed space into a ringed space if is a continuous mapping and is a homomorphism of sheaves of rings over which transfers units in the stalks to units. Ringed spaces and their morphisms constitute a category. Giving a homomorphism is equivalent to giving a homomorphism

which transfers unit elements to unit elements.

A ringed space is called a local ringed space if is a sheaf of local rings (cf. Local ring). In defining a morphism between local ringed spaces it is further assumed that for any , the homomorphism

is local. Local ringed spaces form a subcategory in the category of all ringed spaces. Another important subcategory is that of ringed spaces over a (fixed) field , i.e. ringed spaces where is a sheaf of algebras over , while the morphisms are compatible with the structure of the algebras.

Examples of ringed spaces.

1) For each topological space there is a corresponding ringed space , where is the sheaf of germs of continuous functions on .

2) For each differentiable manifold (e.g. of class ) there is a corresponding ringed space , where is the sheaf of germs of functions of class on ; moreover, the category of differentiable manifolds is a full subcategory of the category of ringed spaces over .

3) The analytic manifolds (cf. Analytic manifold) and analytic spaces (cf. Analytic space) over a field constitute full subcategories of the category of ringed spaces over .

4) Schemes (cf. Scheme) constitute a full subcategory of the category of local ringed spaces.

References

[1]	I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian)
[2]	R. Hartshorne, "Algebraic geometry" , Springer (1977)

Comments

If is a sheaf over a topological space and is a mapping of topological spaces, then the induced sheaf over is the sheaf defined by for all open .