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 .
4) Schemes (cf. Scheme) constitute a full subcategory of the category of local ringed spaces.
|||I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) MR0447223 Zbl 0362.14001|
|||R. Hartshorne, "Algebraic geometry" , Springer (1977) MR0463157 Zbl 0367.14001|
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 .
Ringed space. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Ringed_space&oldid=30763