In mathematics, a ringed space is a family of (commutative) ringss parametrized by open subsets of a topological space together with ring homomorphisms that play roles of restrictions. Precisely, it is a topological space equipped with a sheaf of rings called a structure sheaf. It is an abstraction of the concept of the rings of continuous (scalar-valued) functions on open subsets.

Among ringed spaces, especially important and prominent is a locally ringed space: a ringed space in which the analogy between the stalk at a point and the ring of germs of functions at a point is valid.

Ringed spaces appear in analysis as well as complex algebraic geometry and the scheme theory of algebraic geometry.

Note: In the definition of a ringed space, most expositions tend to restrict the rings to be commutative rings, including Hartshorne and Wikipedia. Éléments de géométrie algébrique, on the other hand, does not impose the commutativity assumption, although the book mostly considers the commutative case.

Definitions

A ringed space is a topological space together with a sheaf of rings is called the structure sheaf.

A locally ringed space such that all stalks are local rings (i.e. they have unique maximal ideals). Note that it is not; in fact, this is almost never the case.

Examples

An arbitrary topological space can be considered a locally ringed space by taking to be the sheaf of real-valued (or complex-valued) continuous functions on open subsets of . The stalk can be thought of as the set of all germss of continuous functions at ; this is a local ring with the unique maximal ideal consisting of those germs whose value at .

If is a manifold with some extra structure, we can also take the sheaf of differentiable, or holomorphic functions. Both of these give rise to locally ringed spaces.

If is an algebraic variety carrying the Zariski topology to be the ring of rational mappings defined on the Zariski-open set . The important generalization of this example is that of the spectrum of any commutative ring; these spectra are also locally ringed spaces. Schemes are locally ringed spaces obtained by "gluing together" spectra of commutative rings.

Morphisms

A morphism is a continuous map is a morphism to the direct image of the structure sheaf of is given by the following data:

• a continuous map

• a family of ring homomorphisms for every open set, then the following diagram must commute (the vertical maps are the restriction homomorphisms):

There is an additional requirement for morphisms between locally ringed spaces:

• between the stalks of and the stalks of must be local homomorphisms, i.e. for every is mapped into the maximal ideal of the local ring at .

Two morphisms can be composed to form a new morphism, and we obtain the category of ringed spaces and the category of locally ringed spaces. Isomorphisms in these categories are defined as usual.

Tangent spaces

Locally ringed spaces have just enough structure to allow the meaningful definition of tangent spaces. Let be a locally ringed space with structure sheaf at the point is a field is a vector space over that field (the cotangent space is defined as the dual of this vector space.

The idea is the following: a tangent vector at should tell you how to "differentiate" "functions" at , i.e. the elements of . Now it is enough to know how to differentiate functions whose value at is zero, since all other functions differ from these only by a constant, and we know how to differentiate constants. So we only need to consider . Furthermore, if two functions are given with value zero at , then their product has derivative 0 at , by the product rule, and this is what the dual space does.

Modules over the structure sheaf

, certain sheaves of modules on occur in the applications, the of abelian groups on is a module over the ring for every open set in an -module. In this case, the stalk of at will be a module over the local ring (stalk) , for every .

A morphism between two such -modules is a morphism of sheaves that is compatible with the given module structures. The category of is an abelian category.

An important subcategory of the category of -modules is the category of quasi-coherent sheaves on . A sheaf of -modules is called quasi-coherent if it is, locally, isomorphic to the cokernel of a map between free -modules. A coherent sheaf is a quasi-coherent sheaf that is, locally, of finite type and for every open subset of the kernel of any morphism from a free -module of finite rank to is also of finite type.

Citations

References

• Section 0.4 of

External links