Ringed space

In mathematics, a ringed space is a family of (commutative) rings 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.

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

Éléments de géométrie algébrique, on the other hand, does not impose the commutativity assumption, although the book mostly considers the commutative case.

are local rings (i.e. they have unique maximal ideals).

be a local ring for every open set

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

can be thought of as the set of all germs of continuous functions at

; this is a local ring with the unique maximal ideal consisting of those germs whose value at

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.

is an algebraic variety carrying the Zariski topology, we can define a locally ringed space by taking

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.

is a continuous map between the underlying topological spaces, and

to the direct image of the structure sheaf of X.

is given by the following data: There is an additional requirement for morphisms between locally ringed spaces: 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.

be a locally ringed space with structure sheaf

Take the local ring (stalk)

is defined as the dual of this vector space.

, and the restriction maps are compatible with the module structure, then we call

will be a module over the local ring (stalk)

-modules is a morphism of sheaves that is compatible with the given module structures.

-modules over a fixed locally ringed space

-modules is called quasi-coherent if it is, locally, isomorphic to the cokernel of a map between free

is a quasi-coherent sheaf that is, locally, of finite type and for every open subset