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.

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.^{[1]}

## Definitions

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

A **locally ringed space** is a ringed space such that all stalks of are local rings (i.e. they have unique maximal ideals). Note that it is *not* required that be a local ring for every open set *;* 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 at a point 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 .*

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, we can define a locally ringed space by taking to be the ring of rational mappings defined on the Zariski-open set ** that do not blow up (become infinite) within . 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 from to is a pair , where is a continuous map between the underlying topological spaces, and is a morphism from the structure sheaf of to the direct image of the structure sheaf of *X*. In other words, a morphism from to is given by the following data:

- a continuous map
- a family of ring homomorphisms for every open set of that commute with the restriction maps. That is, if are two open subsets of , then the following diagram must commute (the vertical maps are the restriction homomorphisms):

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

- the ring homomorphisms induced by between the stalks of
*and the stalks of**must be**local homomorphisms*, i.e. for every*the maximal ideal of the local ring (stalk) at 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 **; we want to define the tangent space at the point **. Take the local ring (stalk) at the point , with maximal ideal . Then is a field and is a vector space over that field (the cotangent space). The tangent 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. So we only need to know how to assign "numbers" to the elements of , and this is what the dual space does.*

## Modules over the structure sheaf

Given a locally ringed space , certain sheaves of modules on * occur in the applications, the **-modules. To define them, consider a sheaf **F* of abelian groups on *. If **F*(*U*) is a module over the ring * for every open set ** in **, and the restriction maps are compatible with the module structure, then we call 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 **-modules over a fixed locally ringed space 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

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