# Ringed space

In mathematics, a **ringed space** can be equivalently thought of either

- (a) a space together with a collection of commutative rings, the elements of which are "functions" on each open set of the space.
- (b) a family of (commutative) rings parametrized by open subsets of a topological space, together with ring homomorphisms coming from the relationships between open sets; in short, the
**sheaf of rings**.

Ringed spaces appear in analysis as well as complex algebraic geometry and scheme theory of algebraic geometry. The point of view (b) is more amenable to generalization; one simply needs to cook up a different way of parametrizing rings (cf. ringed topos).

**Note**: Many expositions tend to restrict the rings to be commutative rings, including Hartshorne and Wikipedia, in the definition of a ringed space. "Éléments de géométrie algébrique", on the other hand, does not impose the commutativity assumption, although the book only considers the commutative case. (EGA, Ch 0, 4.1.1.)

## Definition

Formally, a **ringed space** (*X*, *O*_{X}) is a topological space *X* together with a sheaf of rings *O*_{X} on *X*. The sheaf *O*_{X} is called the **structure sheaf** of *X*.

A **locally ringed space** is a ringed space (*X*, *O*_{X}) such that all stalks of *O*_{X} are local rings (i.e. they have unique maximal ideals). Note that it is *not* required that *O*_{X}(*U*) be a local ring for every open set *U.* In fact, that is almost never going to be the case.

## Examples

An arbitrary topological space *X* can be considered a locally ringed space by taking *O _{X}* to be the sheaf of real-valued (or complex-valued) continuous functions on open subsets of

*X*(there may exist continuous functions over open subsets of X that are not the restriction of any continuous function over X). The stalk at a point

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

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

*x*is 0.

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

If *X* is an algebraic variety carrying the Zariski topology, we can define a locally ringed space by taking *O _{X}*(

*U*) to be the ring of rational mappings defined on the Zariski-open set

*U*that do not blow up (become infinite) within U. 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 (*X*, *O _{X}*) to (

*Y*,

*O*) is a pair (

_{Y}*f*,

*φ*), where

*f*:

*X*→

*Y*is a continuous map between the underlying topological spaces, and

*φ*:

*O*→

_{Y}*f*

_{*}

*O*is a morphism from the structure sheaf of

_{X}*Y*to the direct image of the structure sheaf of

*X*. In other words, a morphism from (

*X*,

*O*) to (

_{X}*Y*,

*O*) is given by the following data:

_{Y}- a continuous map
*f*:*X*→*Y* - a family of ring homomorphisms φ
_{V}:*O*(_{Y}*V*) →*O*(_{X}*f*^{ -1}(*V*)) for every open set*V*of*Y*which commute with the restriction maps. That is, if*V*_{1}⊂*V*_{2}are two open subsets of*Y*, 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 φ between the stalks of
*Y*and the stalks of*X*must be*local homomorphisms*, i.e. for every*x*∈*X*the maximal ideal of the local ring (stalk) at*f*(*x*) ∈*Y*is mapped to the maximal ideal of the local ring at*x*∈*X*.

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 *X* be locally ringed space with structure sheaf *O _{X}*; we want to define the tangent space

*T*at the point

_{x}*x*∈

*X*. Take the local ring (stalk)

*R*at the point

_{x}*x*, with maximal ideal

*m*

_{x}. Then

*k*

_{x}:=

*R*/

_{x}*m*is a field and

_{x}*m*/

_{x}*m*is a vector space over that field (the cotangent space). The tangent space

_{x}^{2}*T*is defined as the dual of this vector space.

_{x}The idea is the following: a tangent vector at *x* should tell you how to "differentiate" "functions" at *x*, i.e. the elements of *R _{x}*. Now it is enough to know how to differentiate functions whose value at

*x*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 worry about

*m*

_{x}. Furthermore, if two functions are given with value zero at

*x*, then their product has derivative 0 at

*x*, by the product rule. So we only need to know how to assign "numbers" to the elements of

*m*/

_{x}*m*, and this is what the dual space does.

_{x}^{2}*O*_{X} modules

_{X}

Given a locally ringed space (*X*, *O _{X}*), certain sheaves of modules on

*X*occur in the applications, the

*O*-modules. To define them, consider a sheaf

_{X}*F*of abelian groups on

*X*. If

*F*(

*U*) is a module over the ring

*O*(

_{X}*U*) for every open set

*U*in

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

*F*an

*O*-module. In this case, the stalk of

_{X}*F*at

*x*will be a module over the local ring (stalk)

*R*

_{x}, for every

*x*∈

*X*.

A morphism between two such *O _{X}*-modules is a morphism of sheaves which is compatible with the given module structures. The category of

*O*-modules over a fixed locally ringed space (

_{X}*X*,

*O*) is an abelian category.

_{X}An important subcategory of the category of *O*_{X}-modules is the category of *quasi-coherent sheaves* on *X*. A sheaf of *O*_{X}-modules is called quasi-coherent if it is, locally, isomorphic to the cokernel of a map between free *O*_{X}-modules. A *coherent* sheaf *F* is a quasi-coherent sheaf which is, locally, of finite type and for every open subset *U* of *X* the kernel of any morphism from a free *O*_{U}-modules of finite rank to *F*_{U} is also of finite type.

## Citations

## References

- Section 0.4 of Grothendieck, Alexandre; Dieudonné, Jean (1960). "Éléments de géométrie algébrique: I. Le langage des schémas".
*Publications Mathématiques de l'IHÉS*.**4**. doi:10.1007/bf02684778. MR 0217083. - Hartshorne, Robin (1977),
*Algebraic Geometry*, Graduate Texts in Mathematics,**52**, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157

## External links

- Onishchik, A.L. (2001), "Ringed space", in Hazewinkel, Michiel,
*Encyclopedia of Mathematics*, Springer, ISBN 978-1-55608-010-4