The Locally ringed space reference article from the English Wikipedia on 24-Jul-2004
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Locally ringed space

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In mathematics, a locally ringed space (or local ringed space) is, intuitively speaking, a space together with, for each of its open sets, a commutative ring the elements of which are thought of as "functions" defined on that open set. Locally ringed spaces appear throughout analysis and are also used to define the schemess of algebraic geometry.

Table of contents
1 Definition
2 Examples
3 Morphisms
4 Tangent spaces
5 OX modules


A locally ringed space is a topological space X, together with a sheaf F of commutative rings on X, such that all stalks of F are local rings. (Note that it is not required that F(U) be a local ring for every open set U -- in fact, that is almost never going to be the case.)

The sheaf F is also called the structure sheaf of the locally ringed space X, and is denoted by OX.


If X is an arbitrary topological space, we can take OX to be the sheaf of continuous functions on X with real values (or alternatively: with complex values). The stalk at xX 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 an algebraic variety carrying the Zariski topology, we can define a locally ringed space by taking OX(U) to be the ring of polynomial functions defined on the Zariski-open set 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.


Given two locally ringed spaces (X, OX) and (Y, OY), a morphism of locally ringed spaces from X to Y is given by the following data:

(1) a continuous map f : XY

(2) for every open set V of Y, a ring homomorphism φV : OY(V) → OX(f -1(V)) such that:

(2a) the ring homomorphisms are compatible with the restriction homomorphisms of the sheaves, i.e. if V1V2 are two open subsets of Y, then the following diagram is commutative (the vertical maps are the restriction homomorphisms):
               OY(V2)  -------->  OX(f -1(V2))
                 |                     |
                 |                     |
                 |                     |
                 V        φV1          V
               OY(V1)  -------->  OX(f -1(V1))

(2b) the ring homomorphisms induced by φ between the stalks of Y and the stalks of X are local homomorphisms, i.e. for every xX the maximal ideal of the local ring (stalk) at f(x) ∈ Y is mapped to the maximal ideal of the local ring at xX.

Two such morphisms can be composed to form a new morphism, and we obtain the category of locally ringed spaces and the notion of isomorphic locally ringed spaces.

Tangent spaces



OX modules

