Locally ringed space

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 schemes of algebraic geometry.

 Contents

Definition

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.

Examples

If X is an arbitrary topological space, we can take OX to be the sheaf of continuous functions on open subsets of X with real values (or alternatively: with complex values) (there are continuous functions over open subsets of X which isn't the restriction of any continuous function over X). 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 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 OX(U) to be the ring of rational functions defined on the Zariski-open set U which does 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.

If X is a smooth supermanifold with |X|, a real smooth manifold being its real projection, then OX is the sheaf of smooth sections of the algebra bundle. This is a sheaf of noncommutative rings.

Morphisms

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):
Missing image
LocallyRingedSpace-01.png
Image:LocallyRingedSpace-01.png

(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

Locally ringed spaces have just enough structure to allow the meaningful definition of tangent spaces. Let X be locally ringed space with structure sheaf OX; we want to define the tangent space Tx at the point xX. Take the local ring (stalk) Rx at the point x, with maximal ideal mx. Then kx := Rx/mx is a field and mx/mx2 is a vector space over that field (the cotangent space). The tangent space Rx is defined as the dual of this vector space.

The idea is the following: a tangent vector at x should tell you how to "differentiate" "functions" at x, i.e. the elements of Rx. 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 mx. 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 mx/mx2, and this is what the dual space does.

OX modules

Given a locally ringed space (X, OX), certain sheaves of modules on X occur in the applications, the OX-modules. To define them, consider a sheaf F of abelian groups on X. If F(U) is a module over the ring OX(U) for every open set U in X, and the restriction maps are compatible with the module structure, then we call F an OX-module. In this case, the stalk of F at x will be a module over the local ring (stalk) Rx, for every xX.

A morphism between two such OX-modules is a morphism of sheaves which is compatible with the given module structures. The category of OX-modules over a fixed locally ringed space (X, OX) is an abelian category.

• Art and Cultures
• Countries of the World (http://www.academickids.com/encyclopedia/index.php/Countries)
• Space and Astronomy