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 schemess of algebraic geometry.

Table of contents

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

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 O_X.

Examples

If X is an arbitrary topological space, we can take O_X to be the sheaf of continuous functions on X with real values (or alternatively: with complex values). The stalk at x∈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 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.

Morphisms

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

(1) a continuous map f : X → Y

(2) for every open set V of Y, a ring homomorphism φ_V : O_Y(V) → O_X(f ^-1(V)) such that:

(2a) the ring homomorphisms are compatible with the restriction homomorphisms of the sheaves, i.e. if V₁ ⊂ V₂ are two open subsets of Y, then the following diagram is commutative (the vertical maps are the restriction homomorphisms):

                          φV2
               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 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 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 O_X; we want to define the tangent space T_x at the point x ∈ X. Take the local ring (stalk) R_x at the point x, with maximal ideal m_x. Then k_x := R_x/m_x is a field and m_x/m_x² is a vector space over that field. The tangent space R_x 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 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_x², and this is what the dual space does.

O_X modules

Given a locally ringed space (X, O_X), certain sheaves of modules on X occur in the applications, the O_X-modules. To define them, consider a sheaf 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_X-module. In this case, the stalk of 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_X-modules over a fixed locally ringed space (X, O_X) is an abelian category.