In mathematics, a 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. Ringed spaces appear throughout analysis and are also used to define the schemes of algebraic geometry. Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ... Analysis is the generic name given to any branch of mathematics which depends upon the concepts of limits and convergence, and studies closely related topics such as continuity, integration, differentiability and transcendental functions. ... In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ...

Definition

Formally, a ringed space is a topological space X together with a sheaf of commutative rings O_X on X. The sheaf O_X is called the structure sheaf of X. Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics, a sheaf F on a topological space X is something that assigns a structure F(U) (such as a set, group, or ring) to each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and... In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ...

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. In mathematics, a sheaf F on a given topological space X gives a set or richer structure F(U) for each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and gluing smaller open sets to obtain... In mathematics, more particularly in abstract algebra, local rings are certain rings that are comparatively simple, and serve to describe the local behavior of functions defined on varieties or manifolds. ... In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring which generalizes important properties of integers. ...

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 which 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. In mathematics, a continuous function is a function in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ... In mathematics, a germ is an equivalence class of continuous functions from one topological space to another (often from the real line to itself), in which one point x0 in the domain has been singled out as privileged. ... In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring which generalizes important properties of integers. ...

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. A manifold is a mathematical space which is constructed, like a patchwork, by gluing and bending together copies of simple spaces. ... In mathematics, the derivative of a function is one of the two central concepts of calculus. ... Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ...

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 functions defined on the Zariski-open set U which 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. In classical algebraic geometry (and to some extent also in modern algebraic geometry), the main objects of study are algebraic varieties. ... This article needs to be cleaned up to conform to a higher standard of quality. ... In mathematics, a rational function in algebra is a function defined as a ratio of polynomials. ... In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by Spec(R), is defined to be the set of all proper prime ideals of R. It is commonly augmented with the Zariski topology and with a structure sheaf, turning it into a locally ringed space. ... In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. ...

Morphisms

A morphism of ringed spaces is simply a morphism of sheaves. Explicitly, a morphism from (X, O_X) to (Y, O_Y) is given by the following data: In mathematics, a morphism is an abstraction of a structure-preserving process between two mathematical structures. ... In mathematics, a sheaf F on a topological space X is something that assigns a structure F(U) (such as a set, group, or ring) to each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and...

• 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 V₁ ⊂ V₂ 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: In topology and related areas of mathematics a continuous function is a morphism between topological spaces; that is, a mapping which preserves the topological structure. ... In abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication. ... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... In mathematics, especially the many applications of category theory, a commutative diagram is a diagram of objects and morphisms such that, when picking two objects, one can follow any path through the diagram and obtain the same result by composition. ... Image File history File links LocallyRingedSpace-01. ...

• 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. In mathematics, categories allow one to formalize notions involving abstract structure and processes which preserve structure. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ...

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 cotangent space). The tangent space R_x is defined as the dual of this vector space. The tangent space of a manifold is a concept which needs to be introduced when generalizing vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ... In differential geometry, one can attach to every point p of a differentiable manifold a vector space called the cotangent space at p. ... In mathematics, the existence of a dual vector space reflects in an abstract way the relationship between row vectors (1×n) and column vectors (n×1). ...

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. In mathematics, the product rule of calculus, which is also called Leibnizs law (see derivation), governs the differentiation of products of differentiable functions. ...

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. In mathematics, a sheaf F on a topological space X is something that assigns a structure F(U) (such as a set, group, or ring) to each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and... In mathematics, an abelian group, also called a commutative group, is a group (G, *) such that a * b = b * a for all a and b in G. Abelian groups are named after Niels Henrik Abel. ... In abstract algebra, the notion of a module over a ring is the common generalizations of two of the most important notions in algebra, vector space (where we take the ring to be a particular field), and abelian group (where we take the ring to be the ring of integers). ...

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. In mathematics, a sheaf F on a topological space X is something that assigns a structure F(U) (such as a set, group, or ring) to each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and... In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have nice properties. ...