In mathematics, a ringed space is, intuitively speaking, a space together with a collection of commutative rings, the elements of which are "functions" on each open set of the space. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Ring theory, a branch of Abstract algebra, a commutative ring is a ring in which the multiplication operation has the commutative property In Metric topology and related fields of Mathematics, a set U is called open if intuitively speaking starting from any point x in Ringed spaces appear throughout analysis and are also used to define the schemes of algebraic geometry. Analysis has its beginnings in the rigorous formulation of Calculus. 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 techniques of Abstract algebra, especially Commutative algebra, with
Contents |
Definition
Formally, a ringed space is a topological space X together with a sheaf of commutative rings OX on X. Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. In Mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the Open sets of a Topological space. In Ring theory, a branch of Abstract algebra, a commutative ring is a ring in which the multiplication operation has the commutative property The sheaf OX is called the structure sheaf of X.
A locally ringed space is a ringed space (X, OX) such that all stalks of OX are local rings (i. In Mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the Open sets of a Topological space. In Mathematics, more particularly in Abstract algebra, local rings are certain rings that are comparatively simple and serve to describe what is called e. they have unique maximal ideals). In Mathematics, more specifically in Ring theory, a maximal ideal is an ideal which is maximal (with respect to set inclusion amongst all proper ideals Note that it is not required that OX(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 OX 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). In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output 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 germ of ( continuous, differentiable or analytic) functions is an Equivalence class of (continuous differentiable In Mathematics, more specifically in Ring theory, a maximal ideal is an ideal which is maximal (with respect to set inclusion amongst all proper ideals
If X is a manifold with some extra structure, we can also take the sheaf of differentiable, or complex-analytic functions. A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change Holomorphic functions are the central object of study of Complex analysis; they are functions defined on an open subset of the complex number plane 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 do not blow up (become infinite) within U. This article is about algebraic varieties For the term "a variety of algebras" and an explanation of the difference between a variety of algebras and an algebraic variety In Mathematics, namely Algebraic geometry, the Zariski topology is a particular Topology chosen for algebraic varieties that reflects the algebraic In Mathematics, a rational function is any function which can be written as the Ratio of two Polynomial functions Definitions In The important generalization of this example is that of the spectrum of any commutative ring; these spectra are also locally ringed spaces. In Abstract algebra and Algebraic geometry, the spectrum of a Commutative ring R, denoted by Spec( R) is defined to be the set of Schemes are locally ringed spaces obtained by "gluing together" spectra of commutative rings. 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. In Mathematics, a morphism is an abstraction derived from structure-preserving mappings between two Mathematical structures The study of morphisms and In Mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the Open sets of a Topological space. Explicitly, a morphism from (X, OX) to (Y, OY) is given by the following data:
- a continuous map f : X → Y
- a family of ring homomorphisms φV : OY(V) → OX(f -1(V)) for every open set V of Y which commute with the restriction maps. In Topology and related areas of Mathematics a continuous function is a Morphism between Topological spaces Intuitively this is a function In Ring theory or Abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication In Metric topology and related fields of Mathematics, a set U is called open if intuitively speaking starting from any point x in 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:
- the ring homomorphisms induced by φ between the stalks of Y and the stalks of X must be local homomorphisms, i. In mathematics and especially in Category theory a commutative diagram is a Diagram of objects also known as vertices, and Morphisms also 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. In Mathematics, a category is a fundamental and abstract way to describe mathematical entities and their relationships Isomorphisms in these categories are defined as usual. In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective
Tangent spaces
Locally ringed spaces have just enough structure to allow the meaningful definition of tangent spaces. In Mathematics, the tangent space of a Manifold is a concept which facilitates the generalization of vectors from Affine spaces to general manifolds since Let X be locally ringed space with structure sheaf OX; we want to define the tangent space Tx at the point x ∈ X. 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). In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Differential geometry, one can attach to every point x of a smooth (or differentiable Manifold a Vector space called the cotangent space The tangent space Tx is defined as the dual of this vector space. In Mathematics, any Vector space V has a corresponding dual vector space (or just dual space for short consisting of all Linear functionals
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. In Calculus, the product rule also called Leibniz's law (see derivation) governs the differentiation of products of differentiable 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. In Mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the Open sets of a Topological space. To define them, consider a sheaf F of abelian groups on X. An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the 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 Abstract algebra, the concept of a module over a ring is a generalization of the notion of Vector space, where instead of requiring the scalars In this case, the stalk of F at x will be a module over the local ring (stalk) Rx, for every x∈X.
A morphism between two such OX-modules is a morphism of sheaves which is compatible with the given module structures. In Mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the Open sets of a Topological space. The category of OX-modules over a fixed locally ringed space (X, OX) is an abelian category. In Mathematics, an abelian category is a category in which Morphisms and objects can be added and in which kernels and Cokernels exist
An important subcategory of the category of OX-modules is the category of quasi-coherent sheaves on X. A sheaf of OX-modules is called quasi-coherent if it is, locally, isomorphic to the cokernel of a map between free OX-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 OU-modules of finite rank to FU is also of finite type.