HOME

TheInfoList



OR:

In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a ringed space is a family of ( commutative) rings parametrized by open subsets of a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
together with ring homomorphisms that play roles of restrictions. Precisely, it is a topological space equipped with a sheaf of rings called a structure sheaf. It is an abstraction of the concept of the rings of
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ...
(scalar-valued) functions on open subsets. Among ringed spaces, especially important and prominent is a locally ringed space: a ringed space in which the analogy between the stalk at a point and the ring of germs of functions at a point is valid. Ringed spaces appear in
analysis Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (3 ...
as well as complex algebraic geometry and the scheme theory of
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
. Note: In the definition of a ringed space, most expositions tend to restrict the rings to be
commutative ring In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...
s, including Hartshorne and Wikipedia. " Éléments de géométrie algébrique", on the other hand, does not impose the commutativity assumption, although the book mostly considers the commutative case.EGA, Ch 0, 4.1.1.


Definitions

A ringed space (X,\mathcal_X) is a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
''X'' together with a sheaf of rings \mathcal_X on X. The sheaf \mathcal_X is called the structure sheaf of X. A locally ringed space is a ringed space (X,\mathcal_X) such that all stalks of \mathcal_X are
local ring In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebrai ...
s (i.e. they have unique maximal ideals). Note that it is ''not'' required that \mathcal_X(U) be a local ring for every open set U'';'' in fact, this is almost never the case.


Examples

An arbitrary topological space ''X'' can be considered a locally ringed space by taking ''\mathcal_X'' to be the sheaf of real-valued (or
complex-valued In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
) continuous functions on open subsets of ''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 the unique maximal ideal consisting of those germs whose value at ''x'' is 0. If ''X'' is a
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
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 Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...
carrying the Zariski topology, we can define a locally ringed space by taking \mathcal_X(U) to be the ring of
rational mapping In mathematics, in particular the subfield of algebraic geometry, a rational map or rational mapping is a kind of partial function between algebraic varieties. This article uses the convention that varieties are irreducible. Definition Formal ...
s defined on the Zariski-open set ''U'' that do not blow up (become infinite) within U. The important generalization of this example is that of the
spectrum A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors ...
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

A morphism from (X,\mathcal_X) to (Y,\mathcal_Y) is a pair (f,\varphi), where f:X\to Y is a continuous map between the underlying topological spaces, and \varphi:\mathcal_Y\to f_*\mathcal_X is a morphism from the structure sheaf of Y to the direct image of the structure sheaf of . In other words, a morphism from (X,\mathcal_X) to (Y,\mathcal_Y) is given by the following data: * a continuous map f:X\to Y * a family of ring homomorphisms \varphi_V : \mathcal_Y(V)\to\mathcal_X(f^(V)) for every
open set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are su ...
V of Y which commute with the restriction maps. That is, if V_1\subseteq V_2 are two open subsets of Y, then the following diagram must
commute Commute, commutation or commutative may refer to: * Commuting, the process of travelling between a place of residence and a place of work Mathematics * Commutative property, a property of a mathematical operation whose result is insensitive to th ...
(the vertical maps are the restriction homomorphisms): There is an additional requirement for morphisms between ''locally'' ringed spaces: *the ring homomorphisms induced by \varphi between the stalks of ''Y'' and the stalks of ''X'' must be '' local homomorphisms'', i.e. for every ''x\in X'' the maximal ideal of the local ring (stalk) at f(x)\in Y is mapped into the maximal ideal of the local ring at ''x\in 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.
Isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
s in these categories are defined as usual.


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 ''\mathcal_X''; we want to define the tangent space T_x(X) at the point ''x\in X''. Take the local ring (stalk) R_x at the point x, with maximal ideal \mathfrak_x. Then k_x := R_x/\mathfrak_x is a field and \mathfrak_x/\mathfrak_x^2 is a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
over that field (the cotangent space). The tangent space T_x(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 consider ''\mathfrak_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 \mathfrak_x/\mathfrak_x^2, and this is what the dual space does.


\mathcal_X-modules

Given a locally ringed space (X,\mathcal_X), certain sheaves of modules on ''X'' occur in the applications, the ''\mathcal_X''-modules. To define them, consider a sheaf ''F'' of
abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
s on ''X''. If ''F''(''U'') is a
module Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Modul ...
over the ring ''\mathcal_X(U)'' for every open set ''U'' in ''X'', and the restriction maps are compatible with the module structure, then we call F an ''\mathcal_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\in X''. A morphism between two such ''\mathcal_X''-modules is a morphism of sheaves which is compatible with the given module structures. The category of ''\mathcal_X''-modules over a fixed locally ringed space (X,\mathcal_X) is an abelian category. An important subcategory of the category of ''\mathcal_X''-modules is the category of ''
quasi-coherent sheaves In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with refe ...
'' on ''X''. A sheaf of ''\mathcal_X''-modules is called quasi-coherent if it is, locally, isomorphic to the cokernel of a map between free ''\mathcal_X''-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 ''\mathcal_U''-modules of finite rank to ''F_U'' is also of finite type.


Citations


References

*Section 0.4 of *


External links

* {{DEFAULTSORT:Ringed Space Sheaf theory Scheme theory