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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 In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...
)
ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
s parametrized by
open subset 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 suff ...
s 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 points ...
together with
ring homomorphism In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function such that ''f'' is: :addition preservi ...
s that play roles of restrictions. Precisely, it is a topological space equipped with a
sheaf Sheaf may refer to: * Sheaf (agriculture), a bundle of harvested cereal stems * Sheaf (mathematics), a mathematical tool * Sheaf toss, a Scottish sport * River Sheaf, a tributary of River Don in England * ''The Sheaf'', a student-run newspaper se ...
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 ...
(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 (38 ...
as well as
complex algebraic geometry In mathematics, complex geometry is the study of geometric structures and constructions arising out of, or described by, the complex numbers. In particular, complex geometry is concerned with the study of spaces such as complex manifolds and c ...
and the
scheme theory In mathematics, a scheme is a mathematical structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations ''x'' = 0 and ''x''2 = 0 define the same algebraic variety but different sc ...
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 sp ...
s, including Hartshorne and Wikipedia. "
Éléments de géométrie algébrique The ''Éléments de géométrie algébrique'' ("Elements of algebraic geometry, Algebraic Geometry") by Alexander Grothendieck (assisted by Jean Dieudonné), or ''EGA'' for short, is a rigorous treatise, in French language, French, on algebraic ge ...
", 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 points ...
''X'' together with a
sheaf Sheaf may refer to: * Sheaf (agriculture), a bundle of harvested cereal stems * Sheaf (mathematics), a mathematical tool * Sheaf toss, a Scottish sport * River Sheaf, a tributary of River Don in England * ''The Sheaf'', a student-run newspaper se ...
of
ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
s \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 algebraic num ...
s (i.e. they have unique
maximal ideal In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring ''R'' if there are no other ideals cont ...
s). 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 In mathematics, value may refer to several, strongly related notions. In general, a mathematical value may be any definite mathematical object. In elementary mathematics, this is most often a number – for example, a real number such as or an i ...
(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
germ Germ or germs may refer to: Science * Germ (microorganism), an informal word for a pathogen * Germ cell, cell that gives rise to the gametes of an organism that reproduces sexually * Germ layer, a primary layer of cells that forms during embry ...
s 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 In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
, 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. Mo ...
carrying the
Zariski topology In algebraic geometry and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in the real or complex analysis; in particular, it is n ...
, 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 de ...
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 i ...
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 In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...
from (X,\mathcal_X) to (Y,\mathcal_Y) is a pair (f,\varphi), where f:X\to Y is a
continuous map In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...
between the underlying topological spaces, and \varphi:\mathcal_Y\to f_*\mathcal_X is a
morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...
from the structure sheaf of Y to the
direct image In mathematics, the direct image functor is a construction in sheaf theory that generalizes the global sections functor to the relative case. It is of fundamental importance in topology and algebraic geometry. Given a sheaf ''F'' defined on a topolo ...
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 In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...
f:X\to Y * a family of
ring homomorphism In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function such that ''f'' is: :addition preservi ...
s \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 suf ...
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 Category, plural categories, may refer to: Philosophy and general uses * Categorization, categories in cognitive science, information science and generally *Category of being * ''Categories'' (Aristotle) *Category (Kant) *Categories (Peirce) * ...
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 is ...
s in these categories are defined as usual.


Tangent spaces

Locally ringed spaces have just enough structure to allow the meaningful definition of
tangent space In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...
s. 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 Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
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 In differential geometry, the cotangent space is a vector space associated with a point x on a smooth (or differentiable) manifold \mathcal M; one can define a cotangent space for every point on a smooth manifold. Typically, the cotangent space, T ...
). 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 In calculus, the product rule (or Leibniz rule or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as (u \cdot v)' = u ' \cdot v + ...
. 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 commut ...
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 * Mo ...
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 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 desirable properties. The motivating prototypical example of an abelian category is the category of ab ...
. An important subcategory of the category of ''\mathcal_X''-modules is the category of '' quasi-coherent sheaves'' 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