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2024-11-04T03:46:54Z

Modules over the structure sheaf

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{{Short description|Sheaf of rings in mathematics}}
In [[mathematics]], a '''ringed space''' is a family of ([[Commutative ring|commutative]]) [[ring (mathematics)|ring]]s parametrized by [[open subset]]s of a [[topological space]] together with [[ring homomorphism]]s that play roles of [[Restriction (mathematics)|restrictions]]. Precisely, it is a topological space equipped with a '''[[sheaf (mathematics)|sheaf]] of rings''' called a '''structure sheaf'''. It is an abstraction of the concept of the rings of [[Continuous_function#Continuous_functions_between_topological_spaces|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 of a sheaf|stalk]] at a point and the ring of [[germ of a function|germs of functions]] at a point is valid.

Ringed spaces appear in [[mathematical analysis|analysis]] as well as [[complex algebraic geometry]] and the [[scheme theory]] of [[algebraic geometry]].

'''Note''': In the definition of a ringed space, most expositions tend to restrict the rings to be [[commutative ring]]s, including [[Algebraic Geometry (book)|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.<ref>''Éléments de géométrie algébrique'', Ch 0, 4.1.1.</ref>

==Definitions==
A '''ringed space''' <math>(X,\mathcal{O}_X)</math> is a [[topological space]] ''<math>X</math>'' together with a [[sheaf (mathematics)|sheaf]] of [[ring (mathematics)|ring]]s <math>\mathcal{O}_X</math> on <math>X</math>. The sheaf <math>\mathcal{O}_X</math> is called the '''structure sheaf''' of <math>X</math>.

A '''locally ringed space''' is a ringed space <math>(X,\mathcal{O}_X)</math> such that all [[stalk of a sheaf|stalks]] of <math>\mathcal{O}_X</math> are [[local ring]]s (i.e. they have unique [[maximal ideal]]s). Note that it is ''not'' required that <math>\mathcal{O}_X(U)</math> be a local ring for every open set <math>U</math>'';'' in fact, this is almost never the case.

==Examples==
An arbitrary topological space ''<math>X</math>'' can be considered a locally ringed space by taking ''<math>\mathcal{O}_X</math>'' to be the sheaf of [[Real number|real-valued]] (or [[Complex number|complex-valued]]) continuous functions on open subsets of ''<math>X</math>''. The [[Stalk (sheaf)|stalk]] at a point <math>x</math> can be thought of as the set of all [[germ (mathematics)|germ]]s of continuous functions at ''<math>x</math>''; this is a local ring with the unique maximal ideal consisting of those germs whose value at ''<math>x</math>'' is <math>0</math>.

If ''<math>X</math>'' is a [[manifold]] with some extra structure, we can also take the sheaf of [[Differentiable function|differentiable]], or [[holomorphic function|holomorphic]] functions. Both of these give rise to locally ringed spaces.

If ''<math>X</math>'' is an [[algebraic variety]] carrying the [[Zariski topology]], we can define a locally ringed space by taking <math>\mathcal{O}_X(U)</math> to be the ring of [[rational mapping]]s defined on the Zariski-open set ''<math>U</math>'' that do not blow up (become infinite) within <math>U</math>. The important generalization of this example is that of the [[spectrum of a ring|spectrum]] of any commutative ring; these spectra are also locally ringed spaces. [[Scheme (mathematics)|Schemes]] are locally ringed spaces obtained by "gluing together" spectra of commutative rings.

==Morphisms==
A [[morphism]] from <math>(X,\mathcal{O}_X)</math> to <math>(Y,\mathcal{O}_Y)</math> is a pair <math>(f,\varphi)</math>, where <math>f:X\to Y</math> is a [[continuous map]] between the underlying topological spaces, and <math>\varphi:\mathcal{O}_Y\to f_*\mathcal{O}_X</math> is a [[Sheaf (mathematics)#Morphisms|morphism]] from the structure sheaf of <math>Y</math> to the [[direct image functor|direct image]] of the structure sheaf of {{math|''X''}}. In other words, a morphism from <math>(X,\mathcal{O}_X)</math> to <math>(Y,\mathcal{O}_Y)</math> is given by the following data:

* a [[continuous function (topology)|continuous map]] <math>f:X\to Y</math>
* a family of [[ring homomorphism]]s <math>\varphi_V : \mathcal{O}_Y(V)\to\mathcal{O}_X(f^{-1}(V))</math> for every [[open set]] <math>V</math> of <math>Y</math> that commute with the restriction maps. That is, if <math>V_1\subseteq V_2</math> are two open subsets of <math>Y</math>, then the following diagram must [[commutative diagram|commute]] (the vertical maps are the restriction homomorphisms):

[[Image:LocallyRingedSpace-01.png|center]]

There is an additional requirement for morphisms between ''locally'' ringed spaces:

*the ring homomorphisms induced by <math>\varphi</math> between the stalks of ''<math>Y</math>'' and the stalks of ''<math>X</math>'' must be ''[[Local ring#Some facts and definitions|local homomorphisms]]'', i.e. for every ''<math>x\in X</math>'' the maximal ideal of the local ring (stalk) at <math>f(x)\in Y</math> is mapped into the maximal ideal of the local ring at ''<math>x\in X</math>''.

Two morphisms can be composed to form a new morphism, and we obtain the [[category (mathematics)|category]] of ringed spaces and the category of locally ringed spaces. [[Isomorphism]]s in these categories are defined as usual.

==Tangent spaces==
{{See also|Zariski tangent space}}

Locally ringed spaces have just enough structure to allow the meaningful definition of [[tangent space]]s. Let ''<math>X</math>'' be a locally ringed space with structure sheaf ''<math>\mathcal{O}_X</math>''; we want to define the tangent space <math>T_x(X)</math> at the point ''<math>x\in X</math>''. Take the local ring (stalk) <math>R_x</math> at the point <math>x</math>, with maximal ideal <math>\mathfrak{m}_x</math>. Then <math>k_x := R_x/\mathfrak{m}_x</math> is a [[field (mathematics)|field]] and <math>\mathfrak{m}_x/\mathfrak{m}_x^2</math> is a [[vector space]] over that field (the [[cotangent space]]). The tangent space <math>T_x(X)</math> is defined as the [[dual space|dual]] of this vector space.

The idea is the following: a tangent vector at ''<math>x</math>'' should tell you how to "differentiate" "functions" at ''<math>x</math>'', i.e. the elements of ''<math>R_x</math>''. Now it is enough to know how to differentiate functions whose value at ''<math>x</math>'' 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 ''<math>\mathfrak{m}_x</math>''. Furthermore, if two functions are given with value zero at ''<math>x</math>'', then their product has derivative 0 at ''<math>x</math>'', by the [[product rule]]. So we only need to know how to assign "numbers" to the elements of <math>\mathfrak{m}_x/\mathfrak{m}_x^2</math>, and this is what the dual space does.

==Modules over the structure sheaf==
{{main|Sheaf of modules}}
Given a locally ringed space <math>(X,\mathcal{O}_X)</math>, certain [[sheaf (mathematics)|sheaves]] of modules on ''<math>X</math>'' occur in the applications, the ''<math>\mathcal{O}_X</math>''-modules. To define them, consider a sheaf <math>\mathcal{F}</math> of [[abelian group]]s on ''<math>X</math>''. If <math>\mathcal{F}(U)</math> is a [[module (mathematics)|module]] over the ring ''<math>\mathcal{O}_X(U)</math>'' for every open set ''<math>U</math>'' in ''<math>X</math>'', and the restriction maps are compatible with the module structure, then we call <math>\mathcal{F}</math> an ''<math>\mathcal{O}_X</math>''-module. In this case, the stalk of ''<math>\mathcal{F}</math>'' at ''<math>x</math>'' will be a module over the local ring (stalk) ''<math>R_x</math>'', for every ''<math>x\in X</math>''.

A morphism between two such ''<math>\mathcal{O}_X</math>''-modules is a [[Morphism of sheaves#Morphisms|morphism of sheaves]] that is compatible with the given module structures. The category of ''<math>\mathcal{O}_X</math>''-modules over a fixed locally ringed space <math>(X,\mathcal{O}_X)</math> is an [[abelian category]].

An important subcategory of the category of ''<math>\mathcal{O}_X</math>''-modules is the category of ''[[quasi-coherent sheaves]]'' on ''<math>X</math>''. A sheaf of ''<math>\mathcal{O}_X</math>''-modules is called quasi-coherent if it is, locally, isomorphic to the [[cokernel]] of a map between free ''<math>\mathcal{O}_X</math>''-modules. A [[Coherent sheaf|''coherent'' sheaf]] ''<math>F</math>'' is a quasi-coherent sheaf that is, locally, of [[algebra of finite type|finite type]] and for every open subset ''<math>U</math>'' of ''<math>X</math>'' the [[Kernel (algebra)|kernel]] of any morphism from a free ''<math>\mathcal{O}_U</math>''-module of finite rank to ''<math>\mathcal{F}_U</math>'' is also of finite type.

==Citations==
{{reflist|2|colwidth=25em}}

==References==
*Section 0.4 of {{EGA|book=I}}
*{{Hartshorne AG}}

== External links ==
*{{springer|title=Ringed space|id=R/r082460|last=Onishchik|first=A.L.}}

{{DEFAULTSORT:Ringed Space}}
[[Category:Sheaf theory]]
[[Category:Scheme theory]]

Ringed space - Revision history

imported>Wataxa: /* Modules over the structure sheaf */