Ringed space

<h2 id="definitions">Definitions</h2>
A ringed space 
 
 
 
 (
 X
 ,
 
 
 
 O
 
 
 
 X
 
 
 )
 
 
 {\displaystyle (X,{\mathcal {O}}_{X})}
 
 is a <a href="/facts/Topological_space/v2ut7CbK">topological space</a> 
 
 
 
 X
 
 
 {\displaystyle X}
 
 together with a <a href="/facts/Sheaf_(mathematics)/MM3hcRw4">sheaf</a> of <a href="/facts/Ring_(mathematics)/JnCUXz63">rings</a> 
 
 
 
 
 
 
 O
 
 
 
 X
 
 
 
 
 {\displaystyle {\mathcal {O}}_{X}}
 
 on 
 
 
 
 X
 
 
 {\displaystyle X}
 
. The sheaf 
 
 
 
 
 
 
 O
 
 
 
 X
 
 
 
 
 {\displaystyle {\mathcal {O}}_{X}}
 
 is called the structure sheaf of 
 
 
 
 X
 
 
 {\displaystyle X}
 
.
A locally ringed space is a ringed space 
 
 
 
 (
 X
 ,
 
 
 
 O
 
 
 
 X
 
 
 )
 
 
 {\displaystyle (X,{\mathcal {O}}_{X})}
 
 such that all <a href="/facts/Stalk_of_a_sheaf/qAD2etdW">stalks</a> of 
 
 
 
 
 
 
 O
 
 
 
 X
 
 
 
 
 {\displaystyle {\mathcal {O}}_{X}}
 
 are <a href="/facts/Local_ring/gRTdg5Qx">local rings</a> (i.e. they have unique <a href="/facts/Maximal_ideal/nf4DnKVS">maximal ideals</a>). Note that it is not required that 
 
 
 
 
 
 
 O
 
 
 
 X
 
 
 (
 U
 )
 
 
 {\displaystyle {\mathcal {O}}_{X}(U)}
 
 be a local ring for every open set 
 
 
 
 U
 
 
 {\displaystyle U}
 
; in fact, this is almost never the case.

<h2 id="examples">Examples</h2>
An arbitrary topological space 
 
 
 
 X
 
 
 {\displaystyle X}
 
 can be considered a locally ringed space by taking 
 
 
 
 
 
 
 O
 
 
 
 X
 
 
 
 
 {\displaystyle {\mathcal {O}}_{X}}
 
 to be the sheaf of <a href="/facts/Real_number/R02gw5Pb">real-valued</a> (or <a href="/facts/Complex_number/2w7mqI7e">complex-valued</a>) continuous functions on open subsets of 
 
 
 
 X
 
 
 {\displaystyle X}
 
. The <a href="/facts/Stalk_(sheaf)/qAD2etdW">stalk</a> at a point 
 
 
 
 x
 
 
 {\displaystyle x}
 
 can be thought of as the set of all <a href="/facts/Germ_(mathematics)/g28mtBu5">germs</a> of continuous functions at 
 
 
 
 x
 
 
 {\displaystyle x}
 
; this is a local ring with the unique maximal ideal consisting of those germs whose value at 
 
 
 
 x
 
 
 {\displaystyle x}
 
 is 
 
 
 
 0
 
 
 {\displaystyle 0}
 
.
If 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is a <a href="/facts/Manifold/rXPERJtu">manifold</a> with some extra structure, we can also take the sheaf of <a href="/facts/Differentiable_function/jQqTgLk1">differentiable</a>, or <a href="/facts/Holomorphic_function/kVzthtEf">holomorphic</a> functions. Both of these give rise to locally ringed spaces.
If 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is an <a href="/facts/Algebraic_variety/Vk7f97vn">algebraic variety</a> carrying the <a href="/facts/Zariski_topology/uolkKBpM">Zariski topology</a>, we can define a locally ringed space by taking 
 
 
 
 
 
 
 O
 
 
 
 X
 
 
 (
 U
 )
 
 
 {\displaystyle {\mathcal {O}}_{X}(U)}
 
 to be the ring of <a href="/facts/Rational_mapping/kqbMfnKn">rational mappings</a> defined on the Zariski-open set 
 
 
 
 U
 
 
 {\displaystyle U}
 
 that do not blow up (become infinite) within 
 
 
 
 U
 
 
 {\displaystyle U}
 
. The important generalization of this example is that of the <a href="/facts/Spectrum_of_a_ring/2k2nDE5w">spectrum</a> of any commutative ring; these spectra are also locally ringed spaces. <a href="/facts/Scheme_(mathematics)/c7uSqGqL">Schemes</a> are locally ringed spaces obtained by "gluing together" spectra of commutative rings.

<h2 id="morphisms">Morphisms</h2>
A <a href="/facts/Morphism/Kdpuyz3S">morphism</a> from 
 
 
 
 (
 X
 ,
 
 
 
 O
 
 
 
 X
 
 
 )
 
 
 {\displaystyle (X,{\mathcal {O}}_{X})}
 
 to 
 
 
 
 (
 Y
 ,
 
 
 
 O
 
 
 
 Y
 
 
 )
 
 
 {\displaystyle (Y,{\mathcal {O}}_{Y})}
 
 is a pair 
 
 
 
 (
 f
 ,
 φ
 )
 
 
 {\displaystyle (f,\varphi )}
 
, where 
 
 
 
 f
 :
 X
 →
 Y
 
 
 {\displaystyle f:X\to Y}
 
 is a <a href="/facts/Continuous_map/sKbl02pB">continuous map</a> between the underlying topological spaces, and 
 
 
 
 φ
 :
 
 
 
 O
 
 
 
 Y
 
 
 →
 
 f
 
 ∗
 
 
 
 
 
 O
 
 
 
 X
 
 
 
 
 {\displaystyle \varphi :{\mathcal {O}}_{Y}\to f_{*}{\mathcal {O}}_{X}}
 
 is a <a href="/facts/Sheaf_(mathematics)/MM3hcRw4">morphism</a> from the structure sheaf of 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 to the <a href="/facts/Direct_image_functor/z3mckxvp">direct image</a> of the structure sheaf of X. In other words, a morphism from 
 
 
 
 (
 X
 ,
 
 
 
 O
 
 
 
 X
 
 
 )
 
 
 {\displaystyle (X,{\mathcal {O}}_{X})}
 
 to 
 
 
 
 (
 Y
 ,
 
 
 
 O
 
 
 
 Y
 
 
 )
 
 
 {\displaystyle (Y,{\mathcal {O}}_{Y})}
 
 is given by the following data:

<ul><li>a <a href="/facts/Continuous_function_(topology)/sKbl02pB">continuous map</a> 
 
 
 
 f
 :
 X
 →
 Y
 
 
 {\displaystyle f:X\to Y}
 
</li>
<li>a family of <a href="/facts/Ring_homomorphism/UptcMpFk">ring homomorphisms</a> 
 
 
 
 
 φ
 
 V
 
 
 :
 
 
 
 O
 
 
 
 Y
 
 
 (
 V
 )
 →
 
 
 
 O
 
 
 
 X
 
 
 (
 
 f
 
 −
 1
 
 
 (
 V
 )
 )
 
 
 {\displaystyle \varphi _{V}:{\mathcal {O}}_{Y}(V)\to {\mathcal {O}}_{X}(f^{-1}(V))}
 
 for every <a href="/facts/Open_set/QfTCIqMu">open set</a> 
 
 
 
 V
 
 
 {\displaystyle V}
 
 of 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 that commute with the restriction maps. That is, if 
 
 
 
 
 V
 
 1
 
 
 ⊆
 
 V
 
 2
 
 
 
 
 {\displaystyle V_{1}\subseteq V_{2}}
 
 are two open subsets of 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
, then the following diagram must <a href="/facts/Commutative_diagram/bdG8xuer">commute</a> (the vertical maps are the restriction homomorphisms):</li></ul>

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

<ul><li>the ring homomorphisms induced by 
 
 
 
 φ
 
 
 {\displaystyle \varphi }
 
 between the stalks of 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 and the stalks of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 must be <a href="/facts/Local_ring/gRTdg5Qx">local homomorphisms</a>, i.e. for every 
 
 
 
 x
 ∈
 X
 
 
 {\displaystyle x\in X}
 
 the maximal ideal of the local ring (stalk) at 
 
 
 
 f
 (
 x
 )
 ∈
 Y
 
 
 {\displaystyle f(x)\in Y}
 
 is mapped into the maximal ideal of the local ring at 
 
 
 
 x
 ∈
 X
 
 
 {\displaystyle x\in X}
 
.</li></ul>
Two morphisms can be composed to form a new morphism, and we obtain the <a href="/facts/Category_(mathematics)/7WzxUThf">category</a> of ringed spaces and the category of locally ringed spaces. <a href="/facts/Isomorphism/pj8KgkxU">Isomorphisms</a> in these categories are defined as usual.

<h2 id="tangent-spaces">Tangent spaces</h2>
See also: <a href="/facts/Zariski_tangent_space/yxljPf4X">Zariski tangent space</a>
Locally ringed spaces have just enough structure to allow the meaningful definition of <a href="/facts/Tangent_space/crvpbSeG">tangent spaces</a>. Let 
 
 
 
 X
 
 
 {\displaystyle X}
 
 be a locally ringed space with structure sheaf 
 
 
 
 
 
 
 O
 
 
 
 X
 
 
 
 
 {\displaystyle {\mathcal {O}}_{X}}
 
; we want to define the tangent space 
 
 
 
 
 T
 
 x
 
 
 (
 X
 )
 
 
 {\displaystyle T_{x}(X)}
 
 at the point 
 
 
 
 x
 ∈
 X
 
 
 {\displaystyle x\in X}
 
. Take the local ring (stalk) 
 
 
 
 
 R
 
 x
 
 
 
 
 {\displaystyle R_{x}}
 
 at the point 
 
 
 
 x
 
 
 {\displaystyle x}
 
, with maximal ideal 
 
 
 
 
 
 
 m
 
 
 
 x
 
 
 
 
 {\displaystyle {\mathfrak {m}}_{x}}
 
. Then 
 
 
 
 
 k
 
 x
 
 
 :=
 
 R
 
 x
 
 
 
 /
 
 
 
 
 m
 
 
 
 x
 
 
 
 
 {\displaystyle k_{x}:=R_{x}/{\mathfrak {m}}_{x}}
 
 is a <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> and 
 
 
 
 
 
 
 m
 
 
 
 x
 
 
 
 /
 
 
 
 
 m
 
 
 
 x
 
 
 2
 
 
 
 
 {\displaystyle {\mathfrak {m}}_{x}/{\mathfrak {m}}_{x}^{2}}
 
 is a <a href="/facts/Vector_space/M1pxMLx2">vector space</a> over that field (the <a href="/facts/Cotangent_space/INNIyG4S">cotangent space</a>). The tangent space 
 
 
 
 
 T
 
 x
 
 
 (
 X
 )
 
 
 {\displaystyle T_{x}(X)}
 
 is defined as the <a href="/facts/Dual_space/4Uw4Knz1">dual</a> of this vector space.
The idea is the following: a tangent vector at 
 
 
 
 x
 
 
 {\displaystyle x}
 
 should tell you how to "differentiate" "functions" at 
 
 
 
 x
 
 
 {\displaystyle x}
 
, i.e. the elements of 
 
 
 
 
 R
 
 x
 
 
 
 
 {\displaystyle R_{x}}
 
. Now it is enough to know how to differentiate functions whose value at 
 
 
 
 x
 
 
 {\displaystyle 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 
 
 
 
 
 
 
 m
 
 
 
 x
 
 
 
 
 {\displaystyle {\mathfrak {m}}_{x}}
 
. Furthermore, if two functions are given with value zero at 
 
 
 
 x
 
 
 {\displaystyle x}
 
, then their product has derivative 0 at 
 
 
 
 x
 
 
 {\displaystyle x}
 
, by the <a href="/facts/Product_rule/MyKxsP5t">product rule</a>. So we only need to know how to assign "numbers" to the elements of 
 
 
 
 
 
 
 m
 
 
 
 x
 
 
 
 /
 
 
 
 
 m
 
 
 
 x
 
 
 2
 
 
 
 
 {\displaystyle {\mathfrak {m}}_{x}/{\mathfrak {m}}_{x}^{2}}
 
, and this is what the dual space does.

<h2 id="modules-over-the-structure-sheaf">Modules over the structure sheaf</h2>
Main article: <a href="/facts/Sheaf_of_modules/grSMBXho">Sheaf of modules</a>
Given a locally ringed space 
 
 
 
 (
 X
 ,
 
 
 
 O
 
 
 
 X
 
 
 )
 
 
 {\displaystyle (X,{\mathcal {O}}_{X})}
 
, certain <a href="/facts/Sheaf_(mathematics)/MM3hcRw4">sheaves</a> of modules on 
 
 
 
 X
 
 
 {\displaystyle X}
 
 occur in the applications, the 
 
 
 
 
 
 
 O
 
 
 
 X
 
 
 
 
 {\displaystyle {\mathcal {O}}_{X}}
 
-modules. To define them, consider a sheaf 
 
 
 
 
 
 F
 
 
 
 
 {\displaystyle {\mathcal {F}}}
 
 of <a href="/facts/Abelian_group/FfWwmx4R">abelian groups</a> on 
 
 
 
 X
 
 
 {\displaystyle X}
 
. If 
 
 
 
 
 
 F
 
 
 (
 U
 )
 
 
 {\displaystyle {\mathcal {F}}(U)}
 
 is a <a href="/facts/Module_(mathematics)/jP0LIhFL">module</a> over the ring 
 
 
 
 
 
 
 O
 
 
 
 X
 
 
 (
 U
 )
 
 
 {\displaystyle {\mathcal {O}}_{X}(U)}
 
 for every open set 
 
 
 
 U
 
 
 {\displaystyle U}
 
 in 
 
 
 
 X
 
 
 {\displaystyle X}
 
, and the restriction maps are compatible with the module structure, then we call 
 
 
 
 
 
 F
 
 
 
 
 {\displaystyle {\mathcal {F}}}
 
 an 
 
 
 
 
 
 
 O
 
 
 
 X
 
 
 
 
 {\displaystyle {\mathcal {O}}_{X}}
 
-module. In this case, the stalk of 
 
 
 
 
 
 F
 
 
 
 
 {\displaystyle {\mathcal {F}}}
 
 at 
 
 
 
 x
 
 
 {\displaystyle x}
 
 will be a module over the local ring (stalk) 
 
 
 
 
 R
 
 x
 
 
 
 
 {\displaystyle R_{x}}
 
, for every 
 
 
 
 x
 ∈
 X
 
 
 {\displaystyle x\in X}
 
.
A morphism between two such 
 
 
 
 
 
 
 O
 
 
 
 X
 
 
 
 
 {\displaystyle {\mathcal {O}}_{X}}
 
-modules is a <a href="/facts/Morphism_of_sheaves/MM3hcRw4">morphism of sheaves</a> that is compatible with the given module structures. The category of 
 
 
 
 
 
 
 O
 
 
 
 X
 
 
 
 
 {\displaystyle {\mathcal {O}}_{X}}
 
-modules over a fixed locally ringed space 
 
 
 
 (
 X
 ,
 
 
 
 O
 
 
 
 X
 
 
 )
 
 
 {\displaystyle (X,{\mathcal {O}}_{X})}
 
 is an <a href="/facts/Abelian_category/4WN473Yc">abelian category</a>.
An important subcategory of the category of 
 
 
 
 
 
 
 O
 
 
 
 X
 
 
 
 
 {\displaystyle {\mathcal {O}}_{X}}
 
-modules is the category of <a href="/facts/Quasi-coherent_sheaves/h2NoUN7e">quasi-coherent sheaves</a> on 
 
 
 
 X
 
 
 {\displaystyle X}
 
. A sheaf of 
 
 
 
 
 
 
 O
 
 
 
 X
 
 
 
 
 {\displaystyle {\mathcal {O}}_{X}}
 
-modules is called quasi-coherent if it is, locally, isomorphic to the <a href="/facts/Cokernel/vYFQRPoh">cokernel</a> of a map between free 
 
 
 
 
 
 
 O
 
 
 
 X
 
 
 
 
 {\displaystyle {\mathcal {O}}_{X}}
 
-modules. A <a href="/facts/Coherent_sheaf/h2NoUN7e">coherent sheaf</a> 
 
 
 
 F
 
 
 {\displaystyle F}
 
 is a quasi-coherent sheaf that is, locally, of <a href="/facts/Algebra_of_finite_type/57a93D5A">finite type</a> and for every open subset 
 
 
 
 U
 
 
 {\displaystyle U}
 
 of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 the <a href="/facts/Kernel_(algebra)/GGSpUPMJ">kernel</a> of any morphism from a free 
 
 
 
 
 
 
 O
 
 
 
 U
 
 
 
 
 {\displaystyle {\mathcal {O}}_{U}}
 
-module of finite rank to 
 
 
 
 
 
 
 F
 
 
 
 U
 
 
 
 
 {\displaystyle {\mathcal {F}}_{U}}
 
 is also of finite type.

<h2 id="citations">Citations</h2>

<ul><li>Section 0.4 of <a href="/facts/Alexander_Grothendieck/jLQGPY7M">Grothendieck, Alexandre</a>; <a href="/facts/Jean_Dieudonn%C3%A9/gWagzMdu">Dieudonné, Jean</a> (1960). <a href="http://www.numdam.org/item/PMIHES_1960__4__5_0">"Éléments de géométrie algébrique: I. Le langage des schémas"</a>. <a href="/facts/Publications_Math%C3%A9matiques_de_l%27IH%C3%89S/2ur5ofxL">Publications Mathématiques de l'IHÉS</a>. 4. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2Fbf02684778">10.1007/bf02684778</a>. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0217083">0217083</a>.</li>
<li><a href="/facts/Robin_Hartshorne/mCBLyIW6">Hartshorne, Robin</a> (1977), <a href="/facts/Algebraic_Geometry_(book)/YcAhHWBj">Algebraic Geometry</a>, <a href="/facts/Graduate_Texts_in_Mathematics/VTIfY29j">Graduate Texts in Mathematics</a>, vol. 52, New York: Springer-Verlag, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-90244-9, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0463157">0463157</a></li></ul>
<h2 id="external-links">External links</h2>
<ul><li>Onishchik, A.L. (2001) [1994], <a href="https://www.encyclopediaofmath.org/index.php?title=Ringed_space">"Ringed space"</a>, <a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Éléments de géométrie algébrique, Ch 0, 4.1.1. <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
</ol>

Ringed space open-in-new

Ringed space