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A [[Topological space|topological space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082460/r0824601.png" /> with a [[Sheaf|sheaf]] of rings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082460/r0824602.png" />. The sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082460/r0824603.png" /> is called the structure sheaf of the ringed space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082460/r0824604.png" />. It is usually understood that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082460/r0824605.png" /> is a sheaf of associative and commutative rings with a unit element. A pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082460/r0824606.png" /> is called a morphism from a ringed space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082460/r0824607.png" /> into a ringed space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082460/r0824608.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082460/r0824609.png" /> is a continuous mapping and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082460/r08246010.png" /> is a homomorphism of sheaves of rings over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082460/r08246011.png" /> which transfers units in the stalks to units. Ringed spaces and their morphisms constitute a category. Giving a homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082460/r08246012.png" /> is equivalent to giving a homomorphism
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{{MSC|14}}
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{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082460/r08246013.png" /></td> </tr></table>
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A ''ringed space'' is a
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[[Topological space|topological space]] $X$ with a
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[[Sheaf|sheaf]] of rings $\def\cO{ {\mathcal O}}\cO_X$. The sheaf $\cO_X$ is called the structure sheaf of the ringed space $(X,\cO_X)$. It is usually understood that $\cO_X$ is a sheaf of associative and commutative rings with a unit element. A pair $(f,f^\sharp)$ is called a morphism from a ringed space $(X,\cO_X)$ into a ringed space $(Y,\cO_Y)$ if $f:X\to Y$ is a continuous mapping and $f^\sharp : f^*\;\cO_Y\to \cO_X$ is a homomorphism of sheaves of rings over $Y$ which transfers units in the stalks to units. Ringed spaces and their morphisms constitute a category. Giving a homomorphism $f^\sharp $ is equivalent to giving a homomorphism
  
which transfers unit elements to unit elements.
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$$f_\sharp :\cO_Y\to f_*\cO_X$$
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which transfers unit elements to unit elements (see the comment below for the definition of $f_*$).
  
A ringed space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082460/r08246014.png" /> is called a local ringed space if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082460/r08246015.png" /> is a sheaf of local rings (cf. [[Local ring|Local ring]]). In defining a morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082460/r08246016.png" /> between local ringed spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082460/r08246017.png" /> it is further assumed that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082460/r08246018.png" />, the homomorphism
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A ringed space $(X,\cO_X)$ is called a local ringed space if $\cO_X$ is a sheaf of local rings (cf.
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[[Local ring|Local ring]]). In defining a morphism $(f,f^\sharp)$ between local ringed spaces $(X,\cO_X)\to (Y,\cO_Y)$ it is further assumed that for any $x\in X$, the homomorphism
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082460/r08246019.png" /></td> </tr></table>
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$$f_X^\sharp : \cO_{Y,f}(x)\to \cO_{X,x}$$
 
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is local. Local ringed spaces form a subcategory in the category of all ringed spaces. Another important subcategory is that of ringed spaces over a (fixed) field $k$, i.e. ringed spaces $(X,\cO_X)$ where $\cO$ is a sheaf of algebras over $k$, while the morphisms are compatible with the structure of the algebras.
is local. Local ringed spaces form a subcategory in the category of all ringed spaces. Another important subcategory is that of ringed spaces over a (fixed) field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082460/r08246020.png" />, i.e. ringed spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082460/r08246021.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082460/r08246022.png" /> is a sheaf of algebras over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082460/r08246023.png" />, while the morphisms are compatible with the structure of the algebras.
 
  
 
===Examples of ringed spaces.===
 
===Examples of ringed spaces.===
  
  
1) For each topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082460/r08246024.png" /> there is a corresponding ringed space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082460/r08246025.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082460/r08246026.png" /> is the sheaf of germs of continuous functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082460/r08246027.png" />.
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1) For each topological space $X$ there is a corresponding ringed space $(X,C_X$, where $C_X$ is the sheaf of germs of continuous functions on $X$.
  
2) For each [[Differentiable manifold|differentiable manifold]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082460/r08246028.png" /> (e.g. of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082460/r08246029.png" />) there is a corresponding ringed space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082460/r08246030.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082460/r08246031.png" /> is the sheaf of germs of functions of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082460/r08246032.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082460/r08246033.png" />; moreover, the category of differentiable manifolds is a full subcategory of the category of ringed spaces over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082460/r08246034.png" />.
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2) For each
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[[Differentiable manifold|differentiable manifold]] $X$ (e.g. of class $C^\infty$) there is a corresponding ringed space $(X,D_X)$, where $D_X$ is the sheaf of germs of functions of class $C^\infty$ on $X$; moreover, the category of differentiable manifolds is a full subcategory of the category of ringed spaces over $\R$.
  
3) The analytic manifolds (cf. [[Analytic manifold|Analytic manifold]]) and analytic spaces (cf. [[Analytic space|Analytic space]]) over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082460/r08246035.png" /> constitute full subcategories of the category of ringed spaces over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082460/r08246036.png" />.
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3) The analytic manifolds (cf.
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[[Analytic manifold|Analytic manifold]]) and analytic spaces (cf.
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[[Analytic space|Analytic space]]) over a field $k$ constitute full subcategories of the category of ringed spaces over $k$.
  
4) Schemes (cf. [[Scheme|Scheme]]) constitute a full subcategory of the category of local ringed spaces.
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4) Schemes (cf.
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[[Scheme|Scheme]]) constitute a full subcategory of the category of local ringed spaces.
  
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) {{MR|0447223}} {{ZBL|0362.14001}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> R. Hartshorne, "Algebraic geometry" , Springer (1977) {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR></table>
 
  
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====Comment====
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If $\def\cF{ {\mathcal F}}\cF$ is a sheaf over a topological space $X$ and $f:X\to Y$ is a mapping of topological spaces, then the induced sheaf $f_*\cF$ over $Y$ is the sheaf defined by $(f_*\cF)(V)=\cF(f^{-1}V)$ for all open $V\in Y$.
  
  
====Comments====
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====References====
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082460/r08246037.png" /> is a sheaf over a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082460/r08246038.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082460/r08246039.png" /> is a mapping of topological spaces, then the induced sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082460/r08246040.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082460/r08246041.png" /> is the sheaf defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082460/r08246042.png" /> for all open <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082460/r08246043.png" />.
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{|
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|-
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|valign="top"|{{Ref|Ha}}||valign="top"| R. Hartshorne, "Algebraic geometry", Springer (1977) {{MR|0463157}} {{ZBL|0367.14001}}
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|-
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|valign="top"|{{Ref|Sh}}||valign="top"| I.R. Shafarevich, "Basic algebraic geometry", Springer (1977) (Translated from Russian) {{MR|0447223}} {{ZBL|0362.14001}}
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|-
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Latest revision as of 15:52, 24 November 2013

2010 Mathematics Subject Classification: Primary: 14-XX [MSN][ZBL]

A ringed space is a topological space $X$ with a sheaf of rings $\def\cO{ {\mathcal O}}\cO_X$. The sheaf $\cO_X$ is called the structure sheaf of the ringed space $(X,\cO_X)$. It is usually understood that $\cO_X$ is a sheaf of associative and commutative rings with a unit element. A pair $(f,f^\sharp)$ is called a morphism from a ringed space $(X,\cO_X)$ into a ringed space $(Y,\cO_Y)$ if $f:X\to Y$ is a continuous mapping and $f^\sharp : f^*\;\cO_Y\to \cO_X$ is a homomorphism of sheaves of rings over $Y$ which transfers units in the stalks to units. Ringed spaces and their morphisms constitute a category. Giving a homomorphism $f^\sharp $ is equivalent to giving a homomorphism

$$f_\sharp :\cO_Y\to f_*\cO_X$$ which transfers unit elements to unit elements (see the comment below for the definition of $f_*$).

A ringed space $(X,\cO_X)$ is called a local ringed space if $\cO_X$ is a sheaf of local rings (cf. Local ring). In defining a morphism $(f,f^\sharp)$ between local ringed spaces $(X,\cO_X)\to (Y,\cO_Y)$ it is further assumed that for any $x\in X$, the homomorphism

$$f_X^\sharp : \cO_{Y,f}(x)\to \cO_{X,x}$$ is local. Local ringed spaces form a subcategory in the category of all ringed spaces. Another important subcategory is that of ringed spaces over a (fixed) field $k$, i.e. ringed spaces $(X,\cO_X)$ where $\cO$ is a sheaf of algebras over $k$, while the morphisms are compatible with the structure of the algebras.

Examples of ringed spaces.

1) For each topological space $X$ there is a corresponding ringed space $(X,C_X$, where $C_X$ is the sheaf of germs of continuous functions on $X$.

2) For each differentiable manifold $X$ (e.g. of class $C^\infty$) there is a corresponding ringed space $(X,D_X)$, where $D_X$ is the sheaf of germs of functions of class $C^\infty$ on $X$; moreover, the category of differentiable manifolds is a full subcategory of the category of ringed spaces over $\R$.

3) The analytic manifolds (cf. Analytic manifold) and analytic spaces (cf. Analytic space) over a field $k$ constitute full subcategories of the category of ringed spaces over $k$.

4) Schemes (cf. Scheme) constitute a full subcategory of the category of local ringed spaces.


Comment

If $\def\cF{ {\mathcal F}}\cF$ is a sheaf over a topological space $X$ and $f:X\to Y$ is a mapping of topological spaces, then the induced sheaf $f_*\cF$ over $Y$ is the sheaf defined by $(f_*\cF)(V)=\cF(f^{-1}V)$ for all open $V\in Y$.


References

[Ha] R. Hartshorne, "Algebraic geometry", Springer (1977) MR0463157 Zbl 0367.14001
[Sh] I.R. Shafarevich, "Basic algebraic geometry", Springer (1977) (Translated from Russian) MR0447223 Zbl 0362.14001
How to Cite This Entry:
Ringed space. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Ringed_space&oldid=30763
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article