# Difference between revisions of "Ringed space"

Ulf Rehmann (talk | contribs) m (MR/ZBL numbers added) |
Ulf Rehmann (talk | contribs) m (tex,MSC, some minor modifications) |
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− | + | {{MSC|14}} | |

+ | {{TEX|done}} | ||

− | + | A ''ringed space'' is a | |

+ | [[Topological space|topological space]] $X$ with a | ||

+ | [[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. | + | $$f_\sharp :\cO_Y\to f_*\cO_X$$ |

+ | which transfers unit elements to unit elements (see the comment below for the definition of $f_*$). | ||

− | A ringed space | + | A ringed space $(X,\cO_X)$ is called a local ringed space if $\cO_X$ is a sheaf of local rings (cf. |

+ | [[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 | ||

− | + | $$f_X^\sharp : \cO_{Y,f}(x)\to \cO_{X,x}$$ | |

− | + | 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. | |

− | 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 | ||

===Examples of ringed spaces.=== | ===Examples of ringed spaces.=== | ||

− | 1) For each topological space | + | 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]] | + | 2) For each |

+ | [[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 | + | 3) The analytic manifolds (cf. |

+ | [[Analytic manifold|Analytic manifold]]) and analytic spaces (cf. | ||

+ | [[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. | + | 4) Schemes (cf. |

+ | [[Scheme|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==== |

− | + | {| | |

+ | |- | ||

+ | |valign="top"|{{Ref|Ha}}||valign="top"| R. Hartshorne, "Algebraic geometry", Springer (1977) {{MR|0463157}} {{ZBL|0367.14001}} | ||

+ | |- | ||

+ | |valign="top"|{{Ref|Sh}}||valign="top"| I.R. Shafarevich, "Basic algebraic geometry", Springer (1977) (Translated from Russian) {{MR|0447223}} {{ZBL|0362.14001}} | ||

+ | |- | ||

+ | |} |

## 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