Ringed space

2020 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