nLab
ringed space

Definition

A ringed space is a pair $(X,O_X)$ where $X$ is a topological space and $O_X$ is a sheaf of unital rings. The sheaf $O_X$ is called the structure sheaf of the ringed space $(X,O_X)$.

If all stalks of the structure sheaf are local rings, it is called a locally ringed space.

A morphism of ringed spaces $(f,f^{♯}):(X,O_X)\to (Y,O_Y)$ is a pair where $f:X\to Y$ is a continuous map and the comorphism $f^{♯}:O_Y\to f_{*}{O}_X$ is a morphism of sheaves of rings over $Y$. Here $f_{*}$ denotes the direct image functor for sheaves. Any sheaf of abelian modules $ℳ$ equipped with actions $O_X(U)\times ℳ(U)\to ℳ(U)$ making $ℳ(U)$ left $O_X$-modules, and such that the actions strictly commute with the restrictions, is called a sheaf of left $O_X$-modules.

Remarks

• Every ringed space induces a ringed site: To a ringed space $(X,O_X)$ assign the ringed site $({\mathrm{Op}}_X,O_X)$ where ${\mathrm{Op}}_X$ is the category of open sets and inclusions equipped with the pretopology of open covers and $O_X$ is just viewed as a sheaf of rings on ${\mathrm{Op}}_X$.

• In toric geometry and sometimes in relation to the “absolute” algebraic geometry over $F_1$, one talks about monoided or monoidal space (Kato; Deitmar); which is a topological space together with a sheaf of monoids. N. Durov on the other hand develops a generalized algebraic geometry based on a notion of generalized ringed space, which is a space equipped with a sheaf of (commutative) generalized rings, which are finitary (= algebraic) monads in $\mathrm{Set}$ with a commutativity condition (which are related to higher analogues of Eckmann-Hilton argument).

Related concepts

• ringed space, locally ringed space,

• ringed site, locally ringed site

• ringed topos, locally ringed topos

• locally algebra-ed topos

• structured (∞,1)-topos

References

Section 6.25 of

• Aise Johan de Jong, The Stacks Project (pdf) (project website)