nLab
locally ringed space

Definition

A locally ringed space is a ringed space $(X,𝒪)$ such that the stalks of the structure sheaf $𝒪$ are local rings.

A morphism of locally ringed space is a morphism of ringed spaces $(f,f^{♯}):(X,{𝒪}_X)\to (Y,{𝒪}_Y)$, where $f:X\to Y$, such that the comorphism $f^{♯}:{𝒪}_Y\to f_{*}{𝒪}_X$ is a morphism of local rings (that is, a map of rings which respects the maximal ideal).

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 19.2 in

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