---
tag: 7Y69FN6M7Y
kind: definition
name: Locally ringed space
msc: [14A15, 18F20]
defines: B7FPT8JTXR
uses: [KNDXDE7K8M, TB6TM6R4MD, JTAKAPTT8H, MZ02BKPHJA, P8YTMGEN9R, 0VGFHTM75Y]
sources:
  - url: https://en.wikipedia.org/wiki/Ringed_space
---

## Definition

Let $(X, \mathcal{O}_X)$ be a [[ringed space|KNDXDE7K8M]], that is, a topological space $X$ with a [[sheaf|TB6TM6R4MD]] $\mathcal{O}_X$ of commutative rings on $X$.
It is a *locally ringed space* if for every point $x \in X$ the [[stalk|JTAKAPTT8H]] $\mathcal{O}_{X,x}$ is a [[local ring|MZ02BKPHJA]].
The maximal ideal of $\mathcal{O}_{X,x}$ is written $\mathfrak{m}_x$.

Let $(X, \mathcal{O}_X)$ and $(Y, \mathcal{O}_Y)$ be locally ringed spaces.
A *morphism of locally ringed spaces* $(X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ is a pair $(f, f^\#)$ such that:

1. $f \colon X \to Y$ is a [[continuous map|0VGFHTM75Y]].
2. $f^\# \colon \mathcal{O}_Y \to f_* \mathcal{O}_X$ is a morphism of sheaves of rings on $Y$.
3. For every $x \in X$, the induced ring map $f^\#_x \colon \mathcal{O}_{Y, f(x)} \to \mathcal{O}_{X,x}$ is a [[local homomorphism|P8YTMGEN9R]], that is, $(f^\#_x)^{-1}(\mathfrak{m}_x) = \mathfrak{m}_{f(x)}$.

## Remarks

Example: let $M$ be a smooth manifold and let $\mathcal{O}_M(U)$ be the ring of smooth real functions on the open set $U$.
A germ $g$ at $p$ is invertible in $\mathcal{O}_{M,p}$ exactly when $g(p) \ne 0$.
So the non-invertible germs form the ideal $\set{g : g(p) = 0}$, and $\mathcal{O}_{M,p}$ is local.
Non-example: the one-point space with the ring $\Z$ as its sheaf is a ringed space.
Its only stalk $\Z$ is not local, since $(2)$ and $(3)$ are two different maximal ideals.
Clause 3 is not automatic: a morphism of ringed spaces between locally ringed spaces need not satisfy it.
It is the clause that makes morphisms respect "vanishing at a point".