Contents

topos theory

Ingredients

Concepts

Constructions

Examples

Theorems

# Contents

## Definition

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

A morphism of locally ringed spaces is a morphism of ringed spaces $(f,f^\sharp)\colon (X,\mathcal{O}_X)\to (Y,\mathcal{O}_Y)$, where $f:X\to Y$, such that $f_\sharp : f^*\mathcal{O}_Y \to \mathcal{O}_X$ is a morphism of local rings (that is, a map of rings which respects the maximal ideal (reflects invertibility)), where $f_\sharp$ is the adjunct of the comorphism $f^\sharp:\mathcal{O}_Y\to f_*\mathcal{O}_X$. Here we are considering $\mathcal{O}_X$ and $\mathcal{O}_Y$ to be local rings defined in the internal logic of the respective sheaf toposes, and $f^*\mathcal{O}_Y$ is an internal local ring because the inverse image functor $f^*$ preserves geometric logic.

## Examples

###### Example

(schemes)
Historically, schemes are thought of as locally ringed spaces and this application of the notion to algebraic geometry motivated much of its development.

(However, already Grothendieck (1973) pointed out that it is often more frutiful to view schemes instead via their functor of points, see at functorial geometry for more.)

###### Example

(smooth manifolds)
The category SmthMfd of smooth manifolds has a canonical embedding into the category of locally ringed spaces:

Indeed, any smooth manifold $M$ comes equipped with a sheaf of real-valued smooth functions $\mathcal{C}^{\infty}_M$, and the maximal ideal in a given stalk consists precisely of germs of smooth functions which vanish at that point. Moreover, a smooth map $f \colon M \to N$ determines a comorphism $f^\sharp: \mathcal{C}^{\infty}_N \to f_* \mathcal{C}^{\infty}_M$ by the usual precomposition.

This inclusion of smooth manifolds into locally ringed spaces is fully faithful. For a proof, see Lucas Braune’s comment at Math.SE:511604.

(In fact, already the embedding of smooth manifolds into formal duals of R-algebras is fully faithful, which means that all smooth manifolds are even “affine” locally ringed spaces, in a sense.)

## On the locality condition

The condition that all stalks $\mathcal{O}_{X,x}$ are local rings can be reformulated without referring to the points of $X$:

• The only open subset $U$ such that $\mathcal{O}_X(U)$ is the zero ring is $U = \emptyset$.
• Let $f, g \in \mathcal{O}_X(U)$ be local sections such that $f + g$ is invertible in $\mathcal{O}_X(U)$. Then there is an open covering $U = V \cup W$ such that $f|_V \in \mathcal{O}_X(V)$ and $g|_W \in \mathcal{O}_X(W)$ are invertible. (It’s allowed that $V$ or $W$ are empty.)

Textbook accounts: