---
tag: MTV9TFKF3X
kind: concept
name: Algebraic space
msc: [14A20]
sources:
  - url: https://en.wikipedia.org/wiki/Algebraic_space
same_as:
  wikidata: Q4724016
  wikipedia: Algebraic space
---

## Summary

An algebraic space is a generalization of a scheme that is glued from schemes along étale maps instead of open immersions.
Formally it is a sheaf on the category of schemes that is the quotient of a scheme by an étale equivalence relation.
Algebraic spaces were introduced by Michael Artin, and they arise as quotients and moduli spaces that do not exist as schemes.

## Explanations

### L1 human

A scheme is glued from affine pieces along open sets.
An algebraic space is glued from schemes along étale maps, which are like local isomorphisms but not open inclusions.
This extra freedom lets one form quotients of schemes by free actions of finite groups, and other constructions, that are not schemes.
Every scheme is an algebraic space, and locally in the étale sense every algebraic space looks like a scheme.