Algebraic space - Knowlpedia

Definition

Let $S$ be a scheme. An algebraic space over $S$ is a sheaf $X$ on the big étale site of $S$ such that:

the diagonal $X\to X\times_SX$ is representable by schemes; and
there is a scheme $U$ and a representable, surjective étale morphism $U\to X$ .

The map $U\to X$ is an étale atlas. Every scheme defines an algebraic space by its functor of points, but an algebraic space need not be a scheme.

Quotients

Algebraic spaces allow many étale equivalence relations to have geometric quotients even when the quotient is not a scheme. They also provide the representability class used for diagonals of algebraic stacks.

References

The Stacks Project Authors, “Algebraic Spaces,” chapter.