Algebraic Spaces - Core Definitions

Algebraic spaces generalize schemes by allowing quotients by étale equivalence relations. They provide the natural setting for many moduli problems and compactifications that cannot be represented by schemes.

DefinitionAlgebraic Space

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

The diagonal morphism $\Delta: X \to X \times_S X$ is representable by schemes
There exists a scheme $U$ and a surjective étale morphism $U \to X$

The first condition ensures that fiber products with schemes exist and are schemes. The second condition says that $X$ is "locally a scheme" in the étale topology.

The key innovation is that we work with sheaves rather than spaces with underlying topological spaces. This flexibility allows gluing that would be impossible for schemes while retaining good geometric properties.

ExampleSchemes as Algebraic Spaces

Every scheme $X$ determines an algebraic space via the Yoneda embedding $h_X: T \mapsto \text{Hom}(T, X)$ . The representability of the diagonal is automatic, and $\text{id}: X \to h_X$ is a surjective étale cover by a scheme. Thus the category of schemes embeds fully faithfully into algebraic spaces.

DefinitionÉtale Equivalence Relation

An étale equivalence relation on a scheme $U$ consists of a scheme $R$ with two étale morphisms $s, t: R \to U$ (source and target) together with:

An identity section $e: U \to R$ with $s \circ e = t \circ e = \text{id}_U$
An inverse morphism $i: R \to R$ swapping $s$ and $t$
A composition morphism $m: R \times_{s,U,t} R \to R$ satisfying associativity

These must satisfy the usual equivalence relation axioms in the category of sheaves.

TheoremQuotients by Étale Equivalence Relations

If $R \rightrightarrows U$ is an étale equivalence relation on a scheme $U$ , then the quotient sheaf $X = U/R$ is an algebraic space. The natural morphism $U \to X$ is surjective and étale, providing the required cover.

This construction is the primary method for producing algebraic spaces that are not schemes. The failure of $X$ to be a scheme reflects the fact that the equivalence relation may not have a geometric quotient in the category of schemes.

ExampleNon-Separated Algebraic Space

Let $U = \mathbb{A}^2 \setminus \{0\}$ and define $R \subset U \times U$ to be the union of the diagonal and the complement of two affine lines. This defines an étale equivalence relation, and the quotient $X = U/R$ is an algebraic space that is not separated (the diagonal is not a closed embedding). This cannot occur for schemes.

Remark

The condition that the diagonal is representable ensures that intersections of "affine opens" in an algebraic space are schemes. This is the minimum requirement for developing a coherent geometric theory. Without it, even basic constructions like fiber products become intractable.

DefinitionMorphism of Algebraic Spaces

A morphism $f: X \to Y$ of algebraic spaces is a morphism of sheaves. We say $f$ has a property $P$ (e.g., proper, smooth, étale, finite type) if for every scheme $T$ and morphism $T \to Y$ , the base change $X \times_Y T \to T$ is representable by schemes and has property $P$ .

This definition extends properties of morphisms of schemes to the more general setting. The representability requirement ensures that properties can be checked on scheme-valued points, making them computable.