Algebraic Spaces - Key Properties

Algebraic spaces inherit many good properties from schemes while providing greater flexibility. Understanding which scheme-theoretic results generalize to algebraic spaces is crucial for applications.

TheoremRepresentability of the Diagonal

For an algebraic space $X$ over $S$ , the diagonal $\Delta: X \to X \times_S X$ being representable implies:

For any schemes $T_1, T_2$ and morphisms $T_i \to X$ , the fiber product $T_1 \times_X T_2$ is representable by a scheme
The graph of any morphism $f: Y \to X$ from a scheme is representable
Automorphism groups and isotropy groups are representable by schemes

This makes many geometric constructions possible without requiring $X$ itself to be a scheme.

DefinitionSeparated and Proper

An algebraic space $X$ is separated if the diagonal $\Delta: X \to X \times_S X$ is a closed embedding. It is proper if the structure morphism $X \to S$ is universally closed, separated, and of finite type.

These definitions generalize the classical notions for schemes and maintain the same essential geometric content.

ExampleProper but Non-Scheme Example

The Hironaka example provides a proper algebraic space over $\mathbb{C}$ that is not a scheme. It arises as a quotient of a scheme by a free action of $\mathbb{Z}/2\mathbb{Z}$ , but the quotient fails to be a scheme due to pathological behavior at a single point. This shows that algebraic spaces genuinely extend the category of schemes.

TheoremQuasi-Coherent Sheaves on Algebraic Spaces

For an algebraic space $X$ with étale cover $U \to X$ , the category $\textbf{QCoh}(X)$ is defined via descent: $\textbf{QCoh}(X) = \text{Desc}(U \to X, \textbf{QCoh})$

This category is well-defined (independent of the choice of $U$ ) and has excellent properties:

It is an abelian category with enough injectives
Pullback functors $f^*: \textbf{QCoh}(Y) \to \textbf{QCoh}(X)$ are exact for flat morphisms
There are well-behaved derived categories $D(\textbf{QCoh}(X))$

Remark

The étale topology is essential here. If we used the Zariski topology, quotients by étale equivalence relations would not generally be sheaves, and the theory would collapse. The étale topology provides exactly the right level of refinement for algebraic spaces.

TheoremCohomological Descent

For a surjective étale morphism $f: U \to X$ of algebraic spaces and $\mathcal{F} \in \textbf{QCoh}(X)$ : $H^i(X, \mathcal{F}) = H^i(\text{Čech}(U/X, f^*\mathcal{F}))$

The right side is the Čech cohomology of the simplicial sheaf obtained from the cover. This allows computation of cohomology on algebraic spaces using scheme-theoretic tools.

DefinitionDeligne-Mumford Stack

An algebraic space can be viewed as a Deligne-Mumford stack with trivial stabilizers. More precisely, an algebraic space is a Deligne-Mumford stack $\mathcal{X}$ such that the diagonal $\mathcal{X} \to \mathcal{X} \times \mathcal{X}$ is representable, separated, and étale, and where all stabilizer groups are trivial.

This perspective embeds algebraic spaces into the larger framework of stacks, showing they are the "stacky" objects without non-trivial automorphisms.