Algebraic Spaces - Main Theorem

The fundamental theorems about algebraic spaces establish their key properties and justify viewing them as the natural generalization of schemes.

TheoremArtin's Representability Theorem

Let $F: (\textbf{Sch}/S)^{\text{op}} \to \textbf{Set}$ be a functor. Then $F$ is representable by an algebraic space if and only if:

Sheaf condition: $F$ is a sheaf for the étale topology
Representable diagonal: The diagonal $\Delta: F \to F \times F$ is representable by algebraic spaces
Openness of versality: There exists a scheme $U$ and a smooth surjection $U \to F$ such that versal deformations exist locally on $U$
Effectiveness: Formal deformations are effective

These conditions are checkable and provide a systematic method for proving that moduli functors are algebraic spaces.

This is one of the most important results in modern algebraic geometry, as it reduces the problem of representability to verifying deformation-theoretic properties.

TheoremKeel-Mori Theorem

Let $\mathcal{X}$ be a separated Deligne-Mumford stack of finite type over a Noetherian base $S$ with finite stabilizers. Then there exists a coarse moduli space $X$ , which is a separated algebraic space of finite type over $S$ , together with a proper morphism $\pi: \mathcal{X} \to X$ such that:

$\pi$ is universal for morphisms from $\mathcal{X}$ to algebraic spaces
$\pi$ induces bijections on geometric points: $|\mathcal{X}| \to |X|$
$\pi_*\mathcal{O}_\mathcal{X} = \mathcal{O}_X$

The coarse moduli space $X$ provides a canonical "classical" geometric object associated to the stack $\mathcal{X}$ .

ExampleModuli of Curves

The moduli stack $\mathcal{M}_g$ of smooth curves of genus $g$ has coarse moduli space $M_g$ , which is a quasi-projective scheme (for $g \geq 2$ ). However, the stack of stable curves $\overline{\mathcal{M}}_g$ has coarse moduli space $\overline{M}_g$ which, while also a scheme, naturally lives in the world of algebraic spaces in general situations.

TheoremChow's Lemma for Algebraic Spaces

Let $X$ be a proper algebraic space over a Noetherian scheme $S$ . Then there exists a proper birational morphism $f: X' \to X$ where $X'$ is a scheme and $f$ is an isomorphism over a dense open subset of $X$ .

This generalizes Chow's lemma for schemes and shows that proper algebraic spaces are "close to" being schemes—they can be dominated by schemes.

Remark

Chow's lemma is essential for extending techniques from scheme theory to algebraic spaces. Many arguments for schemes can be reduced to the projective case via Chow's lemma, and the same strategy works for algebraic spaces.

TheoremNagata Compactification for Algebraic Spaces

Let $X$ be a separated algebraic space of finite type over a Noetherian scheme $S$ . Then there exists a proper algebraic space $\overline{X}$ over $S$ and an open immersion $X \hookrightarrow \overline{X}$ .

This is the analog of Nagata's compactification theorem for schemes, showing that algebraic spaces admit compactifications just as schemes do.

TheoremGrothendieck's Existence Theorem

Let $X$ be a proper algebraic space over a complete Noetherian local ring $(A, \mathfrak{m})$ , and let $X_n = X \times_{\text{Spec}(A)} \text{Spec}(A/\mathfrak{m}^n)$ . Then the category of coherent sheaves on $X$ is equivalent to the category of compatible systems of coherent sheaves on the $X_n$ .

This formal GAGA result shows that coherent sheaves on proper algebraic spaces over complete local rings are determined by their formal completions.

These theorems establish that algebraic spaces behave like schemes in most essential respects while providing the flexibility needed for quotients and moduli problems.