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algebraic space

Idea

An algebraic space is an object in the sheaf topos over the fppf-site, that has representable diagonal and an étale cover(atlas) by a scheme.

In algebraic geometry one can glue affine schemes in various topologies; this way one obtains various kinds of locally affine ringed spaces. For example, schemes locally affine in Zariski topology. Étale topology is finer than Zariski, hence the category of locally affine (ringed) spaces in étale topology is larger than the category of schemes. Algebraic spaces make a category which includes the category of all schemes and is close to the category of locally affine spaces in étale topology, namely it consists of those ringed spaces which may be obtained as a quotient of a scheme $S$ by an equivalence relation $R\subset S\times S$ which is a closed subscheme, and whose projections $p_1,p_2: R\to S$ are étale morphisms of schemes.

Definition

Write $C_{fppf}$ for the fppf-site (over some scheme, as desired).

Definition

An algebraic space is

• an object $X \in Sh(C_{fppf})$ in the sheaf topos;

• whose diagonal morphism $X \to X \times X$ is representable;

• and for which there exists $U \in C$ and a morphism $U \to X$ which is

  • surjective;

  • étale.

In this form this appears as de Jong, def. 35.6.1.

Properties

Characterization as presheaf on affine schemes

• Artin representability theorem

References

Algebraic spaces are the topic of chapter 35 in

• Aise Johan de Jong, The Stacks Project

A standard monograph on algebraic spaces is

• D. Knutson, Algebraic spaces , LNM 203, Springer 1971.

Lecture notes include

• James Milne, section 7 of Lectures on Étale Cohomology
• G. B. Winters, An elementary lecture on algebraic spaces, in: P. Salmon (eds) Categories and Commutative Algebra. C.I.M.E. Summer Schools 58 (Varena 1971), C.I.M.E., Ed. Cremonese, Roma 1973; reprint Springer 2010, doi 2010

Definition in E-∞ geometry is in

• Jacob Lurie, section 1.3 of Quasi-Coherent Sheaves and Tannaka Duality Theorems

Some related MO questions:

• why-is-this-not-an-algebraic-space,

• Can an algebraic space fail to have a universal map to a scheme?

• What are the Benefits of Using Algebraic Spaces over Schemes?

• Commutative rings to algebraic spaces in one jump?