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Canonical static concept record

Algebraic space

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Summary

A sheaf of sets on the big fppf site over $S$ with representable diagonal and an étale surjective cover by a scheme.

Record metadata

Canonical name: Algebraic space
Concept ID: algebraic_space
Content version: 2.3.0
Primary field: Mathematics
Fields: Mathematics
Primary domain: Algebraic geometry
Domains: Algebraic geometry, Category theory
Alternate terminology: No alternate terminology is currently authored for this record.

Carrier(s)

base scheme $S$
presheaf $F\colon(\mathrm{Sch}/S)_{\mathrm{fppf}}^{\mathrm{op}}\to\mathbf{Set}$

Axioms / constraints

$F$ is a sheaf for the fppf topology
the diagonal $F\to F\times F$ is representable by schemes
there exists a scheme $U$ and a surjective étale morphism $h_U\to F$

Notes

The existence of an étale cover is a property; no atlas is selected as part of the algebraic-space structure. Equivalent étale-site formulations are standard.

Concept sources

Wikipedia — Algebraic space — Algebraic space · encyclopedic reference · source ID wp-algebraic-space
nLab — Algebraic space — Algebraic space · mathematical reference · source ID nlab-algebraic-space
Stacks Project — Algebraic spaces — Algebraic spaces · reference · source ID stacks-algebraic-space

Incoming relations (arrows to this concept)

Each relation below ends at this concept.

Sheaf Algebraic space

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: e_sheaf_algspace
Relation type: Impose axiom impose-axiom
Direction: source → target
Endpoint roles: source: Builds toward; target: Built from
Authored annotation: + representable diagonal and existence of étale cover

Authored explanation

Specialize to an fppf sheaf $F$ on $(\mathrm{Sch}/S)_{\mathrm{fppf}}$ , require a representable diagonal, and require that some scheme $U$ admit a surjective étale morphism $h_U\to F$ .

How to interpret this relation type

Keep the existing data and select the subclass satisfying an additional law, existence condition, finiteness condition, or other property.

Relation sources

Stacks Project — Algebraic spaces — Algebraic spaces · reference · source ID stacks-algebraic-space

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No direct relations are authored in this direction.

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