Math 2.0

Algebraic geometry Definition

Algebraic space

Builds on SchemeFunctor of pointsFiber product of schemesÉtale morphismSheaf

Definition

Let SS be a scheme, and let 𝒞\mathcal{C} be the category of schemes over SS with the fppf topology. For an SS-scheme UU, let hUh_U be its functor of points. An algebraic space over SS is a functor F ⁣:𝒞op→SetsF \colon \mathcal{C}^{\mathrm{op}} \to \mathrm{Sets} such that:

  1. FF is a sheaf for the fppf topology.
  2. For all SS-schemes T1,T2T_1, T_2 and maps hT1→Fh_{T_1} \to F, hT2→Fh_{T_2} \to F, the fiber product hT1×FhT2h_{T_1} \times_F h_{T_2} is isomorphic to hTh_T for some scheme TT.
  3. There are an SS-scheme UU and a map hU→Fh_U \to F with the following property. For every SS-scheme TT and map hT→Fh_T \to F, the induced morphism U×FT→TU \times_F T \to T is surjective and étale.

Remarks

Every SS-scheme XX gives an algebraic space hXh_X. Clause 1 holds because representable functors are fppf sheaves, clause 2 because hT1×hXhT2=hT1×XT2h_{T_1} \times_{h_X} h_{T_2} = h_{T_1 \times_X T_2}, and clause 3 holds with U=XU = X. A functor that is not a sheaf is not an algebraic space. For example, the constant presheaf with value {0,1}\set{0, 1} fails clause 1: on the disjoint union of two copies of SS it should have four sections, but it has two. Clause 2 says that the diagonal F→F×FF \to F \times F is representable by schemes, and clause 3 then makes sense. Some authors use the étale topology in clause 1 and add a quasi-compactness condition on the diagonal, as in Knutson's book.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

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Fiber product of schemes, concept, no evidence levelFiber product of sche…conceptFunctor of points, concept, no evidence levelFunctor of pointsconceptScheme, concept, no evidence levelSchemeconceptSheaf, concept, no evidence levelSheafconceptÉtale morphism, concept, no evidence levelÉtale morphismconceptAlgebraic space, definition, no evidence levelAlgebraic spacedefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Fiber product of schemes concept
  2. Functor of points concept
  3. Scheme concept
  4. Sheaf concept
  5. Étale morphism concept
  6. Algebraic space definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Algebraic_space
About this page
TagHQHP2J7F3M
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Kinddefinition
WritingW0 stub
MSC14A20
DefinesAlgebraic space
UsesScheme, Functor of points, Fiber product of schemes, Étale morphism, Sheaf
Changed2 November 2026

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