Math 2.0

Algebraic geometry Definition

Group scheme

Builds on SchemeMorphism of schemesFiber product of schemesFunctor of pointsGroup

Definition

Let SS be a scheme, and let π ⁣:G→S\pi \colon G \to S be a scheme over SS. Products below are fiber products over SS, and all maps are morphisms over SS. A group scheme over SS is such a GG with morphisms m ⁣:G×SG→Gm \colon G \times_S G \to G, e ⁣:S→Ge \colon S \to G and i ⁣:G→Gi \colon G \to G such that:

  1. m∘(m×id⁡G)=m∘(id⁡G×m)m \circ (m \times \id_G) = m \circ (\id_G \times m) as morphisms G×SG×SG→GG \times_S G \times_S G \to G.
  2. m∘(e∘π,id⁡G)=id⁡G=m∘(id⁡G,e∘π)m \circ (e \circ \pi, \id_G) = \id_G = m \circ (\id_G, e \circ \pi).
  3. m∘(i,id⁡G)=e∘π=m∘(id⁡G,i)m \circ (i, \id_G) = e \circ \pi = m \circ (\id_G, i).

Remarks

Equivalently, for every SS-scheme TT, the set G(T)G(T) of SS-morphisms T→GT \to G carries a group structure, and the maps G(T′)→G(T)G(T') \to G(T) induced by SS-morphisms T→T′T \to T' are group homomorphisms. This is the functor of points form of the definition. For example, the multiplicative group 𝔾m,S=Spec⁡ℤ[t,t−1]×Spec⁡ℤS\mathbb{G}_{m,S} = \operatorname{Spec} \Z[t, t^{-1}] \times_{\operatorname{Spec} \Z} S is a group scheme over SS. For an affine SS-scheme T=Spec⁡RT = \operatorname{Spec} R, its points are 𝔾m(T)=R×\mathbb{G}_m(T) = R^\times, the units of RR under multiplication. The affine line 𝔸S1\mathbb{A}^1_S with multiplication (a,b)↦ab(a, b) \mapsto ab is not a group scheme. Clause 3 fails, since the point 0∈R0 \in R has no multiplicative inverse in a nonzero ring RR.

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 pointsconceptGroup, concept, no evidence levelGroupconceptMorphism of schemes, concept, no evidence levelMorphism of schemesconceptScheme, concept, no evidence levelSchemeconceptGroup scheme, definition, no evidence levelGroup schemedefinition
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  1. Fiber product of schemes concept
  2. Functor of points concept
  3. Group concept
  4. Morphism of schemes concept
  5. Scheme concept
  6. Group scheme definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Group_scheme
About this page
TagCRFGFCNG3Q
URIm2://CRFGFCNG3Q
Kinddefinition
WritingW0 stub
MSC14L15
DefinesGroup scheme
UsesScheme, Morphism of schemes, Fiber product of schemes, Functor of points, Group
Changed2 November 2026

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