Math 2.0

Algebraic geometry Algebraic groups Theorem

Chevalley's structure theorem

Builds on Algebraic groupLinear algebraic groupAbelian varietyNormal subgroupQuotient group

Statement

Let kk be a perfect field. Let GG be a smooth connected algebraic group over kk. Then there is a unique smooth connected normal subgroup HH of GG such that:

  1. HH is a linear algebraic group;
  2. the quotient G/HG / H is an abelian variety.

Remarks

Uniqueness is the easy part. If HH and H′H' both satisfy the conclusion, the image of HH in G/H′G / H' is a connected linear algebraic group and also a closed subvariety of an abelian variety. So it is both affine and complete, hence a point, and H⊆H′H \subseteq H'. By symmetry H=H′H = H'. Existence is the deep part of the theorem.

Example: let EE be an elliptic curve over kk and G=E×𝔾mG = E \times \mathbb{G}_m. Then H={0}×𝔾mH = \set{0} \times \mathbb{G}_m is linear, and G/H≅EG / H \cong E is an abelian variety.

Over imperfect fields the statement can fail, so the hypothesis that kk is perfect cannot simply be dropped.

Going deeper

For a student

The theorem reduces many questions about algebraic groups to the two separate theories of linear algebraic groups and abelian varieties. For example, a connected algebraic group that is affine has trivial abelian quotient, and one that is complete has trivial linear part. A standard family of examples are the generalized Jacobians of singular curves. Such a group is an extension of the Jacobian of the normalized curve by a linear group.

Reading path

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

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Abelian variety, concept, no evidence levelAbelian varietyconceptAlgebraic group, concept, no evidence levelAlgebraic groupconceptLinear algebraic group, concept, no evidence levelLinear algebraic groupconceptNormal subgroup, concept, no evidence levelNormal subgroupconceptQuotient group, concept, no evidence levelQuotient groupconceptChevalley's structure theorem, statement, no evidence levelChevalley's structure…statement
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.
  • no evidence level
  1. Abelian variety concept
  2. Algebraic group concept
  3. Linear algebraic group concept
  4. Normal subgroup concept
  5. Quotient group concept
  6. Chevalley's structure theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Chevalley%27s_structure_theorem
About this page
TagMZ6XDATQKR
URIm2://MZ6XDATQKR
Kindstatement, theorem
WritingW0 stub
MSC14L10, 14K05, 20G15
StatusThis copy of the site has no build data. Run m2 status MZ6XDATQKR.
AboutAlgebraic group
UsesAlgebraic group, Linear algebraic group, Abelian variety, Normal subgroup, Quotient group
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Changed2 November 2026

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