Chevalley's structure theorem

In algebraic geometry, Chevalley's structure theorem states that a connected algebraic group over a perfect field has a unique normal affine algebraic subgroup such that the quotient is an abelian variety. It was proved by Chevalley (1960) (though he had previously announced the result in 1953), Barsotti (1955), and Rosenlicht (1956).

Chevalley's original proof, and the other early proofs by Barsotti and Rosenlicht, used the idea of mapping the algebraic group to its Albanese variety. The original proofs were based on Weil's book Foundations of algebraic geometry, but Conrad (2002) later gave an exposition of Chevalley's proof in scheme-theoretic terminology.

References

• Barsotti, Iacopo (1955), "Structure theorems for group-varieties", Annali di Matematica Pura ed Applicata. Serie Quarta 38: 77–119, doi:10.1007/bf02413515, ISSN 0003-4622, MR 0071849
• Barsotti, Iacopo (1955), "Un teorema di struttura per le varietà gruppali", Atti della Accademia Nazionale dei Lincei. Rendiconti. Classe di Scienze Fisiche, Matematiche e Naturali 18: 43–50, MR 0076427
• Chevalley, C. (1960), "Une démonstration d'un théorème sur les groupes algébriques", Journal de Mathématiques Pures et Appliquées. Neuvième Série 39: 307–317, ISSN 0021-7824, MR 0126447
• Conrad, Brian (2002), "A modern proof of Chevalley's theorem on algebraic groups" (PDF), Journal of the Ramanujan Mathematical Society 17 (1): 1–18, ISSN 0970-1249, MR 1906417
• Rosenlicht, Maxwell (1956), "Some basic theorems on algebraic groups", American Journal of Mathematics 78: 401–443, doi:10.2307/2372523, ISSN 0002-9327, MR 0082183