Chevalley's structure theorem

In algebraic geometry, Chevalley's structure theorem states that a smooth connected algebraic group over a perfect field has a unique normal smooth connected 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 and are hard to follow for anyone unfamiliar with Weil's foundations, but Conrad (2002) later gave an exposition of Chevalley's proof in scheme-theoretic terminology.

Over non-perfect fields there is still a smallest normal connected linear subgroup such that the quotient is an abelian variety, but the linear subgroup need not be smooth.

A consequence of Chevalley's theorem is that any algebraic group over a field is quasi-projective.

There are several natural constructions that give connected algebraic groups that are neither affine nor complete.

Chevalley's structure theorem is used in the proof of the Néron–Ogg–Shafarevich criterion.

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