algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...

, Chevalley's structure theorem states that a smooth connected

algebraic group In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Man ...

over a

perfect field In algebra, a field ''k'' is perfect if any one of the following equivalent conditions holds: * Every irreducible polynomial over ''k'' has distinct roots. * Every irreducible polynomial over ''k'' is separable. * Every finite extension of ''k'' is ...

has a unique normal smooth connected affine algebraic subgroup such that the quotient is an

abelian variety In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular func ...

. It was proved by (though he had previously announced the result in 1953), , and . 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 In mathematics, the Albanese variety A(V), named for Giacomo Albanese, is a generalization of the Jacobian variety of a curve. Precise statement The Albanese variety is the abelian variety A generated by a variety V taking a given point of V to t ...

. The original proofs were based on Weil's book

Foundations of algebraic geometry ''Foundations of Algebraic Geometry'' is a book by that develops algebraic geometry over field (mathematics), fields of any characteristic (algebra), characteristic. In particular it gives a careful treatment of intersection theory by defining th ...

and are hard to follow for anyone unfamiliar with Weil's foundations, but 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.

Examples

There are several natural constructions that give connected algebraic groups that are neither affine nor complete. *If ''C'' is a curve with an effective divisor ''m'', then it has an associated

generalized Jacobian In algebraic geometry a generalized Jacobian is a commutative algebraic group associated to a curve with a divisor, generalizing the Jacobian variety of a complete curve. They were introduced by Maxwell Rosenlicht in 1954, and can be used to study ...

''J''_''m''. This is a commutative algebraic group that maps onto the Jacobian variety ''J''₀ of ''C'' with affine kernel. So ''J'' is an extension of an abelian variety by an affine algebraic group. In general this extension does not split. *The reduced connected component of the relative Picard scheme of a proper scheme over a perfect field is an algebraic group, which is in general neither affine nor proper. *The connected component of the closed fiber of a Neron model over a discrete valuation ring is an algebraic group, which is in general neither affine nor proper. *For analytic groups some of the obvious analogs of Chevalley's theorem fail. For example, the product of the additive group ''C'' and any elliptic curve has a dense collection of closed (analytic but not algebraic) subgroups isomorphic to ''C'' so there is no unique "maximal affine subgroup", while the product of two copies of the multiplicative group C* is isomorphic (analytically but not algebraically) to a non-split extension of any given elliptic curve by ''C''.

Applications

Chevalley's structure theorem is used in the proof of the

Néron–Ogg–Shafarevich criterion In mathematics, the Néron–Ogg–Shafarevich criterion states that if ''A'' is an elliptic curve or abelian variety over a local field ''K'' and ℓ is a prime not dividing the characteristic of the residue field of ''K'' then ''A'' has good re ...

References

* * * * *{{Citation , last1=Rosenlicht , first1=Maxwell , title=Some basic theorems on algebraic groups , jstor=2372523 , mr=0082183 , year=1956 , journal=

American Journal of Mathematics The ''American Journal of Mathematics'' is a bimonthly mathematics journal published by the Johns Hopkins University Press. History The ''American Journal of Mathematics'' is the oldest continuously published mathematical journal in the United ...

, issn=0002-9327 , volume=78 , pages=401–443 , doi=10.2307/2372523 Algebraic groups Theorems in algebraic geometry