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A MODERN PROOF OF CHEVALLEY’S THEOREM ON ALGEBRAIC GROUPS

Let k be a field, and let G be an algebraic group over k, by which we mean a connected smooth k-group scheme (not necessarily affine). Recall that such a G is automatically separated, finite type, and geometrically integral over k [4, Exp VIA, 0.3, 2.1.2, 2.4]. The most important classes of algebraic groups are the affine algebraic groups (also called linear algebraic groups, since affine algebraic groups coincide with the closed algebraic subgroups of the matrix groups GL(n)/k [15, Thm 3.4]) and the proper algebraic groups (also called abelian varieties). Other types of algebraic groups do arise naturally. For example, if X is a proper (possibly singular) scheme over a perfect field k then the reduced connected component (PicX/k)red of its relative Picard scheme is a typically non-proper non-affine algebraic group over k; when X is an algebraic curve then such algebraic groups are the so-called generalized Jacobians of geometric class field theory [13]. In another direction, if R is a discrete valuation ring with fraction field K and residue field k and if A is an abelian variety over K with Neron model A over R, the connected component of the closed fiber of A is an algebraic group over k which usually is neither affine nor proper (properness of the connected component of the closed fiber is equivalent to good reduction, by EGA IV3, 15.6.7,15.6.8 and [1, 1.2/8]). Thus, it is important to understand something about the general structure of general algebraic groups. Chevalley’s Theorem asserts that every algebraic group over a perfect field is ‘built up’ from a linear algebraic group and an abelian variety (in a way we will make precise shortly). This is an extremely important result. For example, the proof of the Neron-Ogg-Shafarevich criterion for good reduction of abelian varieties (see [14, Thm 1] and [1, 7.4/5]) relies heavily on this general structure theorem. Geometric class field theory (as in [13]), which classifies rational maps from an algebraic curve to commutative algebraic groups, also depends on Chevalley’s Theorem. It came as somewhat of a surprise to the author to find that in the published literature there does not exist a proof of Chevalley’s Theorem in modern language, even though the theorem is widely used. The purpose of this note is to present a proof based on scheme theory rather than Weil’s Foundations [16]. The published proofs of Chevalley’s Theorem, by both Chevalley [2] and Rosenlicht [12], are written with very archaic terminology which is no longer used. For example, [12] seems to be unreadable for those not familiar with Weil’s Foundations. However, the underlying method in [2] is almost penetrable, so below we present a modern translation of this proof. There are two sources of technical difficulties with Chevalley’s proof. First, he employs an old-style notion of “algebraic families of divisors” which sits somewhere between the notions of Weil divisor and relative effective Cartier divisor. It requires a little care to translate this into modern terminology while bypassing Chevalley’s papers on the old theory of divisor families. Basic results on invertible sheaves (such as in [9]) will supply the tools we need.