Brauer Groups of Linear Algebraic Groups with Characters

Let G be a connected linear algebraic group over an algebraically closed field of characteristic zero. Then the Brauer group of G is shown to be C X (Q/Z)<n) where C is finite and n = d(d l)/2, with d the Z-Ta.nk of the character group of G. In particular, a linear torus of dimension d has Brauer group (Q/Z)(n) with n as above. In [6], B. Iversen calculated the Brauer group of a connected, characterless, linear algebraic group over an algebraically closed field of characteristic zero: the Brauer group is finite-in fact, it is the Schur multiplier of the fundamental group of the algebraic group [6, Theorem 4.1, p. 299]. In this note, we extend these calculations to an arbitrary connected linear group in characteristic zero. The main result is the determination of the Brauer group of a ¿/-dimensional affine algebraic torus, which is shown to be (Q/Z)(n) where n = d(d — l)/2. (This result is noted in [6, 4.8, p. 301] when d = 2.) We then show that if G is a connected linear algebraic group whose character group has Z-rank d, then the Brauer group of G is C X (Q/Z)<n) where n is as above and C is finite. We adopt the following notational conventions: 7ms an algebraically closed field of characteristic zero, and T = (F*)id) is a ^-dimensional affine torus over F. We use 77*„ 77*„ and 77^ to denote étale, singular, and group cohomology. If G is an abelian group and m a positive integer, mG denotes the w-torsion in G. If X is an affine F-variety, F[X] is its coordinate ring. Br(-) denotes Brauer group and [•] denotes the class in the Brauer group of an Azumaya algebra. X(G) is the character group of the algebraic group G and U(A) denotes the units group of the ring A. We let Gm denote GLX(F). Proposition 1. Let A be a finite abelian group and let X be a smooth F-variety such that H'el(X, A) « 0 for i = 1, 2. Then H2t(X X T, A) = A(n) where n = d(d — l)/2. Proof. By the Lefschetz principle and smooth base change [1, Corollary 1.6, p. 211] we may assume F = C. By the comparison theorem for classical and étale cohomology [1, Theorem 4.4, p. 74] H2X(X X T,A) = H2X(X X T,A). Received by the editors November 28, 1977. AMS (MOS) subject classifications (1970). Primary 13A20, 20G10; Secondary 14F20.