Summary: Bull. London Math. Soc. 39 (2007) 731740 Ce2007 London Mathematical Society
EVERY PROJECTIVE SCHUR ALGEBRA IS BRAUER EQUIVALENT
TO A RADICAL ABELIAN ALGEBRA
ELI ALJADEFF and ŽANGEL DEL RŽIO
We prove that any projective Schur algebra over a field K is equivalent in Br(K) to a radical abelian algebra.
This was conjectured in 1995 by Sonn and the first author of this paper. As a consequence, we obtain a
characterization of the projective Schur group by means of Galois cohomology. The conjecture was known for
algebras over fields of positive characteristic. In characteristic zero the conjecture was known for algebras over
fields with a Henselian valuation over a local or global field of characteristic zero.
One of the main theorems on the Schur subgroup of the Brauer group is the BrauerWitt
Theorem (BrauerWitt). Every Schur algebra is Brauer equivalent to a cyclotomic algebra.
Recall that a (finite-dimensional) central simple K-algebra A is called a Schur algebra if it
is spanned over K by a finite group of unit elements. Equivalently, A is a Schur algebra if
it is the homomorphic image of a group algebra KG for some finite group G. The subgroup
of Br(K) generated by (in fact consisting of) classes that are represented by Schur algebras
is called the Schur group of K and is denoted by S(K); see . There is a natural way