 
Summary: Z .Journal of Algebra 239, 356 364 2001
doi:10.1006rjabr.2000.8656, available online at http:rrwww.idealibrary.com on
Exponent Reduction for Projective Schur Algebras1
Eli Aljadeff and Jack Sonn
Department of Mathematics, TechnionIsrael Institute of Technology, 32000 Haifa, Israel
Email: aljadeff@math.technion.ac.il, sonn@math.technion.ac.il
Communicated by Alexander Lubotzky
Received July 10, 2000
In this paper it is proved that the ``exponent reduction property'' holds for all
projective Schur algebras. This was proved in an earlier paper of the authors for a
special class, the ``radical abelian algebras.'' The precise statement is as follows: let
Z .A be a projective Schur algebra over a field k and let k denote the maximal
Z .cyclotomic extension of k. If m is the exponent of A m k , then k contains ak
primitive mth root of unity. One corollary of this result is a negative answer to the
Z .question of whether or not the projective Schur group PS k is always equal to
Z .Br Lrk , where L is the composite of the maximal cyclotomic extension of k and
the maximal Kummer extension of k. A second consequence is a proof of the
Z .``Brauer Witt analogue'' in characteristic p: if char k s p / 0, then every pro
jective Schur algebra over k is Brauer equivalent to a radical abelian algebra.
2001 Academic Press
