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 A be a projective Schur algebra over a field k and let k(μ) denote the maximal cyclotomic extension of k. If m is the exponent of A⊗kk(μ), then k contains a primitive mth root of unity. One corollary of this result is a negative answer to the question of whether or not the projective Schur group PS(k) is always equal to Br(L/k), 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 "Brauer-Witt analogue" in characteristic p: if char(k)=p≠0, then every projective Schur algebra over k is Brauer equivalent to a radical abelian algebra.
|Number of pages||9|
|Journal||Journal of Algebra|
|State||Published - 1 May 2001|