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Exponent Reduction for Projective Schur Algebras

✍ Scribed by Eli Aljadeff; Jack Sonn

Publisher: Elsevier Science
Year: 2001
Tongue: English
Weight: 88 KB
Volume: 239
Category: Article
ISSN: 0021-8693
DOI: 10.1006/jabr.2000.8656

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✦ Synopsis

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 Ž . A be a projective Schur algebra over a field k and let k denote the maximal Ž . cyclotomic extension of k. If m is the exponent of A m k , then k contains a k primitive mth root of unity. One corollary of this result is a negative answer to the Ž . question of whether or not the projective Schur group PS k is always equal to Ž . 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 Ž . ''Brauer᎐Witt analogue'' in characteristic p: if char k s p / 0, then every projective Schur algebra over k is Brauer equivalent to a radical abelian algebra.

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