Exponent reduction for projective Schur algebras

Eli Aljadeff; Jack Sonn

doi:10.1006/jabr.2000.8656

Exponent reduction for projective Schur algebras

Eli Aljadeff^*, Jack Sonn

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

4 Scopus citations

Abstract

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.

Original language	English
Pages (from-to)	356-364
Number of pages	9
Journal	Journal of Algebra
Volume	239
Issue number	1
DOIs	https://doi.org/10.1006/jabr.2000.8656
State	Published - 1 May 2001
Externally published	Yes

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title = "Exponent reduction for projective Schur algebras",

abstract = "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.",

author = "Eli Aljadeff and Jack Sonn",

note = "Funding Information: 1 The research was supported by the Fund for the Promotion of Research at the Technion.",

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T1 - Exponent reduction for projective Schur algebras

AU - Aljadeff, Eli

AU - Sonn, Jack

N1 - Funding Information: 1 The research was supported by the Fund for the Promotion of Research at the Technion.

PY - 2001/5/1

Y1 - 2001/5/1

N2 - 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.

AB - 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.

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SN - 0021-8693

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JO - Journal of Algebra

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Exponent reduction for projective Schur algebras

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