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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Induction theorems on the stable rationality
of the center of the ring of generic matrices

Author: Esther Beneish
Journal: Trans. Amer. Math. Soc. 350 (1998), 3571-3585
MSC (1991): Primary 13A50, 20C10
MathSciNet review: 1475675
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Abstract: Following Procesi and Formanek, the center of the division ring of $n\times n$ generic matrices over the complex numbers $\mathbf C$ is stably equivalent to the fixed field under the action of $S_n$, of the function field of the group algebra of a $ZS_n$-lattice, denoted by $G_n$. We study the question of the stable rationality of the center $C_n$ over the complex numbers when $n$ is a prime, in this module theoretic setting. Let $N$ be the normalizer of an $n$-sylow subgroup of $S_n$. Let $M$ be a $ZS_n$-lattice. We show that under certain conditions on $M$, induction-restriction from $N$ to $S_n$ does not affect the stable type of the corresponding field. In particular, $\mathbf C (G_n)$ and $\mathbf C(ZG\otimes _{ZN}G_n)$ are stably isomorphic and the isomorphism preserves the $S_n$-action. We further reduce the problem to the study of the localization of $G_n$ at the prime $n$; all other primes behave well. We also present new simple proofs for the stable rationality of $C_n$ over $\mathbf C$, in the cases $n=5$ and $n=7$.

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Esther Beneish
Affiliation: Department of Mathematics, University of Wisconsin-Parkside, Kenosha, Wisconsin 53141-2000

Received by editor(s): September 22, 1996
Article copyright: © Copyright 1998 American Mathematical Society