Transactions of the American Mathematical Society

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



Lattice invariants and the center of the generic division ring

Author: Esther Beneish
Journal: Trans. Amer. Math. Soc. 356 (2004), 1609-1622
MSC (2000): Primary 20C10, 16R30, 13A50, 16K20
Published electronically: October 21, 2003
MathSciNet review: 2034321
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Abstract: Let $G$ be a finite group, let $M$ be a $ZG$-lattice, and let $F$ be a field of characteristic zero containing primitive $p^{{th}}$ roots of 1. Let $F(M)$ be the quotient field of the group algebra of the abelian group $M$. It is well known that if $M$ is quasi-permutation and $G$-faithful, then $F(M)^G$ is stably equivalent to $F(ZG)^G$. Let $C_n$ be the center of the division ring of $n\times n$ generic matrices over $F$. Let $S_n$ be the symmetric group on $n$symbols. Let $p$ be a prime. We show that there exist a split group extension $G'$of $S_p$ by a $p$-elementary group, a $G'$-faithful quasi-permutation $ZG'$-lattice $M$, and a one-cocycle $\alpha$ in $\operatorname{Ext}_{G'}^1(M,F^*)$ such that $C_p$ is stably isomorphic to $F_\alpha(M)^{G'}$. This represents a reduction of the problem since we have a quasi-permutation action; however, the twist introduces a new level of complexity. The second result, which is a consequence of the first, is that, if $F$ is algebraically closed, there is a group extension $E$ of $S_p$ by an abelian $p$-group such that $C_p$ is stably equivalent to the invariants of the Noether setting $F(E)$.

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Esther Beneish
Affiliation: Department of Mathematics, Central Michigan University, Mount Pleasant, Michigan 48859

Received by editor(s): May 13, 2002
Received by editor(s) in revised form: March 7, 2003
Published electronically: October 21, 2003
Additional Notes: This work was partially supported by NSF grant #DMS-0070665
Article copyright: © Copyright 2003 American Mathematical Society