Abstract. Following Procesi and Formanek, the center of the division ring of n × n generic matrices over the complex numbers C is stably equivalent to the fixed field under the action of Sn, of the function field of the group algebra of a ZSn-lattice, denoted by Gn. We study the question of the stable rationality of the center Cn 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 Sn. Let M be a ZSn-lattice. We show that under certain conditions on M , inductionrestriction from N to Sn does not affect the stable type of the corresponding field. In particular, C(Gn) and C(ZG ⊗ ZN Gn) are stably isomorphic and the isomorphism preserves the Sn-action. We further reduce the problem to the study of the localization of Gn at the prime n; all other primes behave well. We also present new simple proofs for the stable rationality of Cn over C, in the cases n = 5 and n = 7.
Abstract. Let p be a prime greater than 3, and let N be the semi-direct product of a group H of order p by a cyclic C group of order p − 1, which acts faithfully on H. Let R be the localization of Z at p. We show that the Krull-Schmidt Theorem fails for the category of invertible RN -lattices.
Let G be a finite group, and let F be an algebraically closed field. The rational extension of F, F(G) = F(xg : g ∈ G) with G action given by hxg = xhg for g, h ∈ G, is referred to as the Noether setting for G. We show that if G is a group with zero Schur multiplier, then for any central extension G′ of G, F(G′)G′ and F(G)G are stably equivalent over F. That is, F(G)G is stably rational over F if and only if F(G′)G′ is stably rational over F. In particular, if G is a cyclic group or an abelian group with zero Schur multiplier, then F(G′) is stably rational over F.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.