The center of the generic division ring and twisted multiplicative group actions

✍ Scribed by Esther Beneish

• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 109 KB
• Volume: 259
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1016/s0021-8693(02)00576-8

✦ Synopsis

Let F be a field and let p be a prime. The problem we study is whether the center, C p , of the division ring of p × p generic matrices is stably rational over F . Given a finite group G and a ZGlattice, we let F (M) be the quotient field of the group algebra of the abelian group M. Procesi and Formanek [Linear Multilinear Algebra 7 (1979) 203-212] have shown that for all n there is a ZS nlattice, G n , such that C n is stably isomorphic to the fixed field under the action of S n of F (G n ). Let H be a p-Sylow subgroup of S p . Let A be the root lattice, and let L = F (ZS p /H ). We show that there exists an action of S p on L(ZS P ⊗ ZH A), twisted by an element α ∈ Ext 1 S p (ZS p ⊗ ZH A, L * ), such that L α (ZS p ⊗ ZH A) S p is stably isomorphic to C p . The extension α corresponds to an element of the relative Brauer group of L over L H . Since ZS p ⊗ ZH A and ZS p /H are quasi-permutations, L(ZS p ⊗ ZH A) S p is stably rational over F . However, it is not known whether L α (ZS p ⊗ ZH A) S p is stably rational over F . Thus the result represents a reduction on the problem since ZS p ⊗ ZH A is quasi-permutation; however, the twist introduces a new level of complexity.