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

Following Procesi and Formanek, the center of the division ring of generic matrices over the complex numbers is stably equivalent to the fixed field under the action of , of the function field of the group algebra of a -lattice, denoted by . We study the question of the stable rationality of the center over the complex numbers when is a prime, in this module theoretic setting. Let be the normalizer of an -sylow subgroup of . Let be a -lattice. We show that under certain conditions on , induction-restriction from to does not affect the stable type of the corresponding field. In particular, and are stably isomorphic and the isomorphism preserves the -action. We further reduce the problem to the study of the localization of at the prime ; all other primes behave well. We also present new simple proofs for the stable rationality of over , in the cases and .