PROJECTIVE RESOLUTIONS AND COHOMOLOGICAL TRIVIALITY OFp-PERIODIC BIMODULES OVER FROBENIUS ORDERS

Abstract

Let Λ be a Frobenius order in a simple algebra over the field of p-adic numbers, . For a finitely-generated p-periodic Λ-bimodule, we establish the existence of a Λe/p-free resolution whose generating function is an associate of the Poincare series in the ring of formal power series with integral coefficients. Our subsequent investigations are restricted to orders of the form described which in addition satisfy a certain disjointness condition modulo p, which is formulated in terms of constraints on the Cartan matrix of the ring Λe/p. We find conditions sufficient for the existence of a p-periodic module with trivial homology (in the sense of Hochschild) and having infinite projective dimension over the ring Λe/p. We prove a Nakayama-type theorem on the triviality of the cohomology groups ot Λ with coefficients in irreducible Λ-bimodules. Bibliography. 12 items.