Projective resolutions and cohomological triviality of $p$-periodic bimodules over Frobenius orders

F. R. Bobovich

Sbornik. Mathematics, Tome 19 (1973) no. 1, pp. 23-34

Résumé

Let $\Lambda$ be a Frobenius order in a simple algebra over the field of $p$-adic numbers, $\dim_{\Lambda^e}\Lambda=0$. For a finitely-generated $p$-periodic $\Lambda$-bimodule, we establish the existence of a $\Lambda^e/p$-free resolution whose generating function is an associate of the Poincaré 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 $\Lambda^e/p$. 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 $\Lambda^e/p$. We prove a Nakayama-type theorem on the triviality of the cohomology groups of $\Lambda$ with coefficients in irreducible $\Lambda$-bimodules. Bibliography: 12 titles.