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This article is cited in 1 scientific paper (total in 1 paper)
Projective resolutions and cohomological triviality of $p$-periodic bimodules over Frobenius orders
F. R. Bobovich
Abstract:
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.
Received: 21.12.1971
Citation:
F. R. Bobovich, “Projective resolutions and cohomological triviality of $p$-periodic bimodules over Frobenius orders”, Math. USSR-Sb., 19:1 (1973), 23–34
Linking options:
https://www.mathnet.ru/eng/sm2992https://doi.org/10.1070/SM1973v019n01ABEH001733 https://www.mathnet.ru/eng/sm/v132/i1/p23
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