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Koszul duality for Iwasawa algebras modulo $ p$

Author: Claus Sorensen
Journal: Represent. Theory 24 (2020), 151-177
MSC (2010): Primary 20C08, 22E35, 13D09
Published electronically: March 24, 2020
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Abstract: In this article we establish a version of Koszul duality for filtered rings arising from $ p$-adic Lie groups. Our precise setup is the following. We let $ G$ be a uniform pro-$ p$ group and consider its completed group algebra $ \Omega =k\llbracket G\rrbracket $ with coefficients in a finite field $ k$ of characteristic $ p$. It is known that $ \Omega $ carries a natural filtration and $ \text {gr} \Omega =S(\frak {g})$ where $ \frak {g}$ is the (abelian) Lie algebra of $ G$ over $ k$. One of our main results in this paper is that the Koszul dual $ \text {gr} \Omega ^!=\bigwedge \frak {g}^{\vee }$ can be promoted to an $ A_{\infty }$-algebra in such a way that the derived category of pseudocompact $ \Omega $-modules $ D(\Omega )$ becomes equivalent to the derived category of strictly unital $ A_{\infty }$-modules $ D_{\infty }(\bigwedge \frak {g}^{\vee })$. In the case where $ G$ is an abelian group we prove that the $ A_{\infty }$-structure is trivial and deduce an equivalence between $ D(\Omega )$ and the derived category of differential graded modules over $ \bigwedge \frak {g}^{\vee }$ which generalizes a result of Schneider for $ \Bbb {Z}_p$.

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Claus Sorensen
Affiliation: Department of Mathematics, University of California, San Diego, La Jolla, California 92093

Received by editor(s): March 18, 2019
Received by editor(s) in revised form: January 7, 2020
Published electronically: March 24, 2020
Article copyright: © Copyright 2020 American Mathematical Society