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Representation Theory

Published by the American Mathematical Society, the Representation Theory (ERT) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.7.

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Koszul duality for Iwasawa algebras modulo $p$
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by Claus Sorensen PDF
Represent. Theory 24 (2020), 151-177 Request permission

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 $\Omega =k\lBrack G\rBrack$ 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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Additional Information
  • Claus Sorensen
  • Affiliation: Department of Mathematics, University of California, San Diego, La Jolla, California 92093
  • Email: csorensen@ucsd.edu
  • Received by editor(s): March 18, 2019
  • Received by editor(s) in revised form: January 7, 2020
  • Published electronically: March 24, 2020
  • © Copyright 2020 American Mathematical Society
  • Journal: Represent. Theory 24 (2020), 151-177
  • MSC (2010): Primary 20C08, 22E35, 13D09
  • DOI: https://doi.org/10.1090/ert/539
  • MathSciNet review: 4079101