Koszul duality for Iwasawa algebras modulo $p$

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 $\Bbb {Z}_p$.