Koszul duality for Iwasawa algebras modulo p

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 $\Omega=k[\![G]\!]$ 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 $\Bbb{Z}_p$.