Koszul duality for Iwasawa algebras modulo 𝑝

Abstract

In this article we establish a version of Koszul duality for filtered rings arising from p p -adic Lie groups. Our precise setup is the following. We let G G be a uniform pro- p p group and consider its completed group algebra \Omega =k\lBrack G\rBrack with coefficients in a finite field k k of characteristic p p . It is known that Ω \Omega carries a natural filtration and gr Ω = S ( g ) \text {gr} \Omega =S(\frak {g}) where g \frak {g} is the (abelian) Lie algebra of G G over k k . One of our main results in this paper is that the Koszul dual gr Ω ! = ⋀ g ∨ \text {gr} \Omega ^!=\bigwedge \frak {g}^{\vee } can be promoted to an A ∞ A_{\infty } -algebra in such a way that the derived category of pseudocompact Ω \Omega -modules D ( Ω ) D(\Omega ) becomes equivalent to the derived category of strictly unital A ∞ A_{\infty } -modules D ∞ ( ⋀ g ∨ ) D_{\infty }(\bigwedge \frak {g}^{\vee }) . In the case where G G is an abelian group we prove that the A ∞ A_{\infty } -structure is trivial and deduce an equivalence between D ( Ω ) D(\Omega ) and the derived category of differential graded modules over ⋀ g ∨ \bigwedge \frak {g}^{\vee } which generalizes a result of Schneider for Z p \Bbb {Z}_p .