2019/02/25 by Claus Sorensen, Sorensen, Claus
Mathematics · #FOS: Mathematics #Number Theory (math.NT) #Representation Theory (math.RT) #math.NT #math.RT
paper · pdf · doi:10.48550/arxiv.1902.09632
25 pages
arxiv created 2019/02/25 · arxiv updated 2019/02/27
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 Ω=k[ [G] ] with coefficients in a finite field k of characteristic p. It is known that Ω carries a natural filtration and gr Ω=S(\frakg) where \frakg is the (abelian) Lie algebra of G over k. One of our main results in this paper is that the Koszul dual gr Ω^!=\bigwedge \frakg\vee can be promoted to an A∞-algebra in such a way that the derived category of pseudocompact Ω-modules D(Ω) becomes equivalent to the derived category of strictly unital A∞-modules D∞(\bigwedge \frakg\vee). In the case where G is an abelian group we prove that the A∞-structure is trivial and deduce an equivalence between D(Ω) and the derived category of differential graded modules over \bigwedge \frakg\vee which generalizes a result of Schneider for ℤp.