Unit L-functions for étale sheaves of modules over noncommutative rings

Abstract

Let $s:X\to \mathrm{Spec}\mathbb{F}$ be a separated scheme of finite type over a finite field $\mathbb{F}$ of characteristic $p$, let $\Lambda$ be a not necessarily commutative ${\mathbb{Z}}_p$-algebra with finitely many elements, and let ${\mathcal{F}}^{\bullet }$ be a perfect complex of $\Lambda$-sheaves on the étale site of $X$. We show that the ratio $L({\mathcal{F}}^{\bullet },T)/L(R{s}_{!}{\mathcal{F}}^{\bullet },T)$, which is a priori an element of $K_1(\Lambda [[T]])$, has a canonical preimage in $K_1(\Lambda [T])$. We use this to prove a version of the noncommmutative Iwasawa main conjecture for $p$-adic Lie coverings of $X$.

18 pages