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Let $s\colon X\rightarrow \operatorname{Spec} \mathbb{F}$ 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 \'etale 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 $X$.

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