Unit L-functions for \'etale sheaves of modules over noncommutative rings

Let $s\text{\hspace{0.17em}⁣}:X\to \mathrm{Spec}\mathbb{F}$ be a separated scheme of finite type over a finite field $\mathbb{F}$ of characteristic $p$, let $\mathrm{\Lambda }$ be a not necessarily commutative ${\mathbb{Z}}_p$-algebra with finitely many elements, and let ${\mathcal{F}}^{\bullet }$ be a perfect complex of $\mathrm{\Lambda }$-sheaves on the 'etale site of $X$. We show that the ratio $L({\mathcal{F}}^{\bullet },T)\mathrm{/}L(R{s}_{!}{\mathcal{F}}^{\bullet },T)$, which is a priori an element of $K_1(\mathrm{\Lambda }[[T]])$, has a canonical preimage in $K_1(\mathrm{\Lambda }[T])$. We use this to prove a version of the noncommmutative Iwasawa main conjecture for $p$-adic Lie coverings of $X$.