Higher Algebraic K-Theory of Non-Commutative Gamma Semirings: The Quillen and Waldhausen Spectra

TL;DR

This paper develops a comprehensive framework for higher algebraic K-theory of non-commutative Gamma semirings, establishing equivalences between different models and connecting to derived categories and non-commutative geometry.

Contribution

It constructs and compares Quillen and Waldhausen models for higher K-theory of non-commutative Gamma semirings, proving their equivalence and relating them to derived geometric invariants.

Findings

Quillen and Waldhausen K-theory spectra are canonically weakly equivalent.

K-theory is identified with the K-theory of perfect complexes on a non-commutative spectrum.

Results establish functoriality, localization, excision, and Morita invariance of the K-theory.

Abstract

In the companion paper~\cite{Gokavarapu_IJPA_2025}, we developed a classical algebraic K-theory for non-commutative $n$-ary $\Gamma$-semirings $(T,\Gamma)$ in terms of finitely generated projective $n$-ary $\Gamma$-modules and their automorphisms, and we identified the low K-groups $K_{0}(T,\Gamma)$ and $K_{1}(T,\Gamma)$ with appropriate Grothendieck and Whitehead groups. The present paper continues this programme by constructing and comparing several models for the higher algebraic K-theory of $(T,\Gamma)$. Starting from the Quillen-exact category $\mathcal{C} := T\text{-Mod}^{\mathrm{bi}}$ of bi-finite, slot-sensitive $n$-ary $\Gamma$-modules introduced earlier, we define the higher K-groups $K_{n}(T,\Gamma)$ via Quillen's Q-construction~\cite{Quillen73} on $\mathcal{C}$ and via Waldhausen's $S_{\bullet}$-construction~\cite{Waldhausen85} on the Waldhausen category of bounded chain complexes in $\mathcal{C}$. We prove a Gillet--Waldhausen type comparison theorem~\cite{GilletGrayson87} showing that the resulting Quillen and Waldhausen K-theory spectra are canonically weakly equivalent. Using dg-enhancements and the derived category of quasi-coherent sheaves on the non-commutative spectrum $\operatorname{Spec}_{T}^{\mathrm{nc}}(T)$~\cite{Gokavarapu_JRMS_2266}, we further identify these spectra with the K-theory of the small stable $\infty$-category of perfect complexes~\cite{Thomason90}. As consequences, we obtain functoriality, localization, and excision sequences~\cite{Weibel13}, and a derived Morita invariance statement for $K_{n}(T,\Gamma)$~\cite{Keller94}. These results show that algebraic K-theory of non-commutative $n$-ary $\Gamma$-semirings is a derived-geometric invariant of $\operatorname{Spec}_{\Gamma}^{\mathrm{nc}}(T)$ and reduce concrete computations to geometric d\'evissage and homological techniques developed in the earlier papers of the series.