Paper
Higher Algebraic K-Theory of Non-Commutative Gamma Semirings: The Quillen and Waldhausen Spectra
Authors
Chandrasekhar Gokavarapu
Abstract
In the companion paper~\cite{Gokavarapu_IJPA_2025}, we developed a classical algebraic K-theory for non-commutative -ary -semirings in terms of finitely generated projective -ary -modules and their automorphisms, and we identified the low K-groups and 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 . Starting from the Quillen-exact category of bi-finite, slot-sensitive -ary -modules introduced earlier, we define the higher K-groups via Quillen's Q-construction~\cite{Quillen73} on and via Waldhausen's -construction~\cite{Waldhausen85} on the Waldhausen category of bounded chain complexes in . 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 ~\cite{Gokavarapu_JRMS_2266}, we further identify these spectra with the K-theory of the small stable -category of perfect complexes~\cite{Thomason90}. As consequences, we obtain functoriality, localization, and excision sequences~\cite{Weibel13}, and a derived Morita invariance statement for ~\cite{Keller94}. These results show that algebraic K-theory of non-commutative -ary -semirings is a derived-geometric invariant of and reduce concrete computations to geometric dévissage and homological techniques developed in the earlier papers of the series.