Higher Algebraic K-Theory of Non-Commutative Gamma Semirings: The Quillen and Waldhausen Spectra
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2025
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| author | Gokavarapu, Chandrasekhar |
| author_facet | Gokavarapu, Chandrasekhar |
| contents | In the companion paper~\cite{Gokavarapu_IJPA_2025}, we developed a classical algebraic K-theory for non-commutative $n$-ary $Γ$-semirings $(T,Γ)$ in terms of finitely generated projective $n$-ary $Γ$-modules and their automorphisms, and we identified the low K-groups $K_{0}(T,Γ)$ and $K_{1}(T,Γ)$ 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 $(T,Γ)$. Starting from the Quillen-exact category $\mathcal{C} := T\text{-Mod}^{\mathrm{bi}}$ of bi-finite, slot-sensitive $n$-ary $Γ$-modules introduced earlier, we define the higher K-groups $K_{n}(T,Γ)$ 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 $K_{n}(T,Γ)$~\cite{Keller94}. These results show that algebraic K-theory of non-commutative $n$-ary $Γ$-semirings is a derived-geometric invariant of $\operatorname{Spec}_Γ^{\mathrm{nc}}(T)$ and reduce concrete computations to geometric dévissage and homological techniques developed in the earlier papers of the series. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_11102 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Higher Algebraic K-Theory of Non-Commutative Gamma Semirings: The Quillen and Waldhausen Spectra Gokavarapu, Chandrasekhar Rings and Algebras K-Theory and Homology 18G80, 19D10, 16Y60, 94B05, 94A60 In the companion paper~\cite{Gokavarapu_IJPA_2025}, we developed a classical algebraic K-theory for non-commutative $n$-ary $Γ$-semirings $(T,Γ)$ in terms of finitely generated projective $n$-ary $Γ$-modules and their automorphisms, and we identified the low K-groups $K_{0}(T,Γ)$ and $K_{1}(T,Γ)$ 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 $(T,Γ)$. Starting from the Quillen-exact category $\mathcal{C} := T\text{-Mod}^{\mathrm{bi}}$ of bi-finite, slot-sensitive $n$-ary $Γ$-modules introduced earlier, we define the higher K-groups $K_{n}(T,Γ)$ 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 $K_{n}(T,Γ)$~\cite{Keller94}. These results show that algebraic K-theory of non-commutative $n$-ary $Γ$-semirings is a derived-geometric invariant of $\operatorname{Spec}_Γ^{\mathrm{nc}}(T)$ and reduce concrete computations to geometric dévissage and homological techniques developed in the earlier papers of the series. |
| title | Higher Algebraic K-Theory of Non-Commutative Gamma Semirings: The Quillen and Waldhausen Spectra |
| topic | Rings and Algebras K-Theory and Homology 18G80, 19D10, 16Y60, 94B05, 94A60 |
| url | https://arxiv.org/abs/2512.11102 |