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

Abstract: 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.

• The paper introduces higher K-groups via both Quillen and Waldhausen constructions, establishing a canonical weak equivalence between the two models.
• It applies homotopy-theoretic and derived methods to define localization sequences and prove derived Morita invariance in non-commutative settings.
• The framework extends algebraic K-theory to non-commutative n-ary Γ-semiring modules, offering new insights for coding theory and cryptography.

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

Introduction and Formal Framework

The paper "Higher Algebraic K-Theory of Non-Commutative Gamma Semirings: The Quillen and Waldhausen Spectra" (2512.11102) systematically constructs and compares models for higher algebraic K-theory in the context of non-commutative $n$-ary $\Gamma$-semirings. This continues a comprehensive project to extend the homotopy-theoretic, categorical, and homological aspects of algebraic K-theory beyond the classical ring-theoretic paradigm to the highly structured, non-commutative, slot-sensitive setting.

The fundamental object is a non-commutative $n$-ary $\Gamma$-semiring $(T, \Gamma)$, where $T$ is an additive commutative monoid equipped with a non-symmetric, $\Gamma$-parameterized $n$-ary multiplication. The ambient category $\mathcal{C} = T\text{-Mod}^{bi}$ consists of bi-finite, slot-sensitive $n$-ary $\Gamma$-modules. The exact and homological structure of $\mathcal{C}$, necessary for higher K-theory, derives from recent foundational work on the subject.

Models of Higher Algebraic K-Theory

Two canonical models for higher algebraic K-theory are constructed and rigorously compared:

• Quillen’s Q-construction: $K_n(T, \Gamma)$ is defined as the homotopy groups of the K-theory spectrum built from the exact category $\mathcal{C}$ via the Q-construction, with morphisms given by isomorphism classes of admissible spans. The identification of $K_0$ and $K_1$ recovers the Grothendieck and Whitehead groups as developed for $\Gamma$-semirings.
• Waldhausen’s $S_\bullet$-construction: On the Waldhausen category of bounded complexes $Ch^b(\mathcal{C})$, with degreewise admissible monomorphisms and quasi-isomorphisms as cofibrations and weak equivalences, the associated K-theory spectrum $K^{Wald}(\mathcal{C})$ is constructed.

The main theorem is a non-trivial extension of the Gillet-Waldhausen comparison: the Quillen and Waldhausen spectra for $(T, \Gamma)$ are canonically weakly equivalent. Consequently, all higher K-groups may be computed in either model, leveraging the formal properties of both exact and Waldhausen categories.

Derived Geometry, Localization, and Morita Invariance

The comparison is deepened by embedding the higher K-theory of $(T, \Gamma)$ into the derived and $\infty$-categorical framework associated with non-commutative geometry:

• Derived and $\infty$-categorical interpretation: Using dg-enhancements and the machinery of stable $\infty$-categories, the spectrum $K_n(T, \Gamma)$ is shown to coincide with the K-theory of compact objects (perfect complexes) in the derived category of quasi-coherent sheaves over the non-commutative spectrum $Spec_{\Gamma}^{nc}(T)$.
• Localization sequences: The Waldhausen model transports exactness, additivity, fibration, and localization theorems into the $\Gamma$-semiring context. Explicitly, there are long exact sequences for pairs of exact/excision-closed subcategories, and geometric localization sequences for closed/open decompositions of $Spec_{\Gamma}^{nc}(T)$, akin to the localization theorems for schemes and ringed spaces.
• Derived Morita invariance: An essential structural result is that higher algebraic K-theory is invariant under derived equivalences between module/sheaf categories; if two non-commutative $\Gamma$-semirings have equivalent derived categories of quasi-coherent sheaves, their K-theory spectra are weakly equivalent. This establishes a robust form of descent and is crucial for applications and future computational reductions.

Connections to Coding Theory and Cryptography

The paper delineates new applications to coding theory and cryptography:

• Coding theory: Exact categories of $n$-ary $\Gamma$-submodules (codes) inside modules like $T^m$ yield stable invariants via their K-theory. This provides new, homological invariants for families of codes that complement and extend classical weight enumerator invariants.
• Cryptography: Admissible subcategories of modules encoding public key sets or protocol classes acquire derived invariants from their higher K-theory, potentially facilitating deeper structural analysis of protocol equivalence or indistinguishability properties.

Synthesis: Main Theoretical Advances

The paper accomplishes the following major outcomes:

• Constructs higher algebraic K-groups $K_n(T, \Gamma)$ for non-commutative $n$-ary $\Gamma$-semirings from the perspective of both exact and Waldhausen categories, establishing canonical equivalence of their homotopy spectra.
• Demonstrates that $K_n(T, \Gamma)$ is a geometric invariant of the non-commutative spectrum $Spec^{nc}_\Gamma(T)$, interpretable as the K-theory of perfect complexes.
• Proves the validity of functoriality, localization, additivity, excision, and derived Morita invariance for the K-theory of these objects.
• Lays the groundwork for carrying over techniques of geometric devissage and homological methods for explicit calculation, as developed in previous papers in the series.

Numerical and Structural Results

The paper’s theorems are precise and structural rather than computational. The key claims include:

• Canonical weak equivalence: $K^{Q}(\mathcal{C}) \simeq K^{Wald}(\mathcal{C})$ (Gillet-Waldhausen type) and $K^{Perf}(\mathcal{C})$ identified as $K$-theory of perfect complexes.
• Functoriality and localization: Natural long exact sequences associated to exact subcategories, geometric subspaces, and derived quotients.
• Strong statement of Morita invariance: If two $\Gamma$-semirings are derived Morita equivalent, their higher K-groups coincide.

Implications and Prospects

The extension of higher algebraic K-theory to non-commutative, $n$-ary, slot-sensitive $\Gamma$-semirings advances the categorical and homological apparatus available for the study of generalized algebraic and geometric structures. Practically, it enhances the toolkit for analyzing invariants in discrete mathematics settings, prominently in code theory and cryptography, and connects the abstract machinery of modern higher category theory to concrete computational methods.

On a theoretical level, this body of work points to the universality of higher K-theory as a derived invariant, encompassing both classical and highly non-classical algebraic contexts. It sets a precedent for future research in the extension of motivic and homotopic techniques to generalizations of non-commutative algebraic geometry, and indicates explicit computational paths for concrete classes of codes or cryptosystems modeled within the $\Gamma$-semiring framework.

Conclusion

This paper rigorously defines and relates Quillen, Waldhausen, and $\infty$-categorical models for higher algebraic K-theory in the non-commutative $n$-ary $\Gamma$-semiring context, establishes their canonical equivalence, proves functoriality, localization, excision, and derived Morita invariance, and highlights their applicability to questions in discrete mathematics and cryptography. The results situate the algebraic K-theory of $(T, \Gamma)$ as a central geometric invariant in non-commutative algebra and open the way for both further structural investigation and computational development.