Exact Categories and Homological Foundations of Non-Commutative n-ary Gamma. Semirings
Chandrasekhar Gokavarapu
TL;DR
The paper addresses the need for a rigorous homological framework for non-commutative $n$-ary $\Gamma$-semirings and their modules, unifying derived $\Gamma$-geometry with non-commutative spectral theory. It develops slot-sensitive left, right, and bi-$\Gamma$-modules and endows their category with a Quillen exact structure, enabling kernels, cokernels, and derived functors $\mathrm{Ext}^{(j,k)}_{\Gamma}$ and $\mathrm{Tor}^{(j,k)}_{\Gamma}$. The work introduces the positional tensor product $M\otimes^{(j,k)}_{\Gamma}N$ and the internal Hom $\underline{\mathrm{Hom}}^{(j,k)}_{\Gamma}(M,N)$, establishing key adjunctions and exactness properties that parallel classical homological algebra while incorporating $n$-ary and non-commutative features. It also connects module-theoretic constructions to the non-commutative $\Gamma$-spectrum \(Spec_Gamma^nc(T)\) and outlines how these foundations support Morita-type analyses and spectral interpretations in the ensuing parts, culminating in a derived category framework for non-commutative $\Gamma$-geometry.
Abstract
This paper establishes the homological and geometric foundations of non-commutative n-ary Gamma-semirings, unifying two previously distinct directions in Gamma-algebra: the derived Gamma-geometry developed for the commutative ternary case and the structural and spectral theory for general non-commutative n-ary systems. We introduce categories of left, right, and bi-Gamma-modules that respect positional asymmetry and prove that they form additive and exact categories in Quillen's sense. Within this setting, we construct projective and injective resolutions, define the derived functors Ext^Gamma and Tor_Gamma, and establish long exact sequences and spectral balance theorems in the n-ary regime. By extending sheaf-theoretic and homological tools to the non-commutative Gamma-spectrum Spec_Gamma^nc(T), we obtain a coherent framework of non-commutative derived Gamma-geometry that parallels the classical paradigms of Grothendieck and Kontsevich in homological algebra and non-commutative geometry. The framework developed here establishes the foundational exact-categorical and homological structures that enable Morita-type analyses and spectral interpretations in the subsequent parts of this series.