Homological and Categorical Foundations of Ternary $Γ$-Modules and Their Spectra

Abstract: Purpose: To develop a unified homological categorical foundation for commutative ternary Gamma semirings by formulating a general theory of ternary Gamma modules that integrates algebraic, geometric, and computational layers, extending the ideal theoretic and algorithmic bases of Papers A [Rao2025A] and B [Rao2025B1]. Methods: We axiomatize ternary Gamma modules and establish the fundamental isomorphism theorems, construct annihilator primitive correspondences, and prove Schur density embeddings. Categorical analysis shows that T Gamma Mod is additive, exact, and monoidal closed, enabling the definition of derived functors Ext and Tor via projective injective resolutions and yielding a tensor Hom adjunction. We develop geometric dualities between module objects and the spectrum Spec Gamma (T) and extend them to analytic, fuzzy, and computational settings. Results: The category T Gamma Mod admits kernels, cokernels, coequalizers, and balanced exactness; monoidal closure ensures internal Homs and coherent tensor Hom adjunctions. Derived functors Ext and Tor are well defined and functorial, with long exact sequences and base change compatibility. Schur density yields faithful embedding criteria, while annihilator primitive correspondences control primitivity and support theory. Geometric dualities provide contravariant equivalences linking submodule spectra with closed sets in Spec Gamma (T), persisting under analytic, fuzzy, and computational enrichments. Conclusion: These results complete the algebraic homological geometric synthesis for commutative ternary Gamma semirings, furnish robust tools for derived and spectral analysis, and prepare the framework for fuzzy and computational extensions developed in Paper D [Rao2025D], extending the algebraic framework first established in [Rao2025]