Derived Functors, Resolutions, and Homological Dualities in n-ary Gamma-Semirings

Abstract

This paper develops the homological backbone of the theory of non-commutative $n$-ary $\Gamma$-semirings. Starting from an $n$-ary $\Gamma$-semiring $(T,+,\tilde{\mu})$ and its $\Gamma$-ideals, we work in the slot-sensitive categories of left, right, and bi-$\Gamma$-modules, and endow the bi-module category with a Quillen exact structure compatible with the $n$-ary multiplication. Within this exact framework we construct bar-type projective resolutions and cofree-based injective resolutions under natural $\Gamma$-Noetherian and $\Gamma$-regular hypotheses on $T$, and we obtain finite projective resolutions for finitely presented bi-modules under $\Gamma$-Noetherian conditions. On this basis we define the derived functors $\ExtG$ and $\TorG$ for bi-$\Gamma$-modules, prove their balance with respect to projective and injective resolutions, establish long exact sequences and a Yoneda interpretation via iterated extensions, and construct K\"unneth-type spectral sequences and base-change isomorphisms. Interpreting bi-$\Gamma$-modules as quasi-coherent sheaves on the non-commutative $\Gamma$-spectrum $\SpecGnC{T}$, these homological invariants provide the appropriate derived language for a non-commutative $\Gamma$-geometry and prepare the ground for the spectral and geometric analysis carried out in the third part of this series.