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

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