Derived $Γ$-Geometry, Sheaf Cohomology, and Homological Functors on the Spectrum of Commutative Ternary $Γ$-Semirings

Abstract: This paper develops a comprehensive geometric and homological framework for derived Gamma-geometry, extending the theory of commutative ternary Gamma-semirings established in our earlier works. Building upon the ideal-theoretic, computational, and categorical foundations of Papers A to D (Rao 2025A, Rao 2025B1, Rao 2025B2, Rao 2025C, Rao 2025D), the present study constructs the algebraic and geometric infrastructure necessary to place Gamma-semirings within the modern language of derived and categorical geometry. We define the affine Gamma-spectrum Spec_Gamma(T) together with its structure sheaf O_{Spec_Gamma(T)}, establishing a Zariski-type topology adapted to ternary Gamma-operations. In this setting, the category of Gamma-modules is shown to be additive, exact, and monoidal-closed, supporting derived functors Ext_Gamma and Tor^Gamma, whose existence is ensured through explicit projective and injective resolutions. The derived category D(T-Gamma-Mod) is then constructed to host homological dualities and Serre-type vanishing theorems, culminating in a categorical analogue of the Serre-Swan correspondence (Swan 1962, Gelfand 1960) for Gamma-modules. Geometric and categorical unification is achieved through fibered and derived Gamma-stacks, which provide a natural environment for studying morphisms of affine Gamma-schemes and cohomological descent. The theory connects with noncommutative geometry and higher-arity (n-ary) generalizations, showing that derived Gamma-geometry forms a coherent homological universe capable of expressing algebraic, geometric, and physical dualities within one categorical law. Computational methods for finite Gamma-semirings and applications to mathematical physics, where ternary operations model triadic couplings, are also discussed.