Sheaf Theory and Derived Gamma Geometry over the Non-Commutative Gamma Spectrum

Abstract: We develop the geometric and homological framework for non-commutative $n$-ary $Γ$-semirings by constructing a sheaf and derived theory over their non-commutative $Γ$-spectrum. Starting with a non-commutative $n$-ary $Γ$-semiring $(T,+,Γ,μ)$ and its bi-$Γ$-modules, we define the space $\Spec_Γ^{\mathrm{nc}}(T)$, equip it with a Zariski-type topology, and build the structure sheaf $\mathcal{O}{\SpecΓ^{\mathrm{nc}}(T)}$ via localization at prime $Γ$-ideals. We introduce quasi-coherent $Γ$-sheaves, show that their category is exact with enough injectives, and interpret the derived functors $\Ext^Γ$ and $\Tor^Γ$ as global cohomological invariants on this non-commutative $Γ$-space. On the derived side, we construct the category $\mathbf{D}(\QCoh(\Spec_Γ^{\mathrm{nc}}(T)))$, establish a local--global principle for $\Ext^Γ$ and $\Tor^Γ$, and prove a non-commutative local duality theorem assuming a dualizing complex. We further introduce derived non-commutative $Γ$-stacks and a dg-enhancement of the spectrum, giving a spectral and motivic interpretation of homological invariants. Structural consequences include a Wedderburn--Artin type decomposition in the $n$-ary $Γ$-setting, a derived Morita theory for semisimple $n$-ary $Γ$-semirings, and a duality between the primitive $Γ$-spectrum and simple objects of the derived category. These results extend our earlier commutative derived $Γ$-geometry to a fully non-commutative $n$-ary context.