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 $\Gamma$-semirings by constructing a sheaf and derived theory over their non-commutative $\Gamma$-spectrum. Starting with a non-commutative $n$-ary $\Gamma$-semiring $(T,+,\Gamma,\mu)$ and its bi-$\Gamma$-modules, we define the space $\Spec_{\Gamma}^{\mathrm{nc}}(T)$, equip it with a Zariski-type topology, and build the structure sheaf $\mathcal{O}{\Spec{\Gamma}^{\mathrm{nc}}(T)}$ via localization at prime $\Gamma$-ideals. We introduce quasi-coherent $\Gamma$-sheaves, show that their category is exact with enough injectives, and interpret the derived functors $\Ext^{\Gamma}$ and $\Tor^{\Gamma}$ as global cohomological invariants on this non-commutative $\Gamma$-space. On the derived side, we construct the category $\mathbf{D}(\QCoh(\Spec_{\Gamma}^{\mathrm{nc}}(T)))$, establish a local--global principle for $\Ext^{\Gamma}$ and $\Tor^{\Gamma}$, and prove a non-commutative local duality theorem assuming a dualizing complex. We further introduce derived non-commutative $\Gamma$-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 $\Gamma$-setting, a derived Morita theory for semisimple $n$-ary $\Gamma$-semirings, and a duality between the primitive $\Gamma$-spectrum and simple objects of the derived category. These results extend our earlier commutative derived $\Gamma$-geometry to a fully non-commutative $n$-ary context.