Sheaf Theory and Derived Gamma Geometry over the Non-Commutative Gamma Spectrum
Chandrasekhar Gokavarapu
TL;DR
The paper develops a comprehensive non-commutative, n-ary Gamma-geometry by introducing the non-commutative Gamma-spectrum and a localization-based structure sheaf, then builds a robust theory of quasi-coherent Gamma-sheaves and their derived invariants ExtΓ and TorΓ. It constructs the derived category of QCoh on SpecΓnc(T), establishes local-global dualities, and develops derived Gamma-stacks with dg-enhancements, culminating in spectral dualities and motivic interpretations. Structural results include Wedderburn–Artin type decompositions and derived Morita theory, plus a duality between the primitive Gamma-spectrum and simple derived objects, showing that the geometry is governed by the derived category rather than presentation. The framework unifies algebra, geometry, and higher-categorical/homotopical perspectives, and points to future directions in ∞-categorical refinements, motivic Gamma-homotopy, and derived tropicalization/quantization.
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.