Derived Functors, Resolutions, and Homological Dualities in n-ary Gamma-Semirings
Chandrasekhar Gokavarapu
TL;DR
<3-5 sentence high-level summary> This work builds a comprehensive homological framework for non-commutative, n-ary Γ-semirings by endowing bi-Γ-modules with a Quillen exact structure and constructing intrinsic projective and injective resolutions. It defines derived bifunctors ExtΓ and TorΓ, proves their balance and long exact sequences, and provides a Yoneda interpretation and higher operations, including Künneth-type spectral sequences and base-change formulas. By embedding bi-Γ-modules into a derived category and interpreting them as quasi-coherent sheaves on the non-commutative spectrum SpecncΓ(T), the paper links homological invariants to non-commutative derived geometry and descent phenomena. These results lay the groundwork for Part III, which will develop quasi-coherent sheaves, t-structures, and Morita-type invariants in the derived Γ-geometry setting.</p>
Abstract
This paper develops the homological backbone of the theory of non-commutative $n$-ary $Γ$-semirings. Starting from an $n$-ary $Γ$-semiring $(T,+,\tildeμ)$ and its $Γ$-ideals, we work in the slot-sensitive categories of left, right, and bi-$Γ$-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 $Γ$-Noetherian and $Γ$-regular hypotheses on $T$, and we obtain finite projective resolutions for finitely presented bi-modules under $Γ$-Noetherian conditions. On this basis we define the derived functors $\ExtG$ and $\TorG$ for bi-$Γ$-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-$Γ$-modules as quasi-coherent sheaves on the non-commutative $Γ$-spectrum $\SpecGnC{T}$, these homological invariants provide the appropriate derived language for a non-commutative $Γ$-geometry and prepare the ground for the spectral and geometric analysis carried out in the third part of this series.