Derived Functors, Resolutions, and Homological Dualities in n-ary Gamma-Semirings
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arXiv
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| Format: | Preprint |
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2025
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| author | Gokavarapu, Chandrasekhar |
| author_facet | Gokavarapu, Chandrasekhar |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_21797 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Derived Functors, Resolutions, and Homological Dualities in n-ary Gamma-Semirings Gokavarapu, Chandrasekhar Rings and Algebras 16Y60, 18G10, 18E30, 16Y90, 14A15, 08A3 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. |
| title | Derived Functors, Resolutions, and Homological Dualities in n-ary Gamma-Semirings |
| topic | Rings and Algebras 16Y60, 18G10, 18E30, 16Y90, 14A15, 08A3 |
| url | https://arxiv.org/abs/2511.21797 |