Fundamental Theorems in the K-Theory of Gamma Semirings: Additivity, Localization, and Dévissage
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911325129539584 |
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| author | Gokavarapu, Chandrasekhar |
| author_facet | Gokavarapu, Chandrasekhar |
| contents | Building on the Waldhausen and Quillen models of higher algebraic $K$-theory for exact categories and Waldhausen categories attached to a non-commutative $n$-ary $\Ga$-semiring $(T,\Ga)$, we establish the fundamental formal properties of $K$-theory in this $\Ga$-parametrised, slot-sensitive setting. For the exact/Waldhausen categories of finitely generated bi-positional $n$-ary $\Ga$-modules, perfect complexes in the derived category, and perfect quasi-coherent complexes on the non-commutative $\Ga$-spectrum $\SpecGnC{T}$, we prove Waldhausen Fibration and Additivity theorems and Quillen-type Localization for Serre and Waldhausen pairs. Under natural hypotheses on $\Ga$-stable filtrations we obtain dévissage and Approximation theorems, together with cofinality and Karoubi invariance, showing that idempotent completion does not change $K$-theory and that cofinal subcategories control $K_n$ in positive degrees. We further derive a Bass--Quillen fundamental triangle for polynomial extensions in the $n$-ary $\Ga$-context and prove nilpotent invariance for two-sided $\Ga$-ideals. In geometric terms, these results yield localization and Mayer--Vietoris sequences for the $K$-theory of $\Perf(X)$ on $X=\SpecGnC{T}$ and its admissible open covers. Altogether, the paper shows that the higher $K$-theory of non-commutative $n$-ary $\Ga$-semirings enjoys the same formal properties as in the classical ring and scheme cases, providing a robust foundation for subsequent computational and homotopy-theoretic applications. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_15839 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fundamental Theorems in the K-Theory of Gamma Semirings: Additivity, Localization, and Dévissage Gokavarapu, Chandrasekhar K-Theory and Homology Rings and Algebras 19D10, 19E08, 16Y60, 18G80 Building on the Waldhausen and Quillen models of higher algebraic $K$-theory for exact categories and Waldhausen categories attached to a non-commutative $n$-ary $\Ga$-semiring $(T,\Ga)$, we establish the fundamental formal properties of $K$-theory in this $\Ga$-parametrised, slot-sensitive setting. For the exact/Waldhausen categories of finitely generated bi-positional $n$-ary $\Ga$-modules, perfect complexes in the derived category, and perfect quasi-coherent complexes on the non-commutative $\Ga$-spectrum $\SpecGnC{T}$, we prove Waldhausen Fibration and Additivity theorems and Quillen-type Localization for Serre and Waldhausen pairs. Under natural hypotheses on $\Ga$-stable filtrations we obtain dévissage and Approximation theorems, together with cofinality and Karoubi invariance, showing that idempotent completion does not change $K$-theory and that cofinal subcategories control $K_n$ in positive degrees. We further derive a Bass--Quillen fundamental triangle for polynomial extensions in the $n$-ary $\Ga$-context and prove nilpotent invariance for two-sided $\Ga$-ideals. In geometric terms, these results yield localization and Mayer--Vietoris sequences for the $K$-theory of $\Perf(X)$ on $X=\SpecGnC{T}$ and its admissible open covers. Altogether, the paper shows that the higher $K$-theory of non-commutative $n$-ary $\Ga$-semirings enjoys the same formal properties as in the classical ring and scheme cases, providing a robust foundation for subsequent computational and homotopy-theoretic applications. |
| title | Fundamental Theorems in the K-Theory of Gamma Semirings: Additivity, Localization, and Dévissage |
| topic | K-Theory and Homology Rings and Algebras 19D10, 19E08, 16Y60, 18G80 |
| url | https://arxiv.org/abs/2512.15839 |