Fundamental comparison, base-change, and descent theorems in the $K$-theory of non-commutative n-ary Gamma-semirings
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| Format: | Preprint |
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2025
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| _version_ | 1866911733994487808 |
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| author | Gokavarapu, Chandrasekhar Rao, Dasari Madhusudhana |
| author_facet | Gokavarapu, Chandrasekhar Rao, Dasari Madhusudhana |
| contents | We develop a comparison, base-change, and descent framework for the algebraic $K$-theory of non-commutative $n$-ary $Γ$-semirings. Working in the Quillen-exact (and Waldhausen) setting of bi-finite, slot-sensitive $Γ$-modules and perfect complexes, we construct functorial maps on $K$-theory induced by extension and restriction of scalars under explicit $Γ$-flatness hypotheses in the relevant positional slots. We prove derived Morita invariance (via tilting bimodule complexes), establish Beck-Chevalley type base-change for cartesian squares, and deduce a projection formula compatible with the multiplicative structure coming from positional tensor products. Passing to the non-commutative $Γ$-spectrum Spec$^{\mathrm{nc}}_Γ(T)$, we show locality for perfect objects and derive Zariski hyperdescent for $\mathbb{K}(\mathrm{Perf})$, together with excision and localization sequences for closed immersions and fpqc descent for $Γ$-flat covers. Finally, we interpret $K_Γ(X)$ geometrically as the $K$-theory of the stable $\infty$-category of $Γ$-perfect complexes, describe its universal property in $Γ$-linear non-commutative motives, and record compatibility with cyclotomic and Chern-type trace maps. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_20807 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fundamental comparison, base-change, and descent theorems in the $K$-theory of non-commutative n-ary Gamma-semirings Gokavarapu, Chandrasekhar Rao, Dasari Madhusudhana K-Theory and Homology Rings and Algebras Primary: 19D10, 19E08, 16Y60. Secondary: 18E30, 55U35 We develop a comparison, base-change, and descent framework for the algebraic $K$-theory of non-commutative $n$-ary $Γ$-semirings. Working in the Quillen-exact (and Waldhausen) setting of bi-finite, slot-sensitive $Γ$-modules and perfect complexes, we construct functorial maps on $K$-theory induced by extension and restriction of scalars under explicit $Γ$-flatness hypotheses in the relevant positional slots. We prove derived Morita invariance (via tilting bimodule complexes), establish Beck-Chevalley type base-change for cartesian squares, and deduce a projection formula compatible with the multiplicative structure coming from positional tensor products. Passing to the non-commutative $Γ$-spectrum Spec$^{\mathrm{nc}}_Γ(T)$, we show locality for perfect objects and derive Zariski hyperdescent for $\mathbb{K}(\mathrm{Perf})$, together with excision and localization sequences for closed immersions and fpqc descent for $Γ$-flat covers. Finally, we interpret $K_Γ(X)$ geometrically as the $K$-theory of the stable $\infty$-category of $Γ$-perfect complexes, describe its universal property in $Γ$-linear non-commutative motives, and record compatibility with cyclotomic and Chern-type trace maps. |
| title | Fundamental comparison, base-change, and descent theorems in the $K$-theory of non-commutative n-ary Gamma-semirings |
| topic | K-Theory and Homology Rings and Algebras Primary: 19D10, 19E08, 16Y60. Secondary: 18E30, 55U35 |
| url | https://arxiv.org/abs/2512.20807 |