Fundamental Theorems in the K-Theory of Gamma Semirings: Additivity, Localization, and D\'evissage

TL;DR

This paper extends fundamental theorems of algebraic K-theory to the setting of non-commutative n-ary Gamma semirings, establishing key properties like additivity, localization, and dévissage.

Contribution

It develops a comprehensive framework for higher K-theory of Gamma semirings, proving core theorems and invariance properties analogous to classical cases.

Findings

Proves Waldhausen Fibration and Additivity theorems for Gamma semirings.

Establishes localization, dévissage, and approximation theorems in this setting.

Shows K-theory invariance under idempotent completion and cofinal subcategories.

Abstract

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 d\'evissage 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.