# Algebraic K-functors for $\Gamma$-rings

## TL;DR

This paper extends algebraic K-theory to $ ext{Gamma}$-rings, introducing new K-functors and proving extended versions of key conjectures related to Witt and Chow groups.

## Contribution

It introduces $ ext{Gamma}$-algebraic K-theory, defines Milnor K-theory and Swan's K-functors for $ ext{Gamma}$-rings, and proves extended conjectures.

## Key findings

- Extended Matsumoto conjecture related to symbol groups
- Proved Milnor conjectures for Witt and Chow groups
- Developed foundational theory for $ ext{Gamma}$-rings in algebraic K-theory

## Abstract

This is an attempt to extend to algebraic K-theory our approach to group actions in homological algebra that could be called an introduction to $\Gamma$-algebraic K-theory. For $\Gamma$-rings the Milnor algebraic K-theory and Swan's algebraic K-functors are introduced and investigated. Particularly the Matsumoto conjecture related to the symbol group, and the Milnor conjectures related to Witt and Chow groups are extended and proven.