relative K1-Group functor from Pairs of Rings

Definition

the relative \(K_1\)-group functor is defined as functor

\begin{align*} K_1: \mathrm{PairsRing} \rightarrow& \mathrm{Grp} \\ (R, \mathfrak{a}) \mapsto& K_1(R, \mathfrak{a}) \\ (f: (R, \mathfrak{a}) \rightarrow (S, \mathfrak{b})) \mapsto& (K_1(f): K_1(R, \mathfrak{a}) \rightarrow K_1(S, \mathfrak{b})) \end{align*}

where

  1. \((R, \mathfrak{a})\) is the category of ideal pairs of rings
  2. \(K_1(f)\) is the ring induced morphism by k1

Proof

follows from

for repeated application of

  1. \(\mathrm{colim}_{\mathbb{N}}\) to go from \(\mathrm{GL}_n(-) \rightarrow \mathrm{GL}(-)\) resp. \(E_n, E\)
  2. \(\mathrm{coker}(-)\) to go from \(E(-) \rightarrow \mathrm{GL}\) to \(K_1(-)\)

Date: nil

Author: Anton Zakrewski

Created: 2025-01-15 Mi 17:47