relative K1-Group functor from Pairs of Rings
Definition
the relative \(K_1\)-group functor is defined as functor
where
- \((R, \mathfrak{a})\) is the category of ideal pairs of rings
- \(K_1(f)\) is the ring induced morphism by k1
Proof
follows from
for repeated application of
- \(\mathrm{colim}_{\mathbb{N}}\) to go from \(\mathrm{GL}_n(-) \rightarrow \mathrm{GL}(-)\) resp. \(E_n, E\)
- \(\mathrm{coker}(-)\) to go from \(E(-) \rightarrow \mathrm{GL}\) to \(K_1(-)\)