Relative K-groups

Summary

Relative K-groups $K_n(R, I)$ attach K-groups to a ring $R$ together with a two-sided ideal $I$. They measure the part of the K-theory of $R$ that is invisible in the quotient ring $R/I$. They fit into a long exact sequence that links $K_n(R, I)$, $K_n(R)$ and $K_n(R/I)$. This sequence is a basic tool for computing K-groups, for instance through Mayer–Vietoris sequences and excision.