Lower K-groups

Summary

The lower K-groups of a ring $R$ are the groups $K_0(R)$, $K_1(R)$ and $K_2(R)$ of algebraic K-theory. They are defined by explicit algebraic constructions before Quillen's higher K-theory. The group $K_0(R)$ comes from projective modules, $K_1(R)$ from invertible matrices modulo elementary ones, and $K_2(R)$ from relations among elementary matrices. Quillen's higher K-groups agree with these three in degrees $0$, $1$ and $2$. Some authors also count the negative K-groups of Bass among the lower K-groups.