SK1

Summary

For a commutative ring $R$, the group $SK_1(R)$ is the kernel of the determinant map $K_1(R) \to R^\times$. Here $K_1(R)$ is K1 of a ring and $R^\times$ is the group of units. It collects the part of $K_1(R)$ that the determinant cannot see, and $K_1(R) \cong R^\times \oplus SK_1(R)$. It vanishes for fields, for local rings and for Euclidean domains. Bass, Milnor and Serre showed that it also vanishes for rings of integers of number fields. A parallel group $SK_1(A)$, defined with the reduced norm, exists for central simple algebras.