Nontriviality of SK1(R[M]) Original Research Article

The main result of this paper is as follows: For any commutative regular (actually even K2-regular) ring R and any finitely generated intermediate monoid Z+r subset ofM subset ofQ+r (for some natural r) the following conditions are equivalent: 1. (a) M ≈ Z+r 2. (b) R[M] is K1-regular 3. (c) M is seminormal and SK1(R) = SK1(R[M]) (i.e. the natural homomorphism SK1(R) → SK1(R[M]) is an isomorphism) and, if in addition ΩR ≠ 0 4. (d) SK1(R) = SK1(R[M]) where ωR is a module of absolute differentials. The implications (a) implies (b) implies (c) are well known. In Sections 8–10 we present examples, further generalizations and applications.