Nontriviality of SK1( R[ M

Abstract

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