[For the entire collection see Zbl 0705.00011.]
In a previous paper [Mat. Sb., Nov. Ser. 135(177), No.2, 169-185 (1988; Zbl 0654.13013)], the author described all commutative cancellative monoids L for which projective R[L]-modules are free whenever R is a principal ideal domain. These monoids are the commutative cancellative torsion-free seminormal monoids (where seminormal means that if m is an element of the group of quotients K(L) of L, and 2m,3m\(\in L\), then \(m\in L)\). This same class of monoids is also the maximal class of commutative cancellative monoids L for which \(K_ 0(R)\to K_ 0(R[L])\) is an isomorphism whenever R is a commutative regular ring.
The present paper generalizes these results to \(K_ 1\) and \(K_ 2\). The author proves: (a) For any Euclidean ring R and any c-divisible commutative cancellative torsion-free monoid L we have \(SL_{\tau}(R[L])=E_{\tau}(R[L])\) for \(\tau\geq 3\). (c is a natural number \(>1\), and c-divisible means that if m is an element of K(L), then there exists \(n\in L\) such that \(cn=m.)\) (b) The maximal class of commutative cancellative torsion-free c-divisible monoids L for which \(K_ 1(R)\to K_ 1(R[L])\) is an isomorphism whenever R is a regular ring is the class of all commutative cancellative torsion-free c- divisible monoids L with trivial subgroup of invertible elements. (c) Analogous statements for \(K_ 2\) for those c-divisible monoids L in which there exists a system of d linearly independent elements \(m_ i\) (1\(\leq i\leq d)\) such that for any \(m\in L\), there exist non-negative \(\lambda_ i\in {\mathbb{Q}}\) (1\(\leq i\leq d)\) such that \(m=\sum^{d}_{i=1}\lambda_ im_ i\) (in \({\mathbb{Q}}\otimes K(L)).\)
The proofs are based on a geometric interpretation of monoids obtained by intersecting with the unit sphere rays spanned (in \({\mathbb{Q}}\otimes K(L))\) by elements of the monoid L. Some hypothesis such as c-divisibility is needed since there exist normal monoids L of rank 2 for which \(SK_ 1({\mathbb{Z}}[L])\neq 0\), or even for which \(SK_ 1({\mathbb{Q}}[L])\neq 0\). (Normal means that \(x\in K(L)\) and nx\(\in L\) for a natural number \(n>1\) imply \(n\in L.)\) Note that c-divisibility and normal each imply seminormal.