Gubeladze, J. Geometric and algebraic representations of commutative cancellative monoids. (English) Zbl 0871.19001 Gegelia, T. (ed.) et al., Collected papers on \(K\)-theory and categorical algebra. Tbilisi: Publishing House GCI, Proc. A. Razmadze Math. Inst. 113, 31-81 (1995). This paper continues the author’s study of commutative cancellative torsionfree monoids \(M\) and the \(K\)-theory of the associated monoid rings \(R[M]\) for \(R\) a commutative ring. When \(M\) is normal, the rank and torsion part of the divisor class group of \(M\) are characterized in terms of combinatorial data of \(M\). He shows for \(i\geq 1\) that if the natural map \(K_i(R)\to K_i (R[M])\) is an isomorphism, then \(K_i(R)\to K_i (R[M]/RI)\) is also an isomorphism, where \(M\) is \(p\)-divisible for some prime \(p\) and \(I\) is a proper radical ideal of \(M\), and that there are natural isomorphisms \(K_i(R)\to K_i (R[M])\) when \(R\) is regular and \(M\) is an intermediate \(p\)-divisible monoid \(\mathbb{Z}^r_+ \subset M\subset \mathbb{Q}^r_+\). He also gives a new proof that all finitely generated projective \(R[M]\)-modules are free when \(R\) is a PID and \(M\) an intermediate seminormal monoid \(\mathbb{Z}^r_+ \subset M\subset \mathbb{Q}^r_+\), and studies when \(E_n(R[M])\) acts transitively on the set of unimodular rows \(Um_n(R[M])\).For the entire collection see [Zbl 0851.00017]. Reviewer: D.F.Anderson (Knoxville) Cited in 1 ReviewCited in 7 Documents MSC: 19D50 Computations of higher \(K\)-theory of rings 20M14 Commutative semigroups 19A13 Stability for projective modules 20M25 Semigroup rings, multiplicative semigroups of rings 13C20 Class groups 13D15 Grothendieck groups, \(K\)-theory and commutative rings 13F45 Seminormal rings 13F50 Rings with straightening laws, Hodge algebras Keywords:projective module; monoid rings; divisor class group; seminormal monoid; unimodular rows PDFBibTeX XMLCite \textit{J. Gubeladze}, in: Collected papers on \(K\)-theory and categorical algebra. Tbilisi: Publishing House GCI. 31--81 (1995; Zbl 0871.19001)