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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].

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
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