Negative \(K\)-theory and Chow group of monoid algebras. (English) Zbl 1442.19010
Cortiñas, Guillermo (ed.) et al., \(K\)-theory in algebra, analysis and topology. ICM 2018 satellite school and workshop, La Plata and Buenos Aires, Argentina, July 16–20 and July 23–27, 2018. Providence, RI: American Mathematical Society (AMS). Contemp. Math. 749, 195-224 (2020).
Summary: We show, for a finitely generated partially cancellative torsion-free commutative monoid \(M\), that \(K_i(R)\cong K_i(R[M])\) whenever \(i\leq -d\) and \(R\) is a quasi-excellent \(\mathbb{Q}\)-algebra of Krull dimension \(d\geq 1\). In particular, \(K_i(R[M])=0\) for \(i<-d\). This is a generalization of Weibel’s \(K\)-dimension conjecture to monoid algebras. We show that this generalization fails for \(X[M]\) if \(X\) is not an affine scheme. We also show that the Levine-Weibel Chow group of 0-cycles \(\text{CH}^{LW}_0(k[M])\) vanishes for any finitely generated commutative partially cancellative monoid \(M\) if \(k\) is an algebraically closed field.
For the entire collection see [Zbl 1441.19002].
For the entire collection see [Zbl 1441.19002].
MSC:
| 19D50 | Computations of higher \(K\)-theory of rings |
| 13F15 | Commutative rings defined by factorization properties (e.g., atomic, factorial, half-factorial) |
| 14F35 | Homotopy theory and fundamental groups in algebraic geometry |
| 14C15 | (Equivariant) Chow groups and rings; motives |
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