Cube invariance of higher Chow groups with modulus

Abstract

The higher Chow group with modulus was introduced by Binda-Saito as a common generalization of Bloch's higher Chow group and the additive higher Chow group. In this paper, we study invariance properties of the higher Chow group with modulus. First, we formulate and prove "cube invariance," which generalizes A 1 -homotopy invariance of Bloch's higher Chow group. Next, we introduce the nilpotent higher Chow group with modulus, as an analogue of the nilpotent algebraic K-group, and define a module structure on it over the big Witt ring of the base field. We deduce from the module structure that the higher Chow group with modulus with appropriate coefficients satisfies A 1 -homotopy invariance. We also prove that A 1 -homotopy invariance implies independence from the multiplicity of the modulus divisors.

Torsion in the 0-cycle group with modulus

“…In [49,Theorem 1.3], an isomorphism of similar indication has been very recently shown after inverting char(k). But it has no implication on Corollary 1.4.…”

Isomorphisms Up to Bounded Torsion Between Relative -Groups and Chow Groups With Modulus

“…Remark 5.2. The groups CH p (X |D top ) and its higher variant have been studied in [12,16] by the name of naïve Chow groups with modulus and Chow groups with topological modulus, respectively. Lemma 5.3.…”

“…Remark 5.2. The groups CH p (X |D top ) and its higher variant have been studied in [12,16] by the name of naïve Chow groups with modulus and Chow groups with topological modulus, respectively. The following is a variant of Theorem 2.2, which is a special case of a more general result in [12].…”

A Moving Lemma for Algebraic Cycles With Modulus and Contravariance

“…One of the two spectral sequences degenerates thanks to the cube invariance of the cubical version ( [Miy16a], extended from the minus cube to pairs (∆ p , −∆ p ∞ )). He observes that the simplicial version satisfies the cube invariance at least in pro; this is because the triangulation maps ∆ p+1 ∼ = − → A 1 ×∆ p are not admissible but they are, up to doubling the divisor on the source.…”