A Moving Lemma for Algebraic Cycles With Modulus and Contravariance

Kai, Wataru

doi:10.1093/imrn/rnz018

Cited by 11 publications

(14 citation statements)

References 25 publications

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“…When X is effective, the relative motivic cohomology is introduced in [6]. If moreover X • is smooth, the relative motivic cohomology is contravariantly functorial with respect to coadmissible left fine (not necessarily flat) morphisms of modulus pairs ( [8]). We do not know whether this is true for non-effective modulus pairs or not.…”

Section: A Results On Relative Motivic Cohomologiesmentioning

confidence: 99%

Cube invariance of higher Chow groups with modulus

Miyazaki

2019

J. Algebraic Geom.

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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.

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Section: A Results On Relative Motivic Cohomologiesmentioning

confidence: 99%

Cube invariance of higher Chow groups with modulus

Miyazaki

2019

J. Algebraic Geom.

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“…For different counter-examples to the descent property, see [43], before Theorem 3. On the bright side, we mention the following important result recently obtained by W. Kai (see [29,Theorem 1.4]). If (X, D) and (Y , E) are two pairs of schemes of finite type over k equipped with effective Cartier divisors, we say that a morphism f : X → Y is an admissible morphism of pairs if f * E is defined as Cartier divisor on X and f restricts to a morphism D → E. Remark 2.13.…”

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confidence: 82%

“…There are some new results which provide a ground for the optimistic choice of the words. Kai [29] established a moving lemma for cycle complexes with modulus which implies an appropriate contravariant functoriality of the Nisnevich version of (1.3) (see Theorem 2.12 for the precise statement). A work by Iwasa and Kai [27] provides Chern classes from the relative K -groups of the pair (X , D) to the Nisnevich motivic cohomology groups H * M,Nis (X |D, Z( * )), while a construction of the first author [3, Theorem 4.4.10] (see also [4,Theorem 3.5]) gives cycle classes from the groups of higher 0-cycles with modulus CH d+n (X |D, n) to the relative K -groups K n (X , D).…”

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confidence: 99%

Relative Cycles With Moduli and Regulator Maps

Binda

Saito²

2017

J. Inst. Math. Jussieu

106

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Let X be a separated scheme of finite type over a field k and D a non-reduced effective Cartier divisor on it. We attach to the pair (X, D) a cycle complex with modulus, those homotopy groups -called higher Chow groups with modulus -generalize additive higher Chow groups of Bloch-Esnault, Rülling, Park and Krishna-Levine, and that sheafified on X Zar gives a candidate definition for a relative motivic complex of the pair, that we compute in weight 1.When X is smooth over k and D is such that D red is a normal crossing divisor, we construct a fundamental class in the cohomology of relative differentials for a cycle satisfying the modulus condition, refining El-Zein's explicit construction of the fundamental class of a cycle. This is used to define a natural regulator map from the relative motivic complex of (X, D) to the relative de Rham complex. When X is defined over C, the same method leads to the construction of a regulator map to a relative version of Deligne cohomology, generalizing Bloch's regulator from higher Chow groups.Finally, when X is moreover connected and proper over C, we use relative Deligne cohomology to define relative intermediate Jacobians with modulus J r X|D of the pair (X, D). For r = dim X, we show that J r X|D is the universal regular quotient of the Chow group of 0-cycles with modulus.generalizing Bloch's computation in weight 1, Z X (1) ∼ = O × X [−1], and proving the first of the expected properties of the relative motivic cohomology groups (see Theorem 4.1). 1.7. Let (X, D) be as in 1.6. The second main technical result of this paper, presented in Section 7, is the construction, using the fundamental class in relative differentials, of regulator maps from the relative motivic complex Z X|D (r) to a the relative de Rham complex of XThe map φ dR is compatible with flat pullbacks and proper push forwards of pairs.When X is a smooth algebraic variety over the field of complex numbers, we can use the same technique to define regulator maps to a relative version of Deligne cohomology (see (8.10)) and to Betti cohomology with compact support, where ǫ is the morphism of sites and j : X → X is the open embedding of the complement of D in X, generalizing Bloch's regulator from higher Chow groups to Deligne cohomology, constructed in [7]. This regulator map is further studied in Section 9 in the case of additive Chow groups. Evaluated on suitable cycles, our regulator recovers Bloch-Esnault additive dilogarithm (introduced in [10]) and gives a refinement of [10, Proposition 5.1] (see (9.5)).1.8. Suppose that X is moreover connected and proper over C, and consider the induced maps in cohomology in degree 2r. We have a commutative diagram (see 10.1.1)and in analogy with the classical situation, we call the kernel J r X|D the r-th relative intermediate Jacobian of the pair (X, D). We note that they admit a description in terms of extensions groups Ext 1 in the abelian category of enriched Hodge structures defined by S.Bloch and V.Srinivas in [12].One can show that J r X|D fits into an exact sequen...

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“…Although recently established results by various authors (see [Kai16], [RS16]) have indicated that the Chow groups with modulus (and, more generally, the relative motivic cohomology groups of [BS16]) have some of the above expected properties, many questions remain widely open.…”

Section: Introductionmentioning

confidence: 99%

Zero cycles with modulus and zero cycles on singular varieties

Binda¹,

Krishna²

2017

Compositio Math.

134

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Given a smooth variety X and an effective Cartier divisor D ⊂ X, we show that the cohomological Chow group of 0-cycles on the double of X along D has a canonical decomposition in terms of the Chow group of 0-cycles CH0(X) and the Chow group of 0cycles with modulus CH0(X|D) on X. When X is projective, we construct an Albanese variety with modulus and show that this is the universal regular quotient of CH0(X|D).As a consequence of the above decomposition, we prove the Roitman torsion theorem for the 0-cycles with modulus. We show that CH0(X|D) is torsion-free and there is an injective cycle class map CH0(X|D) ֒→ K0(X, D) if X is affine. For a smooth affine surface X, this is strengthened to show that K0(X, D) is an extension of CH1(X|D) by CH0(X|D). ± 17 4.1. The map ∆ * 17 4.2. The maps ι * ± 18 5. 0-Cycles on S X and 0-cycles with modulus on X 20 5.1. Overture 20

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A Moving Lemma for Algebraic Cycles With Modulus and Contravariance

Cited by 11 publications

References 25 publications

Cube invariance of higher Chow groups with modulus

Cube invariance of higher Chow groups with modulus

Relative Cycles With Moduli and Regulator Maps

Zero cycles with modulus and zero cycles on singular varieties

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