A cycle class map from Chow groups with modulus to relative $K$-theory

Binda, Federico

doi:10.48550/arxiv.1706.07126

Cited by 3 publications

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“…W. 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 R. Iwasa and W. 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 F. Binda [3,Theorem 4.4.10] (see also [4]) 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). Other positive results are obtained in [32], [43] and [5].…”

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confidence: 99%

Relative Cycles With Moduli and Regulator Maps

Binda

Saito²

2017

J. Inst. Math. Jussieu

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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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confidence: 99%

Relative Cycles With Moduli and Regulator Maps

Binda

Saito²

2017

J. Inst. Math. Jussieu

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106

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“…A cycle class map of the kind given in Corollary 7.23 was constructed in [1]. In that construction, Binda uses a different definition for the Chow groups with modulus compared to the ones described in § 2.5.…”

Section: Chow Group With Modulus and Relative Motivic Cohomologymentioning

confidence: 99%

Motivic spectral sequence for relative homotopy K-theory

Krishna¹,

Pelaez²

2020

Annali Scuola Normale Superiore - Classe Di Scienze

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We construct a motivic spectral sequence for the relative homotopy invariant K-theory of a closed immersion of schemes D ⊂ X. The E 2 -terms of this spectral sequence are the cdhhypercohomology of a complex of equidimensional cycles.Using this spectral sequence, we obtain a cycle class map from the relative motivic cohomology group of 0-cycles to the relative homotopy invariant K-theory. For a smooth scheme X and a divisor D ⊂ X, we construct a canonical homomorphism from the Chow groups with modulus CH i (X|D) to the relative motivic cohomology groups H 2i (X|D, Z(i)) appearing in the above spectral sequence. This map is shown to be an isomorphism when X is affine and i = dim(X).

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“…They have also shown that the cycle class map is injective if X is affine and the base filed is algebraically closed. Also, Binda [Bi18] constructed a cycle class map for higher zero cycles with modulus using a slightly different (stronger) modulus condition.…”

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Relative $K_0$ and relative cycle class map

Iwasa

2017

Preprint

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A → B be an exact functor between small exact categories. We study the zeroth homotopy group K 0 (F ) of the homotopy fiber of the map K(A) → K(B) between K-theory spectra. Under the assumption that F is a cofinal and that B is split exact, we give an explicit description of K 0 (F ) in terms of the triangulated functor D b (A) → D b (B) between the derived categories.We apply it to the pair (X, D) of a scheme X and an affine closed subscheme D of X, and get a description of the relative K 0 -group K 0 (X, D) in terms of perfect complexes; it is generated by pairs of two perfect complexes of X together with quasi-isomorphisms along D. This description makes it possible to assign a cycle class in K 0 (X, D) to a cycle on X not meeting D in an intuitive way. When X is a separated regular scheme of finite type over a field and D is an affine effective Cartier divisor on X, we prove that the cycle classes induce a surjective group homomorphism from the Chow group with modulus CH * (X|D) defined by Binda-Saito to a suitable subquotient of K 0 (X, D).

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A cycle class map from Chow groups with modulus to relative $K$-theory

Cited by 3 publications

References 22 publications

Relative Cycles With Moduli and Regulator Maps

Relative Cycles With Moduli and Regulator Maps

Motivic spectral sequence for relative homotopy K-theory

Relative $K_0$ and relative cycle class map

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