Ideles in Higher Dimension

Kerz, Moritz

doi:10.4310/mrl.2011.v18.n4.a9

Cited by 11 publications

(6 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Let C(X, D) denote the idele class group with modulus due to Kerz [25] (see [16, § 3]). Recall from [25,Theorem 8.2] (or [16,Theorem 3.8]) that when X ∖ D is regular, there are canonical maps (2.10)…”

Section: Idele Class Group Of a Surfacementioning

confidence: 99%

Idele class groups with modulus

Gupta¹,

Krishna²

2021

Preprint

View full text Add to dashboard Cite

We prove Bloch's formula for the Chow group of 0-cycles with modulus on smooth projective varieties over finite fields. This shows that the K-theoretic idele class group with modulus due to Kato-Saito and the cycle-theoretic idele class group with modulus due to Kerz-Saito coincide for such varieties. The proof relies on new results in ramification theory. Contents 1. Introduction 1 2. Idele class group of a surface 5 3. Generic fibration over a curve 11 4. The finiteness theorem 18 5. The reciprocity theorem 24 6. Reciprocity theorem for π ab 1 (X, D) 32 7. A moving lemma for 0-cycles with modulus 36 8. Proof of the moving lemma 45 9. Comparison of the idele class groups 49 References 55

show abstract

Section: Idele Class Group Of a Surfacementioning

confidence: 99%

Idele class groups with modulus

Gupta¹,

Krishna²

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…Let us now assume that k is any field and X ∈ Sch k is an integral scheme of dimension d ≥ 1. We shall endow X with its canonical dimension function d X (see [28]). Assume that D ⊂ X is an effective Cartier divisor.…”

Section: Regular and Equicharacteristic Or A Henselian Discrete Valua...mentioning

confidence: 99%

Ramified class field theory and duality over finite fields

Gupta¹,

Krishna²

2021

Preprint

View full text Add to dashboard Cite

We prove a duality theorem for the p-adic étale motivic cohomology of a variety U which is the complement of a divisor on a smooth projective variety over F p . This extends the duality theorems of Milne and Jannsen-Saito-Zhao. The duality introduces a filtration on H 1 ét (U, Q Z). We identify this filtration to the classically known Matsuda filtration when the reduced part of the divisor is smooth. We prove a reciprocity theorem for the idele class groups with modulus introduced by Kerz-Zhao and Rülling-Saito. As an application, we derive the failure of Nisnevich descent for Chow groups with modulus.

show abstract

“…To avoid unnecessarily restricting to rings with infinite residue fields, the notation K M n (A) will be used to denote the improved Milnor K-theory of a ring A, as defined by Gabber and Kerz [28]. Assuming that A is local, recall that this is a certain quotient of the usual Milnor K-group, and that the two coincide if A is field or if the residue field of A has > M n elements, where M n is some constant depending only on n (e.g., M 2 = 5).…”

Section: Preliminaries On K-theorymentioning

confidence: 99%

$K$-theory and logarithmic Hodge-Witt sheaves of formal schemes in characteristic $p$

Morrow

2015

Preprint

View full text Add to dashboard Cite

We describe the mod p r pro K-groups {K n (A/I s )/p r } s of a regular local F p -algebra A modulo powers of a suitable ideal I, in terms of logarithmic Hodge-Witt groups, by proving pro analogues of the theorems of Geisser-Levine and Bloch-Kato-Gabber. This is achieved by combining the pro Hochschild-Kostant-Rosenberg theorem in topological cyclic homology with the development of the theory of de Rham-Witt complexes and logarithmic Hodge-Witt sheaves on formal schemes in characteristic p.Applications include the following: the infinitesimal part of the weak Lefschetz conjecture for Chow groups; a p-adic version of Kato-Saito's conjecture that their Zariski and Nisnevich higher dimensional class groups are isomorphic; continuity results in K-theory; and criteria, in terms of integral or torsion étale-motivic cycle classes, for algebraic cycles on formal schemes to admit infinitesimal deformations.Moreover, in the case n = 1, we compare the étale cohomology of W r Ω 1 log and the fppf cohomology of µ p r on a formal scheme, and thus present equivalent conditions for line bundles to deform in terms of their classes in either of these cohomologies.

show abstract

Ideles in Higher Dimension

Abstract: Abstract. We propose a notion of idele class groups of finitely generated fields using the concept of relative Parshin chains. This new class group allows us to give an idelic interpretation of the higher class field theory of Kato and Saito.

Cited by 11 publications

References 16 publications

Idele class groups with modulus

Idele class groups with modulus

Ramified class field theory and duality over finite fields

$K$-theory and logarithmic Hodge-Witt sheaves of formal schemes in characteristic $p$

Contact Info

Product

Resources

About