Comparison of dualizing complexes

Zhong, Changlong

doi:10.1515/crelle-2012-0088

Cited by 12 publications

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References 21 publications

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“…where K /X is the complex constructed by Spiess in [33] and the middle isomorphism is [38,Theorem 3.8]. Pairing (37), combined with map (38) and taken modulo p ν , can also be characterised as the unique pairing constructed by the method of Sato in [31]. Even though Sato assumes X to have semistable reduction, his arguments work in our situation where X is a relative curve and n = m = 1.…”

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confidence: 99%

Special Values of the Zeta Function of an Arithmetic Surface

Flach¹,

Siebel²

2021

J. Inst. Math. Jussieu

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We prove that the special-value conjecture for the zeta function of a proper, regular, flat arithmetic surface formulated in [6] at $s=1$ is equivalent to the Birch and Swinnerton-Dyer conjecture for the Jacobian of the generic fibre. There are two key results in the proof. The first is the triviality of the correction factor of [6, Conjecture 5.12], which we show for arbitrary regular proper arithmetic schemes. In the proof we need to develop some results for the eh-topology on schemes over finite fields which might be of independent interest. The second result is a different proof of a formula due to Geisser, relating the cardinalities of the Brauer and the Tate–Shafarevich group, which applies to arbitrary rather than only totally imaginary base fields.

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confidence: 99%

Special Values of the Zeta Function of an Arithmetic Surface

Flach¹,

Siebel²

2021

J. Inst. Math. Jussieu

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“…Taking mapping cones of multiplication by p • we obtain the diagram (143)ǫ * i * Z(n − 1) Zar /p • [−2] − −−− → ǫ * Z(n) Zar /p • − −−− → ǫ * j * Z(n) Zar /p • − −−− →     i * Ri ! Z(n)/p • − −−− → Z(n)/p • − −−− → Rj * Z(n)/p • − −−− → .By the Rost-Voevodsky theorem (previously Beilinson-Lichtenbaum conjecture, see e.g [91][91][Lemma 2.4] we obtain a quasi-isomorphismτ ≤n Rj * Z(n) Zar /p • ∼ = τ ≤n Rj * τ ≤n Rǫ * Z(n)/p • ∼ = τ ≤n Rj * Rǫ * Z(n)/p • = τ ≤n Rǫ * Rj * Z(n)/p •and hence an isomorphismτ ≤n ǫ * Rj * Z(n) Zar /p • ∼ = τ ≤n ǫ * Rǫ * Rj * Z(n)/p • ∼ = τ ≤n Rj * Z(n)/p • ,i.e.…”

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confidence: 87%

“…Z(n)/p • − −−− → Z(n)/p • − −−− → Rj * Z(n)/p • − −−− → .By the Rost-Voevodsky theorem (previously Beilinson-Lichtenbaum conjecture, see e.g [91][91][Lemma 2.4] we obtain a quasi-isomorphismτ ≤n Rj * Z(n) Zar /p • ∼ = τ ≤n Rj * τ ≤n Rǫ * Z(n)/p • ∼ = τ ≤n Rj * Rǫ * Z(n)/p • = τ ≤n Rǫ * Rj * Z(n)/p •and hence an isomorphismτ ≤n ǫ * Rj * Z(n) Zar /p • ∼ = τ ≤n ǫ * Rǫ * Rj * Z(n)/p • ∼ = τ ≤n Rj * Z(n)/p • ,i.e. the right vertical map in (143) is an isomorphism in degrees ≤ n. From the Five Lemma and τ ≤n Z(n)/p • ∼ = Z(n)/p • it follows that the truncation of the left vertical map in (143)τ ≤n+1 ǫ * i * Z(n − 1) Zar /p • [−2] → τ ≤n+1 i * Ri !…”

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confidence: 87%

“…but it only induces an isomorphism in degrees i ≤ n. Recall that if Z is a separated, finite type scheme over a perfect field κ of characteristic p all of whose irreducible components are of dimension d−1 and r ≥ 0 there is a quasi-isomorphism on Z et [91] Note that these complexes are identical for Z and Z red . If X s is a normal crossing divisor, i.e.…”

Section: Localization Trianglesmentioning

confidence: 99%

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Weil-étale cohomology and Zeta-values of proper regular arithmetic schemes

Flach¹,

Morin²

2016

Preprint

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We give a conjectural description of the vanishing order and leading Taylor coefficient of the Zeta function of a proper, regular arithmetic scheme X at any integer n in terms of Weil-étale cohomology complexes. This extends work of Lichtenbaum [65] and Geisser [36] for X of characteristic p, of Lichtenbaum [66] for X = Spec(OF ) and n = 0 where F is a number field, and of the second author for arbitrary X and n = 0 [72]. We show that our conjecture is compatible with the Tamagawa number conjecture of Bloch, Kato, Fontaine and Perrin-Riou [31] if X is smooth over Spec(OF ), and hence that it holds in cases where the Tamagawa number conjecture is known. Contents 1. Introduction 2. Motivic cohomology of proper regular schemes and the Beilinson conjectures 2.1. The Beilinson regulator on the level of complexes 2.2. The Beilinson conjectures and arithmetic duality with R-coefficients 2.3. Motivic cohomology of the Artin-Verdier compactification 3. Weil-étale cohomology of proper regular schemes 3.1. Notations 3.2. Assumptions 3.3. The complex RΓ W (X , Z(n)) 3.4. Rational coefficients 3.5. Torsion coefficients 3.6. Relationship with the Lichtenbaum-Geisser definition over finite fields 3.7. Weil-étale duality 3.8. The complex RΓ W,c (X , Z(n)) 4. Weil-Arakelov cohomology of proper regular schemes 4.1. Weil-Arakelov cohomology with R(n)-coefficients 4.2. Weil-Arakelov cohomology of X 4.3. Weil-Arakelov duality for X 4.4. Weil-Arakelov cohomology with compact support 5. Special values of zeta functions 5.1. De Rham cohomology 5.2. The fundamental line 5.3. The complex RΓ ′ eh (X Fp , Z p (n)) and Milne's correcting factor 5.4. The local factor c p (X , n) 5.5. The main conjecture 5.6. Compatibility with the Tamagawa Number Conjecture 2010 Mathematics Subject Classification. 14F20 (primary) 14F42 11G40 . 1 2 M. FLACH AND B. MORIN5.7. Relationship with the functional equation 5.8. Proven cases and examples 6. Appendix A: Artin-Verdier duality 6.1. Introduction 6.2. The motivic complex Z(n) X 6.3. The Artin-Verdier étale topos X et 6.4. Tate cohomology and the functor R π * 6.5. The motivic complex Z(n) X 6.6. Functoriality 6.7. Relationship with Milne's cohomology with compact support 6.8. Products 6.9. Artin-Verdier Duality 6.10. The conjecture AV(f, n) 6.11. The projective bundle formula 7. Appendix B: Motivic and syntomic cohomology 7.1. l-adic cohomology 7.2. p-adic cohomology ReferencesIn particular, the complex RΓ ar,c (X , R(n)) has vanishing Euler characteristic:b) The groups H i ar,c (X , R/Z(n)) are compact, commutative Lie groups for all i. Note here that the cohomology groups of a complex of locally compact abelian groups carry an induced topology which however need not be locally compact. Indeed, the groups H i ar,c (X , Z(n)) will not always be locally compact.The conjectural relation to the Zeta-function of X is given by the following two statements.c) The function ζ(X , s) has a meromorphic continuation to s = n and Motivic cohomology of proper regular schemes and the Beilinson conjecturesThroughout this...

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“…Section 2). In this article we define a complex K n,X,log via Grothendieck's duality theory of coherent sheaves following the idea in [Kat87] and build up a quasi-isomorphism from the Kato-Moser complex of logarithmic de Rham-Witt sheaves ν n,X (namely the Gersten complex of logarithmic de Rham-Witt sheaves, which is introduced and studied in [Kat86a, §1][Mos99, (1.3)-(1.5)]) to K n,X,log for the étale topology and also for the Zariski topology under the extra assumption k = k. Combined with Zhong's quasi-isomorphism from Bloch's cycle complex Z c X to ν n,X [Zho14,2.16], we deduce certain vanishing and finiteness properties as well as invariance under rational resolutions for higher Chow groups of 0-cycles with Z/p n -coefficients. The proofs in this article are self-contained in respect to Kato's work [Kat87].…”

Section: Introductionmentioning

confidence: 99%

Bloch's cycle complex and coherent dualizing complexes in positive characteristic

Ren¹

2021

Preprint

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Let X be a separated scheme of finite type over k with k being a perfect field of positive characteristic p. In this work we define a complex K n,X,log via Grothendieck's duality theory of coherent sheaves following [Kat87] and build up a quasi-isomorphism from the Kato-Moser complex of logarithmic de Rham-Witt sheaves ν n,X to K n,X,log for the étale topology, and also for the Zariski topology under the extra assumption k = k. Combined with Zhong's quasi-isomorphism from Bloch's cycle complex Z c X to ν n,X [Zho14, 2.16], we deduce certain vanishing, étale descent properties as well as invariance under rational resolutions for higher Chow groups of 0-cycles with Z/p n -coefficients. FEI REN 8.2. Étale descent 46 8.3. Birational geometry and rational singularities 47 8.4. Galois descent 48 Appendix 49 A. Semilinear algebra 49 References 51 ≃ − → K n,X,log in the singular setting, such that when pre-composed with Zhong's quasi-isomorphism ψ : Z c X → ν n,X [Zho14, 2.16], it gives

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Comparison of dualizing complexes

Cited by 12 publications

References 21 publications

Special Values of the Zeta Function of an Arithmetic Surface

Special Values of the Zeta Function of an Arithmetic Surface

Weil-étale cohomology and Zeta-values of proper regular arithmetic schemes

Bloch's cycle complex and coherent dualizing complexes in positive characteristic

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