Arithmetic cohomology over finite fields and special values of ζ-functions

Geisser, Thomas

doi:10.1215/s0012-7094-06-13312-4

“…Using Prop. 4.2 and the argument of [7,Thm.7.1] we can reduce to the case where X is smooth, projective, and connected. In this case ord s=0 ζ(X, s) = ρ 0 = −1, and by Prop.4.1b),…”

Section: Special Values Of Zeta-functionsmentioning

confidence: 99%

“…The results of this section could be formulated independently of Conjecture P 0 (X) using the dual of arithmetic cohomology of [7]: which induces an isomorphism on completions. There are short exact sequences…”

Section: Class Field Theorymentioning

confidence: 99%

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Arithmetic homology and an integral version of Kato's conjecture

Geisser¹

2010

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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“…Is there any reason why the maps H p Zar (X, B/2 ν (−q/2)) → H p cdh (X, θ * B/2 ν (−q/2)) should be isomorphisms? A moment of reflection suggests that there might be a base change theorem betweenétale and cdh topology (involving Geisser'séh topology [60]) which should be closely related to Gabber's affine analogue of proper base change [55].…”

Section: Borel-mooreétale Motivic Homologymentioning

confidence: 99%

Algebraic K-Theory, Algebraic Cycles and Arithmetic Geometry

Kahn

¹

2005

Handbook of K-Theory

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IntroductionWarning: This paper is full of conjectures. If you are allergic to them it may be harmful to your health. Parts of them are proven, though.In algebraic geometry, one encounters two important kinds of objects: vector bundles and algebraic cycles. The first lead to algebraic K-theory while the second lead to motivic cohomology. They are related via the Chern character and Atiyah-Hirzebruch-like spectral sequences.Vector bundles and algebraic cycles offer very different features. On the one hand, it is often more powerful and easier to work with vector bundles: for example Quillen's localisation theorem in algebraic K-theory is considerably easier to prove than the corresponding localisation theorem of Bloch for higher Chow groups. In short, no moving lemmas are needed to handle vector bundles. In appropriate cases they can be classified by moduli spaces, which underlies the proof of finiteness theorems like Tate's theorem [189] and Faltings' proof of the Mordell conjecture, or Quillen's finite generation theorem for K-groups of curves over a finite field [66]. They also have a better functoriality than algebraic cycles, and this has been used for example by Takeshi Saito in [163, Proof of Lemma 2.4.2] to establish functoriality properties of the weight spectral sequences for smooth projective varieties over Q p .On the other hand, it is fundamental to work with motivic cohomology: as E 2 -terms of a spectral sequence converging to K-theory, the groups involved contain finer torsion information than algebraic K-groups (see Remark 2.2.2), they appear naturally as Hom groups in triangulated categories of motives and they appear naturally, rather than K-groups, in the arithmetic conjectures of Lichtenbaum on special values of zeta functions.In this survey we shall try and clarify for the reader the interaction between these two mathematical objects and give a state of the art of the (many) 2 Bruno Kahn conjectures involving them, and especially the various implications between these conjectures. We shall also explain some unconditional results.Sections 1 to 4 are included for the reader's convenience and are much more developed in other chapters of this Handbook: the reader is invited to refer to those for more details. These sections are also used for reference purposes. The heart of the chapter is in Sections 6 and 7: in the first we try and explain in much detail the conjectures of Soulé and Lichtenbaum on the order of zeroes and special values of zeta functions of schemes of finite type over Spec Z, and an approach to prove them, in characteristic p (we don't touch the much more delicate Beilinson conjectures on special values of L-functions of Q-varieties and their refinements by Bloch-Kato and Fontaine-Perrin-Riou; these have been excellently exposed in many places of the literature anyway). In the second, we indicate some cases where they can be proven, following [95].There are two sources for the formulation of (ii) in Theorem 6.7

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“…1) This approach does not handle the missing power of p. This has recently been achieved by Geisser [60]: his point of view is to define a compactly supported version of Lichtenbaum's cohomology. To get the right groups he refines Lichtenbaum's topology by adding cdh coverings to it, which unfortunately forces him to assume resolution of singularities.…”

mentioning

confidence: 99%

Algebraic K-Theory, Algebraic Cycles and Arithmetic Geometry

Kahn¹

Handbook of K-Theory

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IntroductionWarning: This paper is full of conjectures. If you are allergic to them it may be harmful to your health. Parts of them are proven, though.In algebraic geometry, one encounters two important kinds of objects: vector bundles and algebraic cycles. The first lead to algebraic K-theory while the second lead to motivic cohomology. They are related via the Chern character and Atiyah-Hirzebruch-like spectral sequences.Vector bundles and algebraic cycles offer very different features. On the one hand, it is often more powerful and easier to work with vector bundles: for example Quillen's localisation theorem in algebraic K-theory is considerably easier to prove than the corresponding localisation theorem of Bloch for higher Chow groups. In short, no moving lemmas are needed to handle vector bundles. In appropriate cases they can be classified by moduli spaces, which underlies the proof of finiteness theorems like Tate's theorem [189] and Faltings' proof of the Mordell conjecture, or Quillen's finite generation theorem for K-groups of curves over a finite field [66]. They also have a better functoriality than algebraic cycles, and this has been used for example by Takeshi Saito in [163, Proof of Lemma 2.4.2] to establish functoriality properties of the weight spectral sequences for smooth projective varieties over Q p .On the other hand, it is fundamental to work with motivic cohomology: as E 2 -terms of a spectral sequence converging to K-theory, the groups involved contain finer torsion information than algebraic K-groups (see Remark 2.2.2), they appear naturally as Hom groups in triangulated categories of motives and they appear naturally, rather than K-groups, in the arithmetic conjectures of Lichtenbaum on special values of zeta functions.In this survey we shall try and clarify for the reader the interaction between these two mathematical objects and give a state of the art of the (many) 2 Bruno Kahn conjectures involving them, and especially the various implications between these conjectures. We shall also explain some unconditional results.Sections 1 to 4 are included for the reader's convenience and are much more developed in other chapters of this Handbook: the reader is invited to refer to those for more details. These sections are also used for reference purposes. The heart of the chapter is in Sections 6 and 7: in the first we try and explain in much detail the conjectures of Soulé and Lichtenbaum on the order of zeroes and special values of zeta functions of schemes of finite type over Spec Z, and an approach to prove them, in characteristic p (we don't touch the much more delicate Beilinson conjectures on special values of L-functions of Q-varieties and their refinements by Bloch-Kato and Fontaine-Perrin-Riou; these have been excellently exposed in many places of the literature anyway). In the second, we indicate some cases where they can be proven, following [95].There are two sources for the formulation of (ii) in Theorem 6.7

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Arithmetic cohomology over finite fields and special values of ζ-functions

Cited by 40 publications

References 23 publications

Arithmetic homology and an integral version of Kato's conjecture

Arithmetic homology and an integral version of Kato's conjecture

Algebraic K-Theory, Algebraic Cycles and Arithmetic Geometry

Algebraic K-Theory, Algebraic Cycles and Arithmetic Geometry

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