Weiletale cohomology

General Introduction
Introduced by Lichtenbaum
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Search results
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Online References
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Paper References
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Definition
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Properties
Morin's thesis: axioms for a Weiletale topos.
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Standard theorems
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Open Problems
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Connections to Number Theory
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Computations and Examples
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History and Applications
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Some Research Articles
Key authors: Flach, Morin, Geisser, Lichtenbaum.
arXiv:1103.6061 Zeta functions of regular arithmetic schemes at s=0 from arXiv Front: math.NT by Baptiste Morin Lichtenbaum conjectured in \cite{Lichtenbaum} the existence of a Weilétale cohomology in order to describe the vanishing order and the special value of the Zeta function of an arithmetic scheme $\mathcal{X}$ at $s=0$ in terms of EulerPoincaré characteristics. Assuming the (conjectured) finite generation of some motivic cohomology groups we construct such a cohomology theory for regular schemes proper over $\mathrm{Spec}(\mathbb{Z})$. In particular, we compute (unconditionally) the right Weilétale cohomology of number rings and projective spaces over number rings. We state a precise version of Lichtenbaum's conjecture, which expresses the vanishing order (resp. the special value) of the Zeta function $\zeta(\mathcal{X},s)$ at $s=0$ as the rank (resp. the determinant) of a single perfect complex of abelian groups $R\Gamma_{W,c}(\mathcal{X},\mathbb{Z})$. Then we relate this conjecture to Soulé's conjecture and to the Tamagawa Number Conjecture. Lichtenbaum's conjecture for projective spaces over the ring of integers of an abelian number field follows.
Geisser: Weiletale cohomology over finite fields
MR2117419 (2005i:11079) Burns, David(4LNDKC) On the values of equivariant zeta functions of curves over finite fields.
arXiv:1103.6061 Zeta functions of regular arithmetic schemes at s=0 from arXiv Front: math.AG by Baptiste Morin Lichtenbaum conjectured in \cite{Lichtenbaum} the existence of a Weilétale cohomology in order to describe the vanishing order and the special value of the Zeta function of an arithmetic scheme $\mathcal{X}$ at $s=0$ in terms of EulerPoincaré characteristics. Assuming the (conjectured) finite generation of some motivic cohomology groups we construct such a cohomology theory for regular schemes proper over $\mathrm{Spec}(\mathbb{Z})$. In particular, we compute (unconditionally) the right Weilétale cohomology of number rings and projective spaces over number rings. We state a precise version of Lichtenbaum's conjecture, which expresses the vanishing order (resp. the special value) of the Zeta function $\zeta(\mathcal{X},s)$ at $s=0$ as the rank (resp. the determinant) of a single perfect complex of abelian groups $R\Gamma_{W,c}(\mathcal{X},\mathbb{Z})$. Then we relate this conjecture to Soulé's conjecture and to the Tamagawa Number Conjecture. Lichtenbaum's conjecture for projective spaces over the ring of integers of an abelian number field follows.
arXiv:1209.4322 The full faithfulness conjectures in characteristic p from arXiv Front: math.AG by Bruno Kahn We present a triangulated version of the conjectures of Tate and Beilinson on algebraic cycles over a finite field. This sheds a new light on Lichtenbaum's Weiletale cohomology.
Title: On the Weilétale cohomology of the ring of $S$integers Authors: YiChih Chiu Categories: math.NT Number Theory Abstract: In this article, we first briefly introduce the history of the Weilétale cohomology theory of arithmetic schemes and review some important results established by Lichtenbaum, Flach and Morin. Next we generalize the Weiletale cohomology to $S$integers and compute the cohomology for constant sheaves $\mathbb{Z}$ or $\mathbb{R}$. We also define a Weilétale cohomology with compact support $Hc(YW, )$ for $Y=Spec \mathcal{O}{F,S}$ where $F$ is a number field, and computed them. We verify that these cohomology groups satisfy the axioms state by Lichtenbaum. As an application, we derive a canonical representation of Tate sequence from $RGammac(YW,\mathbb{Z})$. Motivated by this result, in the final part, we define an étale complex $RGm$, such that the complexes $\mathbb{Z}$dual of the complex $\RG(U{et},R\Gm),\,\mathbb{Z})[2]$ is canonically quasiisomorphic to $\tau^{\leq 3}\RGc(UW,\mathbb{Z})$ for arbitrary étale $U$ over $Spec \mathcal{O}_{F}$. This quasiisomorphism provides a possible approach to define the Weiletale cohomology for higher dimensional arithmetic schemes, as the Weil groups are not involved in the definition of $R\Gm$.
Title: Weilétale Cohomology over $p$adic Fields Authors: David A. Karpuk Categories: math.NT Number Theory (math.AG Algebraic Geometry) Abstract: We establish duality results for the cohomology of the Weil group of a $p$adic field, analogous to, but more general than, results from Galois cohomology. We prove a duality theorem for discrete Weil modules, which implies TateNakayama Duality. We define Weilsmooth cohomology for varieties over local fields, and prove a duality theorem for the cohomology of $\G_m$ on a smooth, proper curve with a rational point. This last theorem is analogous to, and implies, a classical duality theorem for such curves.
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Other Information
See intro to FlachMorin for a general introduction. In intro to Morin's thesis there are some more conjectural properties, notably the statement that integral coeff groups should vanish for i>2d+1, and a different formulation of the Taylor coefficient formula, in which the role of the torsion parts is more clear.
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