Cohomology of tori over \(p\)-adic curves
type: Item
instance of: scholarly article
title: Cohomology of tori over \(p\)-adic curves
zbMATH Open document ID: 1050.14016
author: Claus Scheiderer
author: J. van Hamel
published in: Mathematische Annalen
publication date: 2003-07-02T00:00:00Z
review text: Let \(V\) be a regular curve over a \(p\)-adic field, \(T\) a torus over \(V\), and \(T'\) the corresponding dual torus. The authors construct a suitable étale cohomology group \(H^q_{cc}(V,T)\) together with a pairing \[ H^q_{cc}(V,T)\times H^{3-q}(V,T')\longrightarrow \mathbb Q/\mathbb Z, \] and prove various duality properties for this pairing. This pairing in closely related to three other pairings: First, the Lichtenbaum-Tate pairing \[ H^q(C,\mathbb G_m)\times H^{3-q}(C,\mathbb G_m)\longrightarrow\mathbb Q/\mathbb Z \] for regular proper curves \(C\) over \(p\)-adic fields [\textit{S. Lichtenbaum}, Invent. Math. 7, 120--136 (1969; Zbl 0186.26402)]. Second, the Poincaré duality pairing \[ H^q_c(V,{}_n T)\times H^{4-q}(V,{}_n T')\longrightarrow \mathbb Q/\mathbb Z \] on torsion points. Third, the Tate-Nakayama pairing \[ H^q(k,T)\times H^{2-q}(k,X^*(T))\longrightarrow \mathbb Q/\mathbb Z \] for tori \(T\) and their character group \(X^*(T)\) over \(p\)-adic fields \(k\) [\textit{J.-P. Serre}, Cohomologie Galoisienne, 5th ed. (1994; Zbl 0812.12002)]. The main problem to define the desired pairing is that the curve \(V\) is not necessarily compact and that the torus \(T\) does not necessarily extend over the compactification \(C\). The authors solve this in an elegant way by introducing the cohomology group \(H^q_{cc}(V,T)\), which is defined as the cohomology group of a sheaf \(j_{!!}T\) on the compactification \(C\). In the special case \(T=\mathbb G_{m,V}\), this sheaf is defined by the exact sequence \[ 0\to j_{!!}\mathbb G_{m,V}\to\mathbb G_{m,C}\to i_*\mathbb G_{m,Z}\to 0, \] where \(j:V\rightarrow C\) is the open embedding, and \(i:Z\rightarrow C\) the closed embedding of the complement \(Z=C-V\).
reviewed by: Stefan Schröer
Mathematics Subject Classification ID: 14G20
Mathematics Subject Classification ID: 14F22
Mathematics Subject Classification ID: 14L15
Mathematics Subject Classification ID: 11S25
zbMATH DE Number: 1945199
zbMATH Keywords: tori
zbMATH Keywords: p-adic fields
zbMATH Keywords: duality
zbMATH Keywords: Lichtenbaum-Tate pairing
zbMATH Keywords: Poincaré duality pairing
zbMATH Keywords: Tate-Nakayama pairing
MaRDI profile type: Publication
full work available at URL: s00208-003-0416-y
OpenAlex ID: W1994785852
DOI: 10.1007/S00208-003-0416-Y
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description: scientific article; zbMATH DE number 1945199