Duality for \(\mathbb Z\)-constructible sheaves on curves over finite fields
type: Item
instance of: scholarly article
title: Duality for \(\mathbb Z\)-constructible sheaves on curves over finite fields
zbMATH Open document ID: 1267.14028
author: Thomas Geisser
published in: Documenta Mathematica
publication date: 2013-04-17T00:00:00Z
full work available at URL: 29.html
review text: Let \(X\) be a curve over a finite field and let \(\mathcal F\) be a \(\mathbb Z\)-constructible sheaf. The author defines Weil-etale Borel-Moore homology groups \(H_i^c(X_{\mathrm{ar}},\mathcal F)\) and constructs a pairing between \(H_i^c(X_{\mathrm{ar}},\mathcal F)\) and the Weil-etale cohomology with compact support \(H_c^i(X_W,\mathcal F)\). For \(0\leq i\leq 2\), we have pairings of free groups of finite rank \[ H_i^c(X_{\mathrm{ar}},\mathcal F)/\mathrm{tor}\times H_c^i(X_W,\mathcal F)/\mathrm{tor}\to \mathbb Z, \] and for \(0\leq i\leq 3\), we have pairings of finite groups \[ H^c_{i-1}(X_{\mathrm{ar}},\mathcal F)_{\mathrm{tor}}\times H_c^{i}(X_W,\mathcal F)_{\mathrm{tor}}\to\mathbb Q/\mathbb Z. \] All other cohomology and homology groups vanish. When \(X\) is smooth, the groups \(H_i^c(X_{\mathrm{ar}},\mathbb Z)\) are isomorphic to Weil-etale cohomology group \(H^i(X_W,\mathbb G_m)\). The pairing between \(H_c^i(X_W,\mathbb Z)\) and \(H^i(X_W,\mathbb G_m)\) is constructed by Lichtenbaum.
reviewed by: Lei Fu
Mathematics Subject Classification ID: 14F20
Mathematics Subject Classification ID: 14F42
Mathematics Subject Classification ID: 11G20
zbMATH DE Number: 6155123
zbMATH Keywords: finite fields
zbMATH Keywords: curves
zbMATH Keywords: duality
zbMATH Keywords: \(\mathbb Z\)-constructible sheaves
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 6155123