Duality in the étale cohomology of one-dimensional proper schemes and generalizations
type: Item
instance of: scholarly article
title: Duality in the étale cohomology of one-dimensional proper schemes and generalizations
zbMATH Open document ID: 0607.14011
DOI: 10.1007/BF01458330
published in: Mathematische Annalen
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164264
review text: On an equidimensional scheme X we consider the complex of étale sheaves in degrees 0 and 1 given by: \({\mathcal G}:\oplus _{y\in X^{(0)}}j_{y*}{\mathbb{G}}_{m/y}\to ^{\partial}\oplus _{x\in X^{(1)}}i_{x*}{\mathbb{Z}}\) and we set \({\mathcal G}(n)=R \underline{Hom}_ X({\mathbb{Z}}/n,{\mathcal G})\in D^ +(X,{\mathbb{Z}}/n)\). For regular X we have quasi-isomorphisms \({\mathbb{G}}_ m\to ^{\sim}{\mathcal G}\) and \(\mu _ n\to ^{\sim}{\mathcal G}(n)\). Results: (a) Let X be a one-dimensional proper \({\mathbb{Z}}\)-scheme. Then for any constructible sheaf F on X the Yoneda-pairing \(H^ i(X,F)\times {\mathbb{E}}xt_ X^{3-i}(F,{\mathcal G})\to {\mathbb{H}}^ 3(X,{\mathcal G})\to ^{tr}{\mathbb{Q}}/{\mathbb{Z}}\) is a perfect pairing of finite groups, \(i\in {\mathbb{Z}}\). If X is irreducible the trace map is an isomorphism. (The 2- primary components may have to be excluded.) This extends the case where X is regular, i.e. Artin-Verdier duality for number and function fields. [cf. \textit{B. Mazur}, Ann. Sci. Ec. Norm. Supér., IV. Sér. 6(1973), 521-552 (1974; Zbl 0282.14004) and the author, Math. Z. 188, 91-100 (1984; Zbl 0585.14014)]. (b) There is also a local version of (a) where X is the spectrum of a henselian one-dimensional noetherian local ring with finite residue field. (c) Assume Y is separated, smooth of relative dimension \(d\) over an open subscheme of X as in (a). Then for constructible F on Y with \(nF=0\), n invertible on Y there is a perfect Yoneda duality on Y in the category of \({\mathbb{Z}}/n\)-sheaves with dualizing complex \({\mathcal G}(n)\otimes \mu _ n^{\otimes d}\). (d) If \(\mu_ n\) is replaced by \({\mathcal G}(n)\) Poincaré duality remains valid for singular proper curves over algebraically closed fields. Remarks: The fibres of a regular arithmetical surface can be singular. Therefore (a) should be useful for establishing arithmetical duality on such surfaces. Moreover (a), (b) suggest that higher dimensional arithmetical duality should not be …
Mathematics Subject Classification ID: 14F20
zbMATH DE Number: 3981330
zbMATH Keywords: complex of étale sheaves
zbMATH Keywords: Yoneda-pairing
zbMATH Keywords: Artin-Verdier duality
zbMATH Keywords: Poincaré duality
zbMATH Keywords: arithmetical duality
Wikidata QID: Q56813079
author: Christopher Deninger
MaRDI profile type: Publication
cites work: On Artin-Verdier duality for function fields
cites work: An extension of Aritn-Verdier duality to non-torsion sheaves.
cites work: Unramified class field theory of arithmetical surfaces
cites work: Notes on étale cohomology of number fields
cites work: Q3871006
cites work: Q3754095
cites work: Anneaux locaux henséliens
OpenAlex ID: W2057800505
description: scientific article; zbMATH DE number 3981330