Local duality for 2-dimensional local ring
type: Item
instance of: scholarly article
title: Local duality for 2-dimensional local ring
zbMATH Open document ID: 1184.13071
author: Belgacem Draouil
published in: Proceedings of the Indian Academy of Sciences. Mathematical Sciences
publication date: 2009-10-15T00:00:00Z
review text: The author generalizes \textit{S. Saito}'s result [Adv. Stud. Pure Math. 12, 343--373 (1987; Zbl 0672.12006), Theorem 5.3]. Let \(A\) be a \(2\)-dimensional normal, complete local domain with quotient field \(K\). Assume that the characteristic of \(A\) is positive and that the residue field is an \(n\)-dimensional local field. He gave the following exact sequence: \[ \begin{multlined} 0 \to N/lN \to H^{3+n}(K, (\mathbb{Z}/l \mathbb{Z})^{\otimes (2+n)}) \to\\ \bigoplus_{\text{\(\mathfrak p\) is a prime ideal of height \(1\)}} H^{2+n}(\kappa(\mathfrak p), (\mathbb{Z}/l \mathbb{Z})^{\otimes (1+n)}) \to \mathbb{Z}/l \mathbb{Z} \to 0\end{multlined} \] for any integer \(l\) prime to the characteristic of \(A\). Here \(N\) is a group independent of the choice of \(l\). Saito's theorem is the case of \(n = 0\). To prove this theorem, the author also generalized Grothendieck's local duality theorem [\textit{A. Grothendieck}, Seminaire de géométrie algébrique du Bois-Marie 1965-66 SGA 5 dirige par A. Grothendieck avec la collaboration de I. Bucur, C. Houzel, L. Illusie, J.-P. Jouanolou et J. -P. Serre. Cohomologie \(\ell\)-adique et fonctions L. Lecture Notes in Mathematics. 589. Berlin-Heidelberg-New York: Springer-Verlag (1977; Zbl 0345.00011), p. 49].
reviewed by: Takesi Kawasaki
Mathematics Subject Classification ID: 13H99
Mathematics Subject Classification ID: 11S31
zbMATH DE Number: 5615369
zbMATH Keywords: Hasse principle
zbMATH Keywords: purity
zbMATH Keywords: local duality
zbMATH Keywords: curves over higher local fields
MaRDI profile type: Publication
full work available at URL: s12044-008-0040-6
OpenAlex ID: W2026804594
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DOI: 10.1007/S12044-008-0040-6
description: scientific article; zbMATH DE number 5615369