On the arithmetic of local rings of dimension 2 and 3
type: Item
instance of: scholarly article
title: On the arithmetic of local rings of dimension 2 and 3
zbMATH Open document ID: 0937.13006
author: Belgacem Draouil
author: Jean-Claude Douai
published in: Journal of Algebra
publication date: 1999-10-21T00:00:00Z
review text: Let \(A\) be a 3-dimensional excellent henselian normal local ring. Assume that the residue field \(K\) of \(A\) is algebraically closed or finite. Suppose also that the unique singular point of \(A\) is the closed point \(x\). Let \(f: X \rightarrow \text{Spec } A \) be a resolution of \(\text{Spec } A\) where \(X\) is a 3-dimensional regular scheme. Denote by \(Y\) the surface \(f^{-1} (x)\) and by \(P\) the set of height 1 prime ideals of \(A\). Denote \(Q_{l} (i) = \bigotimes_{j=1}^{i} Q_{j}\), \(Q_{j}\) being the Tate module \(Q_{l}\) for each \(j\). The paper is basically devoted to prove two results. Firstly, when \(K\) is algebraically closed and of characteristic 0, the mapping \(H^{3} (K, Q_{l} (2)) \rightarrow \bigoplus_{v \in P} H^{3} (K_{v}, Q_{l} (2))\) contains a subgroup isomorphic to \(Q_{l}^{r_{1} (A)}\), where \(l\) is a prime number, \(K_{v}\) the completion of \(K\) in \(v\) and \(r_{1} (A)\) the dimension of the set of elements in \(H^{2} (Y, Q_{l})\) with weight \(\leq 1\). Secondly, if \(K\) is a finite field of characteristic \(p \neq 2, 3, 5\), the authors consider the mapping \(H^{4} (K, Q_{l} (3)) \rightarrow \bigoplus_{v \in P} H^{4} (K_{v}, Q_{l} (3))\) and prove that its kernel contains a subgroup of type \((Q_{l})^{r_{1}'} (A) \) where \(r_{1}' (A) = \text{rank}_{Q_{l}} H^{2} ( |\Gamma |, Q_{l})\), \(|\Gamma |\) being the geometric realization of the exceptional fiber \(Y\). Moreover, the data \(r_{1}' (A) \) and \(r_{1} (A)\) are independent of the resolution \(f\) of \(X\). Similar results when \(\dim A = 2\) are also given generalizing a previous result by \textit{S. Saito} [in: Galois representations and arithmetic algebraic geometry, Proc. Symp., Kyoto 1985 and Tokyo 1986, Adv. Stud. Pure Math. 12, 343-373 (1987; Zbl 0672.12006)].
Mathematics Subject Classification ID: 13J15
Mathematics Subject Classification ID: 12G05
Mathematics Subject Classification ID: 14M05
Mathematics Subject Classification ID: 13B40
zbMATH DE Number: 1270862
zbMATH Keywords: excellent henselian normal local ring
zbMATH Keywords: Tate modules
zbMATH Keywords: exceptional fibre
reviewed by: Carlos Galindo
MaRDI profile type: Publication
full work available at URL: jabr.1998.7672
OpenAlex ID: W2001933522
cites work: Q5525147
cites work: La conjecture de Weil. II
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cites work: Resolution of singularities of an algebraic variety over a field of characteristic zero. I
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cites work: Über die lokale Zetafunktion von Shimuravarietaeten. Monodromiefiltration und verschwindende Zyklen in ungleicher Charakteristik
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DOI: 10.1006/JABR.1998.7672
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description: scientific article; zbMATH DE number 1270862