Local cohomology modules of a smooth \(\mathbb{Z}\)-algebra have finitely many associated primes
type: Item
instance of: scholarly article
title: Local cohomology modules of a smooth \(\mathbb{Z}\)-algebra have finitely many associated primes
zbMATH Open document ID: 1318.13025
author: Bhargav Bhatt
author: Manuel Blickle
author: Gennady Lyubeznik
author: Anurag K. Singh
author: Wenliang Zhang
published in: Inventiones Mathematicae
publication date: 2014-10-17T00:00:00Z
full work available at URL: https://arxiv.org/abs/1304.4692
review text: Let \(R\) be a Noetherian ring and \(\mathfrak a\) an ideal. The local cohomology module \(H_{\mathfrak a}^k(R)\) has infinitely many associated primes in general. However, it is known that \[ \# \text{Ass}_R H_{\mathfrak a}^k(R) < \infty \tag{1} \] if \(R\) is a regular local ring of equicharacteristic of if \(R\) is an unramified regular local ring of mixed characteristic. In the present paper, authors want to generalize this result. Assume \(R\) is a smooth algebra over a ring \(A\). If \(A\) is a field, then \(R\) is regular and hence (1) is fulfilled. The main theorem of this paper says that (1) is fulfilled if \(A = \mathbb{Z}\) or if \(A\) is a Dedekind domain such that \(A_{\mathfrak p}\) has mixed characteristic for any maximal ideal \(\mathfrak p\) of \(A\).
reviewed by: Takesi Kawasaki
Mathematics Subject Classification ID: 13D45
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 14B05
Mathematics Subject Classification ID: 13N10
Mathematics Subject Classification ID: 13A35
zbMATH DE Number: 6358025
zbMATH Keywords: associated prime
zbMATH Keywords: local cohomology
zbMATH Keywords: regular local ring
zbMATH Keywords: smooth algebra
MaRDI profile type: Publication
OpenAlex ID: W2951924956
arXiv ID: 1304.4692
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DOI: 10.1007/S00222-013-0490-Z
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description: scientific article; zbMATH DE number 6358025