On finiteness properties of local cohomology modules over Cohen-Macaulay local rings
type: Item
instance of: scholarly article
title: On finiteness properties of local cohomology modules over Cohen-Macaulay local rings
zbMATH Open document ID: 1174.13025
published in: Illinois Journal of Mathematics
publication date: 2009-10-15T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.ijm/1254403711
review text: Let \(\phi:(R,m)\to(A,n)\) be a local morphism of local rings which is module-finite and flat. Let \(I\) be an ideal of \(A\) such that, if \(J=I\cap R,\) then \(I=JA.\) Let \(i\) be an integer. The following facts are proved: \noindent a) if the set \(\mathrm{Ass}(H_J^i(R))\) is finite, then the set \(\mathrm{Ass}(H_I^i(A))\) is also finite; \noindent b) if all the Bass numbers of \(H^i_J(R)\) are finite, then all the Bass numbers of \(H^i_I(A)\) are also finite. Assume moreover that \(R\) and \(A\) are regular local rings and that \(R\) is unramified. Then: a) \(\mathrm{Ass}(H^i_I(A))\) is a finite set; b) all the Bass numbers of \(H^i_I(A)\) are finite. As a corollary, it is proved that if \(A\) is a Cohen-Macaulay local ring containing a field \(k\) such that the residue field of \(A\) is separable over \(k\) and \(I\) is an ideal of \(A\) generated by polynomials with coefficients in \(k\) in a system of parameters of \(A\), all the Bass numbers of \(H^i_I(A)\) are finite.
reviewed by: Cristodor-Paul Ionescu
Mathematics Subject Classification ID: 13D45
Mathematics Subject Classification ID: 13H10
zbMATH DE Number: 5615665
zbMATH Keywords: local cohomology
zbMATH Keywords: Bass numbers
zbMATH Keywords: Cohen-Macaulay ring
author: Ken-Ichiroh Kawasaki
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5615665