Applications of closure operations on big Cohen-Macaulay algebras
type: Item
instance of: scholarly article
zbMATH Open document ID: 1255.13008
DOI: 10.1142/S0219498812500752
author: Mohsen Asgharzadeh
author: Rajsekhar Bhattacharyya
publication date: 2012-09-06T00:00:00Z
full work available at URL: https://arxiv.org/abs/1009.1454
Mathematics Subject Classification ID: 13C14
Mathematics Subject Classification ID: 13A35
zbMATH DE Number: 6078484
zbMATH Keywords: Cohen-Macaulyness
zbMATH Keywords: perfect closure
zbMATH Keywords: closure operations
zbMATH Keywords: non-Noetherian rings
zbMATH Keywords: balanced big Cohen-Macaulay
MaRDI profile type: Publication
arXiv ID: 1009.1454
cites work: Homological properties of the perfect and absolute integral closures of noetherian domains
cites work: On the notion of Cohen-Macaulayness for non-Noetherian rings
cites work: Big Cohen-Macaulay algebras and seeds
cites work: A characterization of closure operations that induce big Cohen-Macaulay modules
cites work: Big Cohen-Macaulay algebras in dimension three via Heitmann's theorem.
cites work: Tight closure and elements of small order in integral extensions
cites work: F-coherent rings with applications to tight closure theory
OpenAlex ID: W2962968194
published in: Journal of Algebra and its Applications
title: Applications of closure operations on big Cohen-Macaulay algebras
review text: The paper under review explores applications of closure operations, such as perfect closure, in establishing the existence of big Cohen-Macaulay modules. For instance, in Lemma~3.1 it is shown that if \(R\) is a noetherian local \(F\)-coherent domain, which is either excellent or homomorphic image of a Gorenstein local ring, then its perfect closure \(R^\infty\) is balanced big Cohen-Macaulay. (cf. Theorem~3.11 in [\textit{K. Shimomoto}, J. Algebra, 338, No. 1, 24--34 (2011; Zbl 1241.13006)].) We recall the definitions of the less familiar terms used in this lemma. The \textsl{perfect closure} \(A^\infty\) of a (not necessarily noetherian) reduced ring \(A\) of characteristic \(p>0\) is obtained by adjoining all the higher \(p\)-power roots of all elements of \(A\) to \(A\). A (not necessarily noetherian) ring is called \textsl{coherent} if each of its finitely generated ideals is finitely presented. A noetherian ring \(R\) of characteristic \(p>0\) is called \(F\)-\textsl{coherent}, if its perfect closure \(R^\infty\) is coherent. (cf. Definition~3.2 in [Zbl 1241.13006].)NEWLINENEWLINEOne of the main results proved in this paper, Theorem~3.3, states that if \(R\) is a noetherian local \(F\)-coherent domain, which is either excellent or homomorphic image of a Gorenstein local ring, then the height of every ideal \(\mathfrak{a}\) of \(R^\infty\) is equal to its polynomial grade, \(\mathrm{p.~grade}_{R^\infty}(\mathfrak{a},R^\infty)\). Recall that if \(\mathfrak{a}\) is an ideal of a (not necessarily noetherian) ring \(A\) and \(M\) is an \(A\)-module, then NEWLINE\[NEWLINE\mathrm{p.~grade}_A(\mathfrak{a},M):=\lim_{m\rightarrow\infty}\mathrm{c.~grade}_{A[t_1,\ldots,t_m]}(\mathfrak{a}A[t_1,\ldots,t_m],A[t_1,\ldots,t_m]\otimes_AM),NEWLINE\]NEWLINE where \(\mathrm{c.~grade}_A(\mathfrak{a},M)\) is the classical grade of \(\mathfrak{a}\) on \(M\), that is, the supremum length of maximal \(M\)-sequences in \(\mathfrak{a}\). (cf. p.~149 in [\textit{D. G. Northcott}, Finite f…
reviewed by: Mahdi Majidi-Zolbanin
description: scientific article; zbMATH DE number 6078484