On Sally modules
type: Item
instance of: scholarly article
title: On Sally modules
zbMATH Open document ID: 1484.13055
author: P. H. Lima
published in: Journal of Pure and Applied Algebra
publication date: 2022-01-21T00:00:00Z
review text: Let \((R, \mathfrak{m})\) be a local Noetherian ring of dimension \(d >0\) with infinite residue field. Let \(I\) be an \(\mathfrak{m}\)-primary ideal and \(J\) a reduction of \(I\). For each pair of integers \((r,k)\) with \(r \geq 0\) and \(0 \leq k \leq d\), the author shows that there exists a largest ideal \(I_k^r \subseteq I^{r+1}\) such that for \(n \gg 0\) the length \[ f(n)=\lambda\left(\frac{ I_k^r J^{n-r}+J^nI}{J^nI} \right) \] is a polynomial function of degree at most \(d-(k+1)\). The author provides structure theorems for these ideals and uses them to study the height of the associated prime ideals of the Sally module \(S_J(I)=\bigoplus_{n \geq 1} I^{n+1}/JI^n\) . (The notation chosen in the paper is a bit unfortunate, as \(I_k^r \) is not a power ideal.) All results are inspired by the original construction of \textit{K. Shah} [Trans. Am. Math. Soc. 327, No. 1, 373--384 (1991; Zbl 0738.13007)] of the coefficient ideals of an \(\mathfrak{m}\)-primary ideal.
reviewed by: Catalin Ciuperca
Mathematics Subject Classification ID: 13H10
Mathematics Subject Classification ID: 13H15
zbMATH DE Number: 7461222
zbMATH Keywords: coefficient ideals
zbMATH Keywords: integral closure
zbMATH Keywords: reduction
zbMATH Keywords: multiplicity
MaRDI profile type: Publication
full work available at URL: j.jpaa.2021.106966
OpenAlex ID: W3216491173
cites work: Q4274969
cites work: Fiber Cones of Ideals with Almost Minimal Multiplicity
cites work: Coefficient Ideals
cites work: Q4680541
cites work: Q3414477
DOI: 10.1016/J.JPAA.2021.106966
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description: scientific article; zbMATH DE number 7461222