Monomial conjecture and complete intersections
type: Item
instance of: scholarly article
title: Monomial conjecture and complete intersections
zbMATH Open document ID: 0792.13010
DOI: 10.1007/BF02568334
author: Jan. R. Strooker
author: J. Stückrad
published in: Manuscripta Mathematica
publication date: 1994-07-24T00:00:00Z
full work available at URL: https://eudml.org/doc/155830
review text: Let \((A,{\mathfrak m})\) be a \(d\)-dimensional local ring. Let \(\{x_ 1, \dots, x_ d\}\) denote a system of parameters of the ring \(A\). The monomial conjecture (MC) [see \textit{M. Hochster}, Nagoya Math. J. 51, 25- 43 (1973; Zbl 0268.13004)], states that \(x_ 1^ t \dots x_ d^ t \notin (x_ 1^{t+1}, \dots, x_ d^{t+1})A\) for all \(t \geq 1\) for all systems of parameters in any local ring \(A\). It is proved by M. Hochster [see the cited paper], that (MC) is true, e.g., whenever \(A\) is an equicharacteristic local ring. Moreover he showed also that (MC) is equivalent to the direct summand conjecture. In the present paper the authors prove an interesting equivalent condition for (MC) which says: For a local complete intersection ring \(R\) and an ideal \({\mathfrak a}\) consisting of zero divisors the annihilator \(0:_ R {\mathfrak a}\) of \({\mathfrak a}\) is not contained in any parameter ideal of \(R\). This follows by elaborating on local cohomology modules.
Mathematics Subject Classification ID: 13H10
Mathematics Subject Classification ID: 14M10
Mathematics Subject Classification ID: 13D45
Mathematics Subject Classification ID: 13C40
Mathematics Subject Classification ID: 14B15
Mathematics Subject Classification ID: 14H99
zbMATH DE Number: 492813
zbMATH Keywords: local ring
zbMATH Keywords: monomial conjecture
zbMATH Keywords: system of parameters
zbMATH Keywords: direct summand conjecture
zbMATH Keywords: complete intersection
zbMATH Keywords: local cohomology modules
Wikidata QID: Q123253301
reviewed by: Peter Schenzel
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 492813