Finite super height of a homogeneous ideal in \({\mathbb{Z}}^ n\)-graded extensions
type: Item
instance of: scholarly article
title: Finite super height of a homogeneous ideal in \({\mathbb{Z}}^ n\)-graded extensions
zbMATH Open document ID: 0624.13009
DOI: 10.1016/0021-8693(87)90143-8
published in: Journal of Algebra
publication date: 1987-01-01T00:00:00Z
review text: The purpose of the paper is to consider big ht I, super ht I and finite super ht I for an ideal I of a noetherian ring R, where big\(ht I=\sup \{ht P| \quad P\) is a prime ideal of \(R\) minimal over \(I\},\) super\(ht I=\sup \{ht IS| \quad S\) is a noetherian R-algebra\}, finite super\(ht I=\sup \{ht IS| \quad S\) is a finitely generated R-algebra\}. The main motivation for the work is the direct summand conjecture. The following theorem is shown in section 1. Theorem: The direct summand conjecture is equivalent to the improved monomial conjecture, which asserts: Let I be the ideal generated by the residue classes of \(X_ 1,...,X_ n\) in the ring \(V[X_ 1,...,X_ n,Y_ 1,...,Y_ n]/(X^ t_ 1...X^ t_ n-\sum^{n}_{i=1}Y_ iX_ i^{t+1})\) with \(t\geq 1\), \(n\geq 3\), indeterminates \(X_ i's\) and \(Y_ i's\), and the ring V of Witt vectors over the algebraic closure of \({\mathbb{Z}}/p{\mathbb{Z}}\). Then, finite super\(ht I<n\). - Zariski's main theorem and its graded version, which is proven in section 2, are used.
Mathematics Subject Classification ID: 13C05
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 14M05
Mathematics Subject Classification ID: 13C15
zbMATH DE Number: 4012775
zbMATH Keywords: big height
zbMATH Keywords: finite super heigth
zbMATH Keywords: direct summand conjecture
zbMATH Keywords: improved monomial conjecture
zbMATH Keywords: Zariski's main theorem
author: Jee Koh
reviewed by: Yoichi Aoyama
MaRDI profile type: Publication
full work available at URL: 0021-8693(87)90143-8
OpenAlex ID: W2094311154
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description: scientific article; zbMATH DE number 4012775