The canonical module of an associated graded ring
type: Item
instance of: scholarly article
title: The canonical module of an associated graded ring
zbMATH Open document ID: 0628.13007
DOI: 10.1007/BF01191357
author: Winfried Bruns
published in: Archiv der Mathematik
publication date: 1986-01-01T00:00:00Z
review text: Herzog, Simis and Vasconcelos have computed, in certain circumstances, the canonical module of the Rees ring \(S=\oplus^{\infty}_{n=0}I^ nt\subseteq R[t]\) of a commutative Noetherian ring R with respect to an ideal I. This paper is concerned with the associated graded ring \(G=S/SI\cong \oplus^{\infty}_{n=0}I^ n/I^{n+1}.\quad Suppose\) that R is Cohen-Macaulay and the ideal I of R has height at least 2; suppose also that S and G are both Cohen-Macaulay rings and that S has a canonical module \(\omega_ S\). The main result of the paper states that G has a canonical module \(\omega_ G\), and if \(\omega_ S\) can be embedded into S in such a way that \(\omega_ S\) (considered as an ideal) is not contained in a minimal prime ideal of SI or SIt, then \(\omega_ G\cong (\omega_ S+SI)/SI\); furthermore, such an embedding exists if and only if all the localizations \(S_ p\) at prime ideals P of S which are minimal over O, SI or SIt are Gorenstein rings.
Mathematics Subject Classification ID: 13D99
Mathematics Subject Classification ID: 13A15
Mathematics Subject Classification ID: 13H10
Mathematics Subject Classification ID: 13E05
zbMATH DE Number: 4021295
zbMATH Keywords: canonical module of the Rees ring
zbMATH Keywords: Noetherian ring
zbMATH Keywords: associated graded ring
zbMATH Keywords: Cohen-Macaulay
zbMATH Keywords: localizations
zbMATH Keywords: Gorenstein rings
MaRDI profile type: Publication
cites work: Der kanonische Modul eines Cohen-Macaulay-Rings. (The canonical moduls of a Cohen-Macaulay-ring)
cites work: On the canonical module of the Rees algebra and the associated graded ring of an ideal
cites work: On the divisor class group of Rees-algebras
cites work: On the associated graded ring of an ideal
Recommended article: On the associated graded modules of canonical modules
Recommended article: On the canonical module of the Rees algebra and the associated graded ring of an ideal
Recommended article: Q3035423
Recommended article: The canonical module of a Stanley-Reisner ring
Recommended article: On the structure of the canonical module of the Rees algebra and the associated graded ring of an ideal
Recommended article: The canonical module of a Cox ring
Recommended article: Embeddings of certain graded rings into their canonical modules
Recommended article: On the depth of the canonical modules of graded rings associated to an ideal
Recommended article: The associated graded ring of an integral group ring
Recommended article: The canonical module of a one-dimensional reduced local ring
description: scientific article; zbMATH DE number 4021295