On the Cohen-Macaulay property of diagonal subalgebras of the Rees algebra
type: Item
instance of: scholarly article
title: On the Cohen-Macaulay property of diagonal subalgebras of the Rees algebra
zbMATH Open document ID: 0897.13013
DOI: 10.1007/BF02678014
author: Olga Lavila-Vidal
published in: Manuscripta Mathematica
publication date: 1998-05-13T00:00:00Z
full work available at URL: https://eudml.org/doc/156359
review text: In chronological order, \textit{A. Simis, Ngo Viet Trung} and \textit{G. Valla} [J. Pure Appl. Algebra 125, No. 1, 305-328 (1998; see the preceding review)], \textit{A. Conca, J. Herzog, Ngo Viet Trung} and \textit{G. Valla} [``Diagonal subalgebras of bigraded algebras and embeddings of blow-ups of projective spaces'', Am. J. Math. 199, No. 4, 859-901 (1997)], and \textit{S. D. Cutkosky} and \textit{J. Herzog} [``Cohen-Macaulay coordinate rings of blowup schemes'', Comment. Math. Helv. 72, No. 4, 605-617 (1997)] all studied the Cohen-Macaulayness of rings of the form \(k[(I^e)_c]\), where \(k\) is a field, \(I\) a homogeneous ideal in the polynomial ring \(R = k[x_1, \ldots, x_n]\), and \((I^e)_c\) the ideal in \(R\) generated by all elements of \(I^e\) of degree at least \(c\). These are homogeneous coordinate rings of projective embeddings of a blowing up of \(\mathbb{P}^{n-1}\) along a closed subscheme defined by \(I\). These authors proved that in many cases, for somewhat explicit \(c\) and \(e\), \(k[(I^e)_c]\) is Cohen-Macaulay. Conca, Herzog, Trung and Valla conjectured in their paper (loc. cit.) that the Cohen-Macaulayness of \(R[It]\) implies that for all large \(e\) and \(c\), \(k[(I^e)_c]\) is Cohen-Macaulay. In the paper under review this conjecture is proved and the author even gives an explicit description of \(c\) in terms of \(e\), degrees and the number of generator of \(I\) and \(n\). The main tools are the multigraded \(a\)-invariants and ``good'' resolutions, all developed in the paper.
Mathematics Subject Classification ID: 13C14
Mathematics Subject Classification ID: 13A30
zbMATH DE Number: 1135224
zbMATH Keywords: Rees algebras
zbMATH Keywords: Cohen-Macaulayness
zbMATH Keywords: blow-up
zbMATH Keywords: homogeneous coordinate rings
zbMATH Keywords: multigraded \(a\)-invariants
reviewed by: Irena Swanson
MaRDI profile type: Publication
cites work: Q4274969
cites work: Diagonal subalgebras of bigraded algebras and embeddings of blow-ups of projective spaces
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cites work: On the Cohen-Macaulay and Gorenstein properties of multigraded Rees algebras
cites work: Q4040095
cites work: The diagonal subring and the Cohen-Macaulay property of a multigraded ring
cites work: On the Ideal of Veronesean Surfaces
cites work: The diagonal subalgebra of a blow-up algebra
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description: scientific article; zbMATH DE number 1135224