A grade five Gorenstein algebra with no minimal algebra resolutions
type: Item
instance of: scholarly article
title: A grade five Gorenstein algebra with no minimal algebra resolutions
zbMATH Open document ID: 0856.13011
author: Hema Srinivasan
published in: Journal of Algebra
publication date: 1997-02-24T00:00:00Z
review text: A free resolution of a cyclic module \(R/I\) over a noetherian local ring \(R\) is called an algebra resolution if it admits a multiplication which makes it an associative, commutative differential graded algebra. It is known that if \(I\) is a Gorenstein ideal of grade less than five, then \(R/I\) has an algebra resolution, and there are examples of Gorenstein ideals of grade six with no minimal algebra resolutions. Let \(R = k[[X_i,Y_{ij},\;1\leq i<j \leq 5]]\) with \(k\) a field of characteristic \(\neq 2\). Put \(Y=(Y_{ij})\) with \(Y_{ij}= -Y_{ji}\), \(X=(X_1, \dots,X_5)\) and let \(I\) be the ideal generated by the \(4\times 4\) Pfaffians of \(Y\) and the five entries of \(XY\). Then the author shows that \(I\) is a grade five Gorenstein ideal of \(R\) and constructs a minimal free resolution \(F_\bullet\) of \(R/I\) which does not admit an algebra structure. -- Moreover, there is an ideal \(J\) generated by a regular sequence contained in \(I\) such that \(F_\bullet\) does not admit a structure of a differential graded module over the Koszul resolution of \(R/J\).
Mathematics Subject Classification ID: 13D03
Mathematics Subject Classification ID: 13H10
zbMATH DE Number: 843871
zbMATH Keywords: differential graded algebra
zbMATH Keywords: algebra resolution
zbMATH Keywords: Gorenstein ideals
reviewed by: Akira Ooishi
MaRDI profile type: Publication
full work available at URL: jabr.1996.0016
OpenAlex ID: W2070005568
DOI: 10.1006/JABR.1996.0016
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description: scientific article; zbMATH DE number 843871