Relations on pfaffians. II: A counterexample
type: Item
instance of: scholarly article
title: Relations on pfaffians. II: A counterexample
zbMATH Open document ID: 0781.13006
author: Kazuhiko Kurano
published in: Journal of Mathematics of Kyoto University
publication date: 1992-09-27T00:00:00Z
review text: Let \(R\) be a commutative ring with unity, and fix integers \(n\geq 2\), \(t\geq 1\) such that \(2t\leq n\). Let \(Pf_{2t}\) denote the ideal generated by the \(2t\)-order pfaffians of the generic \(n\) by \(n\) antisymmetric matrix in the polynomial ring over \(R\) in \(n(n-1)/2\) variables. In a previous paper [J. Math. Kyoto Univ. 31, No. 3, 713-731 (1991; see the preceding review)] the author has proved that the module of relations between the pfaffians is generated by relations of degree 1 when \(n\leq 2t+3\). In the present paper, he proves that, when \(n=8\), \(t=2\) and \(R\) is a field of characteristic 2, the module of relations between the pfaffians has a minimal generator which is a relation of degree 2. Since in characteristic 0 this module is generated by relations of degree 1, it follows that \(Pf_{2t}\) does not have, in general, generic minimal free resolutions.
Mathematics Subject Classification ID: 13C40
Mathematics Subject Classification ID: 13D02
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 13P10
Mathematics Subject Classification ID: 15A72
zbMATH DE Number: 59739
zbMATH Keywords: syzygies
zbMATH Keywords: Schur functors
zbMATH Keywords: Gröbner basis
zbMATH Keywords: pfaffians
zbMATH Keywords: polynomial ring
zbMATH Keywords: module of relations
reviewed by: Iustin Coandă
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250519726
OpenAlex ID: W1572799132
Wikidata QID: Q124819791
DOI: 10.1215/KJM/1250519726
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description: scientific article; zbMATH DE number 59739