Sheaves with canonical determinant on Cohen-Macaulay schemes.
type: Item
instance of: scholarly article
title: Sheaves with canonical determinant on Cohen-Macaulay schemes.
zbMATH Open document ID: 1056.14022
DOI: 10.1016/S0021-8693(02)00134-5
author: Jan Oddvar Kleppe
author: Chris Peterson
published in: Journal of Algebra
publication date: 2003-02-12T00:00:00Z
review text: The paper is devoted to the study of degeneracy loci corresponding to triples \((\phi, \mathcal A, \mathcal B),\) where \(\mathcal A\) is a line bundle on an arithmetically Cohen-Macaulay scheme \(S\), \(\mathcal B\) is a sheaf on \(S,\) the determinant of \(\mathcal B^{\star} \otimes \mathcal A\) is a twist of the canonical sheaf \(\mathcal K_S\) of \(S,\) and \(\phi\) is assumed to be regular. The authors describe a class of coherent sheaves on arithmetically Cohen-Macaulay schemes with the property that regular sections degenerate on arithmetically Gorenstein schemes. Let \(R\) be a graded Gorenstein \(k\)-algebra. The authors and \textit{J. C. Migliore}, \textit{R. Miro-Roig} and \textit{U. Nagel} [''Gorenstein liaison, complete intersection liaison invariants and unobstructedness'', Mem. Am. Math. Soc. 732 (2001; Zbl 1006.14018)] utilised the sheaf \(n\mathcal H - \mathcal K_S\) on an arithmetically Cohen-Macaulay scheme \(S \subset \text{Proj\,} R\), \(\mathcal H\) a hyperplane divisor, to construct divisors on \(S\) which are arithmetically Gorenstein. Results of this type are generalized by the authors [J. Algebra 238, 776--800 (2001; Zbl 1028.13007)], in relation with the use of maximal Cohen-Macaulay modules. There are several features common to all the sheaves occuring in these constructions and several other generalizations. The present paper is devoted to a unified approach that could encompass all of the sheaves appearing in the aforementioned work. The authors realized that if \(S\) is an arithmetically Cohen-Macaulay scheme and if \(\mathcal E\) is a sheaf on \(S\) whose determinant is a twist of the canonical sheaf, then one could give weak sufficient conditions on \(\mathcal E\) such that the top dimensional component of the degeneracy locus of any regular section of \({\mathcal E}^{\star}(n)\) is an arithmetically Gorenstein scheme. It turns out that all of the various sheaves derived from the authors paper [loc. cit.], satisfy this cohomological …
reviewed by: Peter Schenzel
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 14M05
zbMATH DE Number: 1868031
zbMATH Keywords: Gorenstein schemes
zbMATH Keywords: special sheaves
zbMATH Keywords: sections of sheaves
zbMATH Keywords: degeneracy loci
MaRDI profile type: Publication
cites work: Components of the space parametrizing graded Gorenstein Artin algebras with a given Hilbert function
cites work: Hilbert function of powers of ideals of low codimension
cites work: Q4317713
cites work: Q5545200
cites work: Local cohomology. A seminar given by A. Grothendieck, Harvard University, Fall 1961. Notes by R. Hartshorne
cites work: Q3312347
cites work: Linkage and the Koszul Homology of Ideals
cites work: Q3356408
cites work: The smoothness and the dimension of \(PGor(H)\) and of other strata of the punctual Hilbert scheme
cites work: Unobstructedness and dimension of families of Gorenstein algebras
cites work: Gorenstein liaison, complete intersection liaison invariants and unobstructedness
cites work: Maximal Cohen-Macaulay modules and Gorenstein algebras
cites work: Determinantal schemes and Buchsbaum-Rim sheaves
cites work: Introduction to liaison theory and deficiency modules
cites work: Buchsbaum-Rim sheaves and their multiple sections
cites work: A construction of codimension three arithmetically Gorenstein subschemes of projective space
cites work: Q4251933
description: scientific article; zbMATH DE number 1868031