Sheaves with canonical determinant on Cohen-Macaulay schemes.
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 condition. In addition there is a large family of sheaves with are not covered by the aformentioned works but which still satisfy the required cohomological conditions. These conditions are expressed in the vanishing of cohomology modules in a certain range.
scientific article; zbMATH DE number 1868031
scientific article