Canonical complexes associated to a matrix

JOURNAL OF ALGEBRA

卷:460

Article

Kustin, Andrew R.¹ 

[1] Univ S Carolina, Dept Math, Columbia, SC 29208 USA

关键词: Buchsbaum-Rim complex;    Depth-sensitivity;    Determinantal ring;    Duality;    Eagon-Northcott complex;    Hooks;    Koszul complex;    Maximal Cohen-Macaulay module;    Perfect module;    Schur module;    Weyl module;   

DOI  :  10.1016/j.jalgebra.2016.03.037

来源: Elsevier

【 摘 要 】

Let Phi be an f x g matrix with entries from a commutative Noetherian ring R, with g <= f. Recall the family of generalized Eagon-Northcott complexes {C-Phi(i)} associated to Phi. (See, for example, Appendix A2 in Commutative Algebra with a View Toward Algebraic Geometry by D. Eisenbud.) For each integer i, C-Phi(i) is a complex of free R-modules. For example, C-Phi(0) is the original Eagon-Northcott complex with zero-th homology equal to the ring R/I-g(Phi) defined by ideal generated by the maximal order minors of Phi; and C-Phi(1) is the Buchsbaum-Rim complex with zero-th homology equal to the cokernel of the transpose of Phi. If Phi is sufficiently general, then each C-Phi(i), with -1 <= i, is acyclic; and, if Phi is generic, then these complexes resolve half of the divisor class group of R/I-g (Phi). The family {C-Phi(i)} exhibits duality; and, if -1 <= i <= f - g + 1, then the complex C-Phi(i) exhibits depth-sensitivity with respect to the ideal I-g(Phi) in the sense that the tail of C-Phi(i) of length equal to grade(I-g (Phi)) is acyclic. The entries in the differentials of C-Phi(i) are linear in the entries of Phi at every position except at one, where the entries of the differential are g x g minors of Phi. This paper expands the family {C-Phi(i)} to a family of complexes {C-Phi(i,a)} for integers i and a with 1 <= a <= g. The entries in the differentials of {C-Phi(i,a)} are linear in the entries of Phi at every position except at two consecutive positions. At one of the exceptional positions the entries are a x a minors of Phi, at the other exceptional position the entries are (g-a+1) x (g-a+1) minors of Phi. The complexes {C-Phi(i)} are equal to {C-Phi(i,1)} and {C-Phi(i,g)}. The complexes {C-Phi(i,a)} exhibit all of the properties of {C-Phi(i)}. In particular, if -1 <= i <= f - g and 1 <= a <= g, then C-Phi(i,a) exhibits depth-sensitivity with respect to the ideal I-g (Phi). (C) 2016 Elsevier Inc. All rights reserved.

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