Canonical complexes associated to a matrix

Abstract

Let Φ be an f×g matrix with entries from a commutative Noetherian ring R, with g≤f. Recall the family of generalized Eagon–Northcott complexes {CΦi} associated to Φ. (See, for example, Appendix A2 in “Commutative Algebra with a View Toward Algebraic Geometry” by D. Eisenbud.) For each integer i, CΦi is a complex of free R-modules. For example, CΦ0 is the original “Eagon–Northcott” complex with zero-th homology equal to the ring R/Ig(Φ) defined by ideal generated by the maximal order minors of Φ; and CΦ1 is the “Buchsbaum–Rim” complex with zero-th homology equal to the cokernel of the transpose of Φ. If Φ is sufficiently general, then each CΦi, with −1≤i, is acyclic; and, if Φ is generic, then these complexes resolve half of the divisor class group of R/Ig(Φ). The family {CΦi} exhibits duality; and, if −1≤i≤f−g+1, then the complex CΦi exhibits depth-sensitivity with respect to the ideal Ig(Φ) in the sense that the tail of CΦi of length equal to grade(Ig(Φ)) is acyclic. The entries in the differentials of CΦi are linear in the entries of Φ at every position except at one, where the entries of the differential are g×g minors of Φ.This paper expands the family {CΦi} to a family of complexes {CΦi,a} for integers i and a with 1≤a≤g. The entries in the differentials of {CΦi,a} are linear in the entries of Φ at every position except at two consecutive positions. At one of the exceptional positions the entries are a×a minors of Φ, at the other exceptional position the entries are (g−a+1)×(g−a+1) minors of Φ.The complexes {CΦi} are equal to {CΦi,1} and {CΦi,g}. The complexes {CΦi,a} exhibit all of the properties of {CΦi}. In particular, if −1≤i≤f−g and 1≤a≤g, then CΦi,a exhibits depth-sensitivity with respect to the ideal Ig(Φ).