Canonical complexes associated to a matrix

Original · EN

Let Phi be an f by g matrix with entries from a commutative Noetherian ring R, with g at most f. Recall the family of generalized Eagon-Northcott complexes Cⁱ associated to Phi. (See, for example, Appendix A2 in "Commutative Algebra with a view toward Algebraic Geometry" by David Eisenbud.) For each integer i, Cⁱ is a complex of free R-modules. For example, C⁰ is the original "Eagon-Northcott" complex with zero-th homology equal to the ring defined by the maximal order minors of Phi; and C¹ 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 Cⁱ, with i at least -1, is acyclic; and, if Phi is generic, then these complexes resolve half of the divisor class group of R/Ig(Phi). The family Cⁱ exhibits duality; and, if -1≤ i≤ f-g+1, then the complex Cⁱ exhibits depth-sensitivity with respect to the ideal Ig(Phi) in the sense that the tail of Cⁱ of length equal to grade(Ig(Phi)) is acyclic. The entries in the differentials of Cⁱ are linear in the entries of Phi at every position except at one, where the entries of the differential are g by g minors of Phi. This paper expands the family Cⁱ to a family of complexes Cⁱ,ᵃ for integers i and a with 1≤ a≤ g. The entries in the differentials of Cⁱ,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 by a minors of Phi, at the other exceptional position the entries are g-a+1 by g-a+1 minors of Phi. The complexes Cⁱ are equal to Cⁱ,¹ and Cⁱ,ᵍ. The complexes Cⁱ,ᵃ exhibit all of the properties of Cⁱ. In particular, if -1≤ i≤ f-g and 1≤ a≤ g, then Cⁱ,ᵃ exhibits depth-sensitivity with respect to the ideal Ig(Phi).