Canonical complexes associated to a matrix

TL;DR

This paper generalizes the family of Eagon-Northcott complexes associated with a matrix over a Noetherian ring, introducing new complexes that preserve key properties like acyclicity and depth-sensitivity, and expand understanding of their duality and resolution capabilities.

Contribution

It introduces a new family of complexes C^{i,a} that extend existing complexes C^{i}, maintaining their properties and providing deeper insights into their algebraic and homological behavior.

Findings

The new complexes C^{i,a} exhibit depth-sensitivity similar to C^{i}.

They maintain duality and acyclicity properties under general conditions.

The complexes resolve significant parts of the divisor class group when the matrix is generic.

Abstract

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 {C^{i}} associated to Phi. (See, for example, Appendix A2 in "Commutative Algebra with a view toward Algebraic Geometry" by David Eisenbud.) For each integer i, C^i is a complex of free R-modules. For example, C^{0} is the original "Eagon-Northcott" complex with zero-th homology equal to the ring defined by the maximal order minors of Phi; and C^{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^{i}, with i at least -1, 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^{i}} exhibits duality; and, if -1\le i\le f-g+1, then the complex C^{i} exhibits depth-sensitivity with respect to the ideal I_g(Phi) in the sense that the tail of C^{i} of length equal to grade(I_g(Phi)) is acyclic. The entries in the differentials of C^i 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 {C^i} to a family of complexes {C^{i,a}} for integers i and a with 1\le a\le g. The entries in the differentials of C^{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 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 {C^i} are equal to {C^{i,1}} and {C^{i,g}}. The complexes {C^{i,a}} exhibit all of the properties of {C^{i}}. In particular, if -1\le i\le f-g and 1\le a\le g, then C^{i,a} exhibits depth-sensitivity with respect to the ideal I_g(Phi).