  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).
