  A border basis scheme is an affine scheme that can be viewed as an open
subscheme of the Hilbert scheme of \mu points of affine n-space. We study
syzygies of the generators of a border basis scheme's defining ideal. These
generators arise as the entries of the commutators of certain matrices (the
"generic multiplication matrices"). We consider two families of syzygies that
are closely connected to these matrices: The first arises from the Jacobi
identity, and the second from the fact that the trace of a commutator is 0.
Several examples of both types of syzygy are presented, including a proof that
the border basis schemes in case n = 2 are complete intersections.
