  I describe the projective resolution of a codimension 4 Gorenstein ideal,
aiming to extend Buchsbaum and Eisenbud's famous result in codimension 3. The
main result is a structure theorem stating that the ideal is determined by its
(k+1) x 2k matrix of first syzygies, viewed as a morphism from the ambient
regular space to the Spin-Hom variety SpH_k in Mat(k+1,2k). This is a general
result encapsulating some theoretical aspects of the problem, but, as it
stands, is still some way from tractable applications.
