Extremality with respect to the estimates of Dubreil-Campanella: Splitting theorems

Campanella's refinements of Dubreil's theorem are sharp upper and lower bounds, formulated in postulational terms, for the number of forms of any given degree in the standard bases of 2-codimensional perfect homogeneous polynomial ideals. This note considers the implications of upper-bound-extremality. Its principal discovery is that such extremality implies a splitting of the syzygy module of any standard basis of such an ideal into syzygy modules of the standard bases of ‘component’ ideals explicitly describable in terms of the standard basis in question.