2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/72174We extend a reciprocity theorem of Stanley about enumeration of integer points in polyhedral cones when one exchanges strict and weak inequalities. The proof highlights the roles played by Cohen-Macaulayness and canonical modules. The extension raises the issue of whether a Cohen-Macaulay complex of dimension d embedded piecewise-linearly in d-space is necessarily a d-ball. This is observed to be true for d at most 3, but false for d=4.CombinatoricsCommutative Algebra16E65; 52B20; 14M25; 13F55text