Macaulay duality and its geometry

Macaulay Duality, between quotients of a polynomial ring over a field, annihilated by powers of the variables, and finitely generated submodules of the ring's graded dual, is generalized over any Noetherian ring, and used to provide isomorphisms between the subschemes of the Hilbert scheme parameterizing various sorts of these quotients, and the corresponding subschemes of the Quot scheme of the dual. Thus notably the locus of recursively compressed algebras of permissible socle type is proved to be covered by open subschemes, each one isomorphic to an open subscheme of a certain affine space. Moreover, the polynomial variables are weighted, the polynomial ring is replaced by a graded module, and attention is paid to induced filtrations and gradings. Furthermore, a similar theory is developed for (relatively) maximal quotients of a graded Gorenstein Artinian algebra.