  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.
