Macaulay duality and its geometry

TL;DR

This paper extends Macaulay Duality to Noetherian rings, establishing geometric isomorphisms between Hilbert and Quot schemes, and explores the structure of recursively compressed algebras and Gorenstein quotients.

Contribution

It generalizes Macaulay Duality over any Noetherian ring and links subschemes of Hilbert and Quot schemes, revealing open subscheme structures for recursively compressed algebras.

Findings

Locus of recursively compressed algebras covered by open subschemes

Isomorphisms between subschemes of Hilbert and Quot schemes

Development of duality theory for Gorenstein Artinian quotients

Abstract

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.