Poset Embeddings of Hilbert Functions

TL;DR

This paper investigates how to embed the poset of Hilbert functions of quotients of a standard graded algebra into the poset of ideals, exploring conditions for such embeddings and their applications in classification.

Contribution

It introduces a method to embed Hilbert function posets into ideal posets, extends this to ring extensions, and establishes conditions for lexicographic segment ideal classification.

Findings

Embeddings do not always exist for all rings.

Embedding can be lifted to certain ring extensions like polarization.

A condition ensures Hilbert functions are classified via lexicographic segment ideals.

Abstract

For a standard graded algebra $R$, we consider embeddings of the the poset of Hilbert functions of quotients of $R$ into the poset of ideals of $R$, as a way of classification of Hilbert functions. There are examples of rings for which such embeddings do not exist. We describe how the embedding can be lifted to certain ring extensions, which is then used in the case of polarization and distraction. A version of a theorem of Clements--Lindstr\"om is proved. We exhibit a condition on the embedding that ensures that the classification of Hilbert functions is obtained with images of lexicographic segment ideals.