Poset Embeddings of Hilbert functions and Betti numbers

TL;DR

This paper investigates inequalities between graded Betti numbers of ideals under embedding maps, extending known results and providing new proofs and methods for understanding Betti tables in ring extensions and the lex-plus-powers conjecture.

Contribution

It introduces conditions under which Betti number inequalities are preserved in ring extensions and extends key theorems, offering new proofs and a reduction of the lex-plus-powers conjecture to Artinian cases.

Findings

Betti number inequalities are preserved under certain embeddings in ring extensions.

Betti tables can be obtained via consecutive cancellations from embedded versions.

The lex-plus-powers conjecture reduces to the Artinian case.

Abstract

We study inequalities between graded Betti numbers of ideals in a standard graded algebra over a field and their images under embedding maps, defined earlier by us in [Math. Z. 274, (2013), no. 3-4, pp. 809-819; arXiv:1009.4488]. We show that if graded Betti numbers do not decrease when we replace ideals in an algebra by their embedded versions, then the same behaviour is carried over to ring extensions. As a corollary we give alternative inductive proofs of earlier results of Bigatti, Hulett, Pardue, Mermin-Peeva-Stillman and Murai. We extend a hypersurface restriction theorem of Herzog-Popescu to the situation of embeddings. We show that we can obtain the Betti table of an ideal in the extension ring from the Betti table of its embedded version by a sequence of consecutive cancellations. We further show that the lex-plus-powers conjecture of Evans reduces to the Artinian situation.