Distractions of Shakin rings

TL;DR

This paper investigates Shakin rings, a class of quotient rings combining piecewise lex-segment and pure powers ideals, extending key algebraic conjectures and theorems to broader ideal classes and analyzing embeddings via distractions.

Contribution

It generalizes important algebraic results from monomial regular sequences to a wider class of ideals called Shakin rings, and studies embeddings induced by distractions.

Findings

Extended Eisenbud-Green-Harris conjecture results to Shakin rings.

Proved extremality of embeddings via distractions for Hilbert functions.

Generalized Betti number bounds and the Lex-Plus-Power inequality to new ideal classes.

Abstract

We study, by means of embeddings of Hilbert functions, a class of rings which we call Shakin rings, i.e. quotients K[X_1,...,X_n]/a of a polynomial ring over a field K by ideals a=L+P which are the sum of a piecewise lex-segment ideal L, as defined by Shakin, and a pure powers ideal P. Our main results extend Abedelfatah's recent work on the Eisenbud-Green-Harris conjecture, Shakin's generalization of Macaulay and Bigatti-Hulett-Pardue theorems on Betti numbers and, when char(K)=0, Mermin-Murai theorem on the Lex-Plus-Power inequality, from monomial regular sequences to a larger class of ideals. We also prove an extremality property of embeddings induced by distractions in terms of Hilbert functions of local cohomology modules.