Bounding Multiplicity by Shifts in the Taylor Resolution

TL;DR

This paper proves a weaker form of the multiplicity conjecture for specific classes of monomial ideals and explores conditions for bounds to be tight, advancing understanding of algebraic multiplicity in these contexts.

Contribution

It establishes a weaker multiplicity conjecture for quadratic and certain squarefree monomial ideals, and analyzes tensor products and unions of Stanley-Reisner ideals.

Findings

Proves a weaker multiplicity conjecture for quadratic monomial ideals.

Shows tensor products and certain unions satisfy the conjecture if components do.

Studies conditions for bounds to be achieved.

Abstract

A weaker form of the multiplicity conjecture of Herzog, Huneke, and Srinivasan is proven for two classes of monomial ideals: quadratic monomial ideals and squarefree monomial ideals with sufficiently many variables relative to the Krull dimension. It is also shown that tensor products, as well as Stanley-Reisner ideals of certain unions, satisfy the multiplicity conjecture if all the components do. Conditions under which the bounds are achieved are also studied.