Powers of monomial ideals with characteristic-dependent Betti numbers

TL;DR

This paper investigates how the Betti numbers of monomial ideals vary with the characteristic of the field, providing constructions and examples demonstrating characteristic-dependent behavior in all powers and asymptotic regularity.

Contribution

It introduces explicit constructions of monomial ideals with characteristic-dependent Betti numbers across all powers and shows how adding variables spreads this dependence.

Findings

Betti numbers differ in characteristic 0 and p for high powers

Adding variables can extend characteristic dependence to all powers

Existence of monomial ideals with characteristic-dependent Kodiyalam polynomial coefficients

Abstract

We explore the dependence of the Betti numbers of monomial ideals on the characteristic of the field. A first observation is that for a fixed prime $p$ either the $i$-th Betti number of all high enough powers of a monomial ideal differs in characteristic $0$ and in characteristic $p$ or it is the same for all high enough powers. In our main results we provide constructions and explicit examples of monomial ideals all of whose powers have some characteristic-dependent Betti numbers or whose asymptotic regularity depends on the field. We prove that, adding a monomial on new variables to a monomial ideal, allows to spread the characteristic dependence to all powers. For any given prime $p$, this produces an edge ideal such that the Betti numbers of all its powers over $\mathbb{Q}$ and over $\mathbb{Z}_p$ are different. Moreover, we show that, for every $r \geq 0$ and $i \geq 3$ there is a monomial ideal $I$ such that some coefficient in a degree $\geq r$ of the Kodiyalam polynomials $\mathfrak P_3(I),\ldots,\mathfrak P_{i+r}(I)$ depends on the characteristic. We also provide a summary of related results and speculate about the behaviour of other combinatorially defined ideals.