The eventual shape of the Betti table of $\mathfrak{m}^kM$

TL;DR

This paper investigates the asymptotic shape of Betti tables of modules formed by multiplying a finitely generated graded module by high powers of the maximal ideal in a polynomial ring, revealing patterns and properties such as linear quotients.

Contribution

It describes the Betti table pattern of m^kM for large k in characteristic zero and shows that m^kI has linear quotients for monomial ideals when k is large.

Findings

Betti table patterns stabilize for large k

m^kI has linear quotients for monomial ideals

Characterization of Betti tables in characteristic zero

Abstract

Let $S$ be the polynomial ring over a field $K$ in a finite set of variables, and let $ \mathfrak{m}$ be the graded maximal ideal of $S$. It is known that for a finitely generated graded $S$-module $M$ and all integers $k\gg 0$, the module $ \mathfrak{m}^kM$ is componentwise linear. For large $k$ we describe the pattern of the Betti table of $ \mathfrak{m}^kM$ when $\mathrm{char}(K)=0$ and $M$ is a submodule of a finitely generated graded free $S$-module. Moreover, we show that for any $k\gg 0$, $ \mathfrak{m}^kI$ has linear quotients if $I$ is a monomial ideal.