The graded Betti numbers of truncation of ideals in polynomial rings

TL;DR

This paper investigates the graded Betti numbers of truncations of ideals in polynomial rings, providing formulas to compute these invariants from known Betti numbers and combinatorial data, with a focus on squarefree monomial ideals.

Contribution

It introduces methods to determine Betti numbers of ideal truncations using existing Betti diagrams and combinatorial invariants, extending previous results to a broader class of ideals.

Findings

Betti numbers of $R/I_k$ can be computed from $R/I$ and the $f$-vector of $ riangle_I$

Betti numbers of $R/I_{ ext{geq}k}$ are determined by Betti numbers of $R/I$ and the Hilbert series

Results generalize known formulas for ideal truncations to arbitrary graded ideals

Abstract

Let $R=\mathbb{K}[x_1,\dots,x_n]$, a graded algebra $S=R/I$ satisfies $N_{k,p}$ if $I$ is generated in degree $k$, and the graded minimal resolution is linear the first $p$ steps, and the $k$-index of $S$ is the largest $p$ such that $S$ satisfies $N_{k,p}$. Eisenbud and Goto have shown that for any graded ring $R/I$, then $R/I_{\geq k}$, where $I_{\geq k}=I\cap M^k$ and $M=(x_1,\dots,x_n)$, has a $k$-linear resolution (satisfies $N_{k,p}$ for all $p$) if $k\gg0$. For a squarefree monomial ideal $I$, we are here interested in the ideal $I_k$ which is the squarefree part of $I_{\geq k}$. The ideal $I$ is, via Stanley-Reisner correspondence, associated to a simplicial complex $\Delta_I$. In this case, all Betti numbers of $R/I_k$ for $k>\min\{\text{deg}(u)\mid u\in I\}$, which of course is a much finer invariant than the index, can be determined from the Betti diagram of $R/I$ and the $f$-vector of $\Delta_I$. We compare our results with the corresponding statements for $I_{\ge k}$. (Here $I$ is an arbitrary graded ideal.) In this case we show that the Betti numbers of $R/I_{\ge k}$ can be determined from the Betti numbers of $R/I$ and the Hilbert series of $R/I_{\ge k}$.