The graded Betti numbers of truncation of ideals in polynomial rings
Chwas Ahmed, Ralf Fröberg, Mohammed Rafiq Namiq
TL;DR
The paper studies how truncations and squarefree truncations of ideals in polynomial rings affect graded Betti numbers and linear resolution properties. It derives an explicit formula for the graded Betti numbers of $R/I_k$ (the squarefree truncation past $k$) in terms of the Betti diagram of $R/I$ and the $f$-vector of the Stanley–Reisner complex $\\\Delta_I$, revealing precise shifts and invariance patterns; it also shows how regularity behaves under truncation, with $\mathrm{reg}(I_k)=\mathrm{reg}(I)$ when $\mathrm{reg}(I)\ge k$ and $\mathrm{reg}(I_k)=k$ otherwise, and that linear resolutions are preserved in this regime. The work then connects polarization with truncation, proving that Betti numbers and linearity properties are preserved under polarization and establishing equivalences for componentwise linearity between $I$ and its polarization. Finally, it provides a general method to recover all graded Betti numbers of $R/I_{ geq k}$ from the Betti numbers of $R/I$ and the Hilbert series of $R/I_{ geq k}$, with several concrete examples including complete intersections and a detailed computation for a nontrivial eight-variable case. These results yield refined invariants for truncated ideals and offer practical tools for computing Betti diagrams in combinatorial and commutative-algebra contexts.
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 $Δ_I$. 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 $Δ_I$. 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 $R/I_{\ge k}$.