Homological shifts of polymatroidal ideals
Antonino Ficarra
TL;DR
This work addresses whether homological shift ideals preserve polymatroidality for polymatroidal ideals. It establishes that the first homological shift $HS_1(I)$ is polymatroidal whenever $I$ is polymatroidal, providing a universal positive result by giving a concrete description $HS_1(I)=\{\mathbf{x}^F u: u\in G(I), F\subseteq {\rm set}(u)\} = (\mathrm{lcm}(u,v): u,v\in G(I), d(u,v)=1)$ and using linear-quotient techniques to verify the exchange property. Consequently, the conjecture is resolved for matroidal (squarefree polymatroidal) ideals, since $HS_{j+1}(I)=HS_1(HS_j(I))_{>j+1}$ and all $HS_j(I)$ remain matroidal in that case. The results illuminate how homological and combinatorial properties transfer along shifts and suggest directions for classifying polymatroidal ideals where the equality $HS_{j+1}(I)=(HS_1(HS_j(I)))_{>j+1}$ holds for all $j<\mathrm{pd}(I)$.
Abstract
We study the homological shifts of polymatroidal ideals. In our main theorem we prove that the first homological shift ideal of any polymatroidal ideal is again polymatroidal, supporting a conjecture of Bandari, Bayati and Herzog that predicts that all homological shift ideals of a polymatroidal ideal are polymatroidal. As a nice consequence, we recover a result of Bayati which proves this conjecture in the squarefree case.