Monomial Ideals with Quasi-Linear Quotients

Abstract

In this paper, we extend the notion of quasi-linear quotients for a pure monomial ideal (not necessarily square-free) of degree d. We introduce the notion of quasi-linear free resolution and show that if a pure monomial ideal I = (u₁, u₂,…, u_m) of degree d in the polynomial ring S = k[x₁,…, x_n] admits quasi-linear quotients then L_q = (u₁,…, u_q−1): u_q admits quasi-linear free resolution for all q ≤ m. Moreover, we show that if a pure monomial ideal I of degree d admits quasi-linear quotients then I_〈t〉 will also have quasi-linear quotients for t ≥ d.