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,…, um) of degree d in the polynomial ring S = k[x1,…, xn] admits quasi-linear quotients then Lq = (u1,…, uq−1): uq 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.
I. Anwar and K. Shabbir, “A note on: quasi–linear quotients and shellability of pure simplicial complexes,” Commun. Algebra 46 (2), 850–850 (2018). doi 10. 1080/00927872. 2017. 1327599CrossRefzbMATHGoogle Scholar