Homological shifts of polymatroidal ideals

Abstract

We study the homological shifts of polymatroidal ideals. It is shown 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. We also study the "socle ideal" soc(I) of a polymatroidal ideal I and relate it to the highest homological shift ideal of I. It is shown that soc(I) is polymatroidal if: (i) I is a polymatroidal ideal generated in degree two, (ii) I is a polymatroidal ideal in at most three variables, (iii) I is matroidal, (iv) I is a principal Borel ideal, (v) I is a PLP-polymatroidal ideal, (vi) I is a LP-polymatroidal ideal.