The minimal free resolution of a class of square-free monomial ideals

Abstract

For positive integers n, b 1⩽ b 2⩽⋯⩽ b n and t⩽ n, let I t be the transversal monomial ideal generated by square-free monomials (∗) y i 1j 1 y i 2j 2 ⋯y i tj t , 1⩽i 1<i 2<⋯<i t⩽n, 1⩽j k⩽b i k , k=1,…,t, where y ij 's are distinct indeterminates. It is observed that the simplicial complex associated to this ideal is pure shellable if and only if b 1=⋯= b n =1, but its Alexander dual is always pure and shellable. The simplicial complex admits some weaker shelling which leads to the computation of its Hilbert series. The main result is the construction of the minimal free resolution for the quotient ring of I t . This class of monomial ideals includes the ideals of t-minors of generic pluri-circulant matrices under a change of coordinates. The last family of ideals arise from some specializations of the defining ideals of generic singularities of algebraic varieties.