Depth of factors of square free monomial ideals

Abstract

Let I I be an ideal of a polynomial algebra over a field generated by r r square free monomials of degree d d . If r r is bigger than (or equal to, if I I is not principal) the number of square free monomials of I I of degree d + 1 d+1 , then d e p t h S I = d \mathrm {depth}_SI= d . Let J ⊊ I J\subsetneq I , J ≠ 0 J\not =0 be generated by square free monomials of degree ≥ d + 1 \geq d+1 . If r r is bigger than the number of square free monomials of I ∖ J I\setminus J of degree d + 1 d+1 or, more generally, the Stanley depth of I / J I/J is d d , then d e p t h S I / J = d \mathrm {depth}_SI/J= d . In particular, Stanley’s Conjecture holds in these cases.