On the Stanley depth of squarefree monomial ideals

2014/09/18 by Fakhari, S. A. Seyed
#Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.1409.5270

Abstract

Let \mathbbK be a field and S=\mathbbK[x₁,…,x_n] be the polynomial ring in n variables over the field \mathbbK. Suppose that C is a chordal clutter with n vertices and assume that the minimum edge cardinality of C is at least d. It is shown that S/I(c_d(C)) satisfies Stanley's conjecture, where I(c_d(C)) is the edge ideal of the d-complement of C. This, in particular shows that S/I satisfies Stanley's conjecture, where I is a quadratic monomial ideal with linear resolution. We also define the notion of Schmitt--Vogel number of a monomial ideal I, denoted by \rm sv(I) and prove that for every squarefree monomial ideal I, the inequalities \rm sdepth(I)≥ n-\rm sv(I)+1 and \rm sdepth(S/I)≥ n-\rm sv(I) hold.

Related

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• Several combinatorial inequalities related to squarefree monomial ideals
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