Abstract
We prove some new rank selection theorems for balanced simplicial complexes. Specifically, we prove that if a balanced simplicial complex satisfies Serre's condition $(S_{\ell})$ then so do all of its rank selected subcomplexes. We also provide a formula for the depth of a balanced simplicial complex in terms of reduced homologies of its rank selected subcomplexes. By passing to a barycentric subdivision, our results give information about Serre's condition and the depth of any simplicial complex. Our results extend rank selection theorems for depth proved by Stanley, Munkres, and Hibi.
Highlights
Let k be a field, A = k[x1, . . . , xn], and I a square-free monomial ideal in A
The motivating example of a balanced simplicial complex is the order complex O(P ) of a finite poset P, whose vertex set is P and whose faces consist of all chains in P ; we partition the vertices of O(P ) by their height in P
Using Proposition C, we provide a formula for depth k[∆] for any balanced simplicial complex ∆
Summary
The motivating example of a balanced simplicial complex is the order complex O(P ) of a finite poset P , whose vertex set is P and whose faces consist of all chains in P ; we partition the vertices of O(P ) by their height in P. As Serre’s condition (S ) generalizes the Cohen-Macaulay property, it is natural to consider if there is any extension of Theorem 1 to (S ) Thereom B shows, if ∆ is a simplicial complex and P its face poset, that the reduced homologies Hi−1(O(P>j); k) pin the largest for which ∆ satisfies (S ) to two possible values. Using Proposition C, we provide a formula for depth k[∆] for any balanced simplicial complex ∆ (see Theorem 26). The last section discusses open problems related to this work and provides examples indicating the sharpness of our results
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