  We prove some new rank selection theorems for balanced simplicial complexes.
Specifically, we prove that rank selected subcomplexes of balanced simplicial
complexes satisfying Serre's condition $(S_{\ell})$ retain $(S_{\ell})$. 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 compex. Our results extend rank selection
theorems for depth proved by Stanley, Munkres, and Hibi.
