Higher Nerves of Simplicial Complexes

Abstract

We investigate generalized notions of the nerve complex for the facets of a simplicial complex. We show that the homologies of these higher nerve complexes determine the depth of the Stanley-Reisner ring $k[\Delta]$ as well as the $f$-vector and $h$-vector of $\Delta$. We present, as an application, a formula for computing regularity of monomial ideals.