Higher Nerves of Simplicial Complexes

Hailong Dao, Joseph Doolittle, Ken Duna, Bennet Goeckner, Brent Holmes, Justin Lyle

Published 2017-10-17Version 1

We investigate generalized notions of the nerve complex for a collection of subsets of a topological space. For a simplicial complex $\Delta$, 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 develop a strengthened version of the higher nerve complexes, called the nervous system of $\Delta$, that allows one to reconstruct $\Delta$. We present, as an application, a formula for computing regularity of monomial ideals and a new definition of "combinatorial depth" for local schemes.