Higher Nerves of Simplicial Complexes

TL;DR

This paper explores advanced nerve complexes of simplicial complexes, revealing their homologies' role in determining algebraic invariants like depth, f-vector, and h-vector, and providing a formula for regularity of monomial ideals.

Contribution

It introduces generalized higher nerve complexes and links their homologies to key algebraic and combinatorial invariants of simplicial complexes.

Findings

Homologies of higher nerve complexes determine the depth of Stanley-Reisner rings.

They also determine the f-vector and h-vector of the simplicial complex.

A new formula for computing the regularity of monomial ideals is presented.

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.