Conditions for virtually Cohen–Macaulay simplicial complexes

Abstract

A simplicial complex Δ is a virtually Cohen–Macaulay simplicial complex if its associated Stanley-Reisner ring S has a virtual resolution, as defined by Berkesch, Erman, and Smith, of length $\mathrm{codim}\left(S\right)$. We provide a sufficient condition on Δ to be a virtually Cohen–Macaulay simplicial complex. We also introduce virtually shellable simplicial complexes, a generalization of shellable simplicial complexes. Virtually shellable complexes have the property that they are virtually Cohen–Macaulay, generalizing the well-known fact that shellable simplicial complexes are Cohen–Macaulay.