Conditions for Virtually Cohen--Macaulay Simplicial Complexes

Abstract

A simplicial complex $\hat{I}”$ 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 $\mathrm{codim}(S)$. We provide a sufficient condition on $\hat{I}”$ 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.