[Reisner's criterion](https://en.wikipedia.org/wiki/Stanley%E2%80%93Reisner_ring) give a complete characterization of Cohen–Macaulay simplicial complexes, based on $link$s of faces of the simplicial complex. Is there a known fact that relate shellability of a simplicial complex to the $link$s of its faces?  

Thank you.



relate shellability of a simplicial complex to the links of its faces
If $\Delta$ is a shellable simplicial complex, then for any face $\sigma \in \Delta$ the link $\mathrm{ln}\_{\Delta} \sigma$ is also a shellable simplicial complex. This is because we can the order the facets of $\mathrm{ln}\_{\Delta} \sigma$ by taking ordering induced by the ordering of the facets of $\Delta$ which gives the shelling. See Proposition 10.14 of Shellable Nonpure Complexes and Posets II by Bjorner and Wachs available [here](http://www.math.miami.edu/~wachs/papers/nonpure2.pdf).


