Stanley–Reisner ring

[1] Its properties were investigated by Richard Stanley, Melvin Hochster, and Gerald Reisner in the early 1970s.

The face ring k[Δ] is a multigraded algebra over k all of whose components with respect to the fine grading have dimension at most 1.

This was soon followed up by more precise homological results about face rings due to Melvin Hochster.

Then Richard Stanley found a way to prove the Upper Bound Conjecture for simplicial spheres, which was open at the time, using the face ring construction and Reisner's criterion of Cohen–Macaulayness.

Then the vanishing of the simplicial homology groups in Reisner's criterion is equivalent to the following statement about the reduced and relative singular homology groups of |Δ|: In particular, if the complex Δ is a simplicial sphere, that is, |Δ| is homeomorphic to a sphere, then it is Cohen–Macaulay over any field.