 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.Stanley's idea of translating difficult conjectures in algebraic combinatorics into statements from commutative algebra and proving them by means of homological techniques was the origin of the rapidly developing field of combinatorial commutative algebra.