Context: I am looking for topics for an final exam talk of a commutative algebra course.
I have come across the notion of Stanley-Reisner rings in the Miller and Sturmfels' book Combinatorial Commutative Algebra and Stanley's book Combinatorics and Commutative Algebra.
What are some somewhat self-contained applications of Stanley-Reisner rings to the study of simplicial complexes?
I am aware that, for example, they were used by Stanley to prove the Upper Bound Conjecture for simplicial spheres. The proof found in Stanley's book however contains a fair amount of non-trivial results. For example, a result by Hatcher that describes a necessary and sufficient topological condition for a Stanley-Reisner ring $k[\Delta]$ to be Cohen-Macaulay is used.
Edit: by "somewhat self-contained" I mean for the prerequisites to be no more than an elementary course in commutative algebra (we have covered most of the topics in Atiyah-Macdonald) and algebraic/simplicial topology.