Stanley–Reisner ring

Summary

The Stanley–Reisner ring of a finite simplicial complex is the quotient of a polynomial ring by the ideal of products of variables over the non-faces. It turns a combinatorial object into a commutative ring. Algebraic invariants of the ring, such as the Hilbert series and the Krull dimension, then encode counts of faces. Its ideal is a squarefree monomial ideal, and every squarefree monomial ideal arises this way. The construction is a main tool in the proof of the upper bound theorem for simplicial spheres.