The Face Ring of a Simplicial Complex

Abstract

Let Δ be a finite simplicial complex on the vertex set V = {x₁, ..., x_n}. Recall that this means that Δ is a collection of subsets of V such that F ⊆ G ϵ Δ ⇒ F ϵ Δ and {x_i} ϵ Δ for all x_i ϵ V. The elements of Δ are called faces. If F ϵ Δ, then define dim F := ∣F∣ − 1 and $\Delta : = \mathop {\max }\limits_{F \in \Delta }$. Let d = dim Δ + 1 Given any field k we now define the face ring (or Stanley-Reisner ring) k[Δ] of the complex Δ.