The Face Ring of a Simplicial Complex

  • Richard P. Stanley
Part of the Progress in Mathematics book series (PM)


Let Δ be a finite simplicial complex on the vertex set V = {x1, ..., xn}. Recall that this means that Δ is a collection of subsets of V such that F ⊆ G ϵ Δ ⇒ F ϵ Δ and {xi} ϵ Δ for all xi 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.


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Copyright information

© Springer Science+Business Media New York 1983

Authors and Affiliations

  • Richard P. Stanley
    • 1
  1. 1.Mathematics Department, 2-375Massachusetts Institute of TechnologyCambridgeUSA

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