Combinatorial Invariance of Stanley-reisner Rings

In this short note we show that Stanley-Reisner rings of simplicial complexes, which have had a ‘dramatic application’ in combinatorics [2, p. 41] possess a rigidity property in the sense that they determine their underlying simplicial complexes. For the readers convenience we recall the notion of a Stanley-Reisner ring (for more information the reader is referred to [1, Ch. 5]). Let V be a finite set, called a vertex set later on. A system ∆ of subsets of V is called an abstract simplicial complex (on the vertex set V ) if the following conditions hold: a) {v} ∈ ∆ for any element v ∈ V , b) σ′ ∈ ∆ whenever σ′ ⊂ σ for some σ ∈ ∆. Elements of ∆ will be called faces. Now assume we are given a field k and an abstract simplicial complex ∆ on a vertex set V . The Stanley-Reisner ring corresponding to these data is defined as the quotient ring of the polynomial ring k[v1, . . . , vn]/I, where n = #(V ), the vi are the elements of V and the ideal I is generated by the set of monomials {vi1 · · · vik |{vi1 , . . . , vik} / ∈ ∆}. This k-algebra will be denoted by k[∆] and will be called the Stanley-Reisner ring of ∆. Further, the image of vi in it will again be denoted by vi (they are all different!) and, hence, again will be thought of as elements of V . Theorem . Let k be a field and ∆ and ∆′ be two abstract simplicial complexes defined on the vertex sets V = {v1, . . . , vn} and U = {u1, . . . , um} respectively. Suppose k[∆] and k[∆′] are isomorphic as k-algebras. Then there exists a bijective mapping Ψ : V → U which induces an isomorphism between ∆ and ∆′. Proof. Let f : k[∆] → k[∆′] be a k-isomorphism. By scalar extension we may assume k is algebraically closed. Let us first show that without loss of generality we may also assume f is an isomorphism of augmented k-algebras, where k[∆] is endowed with an augmented 1991 Mathematics Subject Classification. 13F20.