GEORGIAN MATHEMATICAL JOURNAL: Vol. 3, No. 4, 1996, 315-318

COMBINATORIAL INVARIANCE OF STANLEY–REISNER RINGS

W. BRUNS AND J. GUBELADZE

Abstract. 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 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 to be called below a vertex set. A system Δ of subsets of V is called an abstract simplicial complex (on the vertex set V) if the following conditions hold:

(a) {v} ∈ Δ for any element v ∈ V,

(b) σ′ ∈ Δ whenever σ′ ⊂ σ for some σ ∈ Δ.

Elements of Δ will be called faces.

Now assume we are given a field k and an abstract simplicial complex Δ on a vertex set V. The Stanley–Reisner ring corresponding to these data is defined as the quotient ring of the polynomial ring k[v_{1}, …, v_{n}]/I, where n = #(V), the v_{i} are the elements of V, and the ideal I is generated by the set of monomials {v_{i_{1}} ⋯ v_{i_{k}}|{v_{i_{1}}, …, v_{i_{k}}} ∉ Δ}. This k-algebra will be denoted by k[Δ] and called the Stanley–Reisner ring of Δ. Further, the image of v_{i} in it will again be denoted by v_{i} (they are all different!) and hence will again be thought of as elements of V.

Theorem. Let k be a field, and Δ and Δ′ be two abstract simplicial complexes defined on the vertex sets V = {v_{1}, …, v_{n}} and U = {u_{1}, …, u_{m}} respectively. Suppose k[Δ] and k[Δ′] are isomorphic as k-algebras. Then there exists a bijective mapping Ψ : V → U which induces an isomorphism between Δ and Δ′.

Proof. Let f : k[Δ] → k[Δ′] be a k-isomorphism. By scalar extension we may assume k is algebraically closed. Let us first show that without loss of

315 generality we may also assume f is an isomorphism of augmented k-algebras, where k[Δ] is endowed with an augmented k-algebra structure induced by v_{i} ↦ 0, and similarly for k[Δ′]. Indeed, if v_{i} is a zero-divisor in k[Δ] for some i ∈ [1, n], then its image in k[Δ′] cannot have a nonzero constant term (with respect to the uniquely determined canonical expansion). So the deviation from “being augmented” for f can appear only at the elements v_{i} ∈ V which are not zero-divisors. It is easy to observe that v_{i} ∈ V is not a zero-divisor in k[Δ] if and only if it is a variable for k[Δ], i.e., k[Δ] = k[Δ^{i}][v_{i}], where Δ^{i} is a simplicial subcomplex of Δ consisting of those faces which do not contain v_{i}, and this v_{i} on the right-hand side is understood as a variable. Let {v_{i_{1}}, …, v_{i_{k}}} be the set of all nonzero-divisor vertices and {c_{i_{1}}, …, c_{i_{k}}} be the set of constant terms in the canonical expansions of f(v_{i_{1}}), …, f(v_{i_{k}}) respectively. Consider the elements w_{i_{1}} = v_{i_{1}} − c_{i_{1}}, …, w_{i_{k}} − c_{i_{k}} ∈ k[Δ]. Clearly, they are all different. Let W = {w_{1}, …, w_{n}} be the set obtained from V by substituting w_{i_{j}} by v_{i_{j}}, respectively, and let Δ^{∘} be the abstract simplicial complex on the vertex set W induced by the natural bijection between V and W. Since all of the v_{i_{j}} are (independent!) variables for k[Δ] (as remarked above), we conclude that k[Δ] = k[Δ^{∘}] and f is an augmented isomorphism between k[Δ^{∘}], considered as an augmented k-algebra with respect to w_{i} ↦ 0, and k[Δ′]. So from the very beginning we can assume f is augmented.

1991 Mathematics Subject Classification. 13F20.

Key words and phrases. Simplicial complexes, Stanley–Reisner rings.

1072-947X/96/0700-0315$09.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1996 Plenum Publishing Corporation

Next we pass to the corresponding graded isomorphism (with respect to the augmentation ideals)

gr(f) : gr(k[Δ]) → gr(k[Δ′]).

But gr(k[Δ]) = k[Δ] and gr(k[Δ′]) = k[Δ′]. This means that we may also assume f is a graded k-isomorphism of graded k-algebras k[Δ] and k[Δ′] where deg(v_{1}) = … = deg(v_{n}) = deg(u_{1}) = … = deg(u_{m}) = 1. Now passing to the geometrical picture (i.e., to the closed points of the corresponding affine schemes) we obtain the following situation: we are given two k-linear spaces

k^{n} = maxSpec(k[v_{1}, …, v_{n}])

and

k^{m} = maxSpec(k[u_{1}, …, u_{m}])

(the v_{i} and u_{j} are considered as variables) and two arrangments of k-linear coordinate subspaces (of appropriate dimensions)

Δ^{∗} = maxSpec(k[Δ]) ⊂ k^{n},
(Δ′)^{∗} = maxSpec(k[Δ′]) ⊂ k^{m}.

More precisely, Δ^{∗} consists of those coordinate subspaces of k^{n} which are spanned by the coordinate directions of v_{i_{1}}, …, v_{i_{k}} whenever {v_{i_{1}}, …, v_{i_{k}}}∈ Δ, and similarly for (Δ′)^{∗}. This claim follows directly from the equality

Δ^{∗} = ⋂_{{v_{i_{1}} ,... ,v_{i_{k}} }∉Δ}(v_{i_{1}}^{∘} ∪ ⋯ ∪ v_{i_{k}}^{∘}),

where v_{i_{j}}^{∘} denotes the coordinate hyperplane of dimension n − 1 avoiding v_{i_{j}}, and the similar one for (Δ′)^{∗}.

So for each maximal face (with respect to the inclusion) σ ∈ Δ we have the corresponding coordinate linear subspace L_{σ} ⊂ k^{n} and

Δ^{∗} = ⋃_{σ a maximal face of Δ} L_{σ}.

Similarly, for each maximal face σ′ ∈ Δ′ we have the corresponding coordinate linear subspace M_{σ′} ⊂ k^{m} and

(Δ′)^{∗} = ⋃_{σ′ a maximal face of Δ′} M_{σ′}.

The corresponding algebraic map

f^{∗} : (Δ′)^{∗} → Δ^{∗}

will be the restriction of the k-linear isomorphism

F^{∗} : k^{m} → k^{n}

contravariantly corresponding to the (uniquely determined) graded k-iso-morphism F from the commutative square

k[v_{1}, …, v_{n}] ⟶^{F} k[u_{1}, …, u_{m}]
↓  ↓
k[Δ]  ⟶^{f}  k[Δ′].

This gives rise to the well defined bijective map

Φ : (maximal faces of Δ′) → (maximal faces of Δ).

Namely, Φ(σ′) = (the maximal face σ of Δ for which L_{σ} = f^{∗}(M_{σ′})).

After this “linear” interpretation it becomes obvious that m = n and #σ′ = #Φ(σ′) for each maximal σ′ ∈ Δ′. Moreover,

#(σ′_{1} ∩ … ∩ σ′_{t}) = #(Φ(σ′_{1}) ∩ ⋯ ∩ Φ(σ′_{t})).  (∗)

Indeed,

#(σ′_{1} ∩ ⋯ ∩ σ′_{t}) = dim_{k}(M_{σ′_{1}} ∩ ⋯ ∩ M_{σ′_{t}})
= dim_{k}(f^{∗}(M_{σ′_{1}}) ∩ ⋯ ∩ f^{∗}(M_{σ′_{t}}))
= dim_{k}(L_{Φ(σ′_{1})} ∩ ⋯ ∩ L_{Φ(σ′_{t})})
= #(Φ(σ′_{1}) ∩ ⋯ ∩ Φ(σ′_{t})).

Now we introduce the following equivalence relations on the vertex sets V and U: for v_{i_{1}}, v_{i_{2}} ∈ V (u_{j_{1}}, u_{j_{2}} ∈ U) we put v_{i_{1}} ∼ v_{i_{2}} if and only if the two sets of maximal faces of Δ containing v_{i_{1}} and v_{i_{2}} respectively coincide (and similarly for u_{j_{1}} and u_{j_{2}}). The equivalence classes in V will be the minimal (with respect to inclusion) nonempty intersections of maximal faces of Δ (and similarly for the vertex set U). Accordingly, these equivalence classes will be in one-to-one correspondence (via Φ) with the minimal nonzero intersections (w.r.t. inclusion) of the linear subspaces L_{σ} ⊂ k^{n} (similarly for the equvalence classes in U and the linear subspaces M_{σ′} ⊂ k^{m}). Since we are given a global linear isomorphism F^{∗}, using Φ we immediately see that the two systems of equivalence classes are in natural bijective correspondence. By (∗) the corresponding equivalence classes have the same quantities of elements. This gives rise in a natural way to the bijective mapping ψ : U → V which satisfies the condition that u ∈ σ′ if and only if ψ(u) ∈ Φ(σ′), where u ∈ U and σ′ ∈ Δ′ is a maximal face. Since any face in an abstract simplicial complex is contained in some maximal face, we finally arrive at the conclusion that Ψ = (ψ)^{−1} : V → U satisfies the desired condition.

Acknowledgment

J. Gubeladze was supported in part by DFG and ISF.

References

1. W. Bruns and J. Herzog, Cohen–Macaulay Rings. Cambridge Studies in Advanced Mathematics 39, Cambridge University Press, Cambridge, 1993.

2. T. Hibi, Algebraic Combinatorics on Convex Polytopes. Carslaw Publications, 1992.

(Received 26.06.1995)

Authors’ addresses:

Winfried Bruns

Universität Osnabrück

Fachbereich

Mathematik/Informatik

49069 Osnabrück, Germany Joseph Gubeladze A. Razmadze Mathematical Institute Georgian Academy of Sciences 1, M. Aleksidze St., Tbibisi 380093 Republic of Georgia