COHEN-MACAULAY AND GORENSTEIN COMPLEXES FROM A TOPOLOGICAL POINT OF VIEW

DIETRICH NOTBOHM

Abstract. The main invariant to study the combinatorics of a simplicial complex K is the associated face ring or Stanley-Reisner algebra. Reisner respectively Stanley explained in which sense Cohen-Macaulay and Gorenstein properties of the face ring are reflected by geometric and/or combinatoric properties of the simplicial complex. We give a new proof for these result by homotopy theoretic methods and constructions. Our approach is based on ideas used very successfully in the analysis of the homotopy theory of classifying spaces.

1. Introduction

The main tool and invariant for understanding the combinatorics of a finite simplicial complex is the associated face ring or Stanley-Reisner algebra which is a quotient of a polynomial algebra generated by the vertices. It is of interest to which extend algebraic properties of the face ring are reflected by combinatorial or geometric properties of the simplicial complex. For example, Reisner characterized all simplicial complexes whose face ring is Cohen-Macaulay [18]. And Stanley proved a similar result for Gorenstein face rings [20]. In this paper we will look at these result with the eyes of a topologist and reprove both results with methods and ideas from homotopy theory, in particular from the homotopy theory of classifying spaces. For the topological proof we introduce and discuss some new spaces associated with simplicial complexes, which, as we feel, deserve interest in their own right.

Let K be an abstract simplicial complex with m vertices given by the set V = {1, ..., m} That is, K = {σ_{1}, ..., σ_{r}} consists of a finite set of faces σ_{i} ⊂ V, which is closed with respect to formation of subsets. The dimension of K is denoted by dim K = n − 1. That is every face σ of K has order ♯σ ≤ n and there exists a face μ of order ♯μ = n. We consider the empty set ∅ as a face of K.

For a commutative ring R with unit we denote by R(K) the associated Stanley-Reisner algebra of K over the ring R. It is the quotient R[V]/(v_{σ} : σ ∉ K), where R[V] =^{def} R[v_{1}, ...v_{m}] is the polynomial algebra on m-generators and v_{σ} =^{def} ∏_{j∈σ} v_{j}. We can think of R(K) as a graded object. Since we want to bring topology into the game we will choose the topological grading and give the generators of R(K) and R[V] the degree 2.

1991 Mathematics Subject Classification. 13F55, 55R35.

Key words and phrases. simplicial complex, Stanley-Reisner algebra, face ring, Cohen-Macaulay, Gorenstein, homology decomposition.

Each abstract simplicial complex K has a geometric realization, denoted by |K|. Let Δ(V) denote the full simplicial complex whose faces are given by all subsets of V. One realization of Δ(V) is given by the standard m−1-dimensional simplex |Δ(V)| =^{def} {Σ_{i=1}^{m}t_{i}e_{i} : 0 ≤ t_{i} ≤ 1, Σ_{i}t_{i} = 1} ⊂ ℝ^{m}, where e_{1}, ..., e_{m} denotes the standard basis. And a topological realization of K is given by the subset |K| ⊂ |Δ(V)| defined by the subset relation K ⊂ Δ(V). We define the homology H_{∗}(K) and cohomology H^{∗}(K) of K as the homology H^{∗}(|K|) and cohomology H^{∗}(|K|) of the topological realization.

Before we can state the theorems of Reisner and Stanley, we have to recall some notions. We call a simplicial complex K Cohen-Macaulay or Gorenstein over a field 𝔽 if 𝔽(K) is a Cohen-Macaulay or Gorenstein algebra over 𝔽. We call K a Gorenstein^{∗} complex if it is Gorenstein and if V equals the union of all minimal missing simplices of K. A subset μ ⊂ V is minimal missing if μ ∉ K and for each σ ⊊ μ we have σ ∈ K. Moreover, for any face σ ∈ K, the link of σ is defined as the simplicial complex

link_{K}(σ) =^{def} {τ \ σ : σ ⊂ τ ∈ K}.

Now the theorems of Reisner and Stanley read as follows: Theorem 1.1. (Reisner [18]) Let 𝔽 be a field and K a simplicial complex. Then, K is Cohen-Macaulay over 𝔽 if and only if for each face σ of K, including the empty face,

H̃^{i}(link_{K}(σ); 𝔽) = 0 for i < dim link_{K}(σ)

Theorem 1.2. (Stanley [20]) Let 𝔽 be a field and K a simplicial complex. Then, K is Gorenstein^{∗} over 𝔽 if and only if

H̃^{i}(link_{K}(σ); 𝔽) ≅ \{ \begin{matrix}𝔽 & for i = dim link_{K}(σ) \\ 0 & for i ≠ dim link_{K}(σ) \end{matrix}

For a definition of Cohen-Macaulay and Gorenstein properties see either [2], [20] or Section 4. There also exists a version of the second statement which deals with general Gorenstein complexes. But for simplification we formulated the result for Gorenstein^{∗} complexes.

Reisner used methods of commutative algebra, in particular the machinery of local cohomology of modules, to prove his theorem. Another proof by Hochster (unpublished, see [20]) is based on similar methods and a detailed analysis of the Poincaré series of R(K). Similar ideas were used by Stanley in his approach towards Gorenstein complexes. In this paper we want to give a different proof for both results. Our proof is based on topological constructions related to and based on topological interpretations of the combinatorial data of K. For example, there exists a topological space c(K) such that H^{∗}(X; ℤ) ≅ ℤ(K) as a graded algebra. These spaces can be constructed as the Borel construction of toric manifolds [4], as a (pointed) colimit of a particular diagram [3] or as the homotopy colimit of the same diagram [15]. This last construction is the most appropriate for doing homotopy theory and is the one which we will use in this paper. Using the homotopy colimit construction, if dim K = n − 1, one can construct a very interesting non trivial map f : c(K) ⟶ BU(n) [17]. The Chern classes of the associated vector bundle are given by the elementary symmetric polynomials in the generators of ℤ(K) (see Section 3). The homotopy fibre X_{K} is a finite CW -complex, which contains a large amount of information about the associated Stanley-Reisner algebra. Theorem 1.3.

(i) A simplicial complex K is Cohen-Macaulay over 𝔽, if and only if H^{∗}(X_{K}; 𝔽) is concentrated in even degrees. (ii) K is Gorenstein over 𝔽 if and only if H^{∗}(X_{K}; 𝔽) is a Poincaré duality algebra and concentrated in even degrees.

This theorem translates Cohen-Macaulay and Gorenstein properties of 𝔽(K) into conditions on X_{K} and is the key result necessary for our proof of the results of Reisner and Stanley.

The paper is organised as follows. In the next two sections we provide the basic topological ingredients necessary for the proof of Theorem 1.1 and Theorem 1.2. In particular, we recall the above mentioned homotopy colimit construction for the space c(K) in Section 2 and discuss the map f_{K} : c(K) ⟶ BU(n) in Section 3. In Section 4 we provide definitions for Cohen-Macaulay and Gorenstein algebras appropriate for our purpose, express these properties in terms of the homotopy fibre X_{K} and prove Theorem 1.3. In Section 5 we discuss homotopy fixed point sets and study them for particular actions of elementary abelian groups on X_{K}. The final three sections are devoted to a proof of Theorem 1.1 and Theorem 1.2.

We will fix the following notation throughout. K always denotes a (n − 1)-dimensional abstract finite simplicial complex with m-vertices. The set of vertices will be denoted by V =^{def} V_{K} =^{def} {1, ..., m}. We denote by ab the category of abelian groups, by Top the category of topological spaces and by Top^{+} the category of pointed topological spaces. Mostly, we are working over commutative rings with unit or fields. In particular, R will always denote such a commutative ring and 𝔽 a field. When we deal with torsion groups, we will use the topological convention for the grading. Projective resolutions are considered as non positively graded cochain complexes and torsion groups are non positively graded objects denoted by Tor_{A}^{j}(M, N), where j ≤ 0.

It is my pleasure to thank N. Ray and T. Panov for introducing me to this subject as well as for plenty of helpful discussions.

2. Pointed and unpointed homotopy colimits

Given a category c and functor F : c ⟶ Top, the colimit colimc F behaves particularly poorly in the context of homotopy theory, since object-wise equivalent diagrams may well have homotopy inequivalent colimits. The standard procedure for dealing with this situation is to introduce the left derived functor, known as the homotopy colimit. Following [11], for example, hocolimc F may be described by the two-sided bar construction. In a similar fashion, we can construct a pointed homotopy colimit hocolimc^{+} G for a functor G : c ⟶ Top^{+}. Composing G with the forgetful functor ϕ : Top^{+} ⟶ Top induces an identity colimc ϕG = colimc^{+} G and a cofibre sequence

Bc ⟶ hocolimc ϕG ⟶ hocolimc^{+} G

[1]. Here, Bc is the classifying space of the category c, that is the topological realization of the nerve N(c). For details of the homotopy colimit construction see [1] [6] or [7].

The cohomology of (pointed) homotopy colimits can be calculated with the help of the Bousfield-Kan spectral sequence [1]. And this tool is more important for our purpose than the actual construction of the homotopy colimit. In both cases, this is a first quadrant spectral sequence and has in the case of the unpointed homotopy colimit the form

E_{2}^{i,j} =^{def} lim c^{op}^{i}H^{j}(F) ⟹ H^{i+j}(hocolimc F)

and in the case of pointed homotopy colimit the form

E_{2}^{i,j} =^{def} lim c^{op}^{i}H̃^{j}(F) ⟹ H̃^{i+j}(hocolimc F)

In both cases, differentials d_{r} : E_{r}^{∗,∗} ⟶ E_{r}^{∗,∗} have the degree (r, 1 − r).

Let ϕ : c^{op} ⟶ ab be a (covariant) functor, e.g. ϕ =^{def} H^{∗}(F). Higher derived limits of ϕ can be thought of as the cohomology groups of a certain cochain complex (C^{∗}(c, F), δ). The groups are defined as

C^{r}(c; F) =^{def} ∏_{c_{o}→c_{1}→...→c_{r}} F (c_{0}) for r ≥ 0.

Here, the morphism are morphisms in c and not in c^{op}. The differential δ : C^{n}(c; F) ⟶ C^{n+1}(c; F) is given by the alternating sum ∑_{k=0}^{r}(−1)^{k}δ^{k} where δ_{k} is defined on u ∈ C^{n}(ϕ) by

δ^{k}(u)(c_{0} → … → c_{n+1}) =^{def} \begin{cases}u(c_{0} → … → ĉ_{k} → … → c_{n+1}) & for k ≠ 0 \\ ϕ(c_{o} → c_{1})u(c_{1} → … → c_{n}) & for k = 0. \end{cases}

We may and will replace C^{∗}(c; F) by it’s quotient N^{∗}(c, F) of normalised cochains, given by the product over chains c_{0} → c_{1}... → c_{r} with distinct objects.

In most cases we consider, the Bousfield-Kan spectral sequence has a particular simple form. All higher limits will vanish. Following Dwyer [7] we turn this property into the following definition. Definition 2.1. Let F : c ⟶ Top be a functor and R a commutative ring with unit. We call a map f : hocolimc F ⟶ Y a sharp R-homology decomposition, if limc^{op}^{i} H^{∗}(F; R) = 0 for i ≥ 1 and if f induces an isomorphism limc^{op} H^{∗}(F; R) ≅ H^{∗}(Y; R). If F takes values in Top^{+}, then we replace cohomology by reduced cohomology.

Using the subset relation on the faces, a simplicial complex K can be interpreted as a poset and therefore as a category which we denote by cat(K). This category contains the empty set ∅ as an initial object. If we want to exclude ∅ we denote this by K^{×} respectively by cat(K^{×}). In these cases the classifying space Bcat(K) is equivalent to the cone of the topological realization |K| and Bcat(K^{×}) ≃ |K|.

For a pointed topological space Y we can define functors

Y^{K} : cat(K) ⟶ Top^{+}, Y^{K} : cat(K) ⟶ Top,

which assigns the Cartesian product Y^{σ} to each face σ of K. The value of Y^{K} on σ ⊂ τ is the inclusion Y^{σ} ⊂ Y^{τ} where the superfluous coordinates are set to ∗. We note that Y^{K}(∅) =^{def} ∗. Moreover, this functor comes with a natural transformation Y^{K} ⟶ 1_{Y^{V}} where 1_{Y^{V}} denotes the constant functor mapping each face to Y^{V} = Y^{m} and each morphism to the identity map.

Since cat(K) has an initial object, the classifying space Bcat(K) is contractible [1] and the above cofibre sequence degenerates to a homotopy equivalence

hocolim_{cat(K)} Y^{K} ⟶^{≃} hocolim_{cat(K)}^{+} Y^{K}.

We want to specialise further. Let T =^{def} S^{1} denote the 1-dimensional torus and BT = ℂP^{∞} the classifying space of T respectively the infinite dimensional complex projective space. For the functor BT^{K} : cat(K) ⟶ Top, in fact for all functors of the form Y^{K}, the higher derived limits of the Bousfield-Kan spectral sequence vanish and the spectral sequence collapses at the E_{2}-term. Only the actual inverse limit contributes something non trivial, namely the associated Stanley-Reisner algebra [15]. Defining c(K) =^{def} hocolim_{cat(K)} BT^{K} we can formulate this as follows. Theorem 2.2. [15]

(i) H^{∗}(hocolim_{cat(K)} BT^{K}; R) ≅ R(K). (ii) hocolim_{cat(K)} BT^{K} ⟶ c(K) is a sharp R-homology decomposition as well as hocolim_{cat(K)}^{+} BT^{K} ⟶ c(K). (iii) The natural map c(K) ⟶ BT^{m} realizes the algebra map ℤ[V] ⟶ ℤ(K).

Proof. Part (i) and (iii) and the first half of Part (ii)are already proven in [15]. The second claim of Part (ii) follows from the equivalence between the pointed and unpointed homotopy colimit, from the fact that reduced cohomology is a natural retract from cohomology and from part (i). □

For our purpose we will also need another homology decomposition for our space c(K). For two simplicial complexes K and L we define the join product by K ∗ L =^{def} {(σ, τ) : σ ∈ K, τ ∈ L}. Then, by construction, we have c(K ∗ L) = c(K) × c(L). We also notice that, for the full simplex Δ(V) =^{def} {σ ⊂ V } on a vertex set V, we have c(Δ(V)) ≃ BT^{V}. This follows from the fact that cat(Δ(V)) has V as a terminal object and that therefore hocolim_{cat(Δ(V))} BT^{Δ(V)} ≃ BT^{K}(V) = BT^{V}. For every face σ ∈ K we denote by st(σ) =^{def} st_{K}(σ) =^{def} {τ ∈ K : σ ∪ τ ∈ K} the star of σ. This is again a simplicial complex and, since st(σ) = Δ(σ) ∗ link(σ) we have c(st(σ)) ≃ BT^{σ} × c(link(σ)).Moreover, for σ ⊂ τ ∈ K, we have st_{K}(τ) ⊂ st_{K}(σ) which induces a pointed map c(st_{K}(τ)) ⟶ c(st_{K}(σ)). This establishes a functor

cst_{K} : cat(K)^{op} ⟶ Top^{+}

defined by cst_{K}(σ) =^{def} c(st_{K}(σ)). Since the category cat(K)^{op} has an terminal object, namely ∅, we have an obvious homotopy equivalence hocolim_{cat(K)^{op}}^{+} cst_{K} ≃ cst_{K}(∅) = c(K). But restricting cst_{K} to the full subcategory cat(K^{×})^{op} produces a map hocolim_{cat(K^{×})^{op}}^{+} cst_{K} ⟶ c(K) which turns out to be an equivalence as well.

Theorem 2.3.

(i) hocolim_{cat(K^{×})^{op}}^{+} cst_{K} ⟶ c(K) is a homotopy equivalence and a sharp R-homology decomposition. (ii) There exists a cofibration

|K| ⟶ hocolim_{cat(K^{×})^{op}} cst_{K} ⟶ c(K).

(iii)

lim_{cat(K^{×})^{op}}^{i} H^{∗}(cst_{K}; R) ≅ \begin{cases}R(K) ⊕ H̃^{0}(K; R) & for i = 0 \\ H^{i}(K; R) & for i > 0. \end{cases}

The rest of this section is devoted to a proof of this theorem. We will compare the two homotopy colimits, hocolim_{cat(K^{×})^{op}}^{+} cst_{K} and hocolim_{cat(K)}^{+} BT^{K} and do this in several steps. First we show that is sufficient to take the pointed homotopy colimit of BT^{K} over the category cat(K^{×}). Proposition 2.4. We have an equivalence

hocolim_{cat(K^{×})}^{+} BT^{K} ⟶^{≃} hocolim_{cat(K)}^{+} BT^{K}.

Moreover, hocolim_{cat(K^{×})}^{+} BT^{K} ⟶ c(K) is also a sharp R-homology decomposition.

Proof. Since H̃^{∗}(∗) = 0, we have an isomorphism

N(cat(K^{×})^{op}, H̃^{∗}(BT^{K})) ≅ N(cat(K)^{op}, H̃^{∗}(BT^{K}))

of normalised cochain complexes. This shows that the map between both pointed homotopy colimits produces an isomorphism between higher limits, an isomorphism between the Bousfield-Kan spectral sequences and therefore an isomorphism in cohomology. Moreover, both homotopy colimits are simply connected. Hence, by the Whitehead theorem, the map also induces an isomorphism between the homotopy groups and is therefore a homotopy equivalence, which proves the second part. □

Now, we construct a category c which contains both, cat(K^{×})^{op} and cat(K^{×}). This will allow to compare the pointed homotopy colimits hocolim_{cat(K^{×})}^{+} BT^{K} and hocolim_{cat(K^{×})^{op}}^{+} cst_{K}. To do this we will distinguish between the objects of cat(K^{×})^{op}, denoted by τ^{op}, and the objects of cat(K^{×}), denoted by τ. The objects of c are given by the union of the objects of cat(K^{×})^{op} and cat(K^{×}). That is each face τ of K generates two objects in c, namely τ and τ^{op}. There exists at most one morphism between two objects. And there are morphism ρ → τ, τ^{op} → ρ^{op} and τ → ρ^{op} if and only if ρ ⊂ τ. We have obvious inclusions ϕ : cat(K^{×})^{op} ⟶ c and ψ : cat(K^{×}) ⟶ c.

Since for ρ ⊂ τ ∈ K, the set τ is a face of st_{K}(ρ), there exists a well defined pointed map BT^{τ} ⟶ c(st_{K}(ρ)). These maps are compatible with the pointed inclusions BT^{α} ⊂ BT^{β} as well as with the pointed maps c(st_{K}(β)) → c(st_{K}(α)) for α ⊂ β. Therefore, we can define a functor F : c ⟶ Top^{+} such that F(τ) =^{def} BT^{τ} and F(τ^{op}) =^{def} cst_{K}(τ). In particular, F ψ = BT^{K} and F ϕ = cst_{K}.

Given a functor Φ : d′ ⟶ d, for each object d ∈ D, we can define the over category Φ ↓ d. The objects are given by morphisms i′ : Φ(d′) → d in D, where d′ is an object of d′. And a morphism (i′ : Φ(d′) → d) → (i′′ : Φ(d′′) → d) is given by a morphism j : d′ → d′′ of d′ such that i′′Φ(j) = i′. The under category d ↓ ρ is defined similarly. As usual, we say that Φ is left cofinal if all over categories Φ ↓ d and right cofinal if all under categories d ↓ ρ are contractible; i.e. the classifying spaces are contractible.

The following series of statement shows how to compare the two homotopy colimits in question. Lemma 2.5.

(i) For each object τ ∈ cat(K^{×}) ⊂ c, the under category τ ↓ ϕ is contractible. (ii) For each object τ^{op} ∈ cat(K^{×})^{op} ⊂ c, there exists an isomorphism of categories cat(link_{K}(τ)) ≅ ψ ↓ τ^{op} induced by ρ ↦ (ρ ∪ τ → τ^{op}).

Proof. The first claim follows from the fact that τ → τ^{op} is a terminal object of τ ↓ ϕ. The second claim follows from an easy straight forward calculation. □

Proposition 2.6.

(i) hocolim_{cat(K^{×})^{op}}^{+} cst_{K} ≃ hocolimc^{+} F. (ii) lim _{cat(K^{×})}^{i}H̃^{∗}(cst_{K}) ≅ lim c^{op}^{i}H̃^{∗}(F)

Proof. Since for every object τ^{op} ∈ c the under category τ^{op} ↓ ϕ is obviously contractible, Lemma 2.5 implies that the inclusion functor cat(K^{×})^{op} → c is right cofinal. Since the restriction of F |_{cat(K^{×})^{op}} = cst_{K}, the equivalence between the pointed homotopy colimits in part (i) follows from [1].

For the isomorphism in (ii) we need that (cat(K^{×})^{op})^{op} → c^{op} is left cofinal [1], i.e. all under categories c ↓ ϕ^{op} ≅ (ϕ ↓ c)^{op} are contractible. But this follows as above. □

Proposition 2.7.

(i) hocolim_{cat(K^{×})}^{+} BT^{K} ≃ hocolimc^{+} F. (ii) lim _{cat(K^{×})^{op}}^{i}H̃^{∗}(BT^{K}) ≅ lim c^{op}^{i}H̃^{∗}(F). In particular, for i ≥ 1,

lim c^{op}^{i}H̃^{∗}(F) = 0.

Proof. The left Kan extension L =^{def} L_{BT^{K}} of the functor BT^{K} : cat(K^{×}) ⟶ Top^{+} along the functor cat(K^{×}) ⟶ c is defined by L(c) =^{def} hocolim_{ψ↓c}^{+} BT^{K} . And hocolimc L ≃ hocolim_{cat(K^{×})} BT^{K} [1]. By Lemma 2.5 and Theorem 2.2, there exists a natural transformation L ⟶ F, which induces a homotopy equivalence at each object. This proves the first part.

For the second part we apply the composition of functor spectral sequence (e.g. see [10] where the dual situation for colimits is discussed in detail). That is there exists a spectral sequence

E_{2}^{i,j} =^{def} lim c^{op}^{i} lim _{c↓ψ^{op}}^{j}H̃^{∗}(BT^{K}) ⟹ lim _{cat(K^{×})^{op}}^{i+j}H̃^{∗}(BT^{K}).

By Theorem 2.2 and Lemma 2.5 lim _{c↓ψ^{op}}^{j}H̃^{∗}(BT^{K}) = 0 for j ≥ 1 and lim_{c↓ψ^{op}} H̃^{∗}(BT^{K}) = H̃^{∗}(F(c)). This proves part (ii). □

Proof of Theorem 2.3. The first part follows from Proposition 2.6 and Proposition 2.7, and the second part from the general relation between pointed and unpointed homotopy colimits as discussed above.

In the rest of the proof all limits are taken over cat(K^{×})^{op}. Let 1_{R} denote the constant functor. Since a category and it’s opposite category have the same geometric realization, we have lim^{i}1_{R} ≅ H^{i}(K). The short exact sequence

0 ⟶ H̃^{∗}(cst_{K}) ⟶ H^{∗}(cst_{K}) ⟶ 1_{R} ⟶ 0

of functors establishes a long exact sequence of the higher limits. By part (i), this long exact sequence splits into a short exact sequence

0 ⟶ H̃^{∗}(c(K)) ≅ lim^{0}H̃^{∗}(cst_{K}) ⟶ lim^{0}H^{∗}(cst_{K}) ⟶ lim^{0}1_{R} ≅ H^{0}(K) ⟶ 0

and isomorphisms lim^{i}H^{∗}(cst_{K}) ≅ lim^{i}1_{R} ≅ H^{i}(K) for i ≥ 1. The short exact sequence can be rewritten as

0 ⟶ R(K) ⟶ lim^{0}R(st_{K}) ⟶ H̃^{0}(K) ⟶ 0,

which proves part (iii). □ Remark 2.8. For later purpose we will calculate the higher limits for particular functors. Let M be an abelian group and ψ_{M} : cat(K)^{op} ⟶ ab be the atomic functor given by ϕ_{M}(∅) =^{def} M and ϕ_{M}(σ) =^{def} 0 for σ ≠ ∅.Let 1_{M} : cat(K)^{op} ⟶ ab denote the constant functor which maps all objects to M and all morphisms to the identity. Then, we have a short exact sequence

0 ⟶ ϕ_{M} ⟶ 1_{M} ⟶ ψ_{M} =^{def} 1_{M}/ϕ_{M} ⟶ 0.

Since lim_{cat(K)^{op}} 1_{M} ≅ M ≅ lim_{cat(K)^{op}} ψ_{M}, the long exact sequence for the higher limits establishes isomorphisms lim _{cat(K)^{op}}^{i}ψ_{M} ≅ lim^{i+1}ϕ_{M}. Since N^{∗}(cat(K)^{op}; ψ_{M}) ≅ N^{∗}(cat(K^{×})^{op}; 1_{M}), we get a sequence of isomorphisms

H^{i}(K) ≅ lim _{cat(K^{×})^{op}}^{i}1_{M} ≅ lim _{cat(K)^{op}}^{i}ψ_{M} ≅ lim _{cat(K)^{op}}^{i+1}ϕ_{M}.

By construction, this composition is natural with respect to maps between simplicial complexes.

3. A vector bundle over c(K)

In [17], Theorem 2.2 was used to construct a particular nontrivial map c(K) ⟶ BU(n), whose construction we recall next.

Let T^{m} ⟶ U(m) denote the maximal torus of the unitary group U(m) given by diagonal matrices. The cohomology H^{∗}(BU(m); ℤ) ≅ ℤ[c_{1}, ..., c_{m}] of the classifying space BU(m) is a polynomial algebra generated by the Chern classes c_{i} and H^{∗}(BT^{m}; ℤ) ≅ ℤ[V] is a polynomial algebra as well which we identify with the polynomial algebra generated by the set V of vertices. The above map induces an isomorphism H^{∗}(BU(m); ℤ) ≅ ℤ[V]^{Σ_{m}} ≅ ℤ[σ_{1}, ..., σ_{m}], where we identify Σ_{m} with the Weyl group of U(m) and where Σ_{m} acts on ℤ[V] by permutations . Here, σ_{j} denotes the j-th elementary symmetric polynomial. We can and will identify the Chern classes c_{j} with the elementary symmetric polynomials σ_{j}.

Since dim K = n − 1, a monomial v_{τ} vanishes in ℤ(K) if ♯τ ≥ n + 1. Hence, the composition ℤ[c_{1}, ..., c_{m}] ⟶ ℤ[V] ⟶ ℤ(K) factors through ℤ[c_{1}, ..., c_{n}] and establishes a commutative diagram

ℤ[c_{1}, ..., c_{m}] ⟶ ℤ[V] ≅ ℤ[v_{1}, ...., v_{m}]
↓  ↓
ℤ[c_{1}, ..., c_{n}] ⟶^{ψ}  ℤ(K)

The left vertical arrow is induced by the canonical inclusion BU(n) ⟶ BU(m), i.e. we put c_{j} = 0 for n+1 ≤ j ≤ m. The following statement concerning a topological realization of ψ is proven in [17]. Theorem 3.1. [17] There exists a topological realization f_{K} : c(K) ⟶ BU(n) of ψ, i.e. H^{∗}(f; ℤ) = ψ, which is unique up to homotopy. Moreover, the diagram

c(K) ⟶^{f_{K}} BU(n)
↓  ↓
BT^{m} ⟶ BU(m)

commutes up to homotopy.

The map f_{K} is constructed as follows. Let T^{n} ⊂ U(n) denote the maximal torus of U(n) given by diagonal matrices. Since ♯τ ≤ n, for each face τ ∈ K, we can think of τ as a subset of the set {1, ..., n} which we also denote by n. Such an inclusion establishes a monomorphism T^{τ} ⟶ T^{n} ⊂ U(n) and, passing two classifying spaces, a map f_{τ} : BT^{τ} ⟶ BU(n). Moreover, for a different inclusion τ ⊂ n, the two monomorphisms T^{τ} ⟶ T^{n} differ only by a permutation. Hence, they are conjugate in U(n) and the induced maps between the associated classifying spaces are homotopic [19]. This establishes a map from the 1-skeleton of the homotopy colimit into BU(n) unique up to homotopy. There exists an obstruction theory for extending this map to a map hocolim BT^{K} ⟶ BU(n) as well as for the uniqueness question of such extensions [21]. The obstruction groups are higher limits of the form lim^{j} π_{i}(map(BT^{τ}, BU(n))_{f_{σ}}). If we pass to completion, i.e. we replace BU(n) by it’s p-adic completion BU(n)_{p}^{∧}, the mapping space can be identified with (BT^{τ}×BU(n\τ))_{p}^{∧} [14]. For j = i, i+1 and target BU(n)_{p}^{∧}, these higher limits do vanish [17], which is sufficient to prove existence and homotopical uniqueness of maps f_{K} : c(K) ⟶ BU(n)_{p}^{∧} realizing ψ for all primes [1] [21]. Rationally, the map ψ can be realized, since the rationalisation BU(n)_{0} ≃ ∏_{i=1}^{n} K(ℚ, 2i) of BU(n) is a product of rational Eilenberg-MacLane spaces in even degrees. An arithmetique square argument then establishes a map f_{K} ⟶ BU(n) and also shows that the homotopy class of this map is uniquely determined (for details see [17]).

As already mentioned in the introduction, we define X_{K} as the homotopy fibre of f_{K} : c(K) ⟶ BU(n). We are particularly interested in the top degrees of H^{∗}(X_{K}).

Proposition 3.2.

(i) X_{K} has the homotopy type of a finite CW -complex of dimension ≤ n^{2} + n. (ii)

H^{i}(X_{K}) ≅ \{ \begin{matrix}0 & for i > n^{2} + n \\ H^{n−1}(K) & for i = n^{2} + n \end{matrix}

(iii) If H̃^{i}(K) = 0 for i < n − 1, then H^{n^{2}+n−1}(X_{k}) = 0 and there exists a short exact sequence

0 ⟶ H^{n^{2}+n−2}(X_{K}) ⟶ ∏_{i∈V} H̃^{n−2}(link_{K}({i})) ⟶ H^{n−1}(K) ⟶ 0

Proof. In the proof cohomology is always taken with coefficients in R. The composition BT^{σ} ⟶ c(K) ⟶^{f_{K}} BU(n) is natural with respect to maps in cat(K). Interpreting this map as the classifying map of a U(n)-principal bundle establishes a diagram of U(n)- principal bundles Y (σ) ⟶ BT^{σ} over cat(K). By construction, Y (σ) ≃ U(n)/T^{σ}. Since U(n) acts freely on Y (σ), the Borel construction Y (σ)_{hU(n)} =^{def} Y (σ) ×_{U(n)} EU(n) projects to Y (σ)/U(n) = BT^{σ} by a homotopy equivalence. These projections are natural with respect to morphisms in cat(K). Since Borel constructions commute with taking homotopy colimits [6] we get a commutative diagram of fibrations

hocolim_{cat(K)} Y (−) ⟶ hocolim_{cat(K)} BT^{K} ⟶ BU(n)
≃↓  ^{≃}↓  ∥
X_{K}  ⟶  c(K)  ⟶ BU(n)

We can calculate H^{∗}(X_{K}) with the help of the Bousfield-Kan spectral sequence. Since the normalised cochain complex N^{i}(cat(K)^{op}, ϕ) vanishes for i > n for any functor ϕ, we have lim _{cat(K)}^{i}ϕ = 0 for i > n. Moreover, H^{i}(U(n)/T^{σ}) = 0 for i > n^{2} = dim U(n). Since lim^{i} H^{j}(Y (−)) is always a finitely generated abelian group, this shows that H^{∗}(X_{K}) vanishes in degrees > n^{2} + n and is a finitely generated abelian group in each degree. By construction, X_{K} is simply connected, and is therefore homotopy equivalent to a finite CW -complex of dimension ≤ n^{2} + n.

The above argument also shows, that the group E_{2}^{n,n^{2}} ≅ lim _{cat(K)}^{n}H^{n^{2}}(Y (−)) is the only possibly non vanishing term of total degree n^{2} + n and survives in the spectral sequence. In particular, E_{2}^{n,n^{2}} ≅ H^{n^{2}+n}(X_{K}). On the other hand, for any σ ≠ ∅ we have dim U(n)/T^{σ} < n^{2}. Hence, the functor H^{n^{2}}(Y (−)) has it’s only non vanishing value for σ = ∅ and H^{n^{2}}(Y (∅)) ≅ H^{n^{2}}(U(n)) ≅ R. Hence, by Remark 2.8, E_{2}^{n,n^{2}} ≅ H̃^{n−1}(K). This proves the second part of the claim.

In fact, Remark 2.8 shows that H̃^{i−1}(K) ≅ lim _{cat(K)^{op}}^{i}H^{n^{2}}(Y (−)) ≅ E_{2}^{i,n^{2}}. Hence, if H̃^{j}(K) vanishes for j ≠ n − 1, the only term of total degree n^{2} + n − 1 is given by lim^{n}H^{n^{2}−1}(Y (−)). Let ϕ =^{def} H^{n^{2}−1}(Y (−)). Since dim U(n)/T^{τ} ≤ n^{2} − ♯τ, the functor ϕ vanishes for all faces τ of order ≥ 2. Moreover, for each vertex i ∈ V the projection U(n) ⟶ U(n)/T^{{i}} induces an isomorphism H^{n^{2}−1}(U(n)/T^{{i}}) ⟶^{≅} H^{n^{2}−1}(U(n)). This follows from an analysis of the Serre spectral sequence of the fibration T^{{i}} ⟶ U(n) ⟶ U(n)/T^{{i}}.

For r = 0, 1 we define functors ϕ_{r} by ϕ_{r}(σ) =^{def} ϕ(σ) if ♯σ = r and ϕ_{r}(σ) =^{def} 0 otherwise. We get short exact sequences of functors

0 ⟶ ϕ_{0} ⟶ ϕ ⟶ ϕ_{1} ⟶ 0

and

0 ⟶ ϕ ⟶ 1_{R} ⟶ ψ =^{def} 1_{R}/ϕ ⟶ 0

where 1_{R} denotes the constant functor. The functor ψ is non trivial only for faces of order ≥ 2.

In [16] higher limits of functors defined on cat(K) are discussed in detail. Those results show, that lim^{j} 1_{R} = 0 for j ≥ 1, that R = lim^{0}1_{R} ≅ lim^{0}ϕ, that lim^{j}ψ ≅ lim^{j+1} ϕ for j ≥ 1 and that lim^{j} ψ = 0 for j ≥ n − 1. In particular, 0 = lim^{n−1}ψ = lim^{n}ϕ. This proves that H^{n^{2}+n−1}(X_{K}) = 0.

The first of the above sequences establishes an exact sequence

0 = lim^{n−1}ϕ_{0} ⟶ lim^{n−1}ϕ ⟶ lim^{n−1}ϕ_{1} ⟶ lim^{n}ϕ_{0} ⟶ lim^{n}ϕ = 0

By part (i) the fourth term can be identified with H̃^{n−1}(K), by [16] the third term with ∏_{i∈V} H̃^{n−2}(link({i})), and the second term with H^{n^{2}+n−2}(X_{K}). This finishes the proof of the third part. □

Corollary 3.3. Let L ⊂ K be a subcomplex of the same dimension. Then, the composition

H^{n−1}(K) ≅ H^{n^{2}+n}(X_{K}) ⟶ H^{n^{2}+n}(X_{L}) ≅ H^{n−1}(L)

is the map induced in cohomology by the inclusion.

Proof. This follows from the above proof and Remark 2.8 □

We are also interested in the top degree of H^{∗}(X_{st({i})}). Lemma 3.4. X_{st({i})} has the homotopy type of a finite CW -complex of dimension ≤ n^{2} + n − 2 and H^{n^{2}+n−2}(X_{st({i})}) ≅ H^{n−2}(link({i})).

Proof. Since c(st({i})) ≃ BT^{{i}} ×c(link({i})) we have a commutative diagram of fibrations

X_{link({i})} ⟶ c(link({i})) × BT^{{i}} ⟶ BU(n − 1) × BT^{{i}}
↓  ^{≃}↓  ↓
X_{st({i})} ⟶  c(st({i}))  ⟶  BU(n)

Since the homotopy fibre of the right vertical arrow is homotopy equivalent to the n − 1dimensional complex projective space P(n − 1), this establishes a fibration X_{link({i})} ⟶ X_{st({i})} ⟶ P(n − 1). This shows that X_{st({i})} is simply connected and has the homotopy type of a finite CW -complex of dimension ≤ (n − 1)^{2} + (n − 1) + 2(n − 1) = n^{2} + n − 2 and that H^{n^{2}+n−2}(X_{st({i})}) ≅ H^{(n−1)^{2}+(n−1)}(link({i})). The last equation follows from an analysis of the Serre spectral sequence of the above fibration. □

For later purpose we need the following lemma. Lemma 3.5. R(K) is a finitely generated R[c_{1}, ..., c_{n}]-module.

Proof. Since R[V] ⟶ R(K) is a surjection and since R[V] is a finitely generated R[c_{1}, ..., c_{m}]- module, the same holds for R(K) as R[c_{1}, ..., c_{m}]-module. By Theorem 3.1 the map R[c_{1}, ..., c_{m}] ⟶ R(K) factors through R[c_{1}, ..., c_{n}], which implies the statement. □

We finish this section with the following observation:

Remark 3.6. The composition c(st (τ)) ⟶ c(K) ⟶^{f_{K}} BU(n) makes R(st(τ)) into a R[c_{1}, ..., c_{n}] ≅ H^{∗}(BU(n); R)-module and, with respect to this structure, all differentials of the normalised chain complex N^{∗}(cat(K); H^{∗}(c(st_{K}); R) become H^{∗}(BU(n); R)-linear. Hence, lim^{i}H^{∗}((cst_{K}); R) = lim^{i}R(st_{K}) is an H^{∗}(BU(n); R)-module. The proof of Theorem 2.3 shows that part (iii) can be refined. There exists a short exact sequence

0 ⟶ R(K) ≅ H^{∗}(c(K); R) ⟶ lim^{0}H^{∗}(cst_{K}); R) ≅ lim^{0}R(st_{K}) ⟶ H̃^{0}(K; R) ⟶ 0

of H^{∗}(BU(n); R)-modules. Here, H^{∗}(BU(n); R) acts on lim^{i}R(st_{K}) ≅ H^{i}(K; R) as well as on H̃^{0}(K; R) via the augmentation H^{∗}(BU(n); R) ⟶ R.

4. Cohen-Macaulay and Gorenstein conditions

In this section we assume that R = 𝔽 is a field and cohomology is always taken with coefficients in 𝔽. In particular, H^{∗}(−) =^{def} H^{∗}(−; 𝔽).

Let A^{∗} be a non negatively graded commutative algebra over 𝔽. We say that A^{∗} is connected if A^{0} ≅ 𝔽 and 𝔽-finite if A^{j} = 0 for j large and A^{j} is a finitely generated 𝔽-module in each degree. We call a finite sequence of elements a_{1}, .., a_{r} ∈ A a homogeneous system of parameters, a hsop for short, if they are homogeneous and algebraically independent and if the quotient A^{∗}/(a_{1}, ..., a_{r}) is 𝔽-finite. We say that the sequence is a regular sequence, if, for all i, a_{i+1} is not a zero divisor in A/(a_{1}, ..., a_{i}).

We call A^{∗} Cohen-Macaulay, if there exists a sequence a_{1}, ..., a_{n} which is both, a hsop and regular. If A^{∗} is Cohen-Macaulay, then it is known, that every hsop is also a regular sequence [2].

A Noetherian local ring S is called Gorenstein if S considered as a module over itself has a finite injective resolution. If A^{∗} is a commutative connected non negatively graded algebra, we can use the following equivalent definition [20, Theorem I.12.4]. That is, A^{∗} is Gorenstein, if A^{∗} is Cohen-Macaulay and if for any hsop a_{1}, ...a_{n} of A^{∗}, we have soc(A^{∗}/a_{1}, ..., a_{n}) ≅ 𝔽. The socle soc(B^{∗}) of a non negatively graded algebra B^{∗} is defined as soc(B^{∗}) =^{def} {b ∈ B^{∗} : B^{+}b = 0} ≅ Hom_{B^{∗}}(𝔽, B^{∗}) where B^{+} denotes the elements of positive degree.

We call A^{∗} a Poincaré duality algebra, or PD-algebra for short, if there exists a class [A] ∈ Hom_{𝔽}(A^{∗}, 𝔽) such that the composition A^{∗} ⊗ A^{∗} ⟶ A^{∗} ⟶^{[A]} 𝔽 is a non degenerate bilinear form. In particular, every PD-algebra is connected. As a straight forward argument shows, the above condition is equivalent to the fact that soc A^{∗} =^{def} Hom_{A^{∗}}(𝔽, A^{∗}) ≅ 𝔽. That is, a 𝔽-finite non negatively graded algebra is a P D-algebra if and only if it is Gorenstein. If A^{∗} ≅ H^{∗}(X; 𝔽) for a topological space, then we call X a Poincaré duality space over 𝔽, a 𝔽-PD-space for short, if A^{∗} is a PD-algebra. In this case [X] =^{def} [A] ∈ H_{∗}(X) is the fundamental class of X.

We want to describe Cohen-Macaulay and Gorenstein properties of the face ring 𝔽(K) in terms of the map f_{K} : c(K) ⟶ BU(n) described in Theorem 3.1 and it’s homotopy fibre X_{K}. The next result contains part (i) of Theorem 1.3. Theorem 4.1. The following conditions are equivalent:

(i) 𝔽(K) is Cohen-Macaulay. (ii) The map c(K) ⟶ BU(n) makes 𝔽(K) into a finitely generated free H^{∗}(BU(n))- module. (iii) The cohomology ring H^{∗}(X_{K}) is concentrated in even degrees. (iv) Tor_{H^{∗}(BU(n))}^{j}(𝔽(K), 𝔽) = 0 for j ≤ −1.

For the proof we need the following lemma. Again we denote by σ_{j} the j-th elementar symmetric polynomial in the generators of 𝔽(K) and identify H^{∗}(BU(n)) with 𝔽[σ_{1}, ..., σ_{n}], that is with image of the map H^{∗}(BU(n)) ⟶ H^{∗}(c(K)). Lemma 4.2. The sequence σ_{1}, .., σ_{n} ∈ 𝔽(K) is a hsop for 𝔽(K).

Proof. By Lemma 3.5 𝔽(K)/(σ_{1}, ..., σ_{n}) is 𝔽-finite. We have only to show that the elements σ_{1}, ..., σ_{n} are algebraic independent in 𝔽(K).

Let μ be a maximal face of K, that is ♯μ = n. The composition BT^{μ} ⟶ c(K) ⟶ BU(n) is induced by a maximal torus inclusion T^{μ} ⟶ U(n) (see Section 3). The images of σ_{1}, ..., σ_{n} in H^{∗}(BT^{μ}) ≅ 𝔽[μ] are given by the elementary symmetric polynomials and therefore are algebraic independent as well as σ_{1}, ...σ_{1} in 𝔽(K). □

Proof of Theorem 4.1: If 𝔽(K) is Cohen-Macaulay, then every hsop is given by a regular sequence. And if we have a hsop of 𝔽(K) given by a regular sequence, then 𝔽(K) is Cohen-Macaulay [2]. In the light of Lemma 4.2 this shows that the first two conditions are equivalent.

If H^{∗}(X_{K}; ℤ) is concentrated in even degrees (part (iii)), then, by degree reasons, the Serre spectral sequence for the fibration X_{K} ⟶ c(K) ⟶ BU(n) collapses at the E_{2} page and H^{∗}(c(K)) ≅ H^{∗}(X_{K}) ⊗_{𝔽} H^{∗}(BU(n)) as H^{∗}(BU(n))-module. This shows that 𝔽(K) is a finitely generated free H^{∗}(BU(n))-module and therefore Cohen-Macaulay, which is part (i).

If 𝔽(K) is Cohen-Macaulay, then, by (ii), it is a finitely generated free module over H^{∗}(BU(n)). In particular, Tor_{H^{∗}(BU(n))}^{j}(𝔽(K), 𝔽) = 0 for j ≤ −1. This is part (iv).

If condition (iv) is satisfied, the Eilenberg-Moore spectral sequence for calculating H^{∗}(X_{K}; 𝔽) collapses at the E_{2}-page and shows that H^{∗}(X_{K}; 𝔽) ≅ 𝔽(K) ⊗_{H^{∗}(BU(n);𝔽)} 𝔽 and that H^{∗}(X_{K}; 𝔽) is concentrated in even degrees, which is condition (iii). This proves the equivalence of (i), (iii) and (iv). □

Proof of Theorem 1.3 (ii): By our definition of Gorenstein, 𝔽(K) is Gorenstein if and only if 𝔽(K) is Cohen-Macaulay and 𝔽(K)⊗_{𝔽[σ_{1},...,σ_{n}]}𝔽 ≅ 𝔽(K)/(σ_{1}, ..., σ_{n}) is a PD-algebra. Hence the equivalence of the two conditions follows from the first part of the Theorem. □

5. Homotopy fixed point sets

For an action of a group G on a space X we can think of the fixed point set as the mapping space X^{G} = map_{G}(∗, X) of G-equivariant maps from a point into X. The notion is not flexible enough for doing homotopy theory, since a homotopy equivalence between G-spaces, which happens to be G-equivariant in addition, does not induce an equivalence between the fixed-point sets in general. We therefore are interested in homotopy fixed point sets which do have this property. They are defined as the equivariant mapping space X^{hG} =^{def} map_{G}(EG, X) where EG is a contractible G-CW-complex with a free G-action. The projection EG ⟶ ∗ induces a map X^{G} ⟶ X^{hG}.

Applying the Borel construction establishes a fibration

X ⟶ X_{hG} =^{def} X ×_{G} EG ⟶^{π} BG.

A straight forward argument shows that we can equivalently define the homotopy fixed point set as the space Γ(X_{hG} → BG) of sections of this fibration. The latter definition also allows to define homotopy fixed point sets in more general situations. A proxy G-action on X is a fibration X ⟶ E ⟶^{π} BG, where we think of E as the Borel construction of this action. We define X^{hG} =^{def} Γ(E → BG). This establishes a fibration

X^{hG} ⟶ map(BG, E)_{{id}} ⟶ map(BG, BG)_{id}

Here, the middle term consist of all lifts of the identity id of BG up to homotopy, i.e. of all maps g : BG ⟶ E such that πf ≃ id. If G is a finite abelian group the base space is homotopy equivalent to BG [12], and composition of maps yields an equivalence BG × X^{hG} ⟶^{≃} map(BG, E)_{{id}}. This equivalence fits into a commutative diagram of fibrations

X^{hG} ⟶ BG × X^{hG} ⟶ BG
↓  ↓  ∥
X ⟶  E  ⟶ BG

where the vertical arrows are induced by evaluation at a basepoint.

Typical examples of proxy actions arise from pull back constructions. Let X ⟶ E ⟶^{π} B be a fibration and f : BG ⟶ B be a map. The pull back construction establishes a commutative diagram

E′ ⟶^{π′} BG
↓  ^{f}↓
E ⟶^{π} B 14

and applying the mapping space functor yields a pull back diagram

map(BG, E′)_{{id}} ⟶^{π′_{∗}} map(BG, BG)_{id}
↓  ^{f}↓
map(BG, E)_{{f}} ⟶^{π_{∗}} map(BG, B)_{f}.

The fibre of both horizontal arrows is given by the homotopy fixed point set X^{hG}. Here, the left mapping space in the bottom row consists of all maps BG ⟶ E which are homotopic to a lift of f. If G is finite and abelian the composition BG ≃ map(BG, BG)_{id} ⟶ map(BG, B)_{f} ⟶^{ev} B equals the map f : BG ⟶ B.

The main goal of this section is to show that, in favourable cases, H^{∗}(X^{hG}; 𝔽_{p}) is concentrated in even degrees or a PD-algebra, if H^{∗}(X; 𝔽_{p}) satisfies these properties. For our method of proof we have to make some restrictions. According to our situation, we have the spaces X_{K} in mind, we will always assume that:

(1) X is 𝔽_{p}-finite and p-complete.

(2) H^{∗}(X; 𝔽_{p}) is concentrated in even degrees. In particular, H^{∗}(X; 𝔽_{p}) is a graded algebra, commutative in the non graded sense. Spaces satisfying both conditions will be called special.

We also restrict ourselves to particular proxy actions. We say that a proxy action X ⟶ E ⟶ BG is orientable if π_{1}(BG) acts trivially on H^{∗}(X; ℤ_{p}^{∧}). Remark 5.1. If G is an elementary abelian p-group, and the G-action extends to a torus action of T^{r}, i.e. the proxy action is induced by a pull back of a fibration X ⟶ E′ ⟶ BT^{r}, the group G always acts trivially on the cohomology of X and the proxy action is orientable.

Moreover, if the proxy action X ⟶ E ⟶ BG is orientable, the Serre spectral sequence for H^{∗}(−; ℤ_{p}^{∧}) collapses at the E_{2}-page by degree reasons. And the same holds for H^{∗}(−; 𝔽_{p}). In particular, H^{∗}(E; 𝔽_{p}) ≅ H^{∗}(BG; 𝔽_{p}) ⊗ H^{∗}(X; 𝔽_{p}) as H^{∗}(BG; 𝔽_{p})-module [5, Corollary 2.5] .

The first part of the next theorem is due to Dehon and Lannes [5, Corollary 2.10]. Theorem 5.2. Let G be an elementary abelian p-group and X ⟶ E ⟶ BG an orientable proxy G-action on a space X.

(i) If X is special, then so is X^{hG}. (ii) If X is special and a 𝔽_{p}-PD-space, then so is each component of X^{hG}.

The proof of the second part needs some preparation. For the rest of this section we make the following assumptions. H^{∗}(−) denotes H^{∗}(−; 𝔽_{p}), G is an elementary abelian p-group and H^{G} =^{def} H^{∗}(BG). Moreover, X is special and the proxy G-action X ⟶ E ⟶ BG is orientable. In particular, H^{∗}(E) ≅ H^{G} ⊗ H^{∗}(X) as H^{G}-module (see Remark 5.1).

For such actions H^{∗}(E) is not concentrated in even degrees and hence not commutative in the non graded sense. To avoid technical difficulties we will use the following construction. We denote by J ⊂ H^{G} the ideal generated by all classes of degree 1. And for an graded H^{G}-module M we denote by M̄ the quotient M/JM. If G ≅ (ℤ/p)^{r}, the composition H^{∗}(BT^{r}) ⟶ H^{G} ⟶ H̄^{G} is an isomorphism, where the first map is induced by the canonical inclusion G ≅ (ℤ/p)^{r} ⊂ T^{r}. In particular, H̄^{G} is a polynomial algebra generated by elements of degree 2. In fact, we can think of it as the polynomial part and as a subalgebra of H^{G}. Hence, every H^{G}-module is naturally an H̄^{G}-module. Lemma 5.3. H^{∗}(X) is a PD-algebra if and only if H̄^{∗}(E) is Gorenstein.

Proof. By assumption, H^{∗}(E) ≅ H^{∗}(X) ⊗ H^{G} as H^{G}-module. And hence, H̄^{∗}(E) ≅ H^{∗}(X) ⊗ H̄^{G} as H̄^{G}-module. Since H̄^{G} is a polynomial algebra, this implies that H̄^{∗}(E) is Cohen-Macaulay. And hence, H̄^{∗}(E) is Gorenstein if and only if H^{∗}(X) ≅ H̄^{∗}(E) ⊗H̄^{G} 𝔽_{p} is a PD-algebra. □

Let S ⊂ H^{G} denote the multiplicative subset generated by all Bockstein images in degree 2 of nontrivial elements of degree 1. That is the subset of all images of non trivial elements of H^{∗}(BT^{r}) of strictly positive degree. For any H^{G}-module M we denote by S^{−1}M the localised module over S^{−1}H^{G}. Let X ⟶ E ⟶ BG be a G-proxy action on Y. Following [8] (Corollary 1.2 and the following remark), there exists a map S^{−1}H^{∗}(E) ⟶ S^{−1}(H^{G} ⊗ H^{∗}(Y^{hG})) between the localised modules. Under favourable circumstances this is an isomorphism, which, as shown in [5, Section 2], always hold if X is special and if the G-action is orientable. We collect this into the following theorem. Theorem 5.4. ([8] [5]) Let X ⟶ E ⟶ BG be an orientable G-proxy action on a special space X. Then there exist isomorphisms

S^{−1}H^{∗}(E) ≅ S^{−1}(H^{G} ⊗ H^{∗}(X^{hG})) ≅ (S^{−1}H^{G}) ⊗ H^{∗}(X^{hG}).

Since the multiplicative subset S ⊂ H^{G} consist of elements of even degree, it also is a multiplicative subset of H̄^{G}. We can think of the localised module S^{−1}H̄^{∗}(E) ≅ H̄^{∗}(E) ⊗H̄^{G} S^{−1}H̄^{G} in a different way. Since H̄^{G} ⟶ H̄^{∗}(E) is a monomorphisms, the set S gives rise to a multiplicative subset S′ ⊂ H̄^{∗}(E). Then multiplication induces an isomorphism H̄^{∗}(E) ⊗H̄^{G} S^{−1}H̄^{G} ≅ (S′)^{−1}H̄^{∗}(E), where the latter localisation is obtained by localising the algebra H̄^{∗}(E) with respect to the subset S′.

Proof of Theorem 5.2 (ii): Let us assume that X is an 𝔽_{p}-PD-space. Then, H̄^{∗}(E) is Gorenstein (Lemma 5.3) as well as S^{−1}H̄^{∗}(E) [2, Proposition 3.1.19]. Since

S^{−1}H̄^{∗}(E) ≅ H^{∗}(X^{hG}) ⊗ S^{−1}H̄^{G} ≅ ⨁_{g} H^{∗}(X_{g}^{hG}) ⊗ S^{−1}H̄^{G}

as algebras, each of the summands is Gorenstein. Here, the direct sum is taken over components of X^{hG}. Let X_{g}^{hG} denotes the component associated to a section g : BG ⟶ E of π : E ⟶ BG. Now we give the algebra H^{∗}(X_{g}^{hG}) ⊗ S^{−1}H̄^{G} a different grading induced by the grading of the first factor. That is elements of S^{−1}H^{G} get degree 0. Since H̄^{G} ≅ 𝔽_{p}[t_{1}, ..., t_{n}], the sequence {t_{1} − 1, ...t_{n} − 1} ⊂ H^{∗}(X_{g}^{hG}) ⊗ S^{−1}H̄^{G} is regular, homogeneous and consists of elements of degree 0. Hence H^{∗}(X_{g}^{hG}) ≅ H^{∗}(X_{g}^{hG})⊗S^{−1}H̄^{G}/(t_{1}−1, ..., t_{n}−1) is a connected 𝔽_{p}-finite Gorenstein algebra and therefore a PD-algebra (see Section 4). □

6. Cohen-Macaulay and Gorenstein properties for links of faces

In this section we want to prove that Cohen-Macaulay or Gorenstein properties are inherited to the Stanley Reisner algebras of links of faces. We call a simplicial complex pure, if all maximal faces have the same dimension. We first consider the case of algebras over 𝔽_{p}.

Theorem 6.1. If 𝔽_{p}(K) is Cohen-Macaulay respectively Gorenstein, then K is pure and, for each face τ ∈ K the algebra 𝔽_{p}(link_{K}(τ)) is also Cohen-Macaulay respectively Gorenstein.

For the proof we will use the space X_{K} given by the fibration X_{K} ⟶ c(K) ⟶ BU(n) described in Section 3. Since BU(n) is simply connected, the p-adic completion maintains the fibration [1]. Since we are working with 𝔽_{p} as coefficients and since for simply connected spaces completion induces an isomorphism in mod-p cohomology, we can and will assume that all spaces are completed. To simplify notation, we will always drop the notation for completion. The proof also relies on the interpretation of 𝔽_{p}(link(τ)) as a certain mapping space given in [16], which we recall next.

For each face τ ∈ K we denote by G^{τ} ⊂ T^{τ} the maximal elementary abelian subgroup of the torus T^{τ}. Let g_{τ} : BG^{τ} ⟶ c(K) denote the composition BG^{τ} ⟶ BT^{τ} ⟶ hocolim_{cat(K)} BT^{K} ≃ c(K). The composition BG^{τ} ⟶^{g_{τ}} c(K) ⟶^{f_{K}} BU(n) establishes a proxy G^{τ}-action X_{K} ⟶ E ⟶ BG^{τ}. Applying the mapping space functor we get a fibration

X^{hG^{τ}} ⟶ map(BG^{τ}, c(K))_{{f_{K}g_{τ} }} ⟶^{(f_{K})_{∗}} map(BG^{τ}, BU(n))_{f_{K}g_{τ}}

(see Section 5). The composition f_{K}g_{τ} is induced from a coordinate-wise inclusion G^{τ} ⊂ T^{τ} ⊂ T^{n} ⊂ U(n) into the set of diagonal matrices (see Section 3). The centraliser of this image equals T^{τ} × U(n \ τ) where we again think of n as the set {1, ..., n} and τ becomes a subset of the set n via the coordinate-wise inclusion. Then, by construction, f_{K}g_{τ} factors through a map id × const : BT^{τ} ⟶ BT^{τ} × BU(n \ τ) where the fist coordinate is the identity and the second the constant map. There also exists a map BT^{τ} × c(link(τ)) = c(st(τ)) ⟶ c(K) as constructed in Section 2 and the map g_{τ} factors through i × const : BG^{τ} ⟶ BT^{τ} × c(link(τ)). Moreover all these maps fit into a diagram

X_{link(τ)} ⟶ BT^{τ} × c(link(τ)) ⟶^{id×f_{link(τ)}} BT^{τ} × BU(n \ τ)
↓  ↓  ↓
X_{K} ⟶  c(K)  ⟶^{f_{K}}  BU(n).

Applying the mapping space functor, there exit the following equivalences (of p-completed)

spaces; BT^{τ} ⟶^{≃} map(BG^{τ}, BT^{τ})_{i}, BT^{τ} × BU(n \ τ) ⟶^{≃} map(BG^{τ}, BU(n)_{f_{K}g_{τ}} [9] and

BT^{τ} × c(link(τ)) ⟶^{≃} map(BG^{τ}, c(K))_{g_{τ}} [16]. Putting all this information together we have a commutative diagram

X_{link(τ))} ⟶ BT^{τ} × c(link(τ)) ⟶^{f_{link(τ))}}  BU(n \ τ)
↓  ^{≃}↓  ^{≃}↓
(X_{K})_{g_{τ}}^{hG^{τ}} ⟶ map(BG^{τ}, c(K))_{g_{τ}} ⟶ map(BG^{τ}, BU(n))_{f_{K}g_{τ}}

of horizontal fibrations. Since BT^{τ} × BU(n \ τ) is simply connected, the map X_{K}^{hG^{τ}} ⟶ map(BG^{τ}, c(K))_{{f_{K}g_{τ} }} induces a bijection between the components of both spaces. Hence, the bottom left space is connected. This proves the following proposition. Proposition 6.2. X_{link(τ)} ≃ (X_{K})_{g_{τ}}^{hG^{τ}}.

Now we are in the position to prove Theorem 6.1.

Proof of Theorem 6.1 If 𝔽_{p}(K) is Cohen-Macaulay, the fibre X_{K} is special (Theorem 1.3(i)). Since the map f_{K}g_{τ} : BG^{τ} ⟶ BU(n) factors through BT^{τ}, the proxy G^{τ} action extends to a torus action and is orientable (see Remark 5.1). We can apply Theorem 5.2. That is that (X_{K})_{g_{τ}}^{hG^{τ}} ≃ X_{link(τ)} is special and that 𝔽_{p}(link(τ)) is Cohen-Macaulay (Theorem 1.3(i)). If 𝔽_{p}(K) is Gorenstein, we use Theorem 1.3(ii) instead of the first part.

Finally we have to show that K is pure. The above argument shows that, if 𝔽_{p}(K) is Cohen-Macaulay, then 𝔽_{p}(link(τ)) is a free H^{∗}(BU(n \ τ))-module. If μ ∈ K is a maximal simplex, then link(μ) is the empty complex and 𝔽_{p}(link(μ)) ≅ 𝔽_{p}. This implies that μ has order n and that K is pure. □

We finally give up the restriction on the coefficients. Corollary 6.3. If 𝔽(K) is Cohen-Macaulay respectively Gorenstein, then K is pure and, for every face τ ∈ K, the Stanley-Reisner algebra 𝔽(link_{K}(τ)) is also Cohen-Macaulay respectively Gorenstein.

Proof. We will use Theorem 6.1 and Theorem 1.3. If 𝔽 is a field of characteristique p > 0, then H^{∗}(X_{K}) ≅ H^{∗}(X_{K}; 𝔽_{p}) ⊗_{𝔽_{p}} 𝔽. This shows that 𝔽(K) is Cohen-Macaulay if and only if 𝔽_{p}(K) is so. The same holds for links. This covers the Cohen-Macaulay case as well as the Gorenstein case and also shows that K is pure.

Let ℤ_{(p)} denote the localisation of ℤ at the prime p. Then, for 𝔽 = ℚ we can argue as follows. Since X_{K} is of the homotopy type of a finite CW -complex (Proposition 3.2), H^{∗}(X_{K}; ℤ) has only finitely many torsion primes. Hence, ℚ(K) is Cohen-Macaulay if and only if H^{∗}(X_{K}; ℚ) is concentrated in even degrees if and only if, for almost all primes, H^{∗}(X_{K}; ℤ_{(p)}) is concentrated in even degrees and torsion free if and only if, for almost all primes, H^{∗}(X_{K}; 𝔽_{p}) is concentrated in even degrees if and only if, for almost all primes, 𝔽_{p}(K) is Cohen-Macaulay. Now, Theorem 6.1 and and the above chain of equivalent statements applied in the case of link(τ) proves the claim for Cohen-Macaulay algebras over 𝔽 = ℚ. In the Gorenstein case, we only have to notice that H^{∗}(X_{K}; ℚ) is a P Dalgebra if and only H^{∗}(X_{K}; 𝔽_{p}) is a P D-algebra for almost all primes.

For a general field of characteristique 0 we deduce the claim from the case 𝔽 = ℚ in the same manner as for fields of characteristique p > 0 from 𝔽 = 𝔽_{p}. □

7. Proof of Theorem 1.1

Theorem 1.1 follows easily from the following statement by induction.

Theorem 7.1. 𝔽(K) is Cohen-Macaulay if and only if 𝔽(link_{K}(τ)) is Cohen-Macaulay for all faces τ ≠ ∅ of K and H̃^{r}(K; 𝔽) = 0 for 0 ≤ r < n − 1.

Proof of Theorem 1.1: If dim K = 0 then 𝔽(K) is always Cohen-Macaulay. In fact, in this case c(K) is the m-fold wedge product of BS^{1}’s and X_{k} the (m−1)-fold wedge product of S^{2}’s, whose cohomology is concentrated in even degrees. On the other hand the set of conditions on the cohomology of the links of K is an empty set. This proves the statement in this case.

The general case follows by induction over the dimension of K and the above theorem. □

In the following cohomology is always with 𝔽-coefficients. We define H^{∗}(−) =^{def} H^{∗}(−; 𝔽) for the rest of this section. Also, for simplification, we set P =^{def} H^{∗}(BU(n)) ≅ 𝔽[σ_{1}, ..., σ_{n}].

The proof of Theorem 7.1 is based on the homological analysis of particular double complexes. To fix notation, we will recall the general concept next. For details see [13]

Let A be a 𝔽-algebra. A differential graded A-module (C^{∗}, d_{C}) is a cochain complex of A-modules such that d_{C} is A-linear. A double complex or differential bigraded module (M^{∗,∗}, d_{h}, d_{v}) over A is a bigraded A- module M^{∗,∗} with two A-linear maps d_{h} : M^{∗,∗} ⟶ M^{∗,∗} and d_{v} : M^{∗,∗} ⟶ M^{∗,∗} of bidegree (1, 0) and (0, 1) such that d_{h}d_{h} = 0 = d_{v}d_{v} and d_{h}d_{v} + d_{v}d_{h} = 0. We think of d_{h} as the horizontal and of d_{v} as the vertical differential. To each double complex M^{∗,∗} we associate a total complex T ot^{∗}(M) which is a differential graded module over A defined by T ot^{n}(M) =^{def} ⊕_{i+j=n}M^{i,j} with differential D =^{def} d_{h} + d_{v}. Examples are given by the tensor products of two differential graded modules (B^{∗}, d_{B}) and (C^{∗}, d_{C}). If we set M^{i,j} =^{def} B^{i} ⊗_{A} C^{j}, d_{h} =^{def} d_{B} ⊗ 1 and d_{v} =^{def} (−1)^{j}1 ⊗ d_{C}, we get a double complex such that T ot^{∗}(M) = B^{∗} ⊗_{A} C^{∗}.

For a double complex (M^{∗,∗}, d_{h}, d_{v}), we can take horizontal or vertical cohomology groups denoted by H_{h}^{∗}(M^{∗,∗}) and H_{v}^{∗}(M^{∗,∗}). The boundary maps d_{h} and d_{v} induce again boundary maps on these cohomology groups. We can consider cohomology groups of the form H_{h}^{∗}(H_{v}^{∗}(M^{∗,∗})) and H_{v}^{∗}(H_{h}^{∗}(M^{∗,∗})).

If (M^{∗,∗}, d_{h}, d_{v}) is bounded below, that is M^{i,j} = 0 if i or j is small enough, there exist two spectral sequences converging towards H^{∗}(T ot(M), D). In one case, we have E_{2}^{i,j} = H_{h}^{i}(H_{v}^{j}(M)) and in the other case E_{2}^{i,j} =^{def} H_{v}^{j}(H_{h}^{i}(M)). In the first case the differential have degree (r, 1 − r,) and in the second case degree (1 − r, r).

Let N^{∗} =^{def} N^{∗}(cat(K^{×})^{op}, H^{∗}(cst_{K})) denote the normalised cochain complex for the functor H^{∗}(cst_{K}) : cat(K^{×})^{op} ⟶ ab considered in Section 2. Actually, the complex N^{∗} is a bigraded object. It inherits an internal degree from the grading of H^{∗}(cst_{K}), which we will not consider in most cases.

We collect the main properties of N^{∗} in the next proposition.

Proposition 7.2.

(i) N^{i} = 0 for i < 0 or i ≥ n. (ii) (N^{∗}, d_{N}) is a differential graded P -module as well as H^{∗}(N^{∗}, d_{N}) ≅ lim_{cat(K^{×})^{op}}^{∗} , 𝔽(st_{K}) (iii) There exist an isomorphism H^{i}(N^{∗}, d_{N}) ≅ H̃^{i}(K) for i ≥ 1 of P -modules and a short exact sequence

0 ⟶ 𝔽(K) ⟶ H^{0}(N^{∗}, d_{L}) ≅ lim_{cat(K^{×})^{op}}^{0} 𝔽(st_{K}) ⟶ H̃^{0}(K) ⟶ 0

of P -modules.

Proof. The first part follows from the fact that N^{s}(cat(K^{×})^{op}, H^{∗}(cst_{k})) = 0 for s ≥ n, the second and the third part from Remark 3.6. □

We can also look at the projective resolution of the trivial P -module 𝔽 given by the Koszul complex Q^{∗} =^{def} Λ^{∗} ⊗_{𝔽} P. According to our convention we make this into a cochain complex and give the generators of the exterior algebra Λ^{∗} =^{def} Λ^{∗}(x_{1}, ..., x_{n}) the degree −1. As usual the differential d_{Q} is defined by d_{Q}(x_{i}) =^{def} σ_{i} and d_{Q}(y) = 0 for y ∈ P and has degree 1. Again, Q^{∗} is a differential graded P -module, bounded below and above by Q^{j} = 0 for j > 0 or j < −n. The differential bigraded P -module N^{∗} ⊗_{P} Q^{∗} is then bounded. In particular, both above mentioned spectral sequences converge towards H^{∗}(Tot(N^{∗}⊗_{P} Q^{∗})).

Proof of Theorem 7.1: If K is the empty complex, there is nothing to show. If K is a 0-dimensional complex, then 𝔽(K) is always Cohen-Macaulay as discussed in the proof of Theorem 1.1 and Theorem 1.2.

Now we assume that dim K ≥ 1. i.e. n ≥ 2. We start with the assumption that 𝔽(K) is Cohen-Macaulay. By Theorem 6.3, we know that K is pure and that for all τ ∈ K, the algebra 𝔽(link(τ)) is also Cohen-Macaulay as well as 𝔽(st(τ)). In particular, since dim st(τ) = dim K, the algebra R(st_{K}(τ)) is a finitely generated free module over P. We have to show that H̃^{i}(K) = 0 for i < n − 1.

We look at the above constructed double complex N^{∗} ⊗_{P} Q^{∗}. All modules N^{i} and Q^{j} are free P -modules. Hence, the functors N^{∗} ⊗_{P} − and − ⊗_{P} Q^{∗} are exact. We get

H_{h}^{i}(H_{v}^{j}(N^{∗} ⊗_{P} Q^{∗})) ≅ H_{h}^{i}(N^{∗} ⊗_{P} H_{v}^{j}(Q^{∗})) ≅ \{ \begin{matrix}0 & for j ≠ 0 \\ H^{i}(N^{∗} ⊗_{P} 𝔽) & for j = 0 \end{matrix}

In particular, the E_{2}-term is concentrated in one horizontal line given by j = 0, the spectral sequence collapses and H^{i}(Tot(N^{∗} ⊗_{P} Q^{∗})) ≅ H^{i}(N^{∗} ⊗_{P} 𝔽) = 0 for i < 0 or i ≥ n.

Considering the second spectral sequence we get H_{v}^{j}(H_{h}^{i}(N^{∗} ⊗_{P} Q^{∗})) ≅ H_{v}^{j}(H_{h}^{i}(N^{∗}) ⊗_{P} Q^{∗})

≅ \{ \begin{matrix}Tor_{P}^{j} (H̃^{i}(K), 𝔽) ≅ Tor_{P}^{j} (𝔽, 𝔽) ⊗_{𝔽} H̃^{i}(K) & for i > 0 \\ H^{∗}(X_{K}) ⊕ Tor_{P}^{0} (𝔽, 𝔽) ⊗ H̃^{0}(K) & for i = 0 and j = 0 \\ Tor_{P}^{0} (𝔽, 𝔽) ⊗ H̃^{0}(K) & for i = 0 and j ≠ 0 \\ 0 & otherwise \end{matrix}

For i = 0 this follows from Proposition 7.2 and the fact that Tor_{P}^{0} (𝔽(K), 𝔽) ≅ H^{∗}(X_{K}). By degree reasons there is no differential ending at or starting from H_{v}^{−n}(H_{h}^{0}(N^{∗} ⊗ P) ≅ Λ^{n}(x_{1}, ..., x_{n}) ⊗ H̃^{0}(K) ≅ H̃^{0}(K). Since n ≥ 2 this group has total degree −n + 1 < 0 and must therefore vanish. Hence, the whole column given by i = 0 and j ≤ −1 vanishes and E_{2}^{0,0} ≅ H^{∗}(X_{K}). By induction we can apply this argument successively for i = 1, ..., n − 2. In these cases the whole columns E_{2}^{i,j} vanish. For i = n − 1 there might be a non trivial differential E_{2}^{n−1,n} ⟶ E_{2}^{0,0} and we cannot conclude anymore that E_{2}^{n−1,j} = 0. This shows that H̃^{i}(K) = 0 for 0 ≤ i ≤ n − 2 and proves one direction of the claim.

Now we assume that, for ∅ ≠ τ ∈ K, all algebras 𝔽(link(τ)) are Cohen-Macaulay and that H̃^{j}(K) = 0 for j < dim K = n − 1 ≥ 1 In particular, K is connected. This implies that for any pair μ, μ′ of maximal faces of K, there exists a chain of maximal faces μ = μ_{1}, ..., μ_{s} = μ′ such that the intersection μ_{j} ∩ μ_{j+1} is non empty. Since link i is pure for all vertices {i} ∈ K (Theorem 6.3), this implies that all maximal simplices of K have the same order, that K is pure, that for all faces τ ∈ K the dimension of link(τ) equals n − 1 − ♯τ and that H(BT^{τ} × c(link(τ))) ≅ 𝔽(st_{K}(τ)) is a finitely generated free module over P. Now we consider again both spectral sequences. In this case, we get for H_{h}^{i}(H_{v}^{j}(N^{∗}⊗_{P} Q^{∗})) the same result as above. And, since H̃^{0}(K) = 0, Proposition 7.2 shows that

H_{v}^{j}(H_{h}^{i}(N^{∗} ⊗_{P} Q^{∗})) ≅ H_{v}^{j}(H_{h}^{i}(N^{∗}) ⊗_{P} Q^{∗}) ≅ \{ \begin{matrix}Tor_{P}^{j} (𝔽(K), 𝔽) & for i = 0 \\ Tor_{P}^{j} (H̃^{n−1}(K), 𝔽) & for i = n \\ 0 & otherwise \end{matrix}

Hence the E_{2}-page is concentrated in two vertical lines given by i = 0, n. Considering again total degrees shows that Tor_{P}^{j} (𝔽(K), 𝔽) = 0 for j ≠ 0 and that 𝔽(K) is a finitely generated free P -module (Theorem 4.1). This proves the other implication of the claim. □

We can draw the following consequence from the above proof. Corollary 7.3. If 𝔽(K) is Cohen-Macaulay, then there exists a short exact sequence

0 ⟶ H̃^{n−1}(K) ⟶ H^{∗}(X_{K}) ⟶ lim H^{∗}(X_{st(−)}) ⟶ Λ^{n−1}(n) ⊗ H̃^{n−1}(K) ⟶ 0

8. Proof of Theorem 1.2

The proof of Theorem 1.2 is an easy consequence of the following statement. Theorem 8.1. 𝔽(K) is Gorenstein^{∗} if and only if 𝔽(link_{K}(τ)) is Gorenstein^{∗} for all faces τ ≠ ∅ of K and

H̃^{i}(K; 𝔽) ≅ \{ \begin{matrix}𝔽 & for i = dim K \\ 0 & for i ≠ dim K \end{matrix}

Proof of Theorem 1.2: We argue as in the proof of Theorem 1.1. If dim K = 0, then K is always Cohen-Macaulay. And K is Gorenstein^{∗} if and only if X_{k} ≃ S^{2} if and only if m = 2 if and only if H̃^{0}(K; 𝔽) = 𝔽.

The general case follows by induction over the dimension of K and the above theorem.

□

The proof of Theorem 8.1 needs some preparation.

Remark 8.2. We call a simplicial complex reduced, if for every vertex i, the inclusion st({i}) ⊂ K is proper. If st({i}) = K then K = {i} ∗ link({i}). Hence, any complex K can be written as a joint product Δ ∗ L of a full simplex Δ and a reduced complex L. Moreover, K is reduced if and only if the union of all minimal missing faces equals the set V of vertices.

We call a simplicial complex K 𝔽-spherical if it satisfies the geometric condition of the Gorenstein property, i.e. if

H̃^{i}(link(τ); 𝔽) ≅ \{ \begin{matrix}𝔽 & for i = dim link(τ) \\ 0 & for i ≠ dim link(τ) \end{matrix}

for every face τ ∈ K in including ∅. Proposition 8.3. Let 𝔽(K) be Gorenstein. Then the following holds:

(i) K is reduced if and only if H̃^{n−1}(K) ≠ 0. In fact, if this is the case, then H̃^{n−1}(K) ≅ 𝔽. (ii) If K is reduced, then, for every vertex i ∈ V, the link link({i}) is also reduced.

Proof. By Corollary 7.3 we have an exact sequence

0 ⟶ H̃^{n−1}(K) ⟶ H^{∗}(X_{K}) ⟶ ∏_{i∈V} H^{∗}(X_{st({i})})

Let d =^{def} dim X_{K}. If H̃^{n−1}(K) = 0 then the second map becomes a monomorphism and there exists an i ∈ V such that 𝔽 ≅ H^{d}(X_{K}) ⟶ H^{d}(X_{st({i})}) is a monomorphisms. On the other hand, if L ⊂ K is a subcomplex of the same dimension as K such that 𝔽(K) and 𝔽(L) are Cohen-Macaulay, then the map H^{∗}(X_{K}) ⟶ H^{∗}(X_{L}) is an epimorphism. Hence, H^{d}(X_{K}) ≅ H^{d}(X_{st({i})}) ≅ 𝔽 for some vertex i ∈ K. Since both algebras are PD-algebras (Theorem 6.3), this implies that H^{∗}(X_{K}) ≅ H^{∗}(X_{st({i})}). Comparing the two fibrations defining X_{K} and X_{st({i})} shows that 𝔽(st({i})) ≅ 𝔽(K), that st({i}) = K, and that K is not reduced.

If K is not reduced, then, as the cone of a subcomplex, |K| is contractible and H̃^{n−1}(K) = 0. This proves the equivalence in part (i). If one of the conditions is satisfied, then, since H^{∗}(X_{K}) is P D-algebra, Lemma 3.2 shows that 𝔽 ≅ H^{n^{2}+n}(X_{K}) ≅ H̃^{n−1}(K).

Since Gorenstein algebras are Cohen-Macaulay, Theorem 7.1 and Proposition 3.2 (ii) tell us that there exists a short exact sequence

0 ⟶ H^{n^{2}+n−2}(X_{K}) ⟶ ∏_{i∈V} H̃^{n−2}(link({i}) ⟶ H̃^{n−1}(K) ≅ 𝔽 ⟶ 0.

Since X_{K} is a P D-space of dimension n^{2} + n , the first term in the above sequence is isomorphic to H_{2}(X_{K}) ≅ 𝔽^{m−1}. Hence, the middle term must be isomorphic to 𝔽^{m}. Since for each vertex i ∈ V, dim_{𝔽} H̃^{n−2}(link({i})) ≤ 1, this shows that H̃^{n−2}(link({i})) ≅ 𝔽 and that link({i}) is reduced. □

For τ ⊂ V we denote by K_{τ} ⊂ K the full subcomplex which consist of all faces ρ ∈ K such that ρ ⊂ V \ τ. If τ = {i} is a vertex, we denote this complex by K_{i}.

Lemma 8.4. If K is 𝔽-spherical, then, for each vertex i ∈ K, the complex K_{i} is Cohen-Macaulay and H̃^{n−1}(K_{i}) = 0.

For the proof we need some preparation. The inclusions K_{i} ⊂ K and st({i}) ⊂ K induce epimorphisms 𝔽(K) ⟶ 𝔽(K_{i}) and β_{i} : 𝔽(K) ⟶ 𝔽(st({i})) of P -modules. The kernel of the first map is the ideal v_{i}𝔽(K) generated by v_{i} ∈ 𝔽(K). And the second epimorphism induces an isomorphism v_{i}𝔽(K) ≅ v_{i}𝔽(st({i})). This follows from the fact that for any face τ ∈ K the monomial v_{i}v_{τ} = 0 in 𝔽(st({i})) if and only if τ ∪ {i} ∉ K. Moreover, since st({i}) = {i} ∗ link({i}), multiplication by v_{i} induces an isomorphism 𝔽(st({i})) ⟶ v_{i}𝔽(st({i})). In fact, all the above maps are 𝔽(K)-linear, where 𝔽(K) acts on 𝔽(K_{i}) and 𝔽(st({i})) via the above projections. Moreover they fit together to a short exact sequence

0 ⟶ 𝔽(st({i})) ⟶ 𝔽(K) ⟶ 𝔽(K_{i}) ⟶ 0

of 𝔽(K)-modules. The first map is given by q ↦ v_{i}q′ where β_{i}(q′) = q. Applying the functor −⊗_{P} establishes an epimorphism α_{i} : H^{∗}(X_{K}) ⟶ H^{∗}(X_{st({i})}) and an H^{∗}(X_{K})-linear map

ψ_{i} : H^{∗}(X_{st({i})}) ⟶ v_{i}H^{∗}(X_{K})

given by ψ_{i}(a) = v_{i}a′ where α_{i}(a′) = a. In fact, we will show that this map is an isomorphism (see Corollary 8.5).

Proof of Lemma 8.4: Since for two vertices i, j ∈ K we have link_{K}({j})_{i} equals link_{K}({j}) if the simplex {i, j} ∉ K and equals link_{K_{i}}({j}) if {i, j} ∈ K, we only have to prove that H̃^{r}(K_{i}) = 0 for r ≤ n − 1. And this claim we prove via an induction over the dimension of K.

For n = 1, 𝔽-spherical implies that K consists only of two vertices. And for n = 2, 𝔽spherical implies that K is a triangulation of S^{1}. In both cases, the claim is straightforward.

Now let us assume that n ≥ 3. By excision, H^{∗}(K, K_{i}) ≅ H^{∗}(st_{K}({i}), link_{K}({i})) ≅ H^{∗}(Σ link_{K}({i})). Here Σ link_{K}({i}) denotes the suspension of link_{K}({i}), actually of the geometric realization of link_{K}({i}). Moreover, the map H^{∗}(K, K_{i}) ⟶ H^{∗}(K) can be identified with H^{∗}(Σ link_{K}({i})) ⟶ H^{∗}(K) induced by the last arrow in the cofibration sequence link_{K}({i}) ⟶ K_{i} ⟶ K ⟶ Σ link_{K}({i}). Since Σ link_{K}({i}) is 𝔽-spherical, it suffices to show that H^{n−1}(K_{i}) = 0.

Let j ∈ V such that τ =^{def} {i, j} ∈ K. In the following, K will also denote the geometric realization of K. Because of the identities link_{K}(τ) = link_{link_{K}({i})}({j}) = link_{link_{K}({j})}({i}), link_{K_{i}}({j}) = link_{K}({j})_{i} and link_{K_{j}}({i}) = link_{K}({i})_{j} all rows and columns in the homotopy commutative diagram

link_{K}(τ) ⟶ link_{K_{j}}({i}) ⟶ link_{K}({i})
↓  ↓  ↓
link_{K_{i}}({j}) ⟶  K_{τ}  ⟶  K_{i}
↓  ↓  ↓
link_{K}({j}) ⟶  K_{j}  ⟶  K

consist of cofibrations. Passing to suspensions we can extend the diagram to the right and to the bottom yielding a homotopy commutative 4×4-diagram, whose bottom right square looks like

K  ⟶ Σ link_{K}({j})
↓  ↓
Σ link_{K}({i}) ⟶ Σ^{2} link_{K}(τ).

In the induced diagram in cohomology in degree n − 1

H^{n−1}(Σ^{2} link_{K}(τ)) ⟶ H^{n−1}(Σ link_{K}({j}))
↓  ↓
H^{n−1}(Σ link_{K}({i})) ⟶  H^{n−1}(K)

the left vertical and top horizontal arrows are isomorphisms by induction hypothesis. Therefore, the other two arrow are either both isomomorphisms or both trivial. And both are isomorphisms if and only if H^{n−1}(K_{i}) = H^{n−1}(K_{j}) = 0.

For n ≥ 2, the complex K is connected and we can connect each pair of vertices by 1-dimensional faces. The above argument now shows that the maps H^{n−1}(K, K_{i}) ⟶ H^{n−1}(K) are either isomorphisms for all vertices or trivial for all vertices. Hence it is sufficient to show this map is at least nontrivial for at least one vertex, or equivalently, that H^{n−1}(K_{i}) = 0 for at least one vertex.

Since H^{∗}(X_{K}) is generated by classes of degree 2, a generator of H^{n^{2}+n}(X_{K}) ≅ 𝔽 can be represented by a monomial a which can be written as v_{i}a′ for a suitable vertex i ∈ K. We fix this vertex. By the above considerations we have an exact sequence

H^{n^{2}+n−2}(X_{st({i})}) ⟶ H^{n^{2}+n}(X_{K}) ⟶ H^{n^{2}+n}(X_{K_{i}}).

The first map is given by multiplication with v_{i} and therefore an isomorphism. By Corollary 3.3, the latter map can be identified with the map H^{n−1}(K) ⟶ H^{n−1}(K_{i}), which, since dim link_{K}({i}) = n − 2, is an epimorphism. Hence, we have 0 = H^{n^{2}+n}(X_{K_{i}}) ≅ H^{n−1}(K_{i}), which completes the proof. □

Corollary 8.5. IF K is 𝔽-spherical, then, for each vertex i ∈ K, multiplication by v_{i} induces an isomorphism H^{∗}(X_{st({i})}) ⟶ v_{i}H^{∗}(X_{K}).

Proof. All terms of the exact sequence

0 ⟶ 𝔽(st({i})) ⟶ 𝔽(K) ⟶ 𝔽(K_{i}) ⟶ 0

are Cohen-Macaulay (Corollary 6.3, Theorem 1.1, Lemma 8.4. Hence, applying the functor ⊗_{P} 𝔽 establishes a short exact sequences

0 ⟶ H^{∗}(X_{st({i})}) ⟶ H^{∗}(X_{K}) ⟶ H^{∗}(X_{K_{i}}) ⟶ 0.

By construction, the first map is given by multiplication by v_{i}. □

Proof of Theorem 8.1: If K is Gorenstein^{∗}, then Proposition 8.3 shows that H^{n−1}(K) ≅ 𝔽 and, together with Corollary 6.3, that for each face τ ∈ K the algebra 𝔽(link(τ)) is Gorenstein^{∗}.

For the opposite conclusion it suffices to show that soc(H^{∗}(X_{K}) ≅ H^{n^{2}+n}(X_{K}) ≅ 𝔽 (Theorem 1.3(ii)). By induction we can assume that K is 𝔽-spherical.

For each vertex i ∈ V the map H^{∗}(X_{K}) ⟶ H^{∗}(X_{st({i})}) is an epimorphism. Hence, this map maps soc H^{∗}(X_{K}) to soc H^{∗}(X_{st({i})}) ≅ H^{n^{2}+n−2}(X_{st({i})}). Moreover, we have an exact sequence 0 ⟶ H^{n−1}(K) ⟶ H^{∗}(X_{K}) ⟶ ∏_{i} H^{∗}(X_{st({i})}) (Corollary 7.3), and hence all elements soc H^{∗}(X_{K}) have degree ≥ n^{2} + n − 2. Let a ∈ H^{n^{2}+n−2}(X_{K}). This class maps to 0 ≠ b ∈ H^{n^{2}+n−2}(X_{st({i})}) ≅ 𝔽 for some vertex i ∈ V. And, by Lemma 8.5, 0 ≠ v_{i}b = v_{i}a ∈ H^{n^{2}+n}(X_{K}). This shows that soc H^{∗}(X_{K}) ≅ H^{n^{2}+n}(X_{K}) ≅ H^{n−1}(K) ≅ 𝔽 and finishes the proof. □

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Department of Mathematics, University of Leicester, University Road, Leicester LE1 7RH, England

  E-mail address: dn8@mcs.le.ac.uk