DEPTH AND HOMOLOGY DECOMPOSITIONS

DIETRICH NOTBOHM

Abstract. Homology decomposition techniques are a powerful tool used in the analysis of the homotopy theory of (classifying) spaces. The associated Bousfield-Kan spectral sequences involve higher derived limits of the inverse limit functor. We study the impact of depth conditions on the vanishing of these higher limits and apply our theory in several cases. We will that the depth of Stanley-Reisner algebras can be characterized in combinatorial terms of the underlying simplicial complexes, the depth of group cohomology in terms of depth of group cohomology of centralizers of elementary abelian subgroups, and the depth of polynomial invariants in terms of depth of polynomial invariants of point-wise stabilizer subgroups. The latter two applications follow from the analysis of an algebraic version of centralizer decompositions in terms of Lannes’ T -functor.

1. Introduction

Homology decompositions are one of the most useful tools in the study of the homotopy theory of topological spaces. Given a cohomology theory h^{∗}, a homology decomposition for a space X is, roughly speaking, a recipe to glue together spaces, desirably of a simpler homotopy type, such that the resulting space maps into X by an h^{∗}-isomorphism.

Technically, this is described by a (covariant) functor F : C ⟶ Top from a (discrete) category C into the category Top of topological spaces together with a natural transformation F ⟶ 1_{X} from F to the constant functor 1_{X} which maps each object to X and each morphism to the identity. Passing to homotopy colimits, this natural transformation induces a map f_{X} : hocolim_{C}F ⟶ X.

In special circumstances we can decide by purely algebraic means, whether the map f_{X} is an h^{∗}-equivalence. The cohomology of the homotopy colimit can be calculated with the help of the Bousfield-Kan spectral sequence [2]. The E_{2}-page of this spectral sequence has the form

E_{2}^{i,j} =^{def} lim _{C^{op}}^{i} h^{∗}(F)

and converges towards h^{i+j}(X). If the derived limits lim _{C^{op}}^{i} h^{∗}(F) of the inverse limit lim_{C^{op}} h^{∗}(F) satisfy the equations

lim _{C^{op}}^{i} h^{∗}(F) ≅ \begin{cases}h^{∗}(X) & for i = 0 \\ 0 & for i ≥ 1, \end{cases}

1991 Mathematics Subject Classification. 55R40, 13A50, 13F55, 20J15, 55S10.

Key words and phrases. cohomology of groups, depth, face ring, homology decomposition, polynomial invariants, Stanley-Reisner algebra.

then f_{X} : hocolim_{C}Φ ⟶ X is an h^{∗}-equivalence. The first isomorphism has to be induced by the natural transformation 1_{h^{∗}(X)} ⟶ h^{∗}(F) respectively by the composition

h^{∗}(X) ⟶^{h^{∗}(f_{X})} h^{∗}(hocolim_{C} F) ⟶ lim_{C^{op}} h^{∗}(F).

We would like to find conditions which ensure the vanishing of the higher derived limits. In particular, we are interested in the impact made by depth conditions.

Let R be a noetherian local ring with maximal ideal m_{R} and let M be a finitely generated R-module. A sequence x_{1}, ..., x_{r} ∈ m_{R} is called regular on M, if for all i, the element x_{i} is not a zero divisor for the quotient M/(x_{1}, ..., x_{i−1})M. And the depth of M, denoted by depth M, is the maximal length of a regular sequence.

Let Φ : C ⟶ R_{fg}−mod be a functor, defined on a discrete category C and taking values in the category of finitely generated R-modules. Let 1_{M} ⟶ Φ be a natural transformation from the constant functor 1_{M} to Φ inducing a R-linear map ρ : M ⟶ lim_{C} Φ. Our technical key result Lemma 2.1 relates depth conditions on M, depth conditions on the values of the functor, on the kernel and cokernel of ρ, and the vanishing of the higher limits of Φ.

We will apply Lemma 2.1 in several cases, namely in the context of Stanley-Reisner algebras, group cohomology, invariant theory and algebras over the Steenrod algebra. In all cases we will describe the algebraic object in question as the inverse limit of a functor defined on a finite category. The first two application have a topological background in terms of homology decompositions.

Example 1.1. (Stanley-Reisner algebras) The main invariant to study the combinatorics of an abstract simplicial complex K is the associated face ring respectively Stanley-Reisner algebra 𝔽(K) (𝔽 a field), which is a quotient of a polynomial algebra generated by the vertices of the complex. Let cat(K^{×}) denote the category given by the non-empty faces of K Then, 𝔽(K) can be described as the inverse limit of a functor Φ_{K} : cat(K^{×}) ⟶ 𝔽−Alg from the category cat(K^{×}) given by the non-empty faces of K into the category of 𝔽-algebras. This algebraic setting has a topological realization. There exists a space c(K) such that H^{∗}(c(K); 𝔽) ≅ 𝔽(K) as well as a functor F_{K} : cat(K^{×})^{op} ⟶ Top and a natural transformation F_{K} ⟶ 1_{c(K)} such that H^{∗}(F_{K}; 𝔽) ≅ Φ_{K} and such that hocolim_{cat(K^{×})^{op}} F_{K} ⟶ c(K) is a homotopy equivalence [15](see also Section 7).

Properties of commutative algebra for 𝔽(K) are reflected in the geometry or combinatorics of the underlying simplicial complex. For example, Reisner proved such a theorem for the Cohen-Macaulay property [21]. With our methods we will generalize this result and characterize the depth of the face ring by the cohomological connectivity of the links of the faces of K (Theorem 7.1).

Example 1.2. (group cohomology) For a compact Lie group G and a fixed prime p, we denote by F_{p}(G) the Frobenius or Quillen category of G [19]. The objects are given by the non-trivial elementary abelian p-subgroups E ⊂ G and the morphism are all monomorphism E ⟶ E′ induced by conjugation by an element of G. For an object E ∈ F_{p}(G) we denote by BC_{G}(E) the classifying space of the centralizer C_{G}(E) of E. These assignments fit together to establish a functor F_{G} : F_{p}(G)^{op} ⟶ Top. The inclusions C_{G}(E) ⊂ G establish a natural transformation F_{G} ⟶ 1_{BG} and the map hocolim_{F_{p}(G)^{op}} F_{G} ⟶ BG induces an isomorphism in mod-p cohomology. Passing to mod-p cohomology gives rise to a functor Φ_{G} : F_{p}(G) ⟶ 𝔽_{p} − Alg defined by Φ_{G}(E) =^{def} H^{∗}(BC_{G}(E); 𝔽_{p}) and to a natural transformation 1_{H^{∗}(BG;𝔽_{p})} ⟶ Φ_{G}, which induces an isomorphism H^{∗}(BG; 𝔽_{p}) ⟶ lim_{F_{p}(G)} Φ_{G}. All these facts can be found in [10] and in [5]. We will show that depth H^{∗}(BG; 𝔽_{p}) equals the minimum of the numbers depth H^{∗}(BC_{G}(E); 𝔽_{p}) taken over all objects in F_{p}(G) (Theorem 5.1).

Example 1.3. (polynomial invariants) For a finite dimensional 𝔽_{p}-vector space V and a representation G ⟶ Gl(V) of a finite group G, we denote by 𝔽_{p}[V]^{G} the ring of polynomial invariants of the G-action on the symmetric algebra 𝔽_{p}[V] of the dual of V. Let S denote the collection of all point wise stabilizer subgroups of non trivial subspaces of V whose order is divisible by p. Let O(G) denote the orbit category of G. That is objects are given by quotients G/H and the morphisms by G-equivariant maps. Let O_{S}(G) ⊂ O(G) denote the full subcategory of the orbit category, whose objects are given by quotients G/H such that H ∈ S. There exists a functor Ψ_{G} : O_{S}(G)^{op} ⟶ 𝔽_{p} −Alg and natural transformation 1_{𝔽_{p}[V]^{G}} ⟶ Ψ_{G}. If p divides the order of G, then this natural transformation establishes an isomorphism 𝔽_{p}[V]^{G} ⟶^{≅} lim_{O_{S}(G)} Ψ_{G} [16]. In this case If p divides the order of G, we will show that depth 𝔽_{p}[V]^{G} equals the minimum of the numbers depth 𝔽_{p}[V]^{H} taken over all subgroups H ⊂ G contained in S (Theorem 6.1). If p does not divide the order of G, then 𝔽_{p}[V]^{G} is Cohen-Macaulay, i.e. depth 𝔽_{p}[V]^{G} equals the dimension of V [14].

The inequality depth 𝔽_{p}[V]^{G} ≤ depth 𝔽_{p}[V]^{H} for H ∈ S was already proven by Kemper [11] as well as by Smith (see [14]).

Example 1.4. (algebraic centralizer decomposition) The second example is a specialization of a more general statement concerning algebras over the Steenrod algebra. We can think of this generalization as an algebraic centralizer decomposition for algebras over the Steenrod algebra.

Let p be a fixed prime. Let A denote the mod-p Steenrod algebra and K the category of unstable algebras over A. Let A be a noetherian object of K. i.e. A is noetherian just as algebra. Rector constructed a category F(A), whose objects (E, ϕ) consist of a non-trivial elementary abelian p-group E and a K-morphism ϕ : A ⟶ H^{E} =^{def} H^{∗}(BE; 𝔽_{p}) into the mod-p cohomology of the classifying space BE of E, such that ϕ makes H^{E} to a finitely generated A-module [20]. With the help of Lannes’ T -functor [12], one can construct a functor Φ_{A} : F(A) ⟶ K and a natural transformation 1_{A} ⟶ ϕ_{A} respectively a map A ⟶ lim_{F(A)} Φ_{A}) (see Section 4). We will establish a relation between the depth of A respectively of the functor values and the vanishing of the higher limits (Theorem 4.1).

Besides Lemma 2.1 there is one further input in all four examples. In all four cases it turns out that depth conditions on 𝔽(K), H^{∗}(BG; 𝔽_{p}), 𝔽_{p}[V]^{G} or on A are inherited to the functor values of teh associated functor. This is a consequence of Theorem 3.1, which states that an application of Lannes’ T -functor may only increase the depth.

The paper is organized as follows. In the next section we establish our technical key result. In Section 3 we analyse the relation between and Lannes’ T -functor. This passes the way to discuss the case of algebraic centralizer decompositions (Section 4), the case of group cohomology (Section 5), the case of polynomial invariants (Section 6) and, finally, the case of Stanley-Reisner algebras (Section 7.

2. The key lemma

Let 𝔽 be a field. To simplify the discussion and since all applications are covered, we suppose that R is either a commutative local noetherian ring with maximal ideal m_{R} and residue field 𝔽 or or a connected commutative graded noetherian 𝔽-algebra. In the graded case, that is that R^{i} = 0 for i < 0 and R^{0} = 𝔽. In particular, R is a also a graded local ring with residue field 𝔽. The maximal ideal m_{R} =^{def} R^{+} is the set of elements of positive degree. In the graded case, the depth of M is defined as the maximal length of a sequence of homogeneous elements in R which is regular on M. Let R_{fg}−mod denote the category of finitely generated R-modules respectively the category of finitely generated non-negatively graded R-modules. Let Φ : C ⟶ R_{fg}−mod be a functor defined on a discrete category C. Let M be an object of R_{fg}−mod and 1_{M} ⟶ Φ a natural transformation from the constant functor 1_{M} to Φ. This establishes a R-linear map ρ : M ⟶ lim_{C} Φ. We measure the vanishing of the higher derived limits and the deviation of ρ being an isomorphism by the R-modules L^{i} =^{def} L^{i}(M, Φ), i ≥ −1. That is L^{−1} is the kernel and L^{0} the cokernel of ρ. And L^{i} =^{def} lim _{C}^{i} Φ_{M} for i ≥ 1.

Higher derived limits of the functor Φ can be defined as the cohomology groups of a certain cochain complex (C^{∗}(C, Φ), δ) over R (for details see [18]). For n ≥ 0, the Rmodule C^{n} =^{def} C^{n}(C, Φ) is defined as

C^{n} =^{def} ∏_{c_{o}→c_{1}→...→c_{r}} F (c_{n}) .

The differential δ : C^{n} ⟶ C^{n+1} is given by the alternating sum ∑_{k=0}^{r}(−1)^{k}δ^{k} where δ_{k} is defined on u ∈ C^{n} by

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

In particular, δ^{k} as well as δ are R-linear. And all the groups L^{i} are in a natural way finitely generated R-modules.

We fix all the above notation through out this section. We say that the sequence L^{i}, i ≥ −1, is almost trivial, if, for all i ≥ −1, either L^{i} = 0 or depth L^{i} = 0. The following result is the key lemma for proving our main results. Lemma 2.1. Let Φ : C ⟶ R_{fg}−mod be a functor. Suppose that depth Φ(c) ≥ r for all objects c ∈ C and that the sequence L^{i}, i ≥ −1, is almost trivial. Then, depth M ≥ r if and only if L^{i} = 0 for i ≤ r − 2.

The rest of this section is devoted to a proof of this lemma. The proof is based on spectral sequences associated to double complexes. To fix notation we recall the basic concept. Details may be found in [24].

A double complex over R or a differential bigraded R-module (D^{∗,∗}, d_{h}, d_{v}) is a bigraded R- module D^{∗,∗} with two R-linear maps d_{h} : D^{∗,∗} ⟶ D^{∗+1,∗} and d_{v} : D^{∗,∗} ⟶ D^{∗,∗+1} 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 D^{∗,∗} we associate a total complex T ot^{∗}(D) which is a differential graded R-module defined by T ot^{n}(D) =^{def} ⊕_{i+j=n}D^{i,j} with differential d =^{def} d_{h} + d_{v}.

If B_{∗} is a chain complex and C^{∗} a cochain complex over R then Hom_{R}(B_{∗}, C^{∗}) can be made into a bigraded R-module. We define D^{i,j} =^{def} Hom_{R}(B_{j}, C^{i}), d_{v} =^{def} Hom_{R}(d_{B}, id) and d_{h} =^{def} (−1)^{j} Hom_{R}(id, d_{C}) as the map D^{i,∗} ⟶ D^{i+1,∗}.

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

If D^{i,j} = 0 for i < 0 or j < 0, there exist two spectral sequences converging towards H^{∗}(T ot(D), d). In one case, we have _{I}E_{2}^{i,j} = H_{h}^{i}(H_{v}^{j}(D)) and in the other case _{II}E_{2}^{i,j} = H_{v}^{j}(H_{h}^{i}(D)). In the first case the differentials on the E_{r}-page have degree (1 − r, r) and in the second case degree (r, 1 − r).

Proof of Lemma 2.1: Since R is local, the depth of a finitely generated R-module N is characterized by the smallest number j, such that Ext_{R}^{j}(𝔽, N) ≠ 0 [24, Theorem 4.4.8]. Hence, we have to relate certain Ext-groups.

If r = 0 there is nothing to show since each R-module has depth ≥ 0. If r = 1 we have an exact sequence

0 ⟶ Ext_{R}^{0}(𝔽, L^{−1}) ⟶ Ext_{R}^{0}(𝔽, M) ⟶ Ext_{R}^{0}(𝔽, ⨁_{c∈C} M(c)) ≅ ⨁_{c∈C} Ext_{R}^{0}(𝔽, M(c)) = 0.

Hence, since L^{−1} = 0 or depth L^{−1} = 0, we have depth M ≥ 1 if and only if depth L^{−1} ≥ 1 if and only if L^{−1} = 0. This proves the statement for r = 1.

Now we assume that depth M ≥ r ≥ 2. In particular, L^{−1} = 0. We want to show that L^{i} = 0 for i ≤ r − 2. Let Q_{∗} be a R-projective resolution of 𝔽 as R-module, let C^{∗} =^{def} C^{∗}(C; Φ) and let D^{∗,∗} denote the differential bigraded R-module Hom_{R}(Q_{∗}, C^{∗}). Since D^{∗,∗} is concentrated in the first quadrant, both spectral sequences converge towards H^{∗}(Tot(D)). We have

IE_{2}^{i,j} = H_{h}^{i}(H_{v}^{j}(D)) ≅ lim^{i} Ext_{R}^{j}(𝔽, Φ),

which vanishes for j ≤ r − 1. On the other hand, we have

IIE_{2}^{i,j} = H_{v}^{j}(H_{h}^{i}(D)) ≅ \begin{cases}Ext_{R}^{j}(𝔽, L^{i}) & for i ≥ 1 \\ Ext_{R}^{j}(𝔽, lim _{C}^{0} Φ) & for i = 0. \end{cases}

The long exact sequence of Ext-groups for the short exact sequence

0 ⟶ M ⟶ lim_{C} Φ ⟶ L^{0} ⟶ 0

shows that Ext_{R}^{i}(𝔽, lim_{C} Φ) ≅ Ext_{R}^{i}(𝔽, L^{0}) for i ≤ r − 2.

By degree reasons and since no differential starts or ends at _{II}E_{2}^{0,0} ≅ Ext_{R}^{0}(𝔽, L^{0}), this group has to vanish. Since L^{0} = 0 or depth L^{0} = 0, this shows that L^{0} = 0, that M ≅ lim _{C}^{0} Φ and that _{II}E_{2}^{0,j} = 0 for j < r. We can repeat the argument successively for i = 1..., r − 2, which implies that L^{i} = 0 for i ≤ r − 2. Since there might be a non-trivial differential

IIE_{2}^{r−1,0} ≅ Ext^{0}(𝔽, L^{r−1}) ⟶ _{II}E_{2}^{0,r} ≅ Ext_{R}^{r}(𝔽, M),

we cannot repeat the argument further.

If L^{i} = 0 for i ≤ r − 2, the E_{2}-page of first spectral sequence is exactly the same, but may fail the vanishing statement. Since M ≅ lim _{C}^{0} Φ, the E_{2}-page of the second spectral sequence satisfies the equations

IIE_{2}^{i,j} ≅ \begin{cases}Ext_{R}^{j}(𝔽, M) & for i = 0 \\ 0 & for 1 ≤ i ≤ r − 2. \end{cases}

A similar degree argument shows that Ext_{R}^{j}(𝔽, M) = 0 for j ≤ r−1 and that depth M ≥ r. This finishes the proof. □

We also will apply Lemma 2.1 in the form of the following corollary. Corollary 2.2. If L^{i} = 0 for all i ≥ −1 then depth M ≥ min{depth Φ_{M}(c) : c ∈ C}.

3. Depth and the T -functor

To fix notation we recall some basic concepts from the T -functor technology. For more details see [22] and [12]

Let p be a fixed prime. As already mentioned in the introduction, A denotes the modp Steenrod algebra, U the category of unstable modules and K the category of unstable algebras over A.

For an elementary abelian p-group E we denote by H^{E} =^{def} H^{∗}(BE; 𝔽_{p}) the mod-p cohomology of the classifying space BE. The functor ⊗H^{E} : U ⟶ U has a left adjoint in U denoted by T_{E} : U ⟶ U. This functor is exact, commutes with tensor products and restricts therefore to a functor T_{E} : K ⟶ K also denoted by T_{E}. The adjoint of id : T_{E}(M) ⟶ T_{E}(M) gives a map M ⟶ T_{E}(M) ⊗ H^{E} and projecting to the first factor via the augmentation H^{E} ⟶ 𝔽_{p} a map e : M ⟶ T_{E}(M).

We can specialize further. For an algebra A ∈ K we denote by A−U the category of A ⊙ A-modules. An A ⊙ A-module is both, an object M ∈ U and an A-module such that the structure map A ⊗ M ⟶ M is A-linear. Here, A acts on the tensor product via the Cartan formula. We also denote by A_{fg}−U ⊂ A−U the full subcategory of all objects which are finitely generated as A-modules.

If M ∈ A−U, then, since T commutes with tensor products, the U-object T_{E}(M) is actually an object of T_{E}(A)−U. And if ϕ : A ⟶ H^{E} is a K-map, we define M(ϕ) =^{def} T_{E}(M; ϕ) =^{def} T_{E}(M) ⊗_{T_{E}^{0}(A)} 𝔽_{p}(ϕ), where 𝔽_{p}(ϕ) denotes the T_{E}(A)-module whose structure map is induced by the adjoint of the K-map ϕ : A ⟶ H^{E}.

Let F (n) ∈ U denote the free object in U generated by one element in degree n. The module F (n) is defined by the equation Hom_{U}(F (n), M) =^{def} M^{n} for any object M ∈ U. Here, M^{n} denotes the set of elements of M of degree n (for details see [22]. For any object M ∈ A−U, we have Hom_{A−U}(A ⊗_{𝔽_{p}} F (n), M) = Hom_{U}(F (n), M) = M^{n}. Therefore, there exists an A−U-epimorphism P ⟶ M such that P is free as A-module as well as a resolution

⟶ P_{n} ⟶ P_{n−1} ⟶ ... ⟶ P_{0} ⟶ M ⟶ 0

of A−U-modules, such that each P_{i} is free as A-module.

In the following we assume that A is noetherian as an algebra, which simplifies the discussion. In particular, Hom_{K}(A, H^{E}) is a finite set and T_{E}^{0}(A) ≅ 𝔽_{p}^{Hom_{K}(A,H^{E})} is a finite dimensional vector space isomorphic to the dual of Hom_{U}(A, H^{E}). And, if M ∈ A−U, then

T_{E}(M) ≅ T_{E}(M) ⊗_{T^{0}(A)} T^{0}(A) ≅ ⨁_{ϕ∈Hom_{K}(A,H^{E})} M(ϕ).

The above map e : M ⟶ T_{E}(M) respects all these additional structures and, restricting to a particular summand of T_{E}(M), establishes a map e_{ϕ} : M ⟶ M(ϕ). This is a map of A-modules, where A acts on M(ϕ) via the K-map A ⟶ A(ϕ), and therefore a A−Umorphism. In fact, if A ∈ K is noetherian and A ⟶ H^{E} a K-map, then the functor T_{E}(−, ϕ) restricts to a functor T_{E}(−, ϕ) : A_{fg}−U ⟶ A_{fg}−U [9]. In particular, if A is noetherian, so is A(ϕ) =^{def} T_{E}(A, ϕ).

In this section we want to prove the following theorem. Theorem 3.1. Let A ∈ K be a noetherian connected algebra and M ∈ A_{fg}−U. Let E be an elementary abelian group and let ϕ : A ⟶ H^{E} be a K-map. Then, depth M ≤ depth M(ϕ).

Using the Auslander Buchsbaum equation (e.g see [24]) we will reduce this statement to a claim about homological dimension.

Let A be a connected commutative graded algebra and M a graded A-module. We say that M has finite homological dimension if there exists a finite resolution

0 ⟶ P_{r} ⟶ P_{r−1} ⟶ ... ⟶ P_{0} ⟶ M ⟶ 0

of graded A-modules, such that P_{i} is projective for 0 ≤ i ≤ r. We say that hom-dim_{A} M = t if the shortest projective resolution has length t. Proposition 3.2. Let A be a noetherian object of K, M and object of A−U, and ϕ : A ⟶ H^{E} a K-map. If M has finite homological dimension over A, then M(ϕ) has finite homological dimension over A(ϕ)and hom-dim_{A} M ≥ hom-dim_{A(ϕ)} M(ϕ).

Proof. Let hom-dim_{A} M = t. We can choose a resolution

0 ⟶ P_{t} ⟶ P_{t−1} ⟶ ... ⟶ P_{0} ⟶ M ⟶ 0

of A−U-moduls, such that P_{i} is free as A-module for 0 ≤ i ≤ t − 1. Since hom-dim_{A} M = t, this implies that P_{t} is a projective A-module. And since A is connected and noetherian, in particular a graded local ring ring, P_{t} is a free A-module.

Applying the exact functor T_{E}(−, ϕ) establishes an exact A(ϕ)−U-sequence

0 ⟶ P_{t}(ϕ) ⟶ P_{t−1}(ϕ) ⟶ ... ⟶ P_{0}(ϕ) ⟶ M(ϕ) ⟶ 0.

Lannes T -functor maintains freeness properties [8]. Hence, for all i, the module P_{i}(ϕ is a free A(ϕ)-module, which proves the claim. □

The proof of Theorem 3.1 needs some further preparation. Let α : A ⟶ B and ϕ : A ⟶ H^{E} be K-maps, A and B noetherian. We denote by e(ϕ, α) the set of all K maps ψ : B ⟶ H^{E} such that ψα = ϕ. This is a finite set. Lemma 3.3. Let α : A ⟶ B and ϕ : A ⟶ H^{E} as above. For each object M of B−U, the T -functor induces an A−U-isomorphism M(ϕ) ≅ ⨁_{ψ∈e(ϕ,α)} M(ψ).

Proof. The algebra A acts on B(ψ) via the composition A ⟶ B ⟶ B(ψ) of K-maps. Since T^{0}(B) ⊗_{T^{0}(A)} 𝔽_{p}(ϕ) ≅ ⨁_{ψ∈e(ϕ,α)} 𝔽_{p}(ψ) as algebras, we have

\begin{aligned}M(ϕ) ≅ T (M) ⊗_{T^{0}(A)} 𝔽_{p}(ϕ) & ≅ & T (M) ⊗_{T^{0}(B)} T^{0}(B) ⊗_{T^{0}(A)} 𝔽_{p}(ϕ) \\ & ≅ & ⨁_{ψ∈e(ϕ,α)} T (M) ⊗_{T^{0}(B)} 𝔽_{p}(ψ) ≅ ⨁_{ψ∈e(ϕ,α)} M(ψ). \end{aligned}

And this composition is obviously a map of A-modules. □

The Dickson algebra D_{r} ≅ 𝔽_{p}[d_{1}, ...d_{r}] ≅ 𝔽_{p}[x_{1}, ..., x_{r}]^{Gl(r,𝔽_{p})} is a polynomial algebra isomorphic to the invariants of the general linear group Gl(r, 𝔽_{p}) acting on the polynomial algebra 𝔽_{p}[x_{1}, ..., x_{r}] with r-generators, which we give the degree 2. Then, this algebra carries an action of A which inherits an action to D_{r}. More details can be found in [14].

For a positive integer l we denote by D_{r}^{l} ⊂ D_{r} the subalgebra of all p^{l}-powers of elements of D_{r}. Then, D_{r}^{l} ≅ 𝔽_{p}[d_{1}^{p^{l}}, ..., d_{r}^{p^{l}}] is again a polynomial algebra , in particular a noetherian object of K [22].

Proof of Theorem 3.1: Since A is noetherian, there exist integers n, l ≥ 1 and a K map α : D =^{def} D_{n}^{l} ⟶ A making A into a finitely generated D-module [1, Appendix]. Hence, M is a finitely generated D-module as well, and for the calculation of depth M we can consider M as an D-module [23]. On the other hand, for each K ϕ : A ⟶ H^{E}, we have M(ϕα) ≅ ⨁_{ψ∈e(ϕα,α)} M(ψ) as D-modules and, since ϕ ∈ e(ϕα, α), we have depth M(ϕ) ≥ depth M(ϕα). Hence, we can and will assume that A = D.

Since D is a polynomial algebra, every finitely generated D-module has finite homological dimension (see [24]. By Proposition 3.2 we have hom-dim_{D} M ≥ hom-dim_{D(ϕ)} M(ϕ). Since D is a polynomial algebra, the same holds for D(ϕ) and D(ϕ) is a free finitely generated D-module [8]. In particular, every projective D(ϕ)-module is a projective D-module and hom-dim_{D(ϕ)} M(ϕ) ≥ hom-dim_{D} M(ϕ). By the Auslander-Buchsbaum equation (see [24]), we have depth N = depth D − hom-dim_{D} N for any finitely generated D-module. Proposition 3.2 shows that depth M ≤ depth M(ϕ). □.

4. Depth and the algebraic centralizer decomposition

We use the same notation as in the last section. For a noetherian object A ∈ K, Rector defined a category F(A) as follows [20]. The objects are given by pairs (E, ϕ), where E is a non-trivial elementary abelian p-group and where ϕ : A ⟶ H^{E} is a K-map such that H^{E} becomes a finitely generated A-module. And a morphism α : (E, ϕ) ⟶ (E′, ϕ′) is a monomorphism i_{α} : E ⟶ E′ such that ϕH^{∗}(Bi_{α}) = ϕ′. Since A is noetherian, this category is finite.

Let M be an object of A−U. Since the T -functor its natural with respect to homomorphism E ⟶ E′, the maps e_{ϕ} : M ⟶ M(ϕ) are compatible with all morphisms in the category F(A). This defines a covariant functor

Φ_{M} : F(A) ⟶ A−U : M ↦ M(ϕ)

as well as a natural transformation 1_{M} ⟶ Φ_{M}. In particular, there exists an A−Umorphism

ρ_{M} : M ⟶ lim_{F(A)} M(ϕ).

For a topological interpretation of these algebraic data and their relation to centralizer decompositions of compact Lie groups see [5] (see also Section 5).

As in Section 2, we define

L^{i}(M) =^{def} \begin{cases}lim _{F(A)}^{i} Φ_{M} & for i ≥ 1 \\ coker(ρ_{M}) & for i = 0 \\ ker(ρ_{M}) & for i = −1 \end{cases}

Theorem 4.1. Let A ∈ K be a connected noetherian algebra and let M be an object of A_{fg}−U. Then, depth M ≥ r if and only if the following two conditions hold:

(i) For all objects (E, ϕ) ∈ F(A), we have depth M(ϕ) ≥ r. (ii) L^{i}(M) = 0 for i ≤ r − 2

Proof. In [9], Henn has shown that the A-modules L^{i} are finite dimensional 𝔽_{p}-vector spaces. In particular, either L^{i}(M) = 0 or depth L^{i} = 0 for all i ≥ −1. That is, the sequence {L^{i} : i ≥ −1} is almost trivial. Moreover, by Theorem 3.1, we have depth M(ϕ) ≥ depth M. An application of Lemma 2.1 finishes the proof. □

5. Depth and group cohomology

Let G be a compact Lie group with classifying space BG. And let p be a fixed prime. In this section, cohomology is always taken with 𝔽_{p}-coefficients and H^{∗}(−) =^{def} H^{∗}(−; 𝔽_{p}).

As in the introduction we denote by F_{p}(G) the Quillen category of G and by F_{G} : F_{p}(G)^{op} ⟶ Top the functor mapping E to BC_{G}(E). The natural transformation F_{G} ⟶ 1_{BG} establishes a map hocolim_{F_{p}(G)^{op}} F_{G} ⟶ BG. Jackowski and McClure showed that this map induces an isomorphism in mod-p cohomology. In fact they showed that H^{∗}(BG) ≅ lim_{F_{p}(G)} H^{∗}(F_{G}), that lim _{F_{p}(G)}^{i} H^{∗}(F_{G}) = 0 for i ≥ 1 [10].

Passing to classifying spaces and mod-p cohomology induces a functor

F_{p}(G)^{op} ⟶ F(H^{∗}(BG)),

which turns out to be a an equivalence of categories [5]. Moreover, for a subgroup i_{e} : E ⊂ G, we have H^{∗}(BC_{G}(E) ≅ T_{E}(H^{∗}(BG), H^{∗}(i_{E})) [12]. Hence, identifying both categories, there exists a natural equivalence H^{∗}(F_{G}) ⟶^{≅} Φ_{G}, where Φ_{G} =^{def} Φ_{H^{∗}(BG)} : F(H^{∗}(BG)) ⟶ H^{∗}(BG)−U denotes the functor constructed in the previous section in the case M = A = H^{∗}(BG). In particular, this implies that L^{i}(F(H^{∗}(BG))) = 0 for all i ≥ −1. and that depth H^{∗}(BG) ≤ depth H^{∗}(BC_{G}(E) (Theorem 3.1).

The inclusion C_{G}(E) ⊂ G induces a K-map H^{∗}(BG) ⟶ H^{∗}(BC_{G}(E)) and makes the target into a finitely generated H^{∗}(BG)-module [19]. Hence, depth_{H^{∗}(BG)} H^{∗}(BC_{G}(E)) = depth_{H^{∗}(BC_{G}(E))} H^{∗}(BC_{G}(E)) [23]).

Applying Corollary 2.2 and fitting all the above arguments together, this proves the following statement. Theorem 5.1. Let G be a compact Lie group. Then,

depth H^{∗}(BG) = min{depth H^{∗}(BC_{G}(E)) : E ∈ F_{p}(G)}.

If E ≅ E′ ×E′′ then we have E ⊂ C_{G}(E′) and C_{G}(E) = C_{C_{G}(E′)}(E). Hence, in the above theorem, we only have to take the minimum over all 1-dimensional elementary abelian psubgroups. Remark 5.2. Results of the Jackowski-McClure type do exist for homotopy theoretic versions of groups. In particular, Theorem 5.1 also holds for p-compact groups and p-local finite groups. For definitions and concepts of these notions see [6] and [7] respectively [3].

6. Depth and polynomial invariants

We use the same notation as in the introduction, V denotes a finite dimensional 𝔽_{p}-vector space, G ⟶ Gl(V) is a (faithful) representation, 𝔽_{p}[V] denotes the ring of polynomial functions on V respectively the symmetric algebra on the dual of V, 𝔽_{p}[V]^{G} ⊂ 𝔽_{p}[V] the subalgebra of polynomial invariants, S the collection of all pointwise stabilizer subgroups of non-trivial subspaces of V whose order is divisible by p, O_{S}(G) ⊂ O(G) the full subcategory of the orbit category associated to the collection S and Ψ_{G} : O_{S} ⟶ 𝔽_{p}[V]^{G}− mod the functor given by G/H ↦ 𝔽_{p}[V]^{H}. For a linear subspace U ⊂ V, we denote by G_{U} the point wise stabilizer of U. Theorem 6.1. If p divides the order of G and G ⟶ Gl(V) is a faithful representation, then

depth𝔽_{p}[V]^{G} = min{depth 𝔽_{p}[V]^{G_{U}} : 0 ≠ U ⊂ V }.

In particular, this says that depth 𝔽_{p}[V]^{G} ≤ depth 𝔽_{p}[V]^{G_{U}} for all non-trivial subspaces U ⊂ V. Since, for U ⊂ W ⊂ V, we have G_{U} = (G_{W})_{U}, we get away in the above statement by taking the minimum over all 1-dimensional subspaces.

Proof. The proof is based on Lemma 2.1 and Theorem 3.1. Before we can apply these results, we have to recall the necessary setting. We make 𝔽_{p}[V] into a graded 𝔽_{p}-algebra by giving all polynomial generators the degree 2. Then, there exists a unique A-action on 𝔽_{p}[V] and 𝔽_{p}[V] becomes an object of K. Moreover, the A-action is inherited to the ring of polynomial invariants 𝔽_{p}[V]^{G}. If U ⊂ V, the composition ϕ : 𝔽_{p}[V]^{G} ⟶ 𝔽_{p}[V] ⟶ 𝔽_{p}[U] ⟶ H^{U} is a K map and T_{U}(𝔽_{p}[V]^{G}, ϕ) ≅ 𝔽_{p}[V]^{G_{U}}. For example, all these constructions and claims can be found in [14]. In particular, this shows that depth 𝔽_{p}[V]^{G} ≤ depth 𝔽_{p}[V]^{G_{U}} for all non-trivial subspaces U ⊂ V (Theorem 3.1.

By [16] we have

lim _{O_{S}(G)}^{i} Φ_{G} ≅ \begin{cases}𝔽_{p}[V]^{G} & for i = 0 \\ 0 & for i ≥ 1. \end{cases}

Since depth 𝔽_{p}[V]^{G} ≤ depth Φ_{G}(G/H) = 𝔽_{p}[V]^{H} for all objects G/H ∈ O_{S}(G), an application of Corollary 2.2 finishes the proof. □

7. Stanley-Reisner algebras

An abstract simplicial complex K with m-vertices given by the set V =^{def} {1, ..., m} consists of a finite set K =^{def} {σ_{1}, ..., σ_{r}} of subsets of V, which is closed with respect to formation of subsets. The subsets σ_{i} ⊂ V are called the faces of K. The dimension of K is denoted by dim K and dim K = n − 1, if 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 field 𝔽 we denote by 𝔽(K) the associated face ring or Stanley-Reisner algebra of K over 𝔽. It is the quotient 𝔽[V]/(v_{σ} : σ ∉ K), where 𝔽[V] =^{def} 𝔽[v_{1}, ..., v_{m}] is a polynomial algebra on m-generators and v_{σ} =^{def} ∏_{j∈σ} v_{j}. We can think of 𝔽(K) as a graded object. It will be convenient to choose the topological grading and give the generators of 𝔽(K) and 𝔽[V] the degree 2.

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 cohomology groups H^{∗}(K) of K as the cohomology groups H^{∗}(|K|) of the topological realization. And H̃^{∗}(K) will denote the reduced cohomology.

For a face σ ∈ K, the star st_{K}(σ) and the link link_{K}(σ) of σ are defined as the simplicial subcomplexes st_{K}(σ) =^{def} {τ ∈ K : σ∪τ ∈ K} and link_{K}(σ) =^{def} {τ ∈ K : σ∪τ ∈ K, σ∩τ = ∅}. Theorem 7.1. Let K be an abstract finite simplicial complex. Let r ≤ dim K + 1. Then the following conditions are equivalent:

(i) depth 𝔽(K) ≥ r.

(ii) For all faces σ ∈ K we have H̃^{i}(link_{K}(σ); 𝔽) = 0 for i ≤ r − ♯σ − 2. (iii) H̃^{i}(K; 𝔽) = 0 = H^{i}(|K|, |K| − x; 𝔽) for i ≤ r − 2 and for all points x ∈ |K|.

The equivalence of the last two conditions was shown by Munkres [13]. In fact, for each face σ ∈ K and each inner point x ∈ σ, he showed that H̃^{i−♯σ}(link_{K}(σ)) ≅ H^{i}(|K|, |K|−x). This shows that the depth condition only depends on the topological structure of |K|, and not on the simplicial structure of K.

Since the Krull dimension of 𝔽(K) equals dim K + 1 = n, the face ring 𝔽(K) is Cohen-Macaulay if and only if depth 𝔽(K) = n. This shows that the above theorem generalizes Reisner’s characterization of Cohen-Macaulay face rings [21].

The proof needs some preparation. The poset structure of K, given by the subset relation between the faces, gives rise to a category denoted by cat(K), and the nonempty faces of K to the full subcategory cat(K^{×}) ⊂ cat(K). There exists a functor Φ_{K} : cat(K^{×}) ⟶ Alg_{𝔽} taking values in the category of 𝔽-algebras. For a face σ ∈ K we define Φ_{K}(σ) =^{def} 𝔽(st(σ)). For an inclusion σ ⊂ τ, we have st_{K}(τ) ⊂ st_{K}(σ), which defines the map 𝔽(st_{K}(σ)) ⟶ 𝔽(st(τ)).

The following theorem is proved in [15]: Theorem 7.2. Let K be an abstract finite simplicial complex. Then,

lim _{cat(K^{×})}^{i} Φ_{K} ≅ \begin{cases}𝔽(K) ⊕ H̃^{0}(K; 𝔽) & for i = 0 \\ H^{i}(K; 𝔽) & for i ≥ 1. \end{cases}

Proposition 7.3. Let K be an abstract finite simplicial complex. For all faces σ ∈ K we have depth 𝔽(link_{K}(σ)) + ♯σ = depth 𝔽(st_{K}(σ) ≥ depth 𝔽(K).

Proof. Since 𝔽(st_{K}(σ) ≅ 𝔽[σ] ⊗ 𝔽(link_{K}(σ)), the first equation always holds. Here, 𝔽[σ] ⊂ 𝔽[V] denotes the polynomial subalgebra generated by all v_{i} such that i ∈ σ.

For the proof of the second equation we first assume that 𝔽 = 𝔽_{p}. Then, 𝔽_{p}(K) is an unstable algebra over the Steenrod algebra. For σ ⊂ V we denote by E^{σ} the ♯σ-fold product of the cyclic group C_{p} of order p. Then, 𝔽_{p}[σ] ⊂ H^{σ} =^{def} H^{E^{σ}} is a subalgebra in the category K. In fact, it is the polynomial part of H^{σ}. Let ϕ_{σ} denote the composition ϕ_{σ} : 𝔽_{p}(K) ⟶ 𝔽_{p}[σ] ⟶ H^{σ}. By [17], there exists an isomorphism T_{E^{σ}}(𝔽_{p}(K), ϕ_{σ}) ≅ 𝔽_{p}(st_{K}(σ)). Now, the second equation follows from Theorem 3.1.

If 𝔽 is a general field of characteristique p > 0, the claim follows from the observation that depth 𝔽_{p}(K) = depth 𝔽_{p}(K) ⊗_{𝔽_{p}} 𝔽 = depth 𝔽(K).

If 𝔽 = ℚ, then we notice that depth ℚ(K) is the maximum of the set {depth 𝔽_{p}(K) : p a prime }. And if 𝔽 is a general field of characteristique 0, then we can argue as in the case of positive characteristique. □

Proof of Theorem 7.1: By Theorem 7.2, the groups L^{i} =^{def} L^{i}(𝔽(K), Φ_{K}) are all finite and vanish for i = −1, and i ≥ dim K + 1. We can apply Lemma 2.1.

If depth 𝔽(K) ≥ r, then depth 𝔽(st_{K}(σ)) = depth 𝔽(Φ(σ)) ≥ r for all σ ∈ K (Proposition 7.3), and H̃^{i}(K; 𝔽) = L^{i}(cat(K^{×}); Φ_{K}) = 0 for i ≤ r − 2. (Lemma 2.1).

If depth 𝔽(st_{K}(σ)) ≥ r for all ∅ ≠ σ ∈ K and if H̃^{i}(K; 𝔽) = 0 for i ≤ r − 2, Lemma 2.1 shows that depth 𝔽(K) ≥ r.

Since depth 𝔽(link_{K}(σ)) = depth 𝔽(st_{K}(σ)) − ♯σ and since dim link_{K}(σ) < dim K, an induction over the dimension of the simplicial complexes proves the claim. □

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

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