A LOCAL METHOD FOR INTEGRAL COHOMOLOGY OF POSETS

ANTONIO DÍAZ RAMOS

1. Introduction and summary

The standard cochain complex [4, Appendix II.3] to compute the integral cohomology of a small category C has as n-cochains a direct product indexed by chains of morphisms in C: c_{0} → c_{1} → … → c_{n}. Nevertheless, in particular cases there exist a simpler cochain complex with the same cohomology. For example, if C is a simplicial complex (viewed as a partially ordered set, a particular kind of category) then the simplicial cochain complex has as n-cochains a direct product indexed by the objects of dimension n. It is natural to wonder which are the special properties of the category associated to a simplicial complex which allow to use a simpler cochain complex.

In this paper we address this question when C is a graded partially ordered set (poset for short). The condition of being graded means that every object of the poset has a dimension associated, as in the case of simplicial complexes (see 2.1 for the precise definition). Roughly, a graded poset can be split in a set of layers, each one corresponding to the objects of a given dimension, and these layers are ordered linearly. We will use the letter P to denote a generic graded poset and dim(p) to denote the dimension of the object p ∈ P. The reader may want to keep in mind the opposite category of a simplicial complex as a model for P.

We find that certain conditions on the subposets P_{≥p} = {q ∈ P|q ≥ p} for every object p of the graded poset P ensure the existence of a simpler cochain complex which gives the integral cohomology of P. In the particular case of the opposite category of a simplicial complex this method gives the standard simplicial cochain complex. Another important example satisfying these conditions is the opposite category of A_{p}(G), the Quillen’s complex of a finite group G [12].

We will assume that the graded poset P is bounded in the sense the dimension of the objects of P attains a minimun value equal to 0, as it ocurrs in the case of simplicial complexes. In general, we denote by P_{n} the objects of dimension n (n ≥ 0). To specify the local conditions on P mentioned above we need first the following

Definition 1.1. Let P be a graded poset. A family of subsets J = {J_{n}^{p}}_{p∈P, 0 ≤ n ≤ dim(p)} with J_{n}^{p} ⊆ (P_{≥p})_{n} is a local covering family if

a) for each p and 0 ≤ n < dim(p) it holds that ⋃_{q∈J_{n+1}^{p}}(P_{≥q})_{n} = (P_{≥p})_{n}, and

b) for each p, 0 ≤ n < dim(p) and q ∈ J_{n+1}^{p} it holds that J_{n}^{q} ⊆ J_{n}^{p}.

Date: November 5, 2008.

Example 1.2. Consider the standard N-simplex △_{N} on the ordered vertices v_{1} < v_{2} < … < v_{N}. Given a face σ of △_{N} denote by v_{σ} the largest vertex in σ. For any 0 ≤ n ≤ dim(σ) define J_{n}^{σ} as those n-dimensional faces of σ which contain v_{σ}. Then J = {J_{n}^{σ}}_{σ∈△_{N}, 0 ≤ n ≤ dim(σ)} is a local covering family for △_{N}^{op}.

Example 1.3. Denote by E_{N} an elementary abelian p-group of rank N, E_{N} = ℤ_{p}× …^{N-times} ×ℤ_{p}, and its Quillen’s complex A_{p}(E_{N}), which coincides with the poset of all its nontrivial subgroups. Here the dimension map is the rank minus 1. Choose a total order for the rank 1 (dimension 0) subgroups of E_{N}. For H ≤ E_{N} denote by ℤ_{p}^{H} the largest rank 1 subgroup contained in H. For any 0 ≤ n ≤ dim(H) define J_{n}^{H} as those n-dimensional subgroups of H which contain ℤ_{p}^{H}. Then J = {J_{n}^{H}}_{H∈A_{p}(E_{N}), 0 ≤ n ≤ dim(H)} is a local covering family for A_{p}(E_{N})^{op}.

We will still require local covering families to satisfy the following extra condition.

Definition 1.4. Let P be a graded poset. For each object p ∈ P define inductively the integers R_{0}^{p} = 1 and R_{n}^{p} = ∑_{q∈(P_{≥p})_{n−1}} R_{n−1}^{q} − R_{n−1}^{p} for 1 ≤ n ≤ dim(p).

Definition 1.5. Let P be a graded poset and J be a local covering family for P. We say that J is adequate if P_{>p} is connected for dim(p) ≥ 2, and if we have the equality

R_{n}^{p} = ∑_{q∈J_{n}^{p}} R_{n}^{q}

for 0 ≤ n < dim(p) and any p ∈ P.

The next result is the main result of this work. It gives a simpler cochain complex to compute H^{∗}(P; ℤ) when P is equipped with an adequate local covering family.

Theorem A. Let P be a graded poset for which there exists an adequate local covering family. Then there exists a co-chain complex

0 → ∏_{p| dim(p)=0} ℤ^{R_{0}^{p}} → ∏_{p| dim(p)=1} ℤ^{R_{1}^{p}} → ∏_{p| dim(p)=2} ℤ^{R_{2}^{p}} → …,

of which cohomology is H^{∗}(P; ℤ).

We will prove this theorem in Section 4. There we also give an explicit description of the differential. This cochain complex, as well as its differential, do not depend on the particular adequate local covering family used, i.e., the existence of at least one adequate local covering family gives rise to the cochain complex described, and any two adequate local covering families produce the same cochain complex.

Example 1.6. Consider the local covering family J for the N-simplex △_{N}^{op} described in Example 1.2. It is easy to see that for any face σ of △_{N} and 0 ≤ n ≤ dim(σ)

R_{n}^{σ} = ∑_{l=0}^{n}(−1)^{(n−l)}(\begin{matrix}dim(σ)+1 \\ l \end{matrix}).

A little combinatorics proves that J is adequate, and the binomial expansion of 0 = (1 − 1)^{dim(σ)+1} shows that R_{dim(σ)}^{σ} = 1 for every σ. Also notice that choosing a different total order of the vertices we obtain different adequate local covering families.

Example 1.7. Consider the local covering family J for A_{p}(E_{N})^{op}, where E_{N} is an elementary abelian p-group of rank N, described in Example 1.3. The number of subgroups of order p^{r} of a given elementary abelian p-group of order p^{n} (1 ≤ r ≤ n) is given by

(1)  ϕ_{n,r} = \frac{(p^{n} − 1)(p^{n−1} − 1) … (p^{n−r+1} − 1)}{(p^{r} − 1)(p^{r−1} − 1) … (p − 1)}.

Using these formulae it is easy to show that J is adequate. Moreover, for any subgroup H ≤ E_{N} it holds that

R_{dim(H)}^{H} = p^{1+2+...+dim(H)} = p^{\frac{dim(H)(dim(H)+1)}{2}} .

Again different total orders for the rank 1 subgroups gives rise to different adequate local covering families.

These two examples are more general than they may appear. The reason is that they easily generalize to the opposite category of a graded poset which is locally as △_{N} (or as A_{p}(E_{N})), i.e., such that for each p ∈ P, the subcategory P_{≤p} is isomorphic to △_{N} (to A_{p}(E_{N}))) for some N ≥ 0. We will show this in 5. In particular, Example 1.6 produces an adequate local covering family for the opposite category of any simplicial complex and, more generally, for the opposite category of any simplex-like poset (see [5, 3.1]). The cochain complex associated to this adequate local covering family by Theorem A is precisely the classical simplicial cochain complex (see 5.4).

In turn, Example 1.7 gives rise to an adequate local covering family for A_{p}(G)^{op}, where A_{p}(G) is the Quillen’s complex of a finite group G, and, more generally, for any poset which is locally as A_{p}(E_{N}). Quillen’s complex A_{p}(G) is the poset of all nontrivial elementary abelian p-subgroups of G. It is graded by the map which assigns the 𝔽_{p}-rank minus 1. We denote the p-rank of G by rk_{p}(G).

Theorem B. Let G be a finite group and let A_{p}(G) be its Quillen’s complex at the prime p. Then there exists a cochain complex

0 → ∏_{A_{p}(G)_{0}} ℤ → ∏_{A_{p}(G)_{1}} ℤ^{p} → ∏_{A_{p}(G)_{2}} ℤ^{p^{3}} → … → ∏_{A_{p}(G)_{rkp(G)−1}} ℤ^{p^{\frac{rkp(G)(rkp(G)−1)}{2}}} → 0,

of which cohomology is H^{∗}(|A_{p}(G)|; ℤ), where A_{p}(G)_{n} consists of the elementary abelian p-subgroups of G of rank n + 1 for 0 ≤ n ≤ rk_{p}(G) − 1.

Brown’s complex S_{p}(G) consists of all the non-trivial p-subgroups of the finite group G. It is well known [12, Proposition 2.1] that the realizations of A_{p}(G) and S_{p}(G) are homotopy equivalent. Using Theorem B we can easily obtain a simple description for the Euler characteristic of |S_{p}(G)|. This homotopy invariant of |S_{p}(G)| was originally studied by Brown in [1], where he introduced the poset S_{p}(G) and he proved the celebrated result that χ(|S_{p}(G)|) = 1 mod |S|, where S is a Sylow p-subgroup of G. We combine all of this in the following

Corollary C. Let G be a finite group and let S_{p}(G) be its Brown’s complex at the prime p. Then

χ(|S_{p}(G)|) = ∑_{n=0}^{rk_{p}(G)−1}(−1)^{n}p^{\frac{n(n+1)}{2}} |A_{p}(G)_{n}| = 1 mod |S|,

where rk_{p}(G) is the p-rank of G, S is a Sylow p-subgroup of G and |A_{p}(G)_{n}| is the number of elementary abelian p-subgroups of G of rank n + 1 for n ≥ 0.

More recently χ(|S_{p}(G)|) has also been studied in [11] and in [8], where they obtain formulas for particular cases of the group G. Hall’s Enumeration Principle [7, Theorem 1.4] for a finite p-group G is deduced from the formulae (1). Corollary C is obtained by gluing the local formulae (1) for the elementary abelian p-subgroups of G into a single formula, and it can thought of an enumeration principle for a finite group G which is not necessarily a p-group.

With the present definition there is not adequate local covering family for cubical complexes, and it would be interesting to investigate if the conditions for adequate local covering family can be relaxed as to provide such family for cubical complexes. Also, Corollary C above suggests that the element ∑_{n=0}^{rk_{p}(G)−1}(−1)^{n}p^{\frac{n(n+1)}{2}} A_{p}(G)_{n} of the Burnside ring of G might coincide with the Leftschetz invariant of S_{p}(G) and A_{p}(G).

Organization of the paper: In Section 2 we introduce preliminaries and we describe a shifting argument in the category of functors Ab^{P}. In Section 3 we further develop this shifting argument for functors which takes as values free abelian groups. Then in Section 4 we specialize to the functor of constant value ℤ, and we obtain Theorem A. Posets which are locally either as simplicial complexes or as Quillen’s complex are studied in Section 5. Here Theorem B is proven.

Notation: If S is a finite set then |S| denotes the number of elements in S. For a small category C we denote its opposite category by C^{op} and its realization by |C|, i.e., the geometrical realization of the (simplicial set) nerve of the category C. By P we denote a poset, by △_{N} the standard N-simplex viewed as a poset and by E_{N} an elementary abelian p-group of rank N.

Acknowledgements: I would to thank my Ph.D. supervisor A. Viruel for his support during the development of this work.

2. Preliminaries

In this section we introduce graded posets P as well as we show a “shifting argument” to compute the cohomology of P with coefficients a functor F : P → Ab. This method will be developed systematically in the next section in case F takes free abelian groups as values. We denote by Ab and Ab^{P} the category of abelian groups and of functors from the poset P to Ab respectively.

We always think of a poset P as a category where the arrows represent the order relations. For a covariant functor F : P → Ab, by the cohomology of P with coefficients F we mean exactly

H^{∗}(P; F) = lim^{∗}F,

where lim^{n}F is the n-th derived functor of the inverse limit functor lim : Ab^{P} → Ab. These derived functors exist as there are enough injectives in the abelian category Ab^{P}. If c_{M} is the constant functor c_{M} : P → Ab of value M ∈ Ab then we write H^{∗}(P; c_{M}) = H^{∗}(P; M). The singular cohomology of the realization |P| with trivial coefficients M satisfies H^{∗}(|P|; M) = H^{∗}(P^{op}; c_{M}) = H^{∗}(P^{op}; M). A functor F ∈ Ab^{P} will be called acyclic if H^{∗}(P; F) = 0 for ∗ > 0.

The definition of a graded poset follows. Our definition is similar to the one in [2, p.32], being the difference that we ask for a “saturation”’ condition. In a poset P we say that p precedes q if p < q and there is no element r satisfying p < r < q.

Definition 2.1. Let P be a poset. P is called graded if there is a function dim : P → ℤ, called the dimension map of P, which is order reversing and such that if p precedes p′ then dim(p) = dim(p′) + 1. If p ∈ P then dim(p) is called the dimension of p.

The reason we require order reversing instead of order preserving is that we will work with the opposite categories of the categories which cohomology we are interested in. The reader may want to keep in mind the standard N-simplex △_{N}: dimension of the simplices is order reversing in △_{N}^{op}. To avoid pathologies we will assume that the graded posets P under consideration satisfy the descending chain condition. In fact, in the applications P will be finite.

Definition 2.2. Let P be a graded poset, let p ∈ P, let n ∈ ℤ and let S ⊆ ℤ. Then we define the following subposets of P: P_{≥p} = {q ∈ P|q ≥ p}, P_{>p} = {q ∈ P|q > p}, P_{n} = {q ∈ P| dim(q) = n} and P_{S} = {q ∈ P| dim(q) ∈ S}

From the homotopical viewpoint restricting to graded poset means no loss: any topological space is weakly homotopy equivalent to a CW -complex which, in turn, is homotopy equivalent to a simplicial complex. This last can be seen as a graded poset in which the dimension function is the dimension of its simplices.

Next we introduce some acyclic objects in Ab^{P}. For any F ∈ Ab^{P} consider the functor F′ ∈ Ab^{P} defined by F′ = ∏_{p∈P} S_{p}(F (p)). Here, for any p ∈ P and M ∈ Ab the representable functor S_{p}(M) is defined by S_{p}(M)(q) = M if q ≤ p and 0 otherwise. It is clear that:

(1) lim^{∗}F′ = 0 for any ∗ > 0, and

(2) for any G ∈ Ab^{P}, Hom_{Ab}^{P}(G, F′) = ∏_{p∈P} Hom_{Ab}(G(p), F (p)).

To finish this section we discuss shortly how to compute higher limits via a “shifting argument”. Fix F ∈ Ab^{P} and consider Ker_{F} ∈ Ab^{P} defined by

Ker_{F} (p_{0}) = ⋂_{p∈P_{>p_{0}}} Ker F (p_{0} → p)

and such that sends non-identity morphisms to zero. A family of maps {τ_{p} : F (p) → Ker_{F} (p)}_{p∈P} gives rise to a natural transformation λ : F ⇒ Ker′_{F}. If λ is object-wise injective then we obtain a short exact sequence in Ab^{P}

0 ⇒ F ⇒ Ker′_{F} ⇒ G ⇒ 0,

where G is the object-wise co-image of λ. As Ker′_{F} is acyclic the long exact sequence of the derived functors lim^{∗} gives lim^{i}F = lim^{i−1}G for i > 1 and lim^{1} F = Coim{lim Ker′_{F} → lim G}. The conditions in the next definition ensure that we can build such a natural transformation λ which is object-wise injective:

Definition 2.3. Let P be a graded poset, let F : P → Ab be a functor and let n ∈ ℤ. We say that F is n-condensed if

(a) F (p) = 0 if dim(p) < n, and

(b) Ker_{F} (p) = 0 if dim(p) > n for all p ∈ P.

If the functor F is n-condensed then we can consider the natural transformation for the maps τ_{p} : F (p) → Ker_{F} (p)

τ_{p} = { 1_{F} (p) if dim(p) = n
0  otherwise.

Notice that we have

Ker_{F} (p) = {

F (p) if dim(p) = n
0  otherwise

by hypotheses (a) and (b) of Definition 2.3. Moreover the functor Ker′_{F} takes values on objects

(2)  Ker′_{F} (p_{0}) = ∏_{p∈(P_{≥p_{0}})_{n}} F (p).

The homomorphism λ_{p} : F (p) → Ker′_{F} (p) is given by

λ_{i} = ∏_{p∈(P_{≥p_{0}})_{n}} F (p_{0} → p) : F (p_{0}) → Ker′_{F} (p_{0}) = ∏_{p∈(P_{≥p_{0}})_{n}} F (p).

So λ_{i} is a kind of “diagonal”. An easy induction argument on dim(p) ∈ {n, n + 1, …} shows that λ is a monic natural transformation and we obtain

Lemma 2.4. Let F : P → Ab be an n-condensed functor. Then there is a short exact sequence

\xymatrix{0 \ar@{=>}[r] & F \ar@{=>}[r]^{λ} & Ker′_{F} \ar@{=>}[r] & G \ar@{=>}[r] & 0.}

On the object p_{0}, G takes the value

(3)  G(p_{0}) = ∏_{p∈(P_{≥p_{0}})_{n}} F (p)/λ_{p_{0}}(F (p_{0})).

It is clear that G satisfies condition (a) of Definition 2.3 for n + 1, but in general condition (b) does not hold for G and n + 1. More precisely, if dim(p_{0}) > n + 1 then Ker_{G}(p_{0}) = 0 is equivalent to the natural map

F (p_{0}) → lim_{P_{>p_{0}}} F

being an isomorphism. This natural map is a monomorphism by condition (b) of F being n-condensed. So, Ker_{G}(p_{0}) = 0 if and only if F (p_{0}) → lim_{P_{>p_{0}}} F is surjective. We summarize these results in the following:

Lemma 2.5. Let F : P → Ab be an n-condensed functor. Then there is a short exact sequence

\xymatrix{0 \ar@{=>}[r] & F \ar@{=>}[r]^{λ} & Ker′_{F} \ar@{=>}[r] & G \ar@{=>}[r] & 0.}

Moreover, G is (n+1)-condensed if and only if for each object p_{0} of dimension greater than n + 1, we have F (p_{0}) →^{≅} lim_{P_{>p_{0}}} F.

Example 2.6. Consider the graded poset P with shape

\xymatrix{·_{2} \ar[r] \ar[dr] & ·_{1} \ar[r] \ar[dr] & ·_{0} \\ ·_{2} \ar[r] \ar[ur] & ·_{1} \ar[r] \ar[ur] & ·_{0},}

where the subindexes denote the dimension of the objects. The geometrical realization |P| has the homotopy type of a two dimensional sphere S^{2}. Consider now the functor F : P → Ab with values

\xymatrix{ℤ_{2} \ar[r]^{×2} \ar[dr]_{×2} & ℤ_{4} \ar[r]^{×3} \ar[dr]^{×6} & ℤ_{12} \\ ℤ_{2} \ar[r]_{×2} \ar[ur]^{×2} & ℤ_{4} \ar[r]_{×6} \ar[ur]_{×3} & ℤ_{12}.}

The functors Ker_{F} and Ker′_{F} have values

\xymatrix{0 \ar[r]^{0} \ar[ddr]_{0} & 0 \ar[r]^{0} \ar[ddr]^{0} & ℤ_{12} &  & ℤ_{12} ⊕ ℤ_{12} \ar[r]^{1} \ar[ddr]_{1} & ℤ_{12} ⊕ ℤ_{12} \ar[r]^{1+0} \ar[ddr]^{0+1} & ℤ_{12} \\  &  &  & and \\ 0 \ar[r]_{0} \ar[uur]^{0} & 0 \ar[r]_{0} \ar[uur]_{0} & ℤ_{12} &  & ℤ_{12} ⊕ ℤ_{12} \ar[r]_{1} \ar[uur]^{1} & ℤ_{12} ⊕ ℤ_{12} \ar[r]_{0+1} \ar[uur]_{1+0} & ℤ_{12}}

respectively. As Ker_{F} is concentrated in dimension 0 the functor F is 0-condensed. The functor G from Lemma 2.4 is given by:

\xymatrix{ℤ_{6} ⊕ ℤ_{12} \ar[r]^{π⊕1} \ar[dr]_{π⊕1} & ℤ_{3} ⊕ ℤ_{12} \ar[r]^{0} \ar[dr]^{0} & 0 \\ ℤ_{6} ⊕ ℤ_{12} \ar[r]^{π⊕1} \ar[ur]^{π⊕1} & ℤ_{3} ⊕ ℤ_{12} \ar[r]^{0} \ar[ur]_{0} & 0,}

where π : ℤ_{6} ↠ ℤ_{3} is the quotient map. Notice that G is not 1-condensed as Ker_{G} takes the value ℤ_{2} on the two objects of dimension 2 of P. In fact, if dim(p_{0}) = 2, then the map

F (p_{0}) = ℤ_{2} → lim_{P_{>p_{0}}} F = ℤ_{4}

clearly is not an isomorphism. A straightforward computation shows that

lim F = ℤ_{2}

and

lim^{1}F = Coim{lim Ker′_{F} → lim G} = Coim{ℤ_{12} ⊕ ℤ_{12} → ℤ_{2} ⊕ ℤ_{6} ⊕ ℤ_{12}} = ℤ_{2}.

The next section is devoted to finding “local” conditions on the shape of the poset P such that G and the subsequent functors obtained by applying Lemma 2.4 to a functor F that takes free abelian groups as values turn out to be condensed.

3. Local covering families

In this section we study a bit further the condition given in Lemma 2.5 when dealing with a functor F which satisfies the following condition

Definition 3.1. Let F : P → Ab be a functor where P is a graded poset. We say that F is free if F (p) is a finitely generated free abelian group for each object p ∈ P (not to be confused with a free object in the abelian category Ab^{P}).

We also shall need the following

Definition 3.2. Let A →^{f} B be a map between free abelian groups. We say that f is pure if Coker(f) is a free abelian group (see [6]).

If A ≅ ℤ^{n} is a finitely generated free abelian group we call rk(A) =^{def} n. The following property of pure maps is straightforward, and will be used repeatedly in what follows,

Lemma 3.3. Let A →^{f} B be a map in Ab between free abelian groups of the same rank. If f is pure and injective then it is an isomorphism.

Remark 3.4. From this section and on we will assume that any graded poset P considered satisfies:

• the dimension map dim : P → ℤ attains the minimum value 0 (this is equivalent to dim(P) bounded below in ℤ by considering the translation dim +k), and

• for any object p ∈ P the set P_{≥p} is finite (this is equivalent to (P_{≥p})_{n} finite for all 0 ≤ n ≤ dim(p)).

Consider an n-condensed F : P → Ab and the condition in Lemma 2.5 again: fix p_{0} of dimension greater than n + 1 and consider the map given by restriction

lim_{P_{>p_{0}}} F = Hom_{P_{>p_{0}}}(c_{ℤ}, F) → ∏_{p∈J} F (p)

over a subset J ⊆ (P_{≥p_{0}})_{n}. If this restriction map turns out to be injective (notice that it is injective for J = (P_{≥p_{0}})_{n} because F is n-condensed) then the composition

F (p_{0}) → lim_{P_{>p_{0}}} F → ∏_{p∈J} F (p)

is also injective. If F is in addition a free functor (Definition 3.1) then both groups F (p_{0}) and ∏_{p∈J} F (p) are free abelian groups (because we are assuming Remark 3.4). If the map

F (p_{0}) → ∏_{p∈J} F (p)

is pure then, by Lemma 3.3, the condition rk F (p_{0}) = ∑_{p∈J} rk F (p) implies that this

composition is an isomorphism and so F (p_{0}) →^{≅} lim_{P_{>p_{0}}} F. Thus we study the subsets J ⊆ P that make this restriction map a pure monomorphism. It turns out that local covering familes (Definition 1.1) are such. For convenience of the reader we repeat here their definition:

Definition 3.5. Let P be a graded poset with dimension function dim. A family of subsets J = {J_{n}^{p_{0}}}_{p_{0}∈P, 0 ≤ n ≤ dim(p_{0})} with J_{n}^{p_{0}} ⊆ (P_{≥p_{0}})_{n} is a local covering family if a) For each p_{0} and 0 ≤ n < dim(p_{0}) it holds that ⋃_{p∈J_{n+1}^{p_{0}}}(P_{≥p})_{n} = (P_{≥p_{0}})_{n}

b) For each p_{0}, 0 ≤ n < dim(p_{0}) and p ∈ J_{n+1}^{p_{0}} it holds that J_{n}^{p} ⊆ J_{n}^{p_{0}}

Notice that the definition does not depend on a functor defined over the category P. Also, we have J_{dim(p_{0})}^{p_{0}} = {p_{0}} by (a). The next definition states the relation we expect between a local covering family and an n-condensed free functor

Definition 3.6. Let P be a graded poset, J be a local covering family and F : P → Ab be an n-condensed free functor. We say that F is J -determined if for any object p_{0} of dimension greater than n + 1 the restriction map

lim_{P_{>p_{0}}} F → ∏_{p∈J_{n}^{p_{0}}} F (p)

is a monomorphism and the map

F (p_{0}) → ∏_{p∈J_{n}^{p_{0}}} F (p)

is pure. If dim(p_{0}) = n+1 then we require the last map above to be a pure monomorphism.

The main feature of local covering families is that they allow freeness plus J - determinacy to pass from F to G. For an object p_{0} with dim(p_{0}) ≥ n + 1 notice that the map

F (p_{0}) → ∏_{p∈J_{n}^{p_{0}}} F (p)

is a pure monomorphism as consequence of Definition 3.6. The condition of Definition 3.6 for dim(p_{0}) = n + 1 is added in order to obtain that G is a free functor. Notice that the following proposition restricts to functors which take free abelian groups as values.

Proposition 3.7. Let P be a graded poset and J a local covering family. Assume that F : P → Ab is n-condensed, free and J -determined and consider the functor G defined by

\xymatrix{0 \ar@{=>}[r] & F \ar@{=>}[r]^{λ} & Ker′_{F} \ar@{=>}[r] & G \ar@{=>}[r] & 0.}

If for each object p_{0} with dim(p_{0}) ≥ n + 1 it holds that rk F (p_{0}) = ∑_{p∈J_{n}^{p_{0}}} rk F (p), then G is (n + 1)-condensed, free and J -determined.

Proof. Notice that the hypothesis implies that for any object p_{0} of dimension dim(p_{0}) > n + 1 the two maps

F (p_{0}) → lim_{P_{>p_{0}}} F → ∏_{p∈J_{n}^{p_{0}}} F (p)

are isomorphisms. In particular, F (p_{0}) →^{≅} lim_{P_{>p_{0}}} F and so G is (n + 1)-condensed. If dim(p_{0}) = n + 1 then the map

F (p_{0}) → ∏_{p∈J_{n}^{p_{0}}} F (p)

is an isomorphism by hypothesis. Next we prove that G is a free functor. Consider any p ∈ P with dim(p) ≥ n + 1 (if dim(p) < n + 1 then G(p) = 0) and the short exact sequence of abelian groups

0 → F (p) →^{λ_{p}} Ker′_{F} (p) →^{π_{p}} G(p) → 0.

Then it is straightforward that the map

s_{p} : Ker′_{F} (p) = ∏_{q∈(P_{≥p})_{n}} F (q) ↠ ∏_{q∈J_{n}^{p}} F (q) →^{≅} F (p)

is a section of λ_{p}, i.e. s_{p} ∘ λ_{p} = 1_{F (p)} (since the restriction map F (p) → ∏_{q∈J_{n}^{p}} F (q) is injective). This implies that the short exact sequence above splits and so G(p) is a subgroup of the free abelian group Ker′_{F} (p), and thus it is free as well.

Next we prove that G is J -determined. Take p_{0} of dimension n = dim(p_{0}) greater than n + 2. We first check that the restriction map

lim_{P_{>p_{0}}} G → ∏_{p∈J_{n+1}^{p_{0}}} G(p)

is injective. Consider any element ψ ∈ lim_{(p_{0}↓P)_{∗}} G = Hom_{Ab}^{P}(ℤ, G) which is in the kernel of the restriction map above. Notice that, as dim(p_{0}) > n + 2, we can consider the subset J_{n+2}^{p_{0}} ⊆ P_{≥p_{0}}_{∗}. If for any q ∈ J_{n+2}^{p_{0}} it holds that ψ_{q}(1) = 0 then ψ = 0 because of Definition 3.5a) and because G is (n + 1)-condensed.

Thus take q ∈ J_{n+2}^{p_{0}}. We want to see that x =^{def} ψ_{q}(1) = 0. Recall the short exact sequence of abelian groups

0 → F (q) →^{λ_{q}} Ker′_{F} (q) →^{π_{q}} G(q) → 0

and take y ∈ Ker′_{F} (q) such that π_{q}(y) = x. Recall that Ker′_{F} (q) = ∏_{p∈(q↓P)_{n}} F (p) and denote by α_{p} : q → p the unique arrow from q to p for p ∈ (q ↓ P)_{n}.

Now consider the restriction y|_{J_{n}^{q}} ∈ ∏_{p∈J_{n}^{q}} F (p). Because dim(q) = n + 2 > n + 1 the map F (q) → ∏_{p∈J_{n}^{q}} F (p) is an isomorphism by hypothesis. Then there exists a unique z ∈ F (q) with F (α_{p})(z) = y_{p} for each p ∈ J_{n}^{q} ⊆ (q ↓ P)_{n}. If we prove that F (α_{p})(z) = y_{p} for each p ∈ (q ↓ P)_{n} then λ_{q}(z) = y. This implies that x = π_{q}(y) = π_{q}(λ_{q}(z)) = 0 and completes the proof.

Thus take p ∈ (q ↓ P)_{n}. By Definition 3.5a) there is β_{p} : p′ → p with p′ ∈ J_{n+1}^{q}. Write β_{p′} : q → p′ for the unique arrow from q to p′. It holds that α_{p} = β_{p} ∘ β_{p′}. By Definition 3.5b) we have that J_{n+1}^{q} ⊆ J_{n+1}^{p_{0}}. Thus G(β_{p′})(x) = G(β_{p′})(ψ_{q}(1)) = ψ_{p′}(1) = 0 as ψ is in the kernel of the restriction map. The short exact sequence

0 → F (p′) →^{λ_{p′}} Ker′_{F} (p′) →^{π_{p′}} G(p′) → 0

implies that there exists t_{p′} ∈ F (p′) such that λ_{p′}(t_{p′}) = Ker′_{F} (β_{p′})(y). Consider z_{p′} = F (β_{p′})(z). We have that z_{p′} and t_{p′} have the same image by the restriction map

lim_{P_{∗}^{p′}} F → ∏_{p∈J_{n}^{p′}} F (p)

because J_{n}^{p′} ⊆ J_{n}^{q}. Because F is J -determined then this restriction map is a monomorphism and so z_{p′} = t_{p′}. This implies that

F (α_{p})(z) = F (β_{p} ∘ β_{p′})(z) = F (β_{p})(z_{p′}) = F (β_{p})(t_{p′}) = y_{p}

and the proof of the restriction map being injective is finished.

Now we check that the map

ω : G(p_{0}) → ∏_{p∈J_{n+1}^{p_{0}}} G(p)

is pure. Take z ∈ ∏_{p∈J_{n+1}^{p_{0}}} G(p) such that there exists x ∈ G(p_{0}) with m · z = ω(x) for some m ≠ 0. We have to check that there exists x′ ∈ G(p_{0}) with z = ω(x′), or equivalently, that x = m · x′ for some x′ ∈ G(p_{0}). Recall once more the short exact sequence of abelian groups

0 → F (p_{0}) →^{λ_{p_{0}}} Ker′_{F} (p_{0}) →^{π_{p_{0}}} G(p_{0}) → 0

and take y ∈ Ker′_{F} (p_{0}) with π_{p_{0}}(y) = x. We are going to build h ∈ F (p_{0}) such that y − λ_{p_{0}}(h) = m · y′, i.e., such that for any p ∈ (P_{≥p_{0}})_{n} the element (y − λ_{p_{0}}(h))_{p} = y_{p} − F (p_{0} → p)(h) ∈ F (p) is divisible by m. This implies that x = m · x′ with x′ = π_{p_{0}}(y′).

Notice that by hypothesis for each q ∈ J_{n+1}^{p_{0}}, G(p_{0} → q)(x) = m · z_{q} ∈ G(q). This implies that there exist h_{q} ∈ F (q) and y_{q} ∈ Ker′_{F} (q) such that Ker′_{F} (p_{0} → q)(y) − λ_{q}(h_{q}) = m · y_{q}, i.e., such that for each p ∈ (q ↓ P)_{n} ⊆ (p_{0} ↓ P)_{n} we have that y_{p} − F (q → p)(h_{q}) = m · (y_{q})_{p} ∈ F (p) (it is enough to take y_{q} with π_{q}(y_{q}) = z_{q}).

To build h we use the map

τ : ∏_{p∈J_{n}^{p_{0}}} F (p) →^{≅} F (p_{0})

given by hypothesis, which is the inverse of the map

F (p_{0}) → ∏_{p∈J_{n}^{p_{0}}} F (p).

For each p ∈ J_{n}^{p_{0}} ⊆ (P_{≥p_{0}})_{n} choose, by Definition 3.5a), q(p) ∈ J_{n+1}^{p_{0}} such that there is an arrow q(p) → p. Then set η_{p} = F (q(p) → p)(h_{q(p)}) ∈ F (p), where h_{q(p)} is built as before. Define h =^{def} τ(η). By construction F (p_{0} → p)(h) = F (q(p) → p)(h_{q(p)}) for each p ∈ J_{n}^{p_{0}} (but not for an arbitrary p ∈ (P_{≥p_{0}})_{n}).

With this definition for h we check now that y_{p} − F (p_{0} → p)(h) is divisible by m for each p ∈ (P_{≥p_{0}})_{n}. This finishes the proof. Fix p ∈ (P_{≥p_{0}})_{n} and q_{p} ∈ J_{n+1}^{p_{0}} such that there is an arrow q_{p} → p (we are not assuming that q_{p} = q(p) if p ∈ J_{n}^{p_{0}}). On the one hand we have by hypothesis that

y_{k} − F (q_{p} → k)(h_{q_{p}}) = m · (y_{q_{p}})_{k}

for each k ∈ P_{n}^{q_{p}}. In particular,

(4)  y_{k} − F (q_{p} → k)(h_{q_{p}}) = m · (y_{q_{p}})_{k}

for each k ∈ J_{n}^{q_{p}}. Set h′ =^{def} F (p_{0} → q_{p})(h). Because q_{p} ∈ J_{n+1}^{p_{0}} then, by Definition 3.5b), J_{n}^{q_{p}} ⊆ J_{n}^{p_{0}} and thus by construction for any k ∈ J_{n}^{q_{p}}

y_{k} − F (p_{0} → k)(h) = y_{k} − F (q(k) → k)(h_{q(k)}) = m · (y_{q(k)})_{k}.

Notice that F (p_{0} → k)(h) = F (q_{p} → k)(F (p_{0} → q_{p})(h)) = F (q_{p} → k)(h′). On the other hand, we have obtained

(5)  y_{k} − F (q_{p} → k)(h′) = m · (y_{q(k)})_{k}

for each k ∈ J_{n}^{q_{p}}.

Now write η_{k} = (y_{q(k)})_{k} − (y_{q_{p}})_{k} for each k ∈ J_{n}^{q_{p}} and write h′′ = τ(η) ∈ F (q_{p}) where τ is the inverse of the map

F (q_{p}) → ∏_{p∈J_{n}^{qp}} F (p).

By Equations (4) and (5) it is straightforward that the elements h_{q_{p}} − m · h′′ and h′ have the same image by this map. Then, as this map is injective by hypothesis, h′ = h_{q_{p}} − m · h′′. As p ∈ J_{n}^{q_{p}} we have

y_{p} − F (p_{0} → p)(h) = y_{p} − F (q_{p} → p)(h′) = y_{p} − F (q_{p} → p)(h_{q_{p}} − m · h′′),

and this equals

m · (y_{q_{p}})_{p} + m · F (q_{p} → p)(h′′).

Thus y_{p} − F (p_{0} → p)(h) is divisible by m.

If dim(p_{0}) = n + 2 we have to see that the map

ω : G(p_{0}) → ∏_{p∈J_{n+1}^{p_{0}}} G(p)

is a pure monomorphism. To prove that ω is a monomorphism use the proof above starting where ψ_{q} is considered for an arbitrary object q of dimension n + 2. The proof of ω being pure is exactly the same as above. □

Remark 3.8. Notice that in the conditions of the proposition we have the following formula for the rank of the free abelian group G(p_{0}) for dim(p_{0}) ≥ n + 1

rk(G(p_{0})) = ∑_{p∈(P_{≥p_{0}})_{n}} rk F (p) − rk F (p_{0}).

This is so because of the short exact sequence of free abelian groups

0 → F (p_{0}) →^{λ_{p_{0}}} Ker′_{F} (p_{0}) →^{π_{p_{0}}} G(p_{0}) → 0.

Remark 3.9. In the conditions of the proposition there are isomorphisms

F (p_{0}) →^{≅} ∏_{p∈J_{n}^{p_{0}}} F (p)

for each p_{0} with dim(p_{0}) ≥ n + 1. Moreover, we have built a map s_{p_{0}} : Ker′_{F} (p_{0}) → F (p_{0}) with s_{p_{0}} ∘ λ_{p_{0}} = 1_{F (p_{0})}. To the homomorphism s_{p_{0}} corresponds the monomorphism

\xymatrix{G(p_{0}) \ar[r]^{δ_{p_{0}}} & Ker′_{F} (p_{0})}

given by

\xymatrix{π_{p_{0}}(x) \ar@{|->}[r] & x − (λ_{p_{0}} ∘ s_{p_{0}})(x)} ,

which satisfies π_{p_{0}} ∘ δ_{p_{0}} = 1_{G(p_{0})}. It is straightforward that, by construction, Im δ_{p_{0}} = ∏_{p∈(P_{≥p_{0}})_{n}\J_{n}^{p_{0}}} F (p), and thus

G(p_{0}) ≅^{δ_{p_{0}}} ∏_{(P_{≥p_{0}})\J_{n}^{p_{0}}} F (p).

Moreover, x = δ_{p_{0}}(y) is the only preimage of y by π_{p_{0}} which verifies x_{p} = 0 for p ∈ J_{n}^{p_{0}}.

The main consequence of the previous proposition is that it reduces the problem of whether G is (n + 1)-condensed to some integral equations. Moreover, this procedure can be applied recursively because the resulting functor G turns out to be (n + 1)- condensed, free and J -determined, and so the proposition applies to G too. Notice again that the ranks of G are given by Remark 3.8.

Example 3.10. Consider the graded poset P with shape

\xymatrix{a_{2} \ar[r] \ar[dr] \ar[ddr] & c_{1} \ar[r] \ar[dr] \ar[ddr] & f_{0} \\  & d_{1} \ar[r] \ar[dr] \ar@{-->}[ur] & g_{0} \\ b_{2} \ar[r] \ar@{-->}[ur] \ar[uur] & e_{1} \ar[r] \ar[ur] \ar[uur] & h_{0},}

where the subindexes denote the dimension of the objects. Notice that P has the homotopy type of a wedge of 4 2-dimensional spheres ⋁_{i=1}^{4} S_{i}^{2}. The sets

J_{0}^{f} = {f}, J_{0}^{g} = {g}, J_{0}^{h} = {h},
J_{1}^{c} = {c}, J_{0}^{c} = {f}, J_{1}^{d} = {d}, J_{0}^{d} = {g}, J_{1}^{e} = {e}, J_{0}^{e} = {h},
J_{2}^{a} = {a}, J_{1}^{a} = {c}, J_{0}^{a} = {f}, J_{2}^{b} = {b}, J_{1}^{b} = {e}, J_{0}^{b} = {h}.

define a local covering family J for P. Consider now the the functor F : P → Ab with values

\xymatrix{ℤ \ar[r] \ar[dr] \ar[ddr] & ℤ \ar[r] \ar[dr] \ar[ddr] & ℤ \\  & ℤ \ar[r] \ar[dr] \ar[ur] & ℤ \\ ℤ \ar[r] \ar[ur] \ar[uur] & ℤ \ar[r] \ar@{-->}[ur] \ar[uur] & ℤ,}

such that all the arrows arriving and departing from c and e are the identity and all the arrows arriving and departing from d are minus the identity. Then F is 0condensed, free and J -determined. Moreover, equations in Proposition 3.7 hold for any object of dimension greater or equal to 1. Thus we obtain a functor G which is 1-condensed, free and J -determined. By Remark 3.8 we know the ranks of the free abelian groups that G takes as values:

\xymatrix{ℤ^{2} \ar[r] \ar[dr] \ar[ddr] & ℤ^{2} \ar[r] \ar[dr] \ar[ddr] & 0 \\  & ℤ^{2} \ar[r] \ar[dr] \ar@{-->}[ur] & 0 \\ ℤ^{2} \ar[r] \ar@{-->}[ur] \ar[uur] & ℤ^{2} \ar[r] \ar@{-->}[ur] \ar[uur] & 0.}

Moreover, equations in Proposition 3.7 applied to G hold for any object of dimension greater or equal to 2, and thus we obtain a functor H which is 2-condensed, free and J -determined:

\xymatrix{ℤ^{4} \ar[r] \ar[dr] \ar[ddr] & 0 \ar[r] \ar[dr] \ar[ddr] & 0 \\  & 0 \ar[r] \ar[dr] \ar[ur] & 0 \\ ℤ^{4} \ar[r] \ar@{-->}[ur] \ar[uur] & 0 \ar[r] \ar[ur] \ar[uur] & 0,}

By Remark 3.9 we can identify G(a) ≅ F (g)⊕F (h) = ℤ_{g}⊕ℤ_{h}. Also G(c) ≅ ℤ_{g}⊕ℤ_{h}, G(d) ≅ ℤ_{f} ⊕ ℤ_{h} and G(e) ≅ ℤ_{f} ⊕ ℤ_{g}. By the definition of G as a co-image it is easy to see that

G(a → c) : ℤ_{g} ⊕ ℤ_{h} → ℤ_{g} ⊕ ℤ_{h}
(g, h) ↦ (g, h).

Also G(a → d)(g, h) = (−g, h − g) and G(a → e)(g, h) = (−h, g − h) with respect to the ordered basis mentioned above. Additional computations lead to

lim F = ℤ,

lim^{1}F = Coim{lim Ker′_{F} → lim G} = Coim{ℤ^{3} → ℤ^{2}} = 0, and
lim^{2}F = Coim{lim Ker′_{G} → lim H} = Coim{ℤ^{6} → ℤ^{8}} = ℤ^{4}.

4. Integral cohomology

In this section we apply the work developed in the preceding sections to compute the cohomology with integer coefficients of the realization of a graded poset P equipped with a local covering family J.

To compute the abelian group H^{n}(P; ℤ) for n ≥ 0 we compute the higher limit lim^{n}c_{ℤ}, where c_{ℤ} : P → Ab is the functor of constant value ℤ which sends every morphism to the identity 1_{ℤ}. We begin studying the extra conditions that the local covering family J must satisfy to apply iteratively the Proposition 3.7 beginning on c_{ℤ}.

First, notice that c_{ℤ} is 0-condensed (we are assuming 3.4) and free (Definition 3.1). By Definition 3.6, c_{ℤ} is J -determined as 0-condensed functor if and only if for each p_{0} ∈ P with dim(p_{0}) ≥ 2 the set J_{0}^{p_{0}} intersects each connected component of P_{>p_{0}}. The dimensional equation in Proposition 3.7 for p_{0} ∈ P with dim(p_{0}) ≥ 1 becomes rk c_{ℤ}(p_{0}) = 1 = |J_{0}^{p_{0}}| = ∑_{p∈J_{0}^{p_{0}}} rk c_{ℤ}(p). Thus, c_{ℤ} is J -determined as a 0-condensed functor if and only if P_{>p_{0}} is connected for dim(p_{0}) ≥ 2 and |J_{0}^{p_{0}}| = 1 for dim(p_{0}) ≥ 1. The successive applications of Proposition 3.7 give, by the dimensional equation in the statement of the Proposition 3.7, the following:

Definition 4.1. Let P be a graded poset. Define, inductively on n, the integer R_{n}^{p_{0}} for each object p_{0} with dim(p_{0}) ≥ n by R_{0}^{p_{0}} = 1 and by

R_{n}^{p_{0}} = ∑_{p∈(P_{≥p_{0}})_{n−1}} R_{n−1}^{p} − R_{n−1}^{p_{0}}

for n ≥ 1.

Definition 4.2. Let P be a graded poset and J be a local covering family for P. We say that J is adequate if P_{>p_{0}} is connected for dim(p_{0}) ≥ 2, and if we have the equality

R_{n}^{p_{0}} = ∑_{p∈J_{n}^{p_{0}}} R_{n}^{p}

for 0 ≤ n < dim(p_{0}) and any p_{0} ∈ P.

Proposition 4.3. Let P be a graded poset and let J be an adequate local covering family. Then there is a sequence of functors F_{0}, F_{1}, F_{2}, … defined by F_{0} =^{def} c_{ℤ} : P → Ab and by the short exact sequence

\xymatrix{0 \ar@{=>}[r] & F_{n−1} \ar@{=>}[r]^{λ_{n−1}} & Ker′_{F_{n−1}} \ar@{=>}[r]^{π_{n}} & F_{n} \ar@{=>}[r] & 0}

for n = 1, 2, 3, . . .. Moreover, F_{n} is n-condensed, free and J -determined for n ≥ 0. For dim(p_{0}) ≥ n we have rk F_{n}(p_{0}) = R_{n}^{p_{0}}.

The local properties of a graded poset P equipped with an adequate local covering family J give rise to a sequence F_{0} = c_{ℤ}, F_{1}, F_{2},… of functors. Now, we study some properties of these functors which are independent of the local covering family J.

The first point to notice is that the sequence of functors from Proposition 4.3 does not depend on the adequate local covering family J. Thus, two or more adequate local covering families can be considered for the the same graded poset and they still give rise to the same sequence of functors. Next we focus on the short exact sequence

\xymatrix{0 \ar@{=>}[r] & F_{n−1} \ar@{=>}[r]^{λ_{n−1}} & Ker′_{F_{n−1}} \ar@{=>}[r]^{π_{n}} & F_{n} \ar@{=>}[r] & 0}

for n ≥ 1 of Proposition 4.3. The beginning of the long exact sequence of derived functors for this short exact sequence is

(6)  \xymatrix{0 \ar[r] & lim F_{n−1} \ar[r]^{ι_{n−1}} & lim Ker′_{F_{n−1}} \ar[r]^{ω_{n}} & lim F_{n} \ar[r] & H^{n}(P; ℤ) \ar[r] & 0,}

where ι_{n−1} = λ̃_{n−1} and ω_{n} = π̃_{n} are the induced maps. Notice that the three inverse limits appearing above are free abelian groups as the corresponding functors take free abelian groups as values. In fact, for the middle term we have the exact description

(7)  lim Ker′_{F_{n−1}} ≅ ∏_{p∈P_{n−1}} F_{n−1}(p)

as it is easily deduced from Section 2. It turns out that there is also a simpler description for lim F_{n}, which can be interpreted as the analogue in the context of CW -complexes to the fact that the cohomology on dimension n depends upon the n + 1 skeleton (recall that F_{n}(p) = 0 if dim(p) < n):

Lemma 4.4. Let P be a graded poset and let J be an adequate local covering family. Let c_{ℤ}, F_{1}, F_{2}, … be the sequence of functors given by Proposition 4.3. Then

lim F_{n} ≅ lim F_{n}|_{P_{{n+1,n}}}

for each n ≥ 0.

Proof. Consider the restriction map

lim F_{n} → lim F_{n}|_{P_{{n+1,n}}}.

This map is clearly a monomorphism because F_{n} is an n-condensed functor. To see that it is surjective take ψ ∈ lim F_{n}|_{P_{{n+1,n}}}. We want to extend ψ to each p ∈ P with dim(p) > n + 1. We do it inductively on dim(p).

Notice that (see the beginning of the proof of Proposition 3.7)

F_{n}(p) → lim_{P_{>p}} F_{n}

is an isomorphism for dim(p) > n + 1. For dim(p) = n + 2 we have that j ∈ P_{>p} implies that dim(j) ≤ n + 1. Then there is a unique way of extending ψ to ψ(p). Once we have extended ψ to P_{{n+2,n+1,n}} we proceed with an induction argument. That the extension that we are building is actually a functor is again due to that F_{n} is n-condensed. □

Also, from Equations (6) and (7), we have the following formula, analogue to that of the Euler characteristic for CW -complexes:

Lemma 4.5. Let P be a graded poset for which exists an adequate local covering family and such that max dim(P) is finite and P_{n} is finite for each 0 ≤ n ≤ max dim(P). Then

∑_{n=0}^{max dim(P)}(−1)^{n} rk H^{n}(P; ℤ) = ∑_{n=0}^{max dim(P)}(−1)^{n} ∑_{p∈P_{n}} R_{n}^{p}.

Take up again Equations (6) and (7). We can form the sequence of abelian groups

0 → ∏_{p∈P_{0}} F_{0}(p) →^{d_{0}} ∏_{p∈P_{1}} F_{1}(p) →^{d_{1}} ∏_{p∈P_{2}} F_{2}(p) →^{d_{2}} …

where d_{n} = ι_{n+1} ∘ w_{n+1} for n ≥ 0. Then it is straightforward that this sequence is a co-chain complex and its cohomology is exactly the cohomology of P with integral coefficients:

Theorem 4.6. Let P be a graded poset for which exists an adequate local covering family. Then there exists a co-chain complex

0 → ∏_{p∈P_{0}} F_{0}(p) →^{d_{0}} ∏_{p∈P_{1}} F_{1}(p) →^{d_{1}} ∏_{p∈P_{2}} F_{2}(p) →^{d_{2}} …

of which cohomology is H^{∗}(P; ℤ).

This proves Theorem A in the introduction. Recall that the rank of F_{dim(p)}(p) is given by Proposition 4.3 as rk F_{dim(p)}(p) = R_{dim(p)}^{p}.

5. Locally-like posets

In this section we show that if the poset P is locally either as △_{N} or as A_{p}(E_{N}) then its opposite category P^{op} is graded and can be equipped with an adequate locally covering family. We start defining posets which locally are like simplicial complexes:

Definition 5.1. Let P be a poset. Then P is simplex-like if for all p ∈ P the category P_{≤p} is isomorphic to △_{n} for some n.

The notion of simplex-like poset is half-way between semi-simplicial complexes (as defined originally by Eilenberg and Zilber in [3]) and simplicial complexes in the classical sense. Moreover, ([5, 3.1]), simplicial complexes are exactly the simplexlike posets such that any two elements which have a lower bound have an infimum, i.e., a greatest lower bound. Another examples of simplex-like posets are barycentric subdivisions and, in general, subdivision categories in the sense of [10]. Now we reach the main result of this section:

Lemma 5.2. If P is a simplex-like poset then P^{op} is graded and there is an adequate local covering family for P^{op}.

Proof. To see that P^{op} is graded recall that for any p ∈ P the subcategory P_{≤p} is isomorphic to △_{n} for some n ≥ 0. Notice that n with this property has to be unique as if n ≠ m then △_{n} and △_{m} are not isomorphic. Define dim(p) = n. Then dim : P^{op} → ℤ is clearly order reversing and if p precedes p′ then dim(p) = dim(p′)+1. Hence, dim is a dimension function which assigns 0 to maximal elements of P^{op}.

Recall that △_{n}^{op} can be equipped with an adequate local covering family for any n ≥ 0 by Example 1.2. To build an adequate local covering family for P^{op},

K = {K_{n}^{p}}_{p∈P^{op}, 0 ≤ n ≤ dim(p)},

we just have to choose appropriately the isomorphisms P_{≤p} ≃ △_{n} for all p ∈ P.

Consider the set T of the maximal elements of P^{op}, i.e., T = {p ∈ P^{op}| dim(p) = 0}. Choose a total order < for T (suppose T is finite or use the Axiom of Choice [9]). Then, given p ∈ P^{op}, consider the subset ((P^{op})_{≥p})_{0} = ((P_{≤p})^{op})_{0} = (P_{≤p})_{0} ⊆ T and the restriction ((P_{≤p})_{0}, <) of the total order from T. There is a unique isomorphism

φ_{p} : P_{≤p} →^{≃} △_{dim(p)}

which induces an order preserving map

((P_{≤p})_{0}, <) →^{≃} (△_{dim(p)})_{0} = {v_{0}, v_{1}, v_{2}, …, v_{dim(p)}}.

Denote by J the adequate local covering family for △_{dim(p)}^{op} of Example 1.2 and define, for 0 ≤ n ≤ dim(p),

K_{n}^{p} = φ_{p}^{−1}(J_{n}^{φ_{p}(p)}).

Then it is straightforward that K is an adequate local covering family for P^{op}.

□

By Example 1.6, R_{dim(p)}^{p} = 1 and, by Proposition 4.3, rk F_{dim(p)}(p) = 1 and therefore F_{dim(p)}(p) = ℤ. This implies that the cohain complex given by Theorem 4.6 for the graded poset P^{op} and the adequate covering family given by the previous lemma has one copy of ℤ at dimension n for each object of that dimension. In particular, we can restate Lemma 4.5 as Lemma 5.3. Let P be a simplex-like poset such max dim(P) is finite and P_{n} is finite for each 0 ≤ n ≤ max dim(P). Then

∑_{n=0}^{max dim(P)}(−1)^{n} rk H^{n}(P; ℤ) = ∑_{n=0}^{max dim(P)}(−1)^{n}|P_{n}|,

where |P_{n}| is the number of objects of dimension 0 ≤ n ≤ max dim(P).

We may identify the object p ∈ P_{n} with its image v_{0}v_{1} … v_{n} under the isomorphism φ_{p} : P_{≤p} →^{≃} △_{dim(p)} built in Lemma 5.2. Then the expression for the differential in Theorem 4.6 becomes the familiar expression for simplicial complexes:

Proposition 5.4. Let P be a simplex-like poset. Then there exists a co-chain complex

0 → ∏_{p∈P_{0}} ℤ →^{d_{0}} ∏_{p∈P_{1}} ℤ →^{d_{1}} ∏_{p∈P_{2}} ℤ →^{d_{2}} …

of which cohomology is H^{∗}(|P|; ℤ). Under the identifications described above we have for p = v_{0}v_{1} … v_{n} ∈ P_{n}, n ≥ 1 and x ∈ ∏_{q∈P_{n−1}} ℤ

d_{n−1}(x)_{p} = ∑_{j=0}^{n}(−1)^{n−j}x_{v_{0}v_{1}...v̂_{j}...v_{n}}.

Proof. The cochain complex in the statement is given by Theorem 4.6 applied to P^{op}. The adequate covering family for P^{op} is given by Lemma 5.2. Now take p and x as in the statement. The short exact sequence considered in Remark 3.8

0 → F_{n−1}(p) →^{λ_{p}} Ker′_{F_{n−1}}(p) →^{π_{p}} F_{n}(p) → 0

takes the form

0 → F_{n−1}(v_{0}v_{1} … v_{n}) →^{λ_{p}} ∏_{j=0}^{n} F_{n−1}(v_{0} … v̂_{j} … v_{n}) →^{π_{p}} F_{n}(v_{0}v_{1} … v_{n}) → 0.

By the definition of the differential d_{n−1} we have that d_{n−1}(x)_{p} = π_{p}(y), where y = x|_{{v_{0}...v̂_{j}...v_{n}, j = 0, …, n}} ∈ ∏_{j=0}^{n} F_{n−1}(v_{0} … v̂_{j} … v_{n}). As in Remark 3.9 we identify π_{p}(y) with its unique pre-image y′ ∈ ∏_{j=0}^{n} F_{n−1}(v_{0} … v̂_{j} … v_{n}) such that y′_{v_{0}...v̂_{j}...v_{n}} = 0 for j = 0, …, n − 1. Thus, if we obtain z ∈ F_{n−1}(v_{0}v_{1} … v_{n}) such that λ_{p}(z)_{v_{0}...v̂_{j}...v_{n}} = y_{v_{0}...v̂_{j}...v_{n}} for j = 0, …, n − 1, we will have π_{p}(y) ≡ y′ = y − λ_{p}(z).

The rest of the proof is by induction on n. For n = 1 the short exact sequence is

0 → ℤ_{v_{0}v_{1}} →^{λ_{p}} ℤ_{v_{0}} × ℤ_{v_{1}} →^{π_{p}} ℤ_{v_{0}v_{1}} → 0,

where λ_{p}(n) = (n, n). Then, if y = (y_{v_{0}}, y_{v_{1}}), we have z = y_{v_{1}} and

y′ = y − λ_{p}(z) = (y_{v_{0}}, y_{v_{1}}) − (y_{v_{1}}, y_{v_{1}}) = (y_{v_{0}} − y_{v_{1}}, 0).

Now we make the inductive step for n ≥ 2. Recall that we want z such that λ_{p}(z)_{v_{0}...v̂_{j}...v_{n}} = y_{v_{0}...v̂_{j}...v_{n}} for j = 0, …, n − 1. Again by Remark 3.9 we identify

(8)  F_{n−1}(v_{0}v_{1} … v_{n}) ≅ ∏_{j=0}^{n−1} F_{n−2}(v_{0}v_{1} … v̂_{j} … v_{n−1}),

and, for j = 0, …, n − 1,

(9)  F_{n−1}(v_{0}v_{1} … v̂_{j} … v_{n}) ≅ F_{n−2}(v_{0}v_{1} … v̂_{j} … v_{n−1}),

and

(10)  F_{n−1}(v_{0}v_{1} … v_{n−1}) ≅ F_{n−2}(v_{0}v_{1} … v_{n−2}).

Recall that the “diagonal” λ_{p} is given by ∏_{j=0}^{n} F_{n−1}(v_{0} … v_{n} → v_{0} … v̂_{j} … v_{n}). For j_{0} ∈ {0, …, n−1}, and with the identifications (8) and (9), the map F_{n−1}(v_{0} … v_{n} → v_{0} … v̂_{j_{0}} … v_{n}) becomes exactly the projection

∏_{j=0}^{n−1} F_{n−2}(v_{0}v_{1} … v̂_{j} … v_{n−1}) → F_{n−2}(v_{0}v_{1} … v̂_{j_{0}} … v_{n−1}).

Thus, z ∈ F_{n−1}(v_{0}v_{1} … v_{n}) is such that z_{v_{0}v_{1}...v̂_{j}...v_{n−1}} = y_{v_{0}v_{1}...v̂_{j}...v_{n}} (again under the identifications (8) and (9)). The only thing left to compute is F_{n−1}(v_{0} … v_{n} → v_{0}v_{1} … v_{n−1})(z) under the identifications (8) and (10). Define q = v_{0}v_{1} … v_{n−1} and consider the short exact sequence given by Remark 3.8

0 → F_{n−2}(v_{0}v_{1} … v_{n−1}) →^{λ_{q}} ∏_{j=0}^{n−1} F_{n−2}(v_{0}v_{1} … v̂_{j} … v_{n−1}) →^{π_{q}} F_{n−1}(v_{0}v_{1} … v_{n−1}) → 0.

Now, by the induction hypothesis, π_{q}(z) = ∑_{j=0}^{n−1}(−1)^{n−1−j}z_{v_{0}v_{1}...v̂_{j}...v_{n−1}}. Thus, finally,

y′ = y − z = (y_{v_{0}...v_{n−1}} − ∑_{j=0}^{n−1}(−1)^{n−1−j}z_{v_{0}v_{1}...v̂_{j}...v_{n−1}}, 0, …, 0)

which equals

y′ = (∑_{j=0}^{n}(−1)^{n−j}y_{v_{0}v_{1}...v̂_{j}...v_{n}}, 0, …, 0).

□

The symmetry of the category △_{n} plays a role in this setting. Identify again the object p ∈ P_{n} with its image v_{0}v_{1} … v_{n} under the isomorphism φ_{p} : P_{≤p} →^{≃} △_{dim(p)} built in Lemma 5.2. By Remark 3.9 there are isomorphisms

F_{n−1}(v_{0}v_{1} … v_{n}) →^{≅} ∏_{j=0}^{n−1} F_{n−1}(v_{0} … v̂_{j} … v_{n})

and

F_{n}(v_{0}v_{1} … v_{n}) →^{≅} F_{n−1}(v_{0}v_{1} … v_{n−1}).

By letting the symmetric group Σ_{n+1} act on △_{n}^{op} we map the given local covering families on (P^{op})_{≥p} to others local covering families on (P^{op})_{≥p}. In particular we can permute v_{n} with any of the objects {v_{0}, …, v_{n−1}}. Recall that the functors F_{·} do not depend on the local covering family chosen. We may summarize this as

Remark 5.5. Let P be a simplex-like poset and let p = v_{0}v_{1} … v_{n} be an object with dim(p) = n ≥ 1. The maps

F_{n−1}(v_{0}v_{1} … v_{n}) → ∏_{j=0,j≠j_{0}}^{n} F_{n−1}(v_{0} … v̂_{j} … v_{n})

and

F_{n}(v_{0}v_{1} … v_{n}) → F_{n−1}(v_{0} … v̂_{j_{0}} … v_{n}).

are both isomorphisms for any j_{0} ∈ {0, …, n} .

To finish this section we state the analogous results for posets which are locally as A_{p}(E_{N}). The proof of Lemma 5.6 below is similar to the proof of Lemma 5.2 above. The property that A_{p}(E_{N}) and △_{N} share is that any object in one of these categories is determined by the objects of dimension 0 it contains, i.e., a simplex is determined by its vertices and a subgroup of E_{N} is determined by its rank 1 subgroups.

Lemma 5.6. Let P be a poset such that for all p ∈ P the category P_{≤p} is isomorphic to A_{p}(E_{n}) for some n. Then P^{op} is a graded poset and there is an adequate local covering family for P^{op}.

Proposition 5.7. Let P be poset such that for all p ∈ P the category P_{≤p} is isomorphic to A_{p}(E_{n}) for some n. Then there exists a co-chain complex

0 → ∏_{P_{0}} ℤ →^{d_{0}} ∏_{P_{1}} ℤ^{p} →^{d_{1}} ∏_{P_{2}} ℤ^{p^{3}} →^{d_{2}} …

of which cohomology is H^{∗}(|P|; ℤ).

The proof of this proposition follows from Theorem 4.6 applied to the adequate local covering family for P^{op} given by Lemma 5.6. Notice that rk F_{dim(p)}(p) = R_{dim(p)}^{p} =

p^{\frac{dim(p)(dim(p)+1)}{2}} by Example 1.7. Applying this theorem to the Quillen’s complex A_{p}(G) of the finite group G we obtain Theorem B in the introduction.

References

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Department of Mathematical Sciences, King’s College, University of Aberdeen, ABERDEEN AB24 3UE U.K.

E-mail address: a.diaz@maths.abdn.ac.uk