Higher inverse Limits

M.Ziegler ^{∗}

June 1999

Let I be a totally ordered set. A projective system is an I-indexed family (A_{α}) of abelian groups together with a commutative system of homomorphisms

π_{αβ} : A_{β} → A_{α} , (α < β ∈ I).

Projective systems forms an abelian category in a natural way. lim_{⟵} is a left exact functor to the category of abelian groups. Since the category of projective systems has enough injectives lim_{⟵} has right derived functors

lim_{⟵} = lim_{⟵}^{0}, lim_{⟵}^{1}, lim_{⟵}^{2} …

Fix a projective system (A_{α}, π_{αβ})_{α<β∈I} and a number n ≥ 0. We call a family

c = (c_{α_{0}...α_{n}}),

indexed by ascending sequences α_{0} < … < α_{n} of elements of I, an n–cochain if each c_{α_{0}...α_{n}} is an element of A_{α_{0}}. The set of n–chains form an abelian group C^{n} under component–wise addition. The coboundary homomorphisms

δ : C^{n} → C^{n+1},

defined by

(δc)_{α_{0}...α_{n+1}} = π_{α_{0}α_{1}}(c_{α_{1}...α_{n}}) + ∑_{i=1}^{n+1}(−1)^{i}c_{α_{0}...α̂_{i}...α_{n+1}},

make C = (C^{n})_{n≥0} into a cochain complex, which means that δ^{2} = 0.

As usual the cohomology groups of C are defined as the quotients

H^{n}(C) = Z^{n}(C)/ B^{n}(C)

of the groups

Z^{n}(C) = {z ∈ C^{n} | δz = 0}

∗Material related to a talk given in Hattingen, July 1999

of n–cocycles and the subgroups

B^{n}(C) = {δc | c ∈ C^{n−1}}

of n–coboundaries.

Theorem 1 ([1, Théorème 4.1]).

lim_{⟵}_{α∈I}^{n}A_{α} = H^{n}(C)

Readers who don’t like derived functors can take H^{n}(C) as the definition of lim_{⟵}_{α∈I}^{n}A_{α}. The content of the last theorem is then that the lim_{⟵}^{n} has the

characterizing properties of the derived functors: They are trivial on injective projective systems and there is a natural long cohomology sequence.

Lemma 2 ([1, p.12]). If J is cofinal in I, the natural restriction map

lim_{⟵}_{α∈I}^{n}A_{α} → lim_{⟵}_{α∈J}^{n}A_{α}

is an isomorphism for all n.

Proof. The lim_{⟵}_{α∈J}^{n}A_{α} (n = 0, 1, …) have the characterizing properties of the

right derived functors of lim_{⟵}_{α∈I}^{0}A_{α} = lim_{⟵}_{α∈J}^{0}A_{α}.

Lemma 3. If I has a last element the projective system (A_{α})_{α∈I} is acyclic. That means that lim_{⟵}_{α∈I}^{n}A_{α} = 0 for all n ≥ 1.

Proof. We begin with a general observation, which will be useful later on. Fix an element λ ∈ I and denote by C_{λ}^{n} the set of n–cochains over I_{λ} = {α ∈ I | α < λ}. Define two homomorphisms, the restriction

t : C^{n} → C_{λ}^{n}

and

h : C^{n} → C_{λ}^{n−1}

by h(c)_{α_{0}...α_{n−1}} = c_{α_{0}...α_{n−1}λ}. h does not commute with δ, but we have for c ∈ C^{n}

hδ(c) = (−1)^{n+1}t(c) + δh(c).  (1)

Now assume n ≥ 1 and z a n–cocycle. Let λ be the last element of I. Define the n − 1–cochain d by

d_{α_{0}...α_{n−1}} = \begin{cases}z_{α_{0}...α_{n−1}λ} & if α_{n−1} < λ \\ 0 & otherwise \end{cases}

Then δ(d) = (−1)^{n}z. This follows from (1) for indices in I_{λ} and

δ(d)_{α_{0}...α_{n−1}λ} = (−1)^{n}d_{α_{0}...α_{n−1}} = (−1)^{n}z_{α_{0}...α_{n−1}λ}.

Jensen proved in [1, Corollaire 3.2] that

lim_{⟵}_{α∈I}^{n+2}A_{α} = 0,

whenever cf(I) ≤ ω_{n}. Furthermore he proved that the result is optimal: For every n there is a projective system (A_{α})_{α∈ω_{n}} such that lim_{⟵}_{α∈ω_{n}}^{n+1}A_{α} ≠ 0 ([1,

Proposition 6.2]).

If we look at epimorphic systems (A_{α}, π_{αβ})_{α<β∈I}, where all the π_{αβ} are surjective, we have a better result:

Theorem 4 ([3, Theorem 3.3]). For epimorphic systems with cf(I) ≤ ω_{n} we have

lim_{⟵}_{α∈I}^{n+1}A_{α} = 0.

Proof. We use induction on n and begin with the case n = 0, where we can assume that I = ℕ. Let a 1–cocycle c be given. We choose recursively elements d_{i} ∈ A_{i} such that π_{i,i+1}(d_{i+1}) = d_{i} + c_{i,i+1}. The relation δc = 0 entails now δd = c.

Now assume n > 0.

We may assume that I is isomorphic to ω_{k} for some k ≤ n. Let c be an (n + 1)–cocycle. We want to write c as the coboundary of an n–cochain d. We construct the components d_{α_{0}...α_{n}} by recursion on α_{n}.

Fix λ ∈ I and assume that d is already constructed up to λ. This means that a d′ ∈ C_{λ}^{n} is given such that δ(d′) = t(c). To extend d′ to a suitable n– cochain d defined on {α ∈ I | α ≤ λ} means that t(d) = d′ and that tδ(d) = t(c) and hδ(d) = h(c). But I_{λ} either has a last element or has a cofinality smaller that ω_{n}, which gives us lim_{⟵}_{α∈I_{λ}}^{n}A_{α} = 0. On the other hand δ(c) = 0 implies

(−1)^{n}t(c)+δh(c) = 0. Therefore (−1)^{n}d′+h(c) is a cocycle, which we may write as δe for some (n − 1)–chain e on I_{λ}. Now extend d′ to d such that t(d) = d′ and h(d) = e. Then tδ(d) = δt(d) = δ(d′) = t(c) and

hδ(d) = (−1)^{n}t(d) + δh(d)
= (−1)^{n+1}d′ + δe
= (−1)^{n+1}d′ + (−1)^{n}d′ + h(c)
= h(c).

Lemma 5 (Todorcevic). Let (B_{ξ})_{ξ∈ω_{1}} be a family of infinite abelian groups. For the projective system A_{α} = ⨁_{ξ<α} B_{ξ} (α ∈ ω_{1}) with the the obvious projection maps we have

lim_{⟵}_{α∈ω_{1}}^{1}A_{α} ≠ 0.

Proof. In ([2, p.70]) an Aronszajn tree is constructed from a sequence (f_{α})_{α<ω_{1}} of injective functions f_{α} : α → ω such that for all α < β the two functions f_{α} and f_{β} ↾ α differ only for finitely many arguments. In each B_{ξ} we choose a copy of ω. Then f_{α} defines an element of A′_{α} = ∏_{ξ<α} B_{ξ}. Define

c_{αβ} = f_{β} ↾ α − f_{α} ∈ A_{α}.

Then c is a 1-cocycle, which is not a coboundary. For otherwise, there would be a sequence d_{α} ∈ A_{α} (α ∈ ω_{1}) such that c_{αβ} = d_{β} ↾ α − d_{α}. But then the functions f_{α} − d_{α} form an ascending sequence and the union f of this sequence is a map defined on ω_{1}, which is finite to one since it is finite to one on every α. This is impossible.

Theorem 6. Let (B_{ξ})_{ξ∈ω_{n}} be a family of countably infinite abelian groups and A_{α} = ⨁_{ξ<α} B_{ξ}. Assume n ≥ 1 and that for each 1 < i ≤ n ◇_{ω_{i}}(E_{i}) holds for E_{i} = {α ∈ ω_{i} | cf(α) = ω_{i−1}}. Then

lim_{⟵}_{α<ω_{n}}^{n}A_{α} ≠ 0.

Proof. ^{1} The proof proceeds by induction on n. The case n = 1 is a special case of Todorcevics lemma. So we assume n ≥ 2. ◇_{ω_{n}}(E_{n}) gives us a sequence (S^{λ})_{λ∈E_{n}} such that

1. each S^{λ} is an (n − 1)–cochain of (A_{α})_{α<λ}

2. for each (n − 1)–cochain d defined on ω_{n} the set

{λ ∈ E_{n} | d ↾ λ = S^{λ}}

is stationary in ω_{n}.

We define the components c_{α_{0}...α_{n}} of an n–cocycle c by induction on α_{n}. We can start the construction anywhere. For example with the zero n–cocycle defined on ω_{0}. Now assume that the c_{α_{0}...α_{n}} are already defined for all α_{n} < λ, giving rise to a cocycle c′ on C_{λ}^{n}.

\underline{Claim} c′ can be extended to a cocycle c defined on λ + 1. Proof: If we let c extend the cocycle c′ we have only to ensure that hδ(c) = 0. By (1) this is equivalent to

δh(c) = (−1)^{n+1}c′.

1I thank Burban Veliskovic for a helpful discussion of this proof

By theorem 4 lim_{⟵}_{α<λ}^{n}A_{α} = 0. Whence there is an e ∈ C_{λ}^{n−1} with δe = (−1)^{n+1}c′

and we can extend c′ by setting h(c) = e. All other extension of c to cocycles on λ + 1 can be obtained by adding an (n − 1)–cocycle (defined over λ) to h(c).

Now if λ is a successor or has cofinality smaller than ω_{n−1} we don’t care and choose an arbitrary extension of c′ to λ + 1.

If cf(λ) = ω_{n−1} we choose c more carefully and ensure that the difference h(c) − (−1)^{n}S^{λ} is not the coboundary of an (n − 2)–cycle over λ. If necessary we change h(c) by a cocycle e′ which is not a coboundary over λ. Such an e′ exists by the induction hypothesis.

To complete the proof we show that c is not a coboundary. For this look at an arbitrary (n − 1)–cochain δd defined on ω_{n}. By the choice of the S^{λ} there is a λ ∈ E_{n} such that d ↾ λ = S^{λ}. By (1) we have

hδ(d) − (−1)^{n}S^{λ} = δh(d).

By our construction hδ(d) ≠ h(c) and thererfore δ(d) ≠ c.

Open Problem: Can one prove the last theorem without diamond?

References

[1] C. U. Jensen. Les Foncteurs Dérivés de lim_{⟵} et leurs Applications en Théorie des Modules. Number 254 in Lecture Notes in Mathematics. Springer Verlag; Berlin, Göttingen, Heidelberg, 1970.

[2] Kenneth Kunen. Set Theory. An Introduction to Independence Proofs. North Holland Publishing Company, 1980.

[3] Martin Ziegler. Divisible uniserial modules over valuation domains. In Manfred Droste and Rüdiger Göbel, editors, Advances in Algebra and Model Theory, volume 9 of Algebra, Logic and Applications Series, pages 433–444. Gordon and Breach Science Publishers, 1997.