Divisible Uniserial Modules over Valuation Domains

Martin Ziegler June 1994

1 Introduction

Let R be a valuation domain with quotient field K. An R–module M is uniserial if for all x, y ∈ M there is an r ∈ R such that

rx = y or ry = x.

M is divisible if for all x ∈ M and all non–zero r ∈ R there is a y ∈ M such that ry = x.

A non–zero torsionfree divisible uniserial R–module is obviously isomorphic to the quotient field K of R. We call a divisible uniserial R–module proper if it is not torsionfree.

The main theorem 4.3 of this paper classifies proper divisible uniserial R–modules by two invariants:

• an element of the cohomology group

lim_{⟵}_{α∈Γ}^{1}U/U_{α},

where Γ is the value semigroup of R, U is the group of units of R and U_{α} is the group of units, which differ from 1 by an element of larger value than α.

• an element of the ideal class semigroup of R

The first invariant of M is zero iff M is a standard, i.e. if M is a quotient of K. In section 2 we give an easy example of a valuation domain R for which lim_{⟵}_{α∈Γ}^{1}U/U_{α} does not vanish, which shows that R has a nonstandard divisible uniserial module M. The part of 4.3 needed to obtain M is given an extra proof in section 2, which thereby contains a self–contained new construction of a nonstandard divisible uniserial module over a valuation domain.

Shelah showed in [7, p.135-150] that the existence of nonstandard divisible uniserial modules is consistent with ZFC. This result was then improved by Fuchs and Salce [2] and Franzen and Göbel [1], who showed that the existence of nonstandard divisible uniserial modules follows from ◇_{ω_{1}} and even from 2^{ω_{0}} < 2^{ω_{1}}.

The existence of nonstandard divisible uniserial modules was finally proved without set– theoretical hypotheses by Fuchs and Shelah [3] using a model theoretic transfer principle and later by B. Osofsky [6] by purely algebraic means.

If M is a nonstandard divisible uniserial module the matrix ring

S = \{ ( \begin{matrix}r & m \\ 0 & r \end{matrix} )| r ∈ R, m ∈ M\}

is a counter example for an old conjecture of Kaplansky, according to which every valuation ring should be a quotient of a valuation domain.

2 Construction of a nonstandard uniserial module

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).

A 0–cochain is a sequence (e_{α})_{α∈I} of elements e_{α} of A_{α}. A 1–cochain is a family (c_{αβ})_{α<β∈I} of elements c_{αβ} of A_{α}. The coboundary δ(e) of a 0-cochain e is the 1–cochain defined by

δ(e)_{αβ} = π_{αβ}(e_{β}) − e_{α}.

Coboundaries are cocycles in the following sense: A 1–cochain c is called a 1–cocycle if

π_{αβ}(c_{βγ}) − c_{αγ} + c_{αβ} = 0

for all α < β < γ. Let us denote the quotient group (1–cocycles)/(coboundaries) by

lim_{⟵}_{α∈I}^{1}A_{α}.

Now let R be valuation domain and v : R → Γ ∪ {∞} the valuation of R. For every α ∈ Γ we define

U_{α} = {u ∈ R | v(1 − u) > α}.

U_{α} is a multiplicative subgroup of U, the group of units of R.

Theorem 2.1 R has a nonstandard uniserial divisible module iff lim_{⟵}_{α∈Γ}^{1}U/U_{α} is non– trivial.

Proof: That a nonstandard uniserial divisible module gives rise to a non–trivial element of lim_{⟵}_{α∈Γ}^{1}U/U_{α} is not needed for our construction. We will prove this in Lemma 4.3.

Now assume that the units (u_{αβ}) represent a cocycle of the projective system (U/U_{α}). We are going to construct a uniserial divisible module M, which can only be standard if the (u_{αβ}) represent a coboundary.

First let us fix some notation. For every α ∈ Γ we choose an element r_{α} with value α. If A is an R–module the multiple r_{α}A does not depend on the choice of r_{α}, so we denote it by αA. If P is the maximal ideal of R we have then U_{α} = 1 + αP.

For all α < β multiplication by r_{α}^{−1}r_{β} defines an embedding from R/(αP) into R/(βP). The direct limit of this system is isomorphic to K/P. To obtain a more interesting limit we use the embeddings

u_{αβ}r_{α}^{−1}r_{β} : R/(αP) → R/(βP).

This is a commutative system since (u_{αβ}r_{α}^{−1}r_{β})(u_{βγ}r_{β}^{−1}r_{γ}) and u_{αγ}r_{α}^{−1}r_{γ} differ only by the factor u_{βγ}u_{αγ}^{−1}u_{αβ} ∈ U_{α} and define therefore the same map R/(αP) → R/(γP). Clearly the direct limit M is uniserial. M is divisible since every element of R/(αP) is divisible by r_{α}^{−1}r_{β} in R/(βP). We will use the notation x_{α} for the coset of 1 in R/(αP).

Now suppose M ≅ K/I and let y_{α} + I be the image of x_{α} under this isomorphism. Multiplication by y_{0}^{−1} shows that we can assume that y_{0} = 1 and I = P. Then y_{α} has value −α and e_{α} = r_{α}y_{α} is a unit. x_{α} = u_{αβ}r_{α}^{−1}r_{β}x_{β} implies y_{α} ≡ u_{αβ}r_{α}^{−1}r_{β}y_{β} (mod P). If we multiply this equation by r_{α} we obtain e_{α} ≡ u_{αβ}e_{β} (mod αP). This shows that u is a coboundary. □

To find a valuation domain with the property of 2.1 we make use of the following Lemma.

Lemma 2.2 (Todorcevic) Let (B_{ξ})_{ξ∈ω_{1}} be a family of infinite abelian groups. For the projective system (A_{ξ}) = ⨁_{η<ξ} B_{η} (ξ ∈ ω_{1}) with the the obvious projection maps we have

lim_{⟵}_{ξ∈ω_{1}}^{1}A_{ξ} ≠ 0.

Proof: The standard construction of an Aronszajn tree ([5, p.70]) yields a sequence (f_{ξ})_{ξ<ω_{1}} of injective functions f_{ξ} : ξ → ω such that for all ξ < ζ the two functions f_{ξ} and f_{ζ} ↾ ξ differ only for finitely many arguments. In each B_{ζ} we choose a copy of ω. Then f_{ξ} defines an element of A′_{ξ} = ∏_{η<ξ} B_{η}. Define

c_{ξζ} = f_{ξ} − f_{ζ} ∈ A_{ξ}.

Then c is a 1-cocycle, which is not a coboundary. Otherwise, there would be a sequence d_{ξ} ∈ A_{ξ} (ξ ∈ ω_{1}) such that c_{ξζ} = d_{ξ} − d_{ζ}. But then the functions f_{ξ} − d_{ξ} form an ascending sequence. The union f of this sequence is a map defined on ω_{1}, which differs from each f_{ξ} only on finitely many values in ξ. Since the f_{ξ} have values in ω, the preimage f^{−1}(ω) is cofinal in ω_{1}. Since the f_{ξ} are injective all f^{−1}(n) are finite. This is impossible. □

Fix a field F. Choose indeterminates t_{ξ}, (ξ < ω_{1}) and let K be the rational function field F ( t_{ξ})_{ξ<ω_{1}}. Order Γ = ⨁_{ξ∈ω_{1}} ℤ lexicographically and let

v : K → Γ ∪ {∞}

be the (uniquely determined) valuation of K which is trivial on F and maps t_{ξ} to 1_{ξ}, the 1 of the ξ–th copy of ℤ. (Note that the 1_{ξ} form an ascending sequence.)

Theorem 2.3 The valuation ring of (K, v) has a nonstandard uniserial divisible module.

Proof: We start the computation of the group U of units with the observation that

U = F^{·} × U_{0},

where F^{·} is the multiplicative group of F. In order to compute U_{0} we enlarge the linear order Γ by cuts

α_{ξ} = sup{n · 1_{η} | η < ξ, n ∈ ℕ}

to Γ̃. By the next lemma we have then

U_{0} = ∏_{ξ<ω_{1}} V_{ξ},

where V_{ξ} = {x ∈ F ( t_{η})_{η≤ξ} | v(1 − x) > α_{ξ}}. It results that

U/U_{α_{ξ}} = F^{·} × (∏_{η<ξ} V_{η}) .

By Lemma 2.2 lim_{⟵}_{ξ<ω_{1}}^{1}U/U_{α_{ξ}} is not trivial. By proposition 3.2 we conclude that

lim_{⟵}_{α<Γ}^{1}U/U_{α} ≅ lim_{⟵}_{α<Γ̃}^{1}U/U_{α} ≅ lim_{⟵}_{ξ<ω_{1}}^{1}U/U_{α_{ξ}}

is not trivial. Now apply Theorem 2.1. □

Avoiding the use of 3.2 one can construct M directly as follows: Choose a family (u_{ξζ})_{ξ<ζ<ω_{1}} of units which represents a non–trivial element of lim_{⟵}_{ξ<ω_{1}}^{1}U/U_{α_{ξ}}. Let M be

the direct limit of the system (R/(t_{ξ}P))_{ξ<ω_{1}} with maps

u_{ξ+1,ζ+1}t_{ξ}^{−1}t_{ζ} : R/(t_{ξ}P) → R/(t_{ζ}P).

Lemma 2.4 Let v : H → G ∪ {∞} be a valued field. Order G × ℤ lexicographically and let α be the cut sup G. Extend v to a valuation v : H(t) → G × ℤ with v(t) = (0, 1). Then the group U_{0} of 1-units of H(t) is the direct product of the 1–units of H and of U_{α}.

Proof: Easy. □

3 The right derived functors of the inverse limit functor

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 right 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+1} → C^{n},

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}..._{α_{i}}...α_{n+1}}, make C = (C^{n})_{n≥0} into a

cochain complex.

Theorem 3.1 ([4, 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.

If J is a subset of I there is an obvious restriction map

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

Jensen proved in [4, p.12] that res is an isomorphism if J is cofinal in I. (As a special case, we have for all I with a last element that lim_{⟵}_{α∈I}^{n}A_{α} = 0 for all n ≥ 1.) The inverse map can be obtained as follows: One chooses a function ϕ : I → J such that always α ≤ ϕ(α). Then to every n–cochain d over J assign the n–cochain ϕ^{∗}(d) over I defined by

ϕ^{∗}(d)_{α_{0}...α_{n}} = π_{α_{0}β}d_{ϕ(α_{0})...ϕ(α_{n})},

where β is the smallest element of ϕ(α_{0}) … ϕ(α_{n}). (We use here the convention that d_{β_{0}...β_{n}} is zero if there is an double index, and is changed by the sign of permutation if the β_{i} are not in ascending order.) It is easy to see that the maps ϕ^{∗} : C^{n} → C^{n} commute with δ and induce a homomorphism

ϕ^{∗} : lim_{⟵}_{α∈J}^{n}A_{α} → lim_{⟵}_{α∈I}^{n}A_{α}.

The next proposition shows that the composition

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

is the identity, giving another proof of Jensen’s result.

Proposition 3.2 Let (A_{α})_{α∈I} be a projective system and ϕ : I → I a function with α ≤ ϕ(α). Let d be an n–cocycle and let c be the n–cochain defined by c_{α_{0}...α_{n}} = π_{αβ}d_{ϕ(α_{0})...ϕ(α_{n})}, where β is the smallest element of ϕ(α_{0}) … ϕ(α_{n}). Then d and c differ by a coboundary.

Proof: Let C be the functor which assigns to every projective system (A_{α}) the corresponding cochain complex (C^{n}). ϕ^{∗} defines a natural transformation C → C and therefore a family of natural transformations ϕ^{∗} : lim_{⟵}^{n} → lim_{⟵}^{n} which for every short exact sequence 0 → (A_{α}) → (B_{α}) →

(C_{α}) → 0 commute with the connecting homomorphisms δ : lim_{⟵}^{n}(C_{α}) → lim_{⟵}^{n+1}(A_{α}). By

the general theory of derived functors ϕ^{∗} is therefore determined by what it does on lim_{⟵}^{0}.

Since ϕ^{∗} is the identity on lim_{⟵}^{0}, as one can easily check, it is the identity on all lim_{⟵}^{n}. □

Jensen proved in [4, Corollaire 3.2] that for all I of cofinality ω_{k}

lim_{⟵}_{α∈I}^{n}A_{α} = 0 (for all n ≥ k + 2).

Furthermore he proved that the result is optimal: For every n ≥ 2 there is a projective system (A_{α})_{α∈ω_{n−1}} such that lim_{⟵}_{α∈ω_{n−1}}^{n}A_{α} ≠ 0 ([4, 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 3.3 For epimorphic systems of cofinality ω_{k} we have

lim_{⟵}_{α∈I}^{n}A_{α} = 0 (for all n ≥ k + 1).

Proof: We use induction on n and begin with the case n = 1, 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 > 1. We begin with a general observation. 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).

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

Fix λ ∈ I and assume that d is already constructed up to λ. This means that a d′ ∈ C_{λ}^{n−1} is given such that δ(d′) = t(c). To extend d′ to a suitable (n − 1)–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_{k}, which gives us lim_{⟵}_{α∈I_{λ}}^{n−1}A_{α} = 0. On

the other hand δ(c) = 0 implies (−1)^{n+1}t(c) + δh(c) = 0. Therefore (−1)^{n+1}d′ + h(c) is a cocycle, which we may write as δe for some (n − 2)–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}d′ + δe
= (−1)^{n}d′ + (−1)^{n+1}d′ + h(c)
= h(c).
□

4 The classification

Let R be a valuation domain. We use the notations of section 2.

Lemma 4.1 Let A be a cyclic non-zero R–module and α a positive element of the valuation semigroup Γ. Then αA is a proper submodule of A.

Proof: This is Nakayama’s Lemma. For a short proof assume that A is generated by a and that v(r) = α. A = rA would imply that a = sra for some s ∈ R. Then (1 − sr)a = 0, which implies a = 0 since 1 − sr is a unit. □

Let M be a proper divisible uniserial R-module. We want to associate to it an element

μ(M) of lim_{⟵}_{α∈Γ}^{1}U/U_{α}.

Fix a non–zero cyclic submodule Z_{0}. Then for every α ∈ Γ there is a unique cyclic submodule Z_{α} with αZ_{α} = Z_{0}. This defines a bijection between Γ and the set of all cyclic submodules which contain Z_{0}. Note that (β − α)Z_{β} = Z_{α} for α ≤ β. If Z_{0} is isomorphic to R/I, Z_{α} is isomorphic to R/(αI).

Now fix isomorphisms f_{α} : R/(αI) → Z_{α}. If α < β the induced embedding R/(αI) → R/(βI)

[figure: r_{αβ}; f_{α}; f_{β}; R/(βI); Z_{β}; R/(αI); Z_{α}]

is given by multiplication with an element r_{αβ} of value β − α.

We define μ(M) from the r_{αβ} as follows: Choose elements r_{α} of value α and write r_{αβ} = u_{αβ}r_{α}^{−1}r_{β} for units u_{αβ}. Since (u_{αβ}r_{α}^{−1}r_{β})(u_{βγ}r_{β}^{−1}r_{γ}) and u_{αγ}r_{α}^{−1}r_{γ} define the same map R/(αI) → R/(βI) they differ by a factor from 1 + αI ⊂ U_{α}. This shows that (u_{αβ}) is a 1–cocycle. We let μ(M) be the class determined by this cocycle.

Lemma 4.2 μ(M) does not depend on the choice of Z_{0}, of the isomorphisms f_{α} and the choice of the r_{αβ} and r_{α}.

Proof: We treat first the case where Z_{0} (and therefore all Z_{α}) remains the same but we have new f′_{α}, r′_{αβ} and r′_{α}. Then there are units v_{α} and w_{α} such that f_{α} = f′_{α}v_{α} and r_{α} = r′_{α}w_{α}.

[figure: r_{αβ}; v_{α}; v_{β}; r′_{αβ}; R/(βI); R/(βI); R/(αI); R/(αI)]

Since r′_{αβ}v_{α} and v_{β}r_{αβ} define the same map R/(αI) → R/(βI) we have v_{α}v_{β}^{−1}r′_{αβ}r_{αβ}^{−1} ∈ 1 + αI ⊂ U_{α}. Therefore

u′_{αβ}u_{αβ}^{−1} = r′_{α}r_{β}^{′−1}r′_{αβ}r_{α}^{−1}r_{β}r_{αβ}^{−1} = w_{α}^{−1}w_{β}r′_{αβ}r_{αβ}^{−1} = (w_{α}v_{α})^{−1}(w_{β}v_{β})(v_{α}v_{β}^{−1}r′_{αβ}r_{αβ}^{−1})

If we define e_{α} = w_{α}v_{α} we have

u′u^{−1}(δ(e))^{−1} ∈ U_{α},

which shows that μ = μ′.

Now assume that we have chosen Z′_{0} to be Z_{α_{0}}. For the computation of u′_{αβ} we can use f′_{α} = f_{α_{0}+α}, r′_{αβ} = r_{α_{0}+α, α_{0}+β} and r′_{α} = r_{α}. It results that u′_{αβ} = u_{α_{0}+α, α_{0}+β}. By Proposition 3.2 we have again μ = μ′. □

The next theorem will describe a proper uniserial divisible module M by two invariants: μ(M) and its ideal class C(M), which is defined as the class of any annihilator of a non-zero element of M in the ideal class semigroup, the multiplicative semigroup of all non-zero ideals of R modulo the principal ideals. Observe that the annihilators of all non-zero elements of M are in the same class.

If R is a field there are no proper divisible uniserial modules. We assume from now on that R is not a field.

Theorem 4.3 M ↦ (μ(M), C(M)) defines a bijection between proper divisible uniserial modules up to isomorphy and pairs of elements of lim_{⟵}_{α∈γ}^{1}U/U_{α} and ideal classes of R.

M is standard iff μ(M) = 1.

Proof: Let us first assume that M and M′ have the same invariants. M and M′ are (isomorphic to) the direct limits of two systems (R/(αI), r_{αβ}) and (R/(αI′), r′_{αβ}). Since C(M) = C(M′) we can assume that I = I′. Choose ring elements r_{α} with value α. That μ(M) = μ(M′) means that (r_{α}r_{β}^{−1}r_{αβ}) and (r_{α}r_{β}^{−1}r′_{αβ}) differ by a coboundary i.e. there is a family (v_{α}) of units such that r′_{αβ}r_{αβ}^{−1} differ from v_{α}^{−1}v_{β} by a factor from U_{α}. Since the (r_{αβ}) and (r′_{αβ}) form commuting systems and are therefore cocycles modI in the sense of the next lemma by part 2 of this lemma this factor belongs to 1 + αI. Then the last diagram is commutative for all α < β and the family of maps v_{α} : R/(αI) → R/(αI) defines an isomorphism between M and M′.

Now assume that an element μ of lim_{⟵}_{α∈Γ}^{1}U/U_{α} and an ideal class C is given. Let μ be represented by (u_{αβ}). Since R is not a field C can be represented by a proper ideal I. By part 1 of the next lemma we can assume that u is a cocycle modI. The direct limit of the commutative system

(R/(αI), u_{αβ}r_{α}^{−1}r_{β}),

(r_{α} elements with value α) is then a proper uniserial divisible module with the desired invariants.

The ideal class of the standard module K/I is the class of I. μ(K/I) = 1 since it is the direct limit of the system (R/(αI), r_{α}^{−1}r_{β}). This proves the last part of the theorem. □

Lemma 4.4 Let the family of units u = (u_{αβ}) represent a 1–cocycle for the projective system (U/U_{α}) and I be a non–zero proper ideal. We call u a cocycle modI if u_{βγ}u_{αγ}^{−1}u_{αβ} ∈ 1 + αI for all α < β < γ and a coboundary modI if there is a family (v_{α}) of units such that v_{α}v_{β}^{−1}u_{αβ} ∈ 1 + αI for all α < β.

1. The class of u in lim_{⟵}_{α∈Γ}U/U_{α} can be represented by a cocycle modI.

2. If u is a cocycle modI and represents a coboundary it is a coboundary modI.

Proof: Let α_{0} be the value of a non–zero element of I. We will use Proposition 3.2 with the map ϕ(α) = α_{0} + α.

Proof: of (1) Define u′ by u′_{αβ} = u_{α_{0}+α, α_{0}+β}. u′ is a cocycle modI and in the same class as u by Proposition 3.2.

Proof: of (2) By 1 we can assume that (u_{αβ}) is a cocycle modα_{0}P, which implies that u′ defined by u′_{αβ} = u_{−α_{0}+α, −α_{0}+β} is a cocycle of the projective system (U/U_{α})_{α_{0}≤α}. By Proposition 3.2 the class of u′ corresponds to the class of u in the isomorphism between lim_{⟵}_{α∈Γ}^{1}U/U_{α} and

lim_{⟵}_{α_{0}≤α}^{1}U/U_{α}. Since u is a coboundary u′ is also a coboundary. So we have a family v′_{α}

of units, such that v′_{α}v_{β}^{′−1}u′_{αβ} ∈ U_{α} for all α_{0} ≤ α < β. If we set v_{α} = v′_{α_{0}+α} we have v_{α}v_{β}^{−1}u_{αβ} ∈ 1 + αI for all α < β. □

Finally we show that in Theorem 4.3 we can replace lim_{⟵}_{α∈Γ}^{1}U/U_{α} by lim_{⟵}_{α∈Γ}^{1}U_{0}/U_{α}

Proposition 4.5 The natural map

lim_{⟵}_{α∈Γ}^{1}U_{0}/U_{α} → lim_{⟵}_{α∈Γ}^{1}U/U_{α}

is an isomorphism

Proof: One checks easily that lim_{⟵}_{α∈I}^{1}A_{α} = 0 if A_{α} = A is a constant sequence. (In fact lim_{⟵}_{α∈I}^{n}A_{α} = 0 for all n ≥ 1). Look at the following part of the long exact cohomology sequence:

lim_{⟵}_{α∈Γ}^{0}U/U_{α} → lim_{⟵}_{α∈Γ}^{0}U/U_{0} → lim_{⟵}_{α∈Γ}^{1}U_{0}/U_{α} → lim_{⟵}_{α∈Γ}^{1}U/U_{α} → lim_{⟵}_{α∈Γ}^{1}U/U_{0} = 0.

Since the first arrow is surjective the third arrow is an isomorphism. □

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