WHEN EXT COMMUTES WITH DIRECT SUMS
SIMION BREAZ AND PHILL SCHULTZ
Abstract. We characterize the abelian groups G for which Ext(G, −) commutes with direct sums.
1. Introduction
It is proved in [Br11] that a right module M over a hereditary ring R has the
property that the canonical homomorphism ⊕i ExtR (M, Mi ) → ExtR (M, ⊕i Mi ) is
an isomorphism for all families (Mi )i∈I if and only if M is a direct sum of a finitely
presented module and a projective module.
In this note we will prove for abelian groups a stronger version of this result,
since it does not require the isomorphism to be natural. If A is an abelian group
such that Ext(A, ⊕i Ai ) ∼
= ⊕i Ext(A, Ai ) for all families (Ai )i∈I , then A has the
same form as in the previous mentioned result, that is, A is the direct sum of a
finite group and a free group. Moreover, it is proved that in this case we can assume
that all the groups Ai are torsion-free.
Let A be an abelian group and C a class of abelian groups. We say that A is
C–extendible if whenever (Ai ) is a family of groups which are in C, Ext(A, ⊕i Ai ) ∼
=
⊕i Ext(A, Ai ). If A is C–extendible for C the class of all abelian groups, we say that
A is extendible.
Throughout the paper, all groups are abelian, and unless noted otherwise, all
notation is that of [F70, F73]. In particular, T (A) is the torsion subgroup of the
group A and A the factor group A/T (A).
2. C–extendible groups
Let us recall a very useful lemma.
Lemma 2.1. [S58, pp. 153, 154], [G70, Lemma 3.1] For every cardinal κ there is
a cardinal λ ≥ κ such that λℵ0 = 2λ .
Using this we first prove that G is an an extendible elementary p-group if and
only if G is finite. This can be deduced from [S11, Theorem 5.3], but we include a
proof for the reader’s convenience.
Lemma 2.2. Let p be a prime number. If κ is an infinite cardinal then there is
λ ≥ κ such that
| Ext(Z(p)(κ) , Z(p)(λ) )| > | Ext(Z(p)(κ) , Z(p))(λ) |.
Date: September 20, 2011.
2000 Mathematics Subject Classification. 20K35, 20K40.
Key words and phrases. Extensions, direct sums, cotorsion groups.
1
2
SIMION BREAZ AND PHILL SCHULTZ
Proof. Using the basic properties of extension groups found in [F70, Section 52],
we observe that for every cardinal λ we have
Ext(Z(p)(κ) , Z(p)(λ) ) ∼
= (Z(p)(λ) )κ ,
= Ext(Z(p), Z(p)(λ) )κ ∼
while
∼ (Ext(Z(p), Z(p))κ )(λ) ) ∼
Ext(Z(p)(κ) , Z(p))(λ) =
= (Z(p)κ )(λ) .
Therefore the first group has the cardinality λκ , while the second group has the
cardinality λ2κ .
If we choose λ ≥ 2κ such that λℵ0 = 2λ , we have λ2κ = λ < 2λ = λℵ0 ≤ λκ , and
the proof is complete.
Since by [F70, Theorem 54.6] all groups Ext(G, H) are cotorsion, to describe
the C–extendible groups (for various classes C) we first need to characterize the
cotorsion groups which can be decomposed as an infinite direct sum.
Proposition 2.3. Let Gi , i ∈ I, be a family of cotorsion groups. Then ⊕i∈I Gi
is cotorsion if and only if there is J ⊆ I such that I \ J is finite and ⊕i∈J Gi is a
direct sum of a divisible group and a bounded group.
Proof. We denote the first Ulm subgroup ([F73, Section 76]) of a group G by G1 .
Suppose that ⊕i∈I Gi is cotorsion. Using [F70, Theorem 54.3] we deduce that
the sum ⊕i∈I Gi /G1i = (⊕i∈I Gi )/(⊕i∈I Gi )1 is algebraically compact. But this is
possible, by [F70, Corollary 39.10], only if there is an n > 0 such that n(Gi /G1i ) = 0
for almost all i. It is not hard to see that for these indices i the groups Gi are direct
sums of divisible groups and groups bounded by n.
The converse implication is obvious.
We obtain the characterization of Σ-cotorsion groups (i.e. groups such that all
self-sums are cotorsion) proved in [BS09, Proposition 1.8]:
Corollary 2.4. A group G is Σ-cotorsion if and only if it is a direct sum of a
divisible group and a bounded group.
The main result follows:
Theorem 2.5. The following are equivalent for a group A:
(1) A is extendible;
(2) A is C–extendible for the class C of torsion–free groups;
(3) A = B ⊕ F , where B is a finite group and F is free.
Proof. We only need to prove (2)⇒(3). Since A is C–extendible, it follows that for
every family (Ki )i∈I of torsion–free groups Ext(A, ⊕i∈I Ki ) ∼
= ⊕i∈I Ext(A, Ki ). But
Ext(A, ⊕i∈I Ki ) is a cotorsion group. By Proposition 2.3 we deduce that there is a
positive integer n such that for every torsion–free group K, the group n Ext(A, K)
is divisible.
From the exact sequence 0 → T (A) → A → A → 0 we obtain for all torsion-free
groups K an exact sequence
Ext(A, K) → Ext(A, K) Ext(T (A), K).
The last term is reduced by [F70, Lemma 55.3], and by the hypothesis it follows that Ext(T (A), K) is bounded by n. In particular, by [F70, Corollary 52.4],
Ext(T (A), Z) ∼
= Hom(T (A), Q/Z), so T (A) is bounded. Hence A = B ⊕ F , where
B = T (A) is bounded and F is torsion-free.
WHEN EXT COMMUTES WITH DIRECT SUMS
3
For every torsion–free group K we have
Ext(A, K) = Ext(B, K) ⊕ Ext(F, K),
and in this direct sum the first group is bounded and the last group is divisible. It
follows that both groups B and F are C–extendible.
Let p be a prime. For every cardinal λ there is an isomorphism
∼ Ext(B, Z(p))(λ) .
Ext(B, Z(p)(λ) ) =
Using a similar proof as in [F70, 52(F)] we deduce that for every cardinal λ we have
an isomorphism
Ext(B[p], Z(p)(λ) ) ∼
= Ext(B[p], Z(p))(λ) .
By Lemma 2.2 it follows that B[p] is finite so B is a finite group.
To prove that F is free, we show that it is a Baer group. Let T be any torsion
group. Then T is an epimorphic image of a torsion–free group K = ⊕p Lp , where
Lp is a direct sum of copies of p-adic groups Jp . Hence Ext(F, T ) is an epimorphic
image of Ext(F, K) ∼
= ⊕p Ext(F, Jp ) = 0. Thus Ext(F, T ) = 0 so F is a Baer group,
and the conclusion follows from [G69].
References
[BS09]
[Br11]
[F70]
[F73]
[G69]
[G70]
[S11]
[S58]
S. Bazzoni, and J. Šťovı́ček: Sigma-cotorsion modules over valuation domains, Forum
Math. 21 (2009), 893–920.
S. Breaz, Modules M such that Ext(M, −) commutes with direct sums–the hereditary
case, preprint 2011, arXiv:1107.0557.
L. Fuchs: Infinite Abelian Groups, Vol I, Academic Press (1970).
L. Fuchs: Infinite Abelian Groups, Vol II, Academic Press (1973).
P. Griffith: A solution to the splitting mixed group problem of Baer, Trans. Amer.
Math. Soc. 139, (1969), 261–269.
P. Griffith: On a subfunctor of Ext, Arch. Math. 21 (1970), 17–22.
P. Schultz: Commuting properties of Ext, preprint 2011.
W. Sierpinski: Cardinal and Ordinal Numbers, Polska Akademia Nauk. Monografie
Matematyczne tom 34, 1958
(Breaz) ”Babeş-Bolyai” University, Faculty of Mathematics and Computer Science,
Str. Mihail Kogălniceanu 1, 400084 Cluj-Napoca, Romania
E-mail address: [email protected]
(Schultz) School of Mathematics and Statistics, The University of Western Australia, Nedlands, 6009, Australia
E-mail address: [email protected]