The pure injectives and pure projectives in the category of totally disconnected, locally compact abelian groups

Abstract

Let ℘ be the category of totally disconnected, locally compact abelian groups. In this paper, we determine the discrete or compact pure injective groups in ℘. Also, we determine the compact pure projective groups in ℘.

Introduction

Throughout, all groups are Hausdorff abelian topological groups and will be written additively. Let £ denote the category of locally compact abelian (LCA) groups with continuous homomorphisms as morphisms. A morphism is called proper if it is open onto its image, and a short exact sequence $0\to A\stackrel{\varphi }{\to }B\stackrel{\psi }{\to }C\to 0$ in £ is said to be proper exact if ϕ and ψ are proper morphisms. In this case, the sequence is called an extension of A by C (in £). A subgroup H of a group C is called pure if $\mathit{\text{nH}}=H\bigcap \mathit{\text{nC}}$ for all positive integers n. An extension $0\to A\stackrel{\varphi }{\to }B\stackrel{\psi }{\to }C\to 0$ is called a pure extension if ϕ(A) is pure in B. Following Fulp and Griffith [1], we let Ext(C, A) denote the (discrete) group of extensions of A by C. The elements represented by pure extensions of A by C form a subgroup of Ext(C, A) which is denoted by P ext(C, A). Assume that ℑ is any subcategory of £ such that whenever $0\to A\stackrel{\varphi }{\to }B\stackrel{\psi }{\to }C\to 0$ is an extension in £, and A and C are in ℑ, then B is in ℑ. Following Fulp [2], G in ℑ is called a pure projective group if and only if P ext(G, X) = 0 for all X in ℑ. Similarly, G is a pure injective group in ℑ if and only if P ext(X, G) = 0 for all X in ℑ. Fulp [2] has described the pure injective and pure projective in some categories such as the category of connected, locally compact abelian groups. Let ℘ be the category of totally disconnected, locally compact abelian groups. In this paper, we determine the discrete or compact groups which are pure injective in ℘. We show that a discrete (torsion or torsion-free) group is pure injective in ℘ if and only if it is divisible (Theorems 1 and 2). We show that a compact group G is pure injective in ℘ if and only if G = 0 (Corollary 1). We also introduce a result on the pure projective of ℘. We show that if a compact dual cotorsion group is a pure projective of ℘, then it is a torsion group (Corollary 2).

The additive topological group of real numbers is denoted by R. Q is the group of rationales, Z is the group of integers, and Z(n) is the cyclic group of order n. By G _d , we mean the group G with discrete topology. tG is the torsion part of G, and G₀ is the identity component of G. The Pontrjagin dual group of a group G is denoted by Ĝ. The topological isomorphism will be denote by ‘ ≅’.

Pure injective in ℘

Let ℘ be xct abelian groups. In this section, we determine the structure of a discrete or compact pure injective group in ℘. Recall that a group B is said to be bounded if n B = 0 for some integer n.

Lemma 1

Suppose B is a discrete bounded group. Then, B is pure injective in ℘ if and only if B = 0.

Proof

Assume that E is a torsion-free group in £. Let E₀ be the identity component of E. Then, the sequence

$$0\to E_0\to E\to E/E_0\to 0$$

is a proper pure exact. Thus, the sequence

$$0=P\text{ext}(E/E_0,B)\to P\text{ext}(E,B)\to P\text{ext}(E_0,B)$$

is exact (Proposition 4 in [2]). By Theorem 2.11 in [3], P ext(E₀,B) = 0. Thus, P ext(E, B) = 0. It follows that B is divisible (Corollary 10 in [4]). Since B is bounded, so B = 0. □

Theorem 1

Let A be a discrete torsion group. Then, A is pure injective in ℘ if and only if A is a divisible group.

Proof

If A is a divisible group, it is clear that P ext(X, A) = 0 for all totally disconnected groups X (Theorem 3.4 in [1]). So, A is pure injective in ℘. Conversely, suppose that A is a discrete torsion, pure injective group in ℘. Then, Ext(Q, A) = 0. Hence, A is a cotorsion group. By Corollary 54.4 in [5], A = B ⊕ D where B and D are bounded and divisible groups, respectively. Clearly B is pure injective in ℘. So by Lemma 1, B = 0. Hence, A = D is a divisible group. □

Theorem 2

Let A be a discrete torsion-free group. Then, A is pure injective in ℘ if and only if A is a divisible group.

Proof

Suppose that A is a discrete torsion-free, pure injective group in ℘. Then, $\text{Ext}(Q\widehat{/}Z,A)=P\text{ext}(Q\widehat{/}Z,A)=0$. By corollary 2.10 in [6], we have the exact sequence

$$\begin{array}{ll}(\ast )\mathit{\text{Hom}}(Q\widehat{/}Z,A)& \to \text{Ext}(\widehat{Z},A)\to \text{Ext}(\widehat{Q},A)\\ \to \text{Ext}(Q\widehat{/}Z,A)=0.\end{array}$$

Since $\widehat{A}$ is connected, so $\mathit{\text{Hom}}(Q\widehat{/}Z,A)\cong \mathit{\text{Hom}}(\widehat{A},Q/Z)=0$. It follows from (∗) that $\text{Ext}(\widehat{Z},A)\cong \text{Ext}(\widehat{Q},A)$. By Proposition 2.17 in [1], $A\cong \text{Ext}(\widehat{Z},A)$. So, $A\cong \text{Ext}(\widehat{Q},A)$. Since $\text{Ext}(\widehat{Q},A)$ is divisible (p. 223(I) in [3]), so A is a divisible group. □

Definition 1

A locally compact abelian group G will be called an £-cotorsion if and only if Ext(X, G) = 0 for each torsion-free group X in £ [4].

Theorem 3

Let G be a compact group. Then, P ext(X, G) = 0 for any totally disconnected group X in £ if and only if G ≅ (R / Z)^σwhere σ is a cardinal number.

Proof

Suppose P ext(X, G) = 0 for any totally disconnected group X. First, we show that G is an £-cotorsion group. Let X be torsion-free in £ and X₀ the component of identity. Since X₀ is pure in X, so X / X₀ is torsion-free. Hence, 0 = P ext(X / X₀, G) = Ext(X / X₀,G). Consider the exact sequence

$$(\ast )0=\text{Ext}(X/X_0,G)\to \text{Ext}(X,G)\to \text{Ext}(X_0,G)\to 0.$$

Recall that since X₀ is a compact torsion-free group, $\widehat{X_0}$ is a discrete divisible group. Consequently, $\text{Ext}(X_0,G)\cong \text{Ext}(\mathit{Ĝ},\widehat{X_0})=0$. By (*), Ext(X, G) = 0. So G is an £-cotorsion. By Corollary 9 in [4], G is connected. It follows that Ext(X, G) = 0 for any totally disconnected group in £. By Theorem 5.1 in [1], G ≅ (R / Z)^σ. The converse is clear. □

Corollary 1

A compact group G is pure injective in ℘ if and only if G = 0.

Proof

Let G be a compact, pure injective group in ℘. Then, G is totally disconnected, and P ext(X, G) = 0 for any totally disconnected group X. So, by Theorem 3, G = 0. □

Pure projective in ℘

In this section, we show that if a compact group is pure projective in ℘, then it is a torsion group.

Lemma 2

A discrete group A is pure projective in ℘ if and only if A is a direct sum of cyclic groups.

Proof

Let A be a discrete pure projective group in ℘. So, P ext(A, X) = 0 for any discrete group X. By Theorem 30.2 in [5], A is a direct sum of cyclic groups. The converse is clear. □

Recall that a discrete group A is said to be a cotorsion if for any discrete torsion-free group B, Ext(B, A) = 0 [5]. A compact group G is called dual cotorsion if and only if the dual group of G is a cotorsion.

Theorem 4

Let G be a compact dual cotorsion group. If P ext(G, X) = 0 for any X ∈ ℘, then G ≅ Π_i∈IZ(b _i ).

Proof

Let G be a compact group such that Ĝ is a cotorsion group and P ext(G, X) = 0 for any totally disconnected group X. By Theorem 2.11 in [3], it is enough to show that P ext(G, F) = 0 for any compact group F in £. We have $P\text{ext}(G,F)=P\text{ext}(\widehat{F},\mathit{Ĝ})$. Consider the exact sequence

$$\begin{array}{ll}(\ast )\dots & \to P\text{ext}(\widehat{F}/t\widehat{F},\mathit{Ĝ})\to P\text{ext}(\widehat{F},\mathit{Ĝ})\\ \to P\text{ext}(t\widehat{F},\mathit{Ĝ})\to 0.\end{array}$$

Since Ĝ is a cotorsion group, $P\text{ext}(\widehat{F}/t\widehat{F},\mathit{Ĝ})=0$. Since $\left(\widehat{t\widehat{F}}\right)$ is totally disconnected, then $P\text{ext}(t\widehat{F},\mathit{Ĝ})=P\text{ext}(G,(\widehat{t\widehat{F}})=0$. It follows from (∗) that $P\text{ext}(\widehat{F},\mathit{Ĝ})=0$. Hence, P ext(G, F) = 0. □

Corollary 2

If a compact dual cotorsion group is pure projective in ℘, then it is a torsion group.

Proof

It is clear by Theorem 4. □

Theorem 5

Let C be a connected group. Then, P ext(C, X) = 0 for all totally disconnected groups X in £ if and only if C is a vector group.

Proof

Since a vector group is a projective object of £, it is clear that P ext(C, X) for all totally disconnected groups X ∈ £. Conversely, assume that C is a connected group. Then, C≅Rⁿ ⊕ K where K is a compact connected group (Theorem 9.14 in [7]). Since the sequence

$$\begin{array}{ll}0& =\mathit{\text{Hom}}(Q\widehat{/}Z,\widehat{K})\to \text{Ext}(\widehat{Z},\widehat{K})\to \text{Ext}(\widehat{Q},\widehat{K})\\ =P\text{ext}(\widehat{Q},\widehat{K})=0\end{array}$$

is exact, so by Proposition 2.17 in [1], $\widehat{K}$ is isomorphic to $\text{Ext}(\widehat{Z},\widehat{K})=0$ and therefore K = 0. It follows that C is a vector group. □

Theorem 6

Let C be a locally connected group in £. Then, P ext(C, X) = 0 for all totally disconnected groups X in £ if and only if C ≅ Rⁿ ⊕ E where E is a discrete direct sum of cyclic groups.

Proof

Let C be a locally connected group in £ and P ext(C, X) = 0 for all totally disconnected groups X in £. By p. 19 in [8] and p. 38 in [9], $C\cong R^n\oplus E\oplus \widehat{D}$ where Rⁿ is a vector group with n ≥ 0, E a discrete group, and D is a discrete torsion-free abelian group in which every subgroup of finite rank is free. Then, P ext(E, X) = 0 for all discrete groups X. Hence, E is a direct sum of cyclic groups. We show that $\widehat{D}=0$. Recall that $\text{Ext}(\widehat{D},Q)=P\text{ext}(\widehat{D},Q)=0$. Consider the exact sequence

$$0=\mathit{\text{Hom}}(\widehat{D},Q/Z)\to \text{Ext}(\widehat{D},Z)\to \text{Ext}(\widehat{D},Q)=0.$$

So $D\cong \text{Ext}(\widehat{Z},D)=\text{Ext}(\widehat{D},Z)=0$, i.e., D = 0. □