Characterisations of $\Sigma$ -pure-injectivity in triangulated categories and applications to endoperfect objects.

Raphael Bennett-Tennenhaus R. Bennett-Tennenhaus
Faculty of Mathematics
Bielefeld University
Universität sstraße 25
33615 Bielefeld
Germany [email protected]

Abstract.

We provide various ways to characterise $\Sigma$ -pure-injective objects in a compactly generated triangulated category. These characterisations mimic analogous well-known results from the model theory of modules. The proof involves two approaches. In the first approach we adapt arguments from the module-theoretic setting. Here the one-sorted language of modules over a fixed ring is replaced with a canonical multi-sorted language, whose sorts are given by compact objects. Throughout we use a variation of the Yoneda embedding, called the resticted Yoneda functor, which associates a multi-sorted structure to each object. The second approach is to translate statements using this functor. In particular, results about $\Sigma$ -pure-injectives in triangulated categories are deduced from results about $\Sigma$ -injective objects in Grothendieck categories. Combining the two approaches highlights a connection between sorted pp-definable subgroups and annihilator subobjects of generators in the functor category. Our characterisation motivates the introduction of what we call endoperfect objects, which generalise endofinite objects.

2020 Mathematics Subject Classification:

18E45, 18G80 (primary), 03C60 (secondary)

1. Introduction.

The model theory of modules refers to the specification of model theory to the module-theoretic setting. Fundamental work, such as that of Baur [2], placed focus on certain formulas in the language of modules, known as pp-formulas. In particular, module embeddings which reflect solutions to pp-formulas, so-called pure embeddings, became of particular interest. This served as motivation to study modules which are pure-injective: that is, injective with respect to pure embeddings.

In famous work of Ziegler [21], a topological space was defined whose points are indecomposable pure-injective modules. The introduction of the Ziegler spectrum proved to be a groudbreaking moment in this branch of model-theoretic algebra, and interest in understanding pure-injectivity has since grown. Specifically, in work such as that of Huisgen-Zimmerman [11], functional results appeared in which pure-injective and so-called $\Sigma$ -pure-injective modules were characterised. These characterisations are well documented, for example, by Jensen and Lenzing [14].

A frequently used tool in these characterisations is the relationship between a module and its image in a certain functor category. To explicate, the functor is given by the tensor product, restricted to the full subcategory of finitely presented modules. For example, a module is pure-injective if and only if the corresponding tensor functor is injective. Subsequently one may convert statements about pure-injective modules into statements about injective objects in Grothendieck categories, and translate problems and solutions back and forth.

For example, Garcia and Dung [7] developed the understanding of $\Sigma$ -injective objects in Grothendieck categories by building on work of Harada [9], which generalised a famous characterisation of $\Sigma$ -injective modules going back to Faith [6]. These authors showed that, as above, such developments helped simplify arguments about $\Sigma$ -pure-injective modules.

In this article we attempt to provide, in a utilitarian manner, some analouges to the previously mentioned characterisations. The difference here is that, instead of working in a category of modules, we work in a triangulated category which, in a particular sense, is compactly generated. Never-the-less, the statements we prove and arguments used to prove them are motivated directly from certain module-theoretic counterparts.

The notion of compactness we refer to comes from work of Neeman [19], where the idea was adapted from algebraic topology. Krause [18] provided the definitions of pure monomorphisms, pure-injective objects and the Ziegler spectrum of a compactly generated triangulated category. Garkusha and Prest [8] subsequently introduced a multi-sorted language for this setting, which mimics the role played by the language of modules. They then gave a correspondence between the pp-formulas in this multi-sorted language and coherent functors. In our main result, Theorem 1.1, we use the following notation and assumptions.

•

$\mathcal{T}$ is a compactly generated triangulated category with all small coproducts.
•

$\mathcal{T}^{c}$ is the full subcategory of $\mathcal{T}$ consisting of compact objects, which is assumed to be skeletally small.
•

$\mathbf{Ab}$ is the category of abelian groups.
•

$\mathbf{Mod}\text{-}\mathcal{T}^{c}$ is the category of contravariant additive functors $\mathcal{T}^{c}\to\mathbf{Ab}$ .
•

$\mathtt{G}$ is a set of generators of $\mathbf{Mod}\text{-}\mathcal{T}^{c}$ where each $\mathscr{G}\in\mathtt{G}$ is finitely presented.
•

$\mathbf{Y}\colon\mathcal{T}\to\mathbf{Mod}\text{-}\mathcal{T}^{c}$ is the restricted Yoenda functor, which takes an object $M$ to the restriction of the corepresentable $\mathcal{T}(-,M)$ .

Recall that, by the Brown representability theorem, since the category $\mathcal{T}$ has all small coproducts, it has all small products; see Remark 4.4.

Theorem 1.1.

For any object $M$ of $\mathcal{T}$ the following statements are equivalent.

(1)

For any set $\mathtt{I}$ the coproduct $M^{(\mathtt{I})}=\coprod_{\mathtt{I}}M$ is pure-injective.
(2)

The countable coproduct $M^{(\mathbb{N})}$ is pure-injective.
(3)

For any generator $\mathscr{G}\in\mathtt{G}$ each ascending chain of $\mathbf{Y}(M)$ -annihilator subobjects of $\mathscr{G}$ must eventually stabilise.
(4)

For any set $\mathtt{I}$ the morphism from $M^{(\mathtt{I})}$ to the product $M^{\mathtt{I}}=\prod_{\mathtt{I}}M$ , given by the universal property, is a section.
(5)

For any object $X$ of $\mathcal{T}^{c}$ each descending chain of pp-definable subgroups of $M$ of sort $X$ must eventually stabilise.
(6)

$M$ is pure injective, and for any set $\mathtt{I}$ , $M^{\mathtt{I}}$ is isomorphic to a coproduct of indecomposable pure-injective objects with local endomorphism rings.

The proof of Theorem 1.1 is at the end of the article. The equivalences of (1), (2), (3) and (4) in Theorem 1.1 follow by directly combining work of Garcia and Dung [7] and work of Krause [18]. The equivalence of (5) and (6) with the previous conditions is more involved. For (5) we adapt ideas going back to Faith [6], whilst applying results due to Harada [9] and Garcia and Dung [7]. For (6) we adapt arguments of Huisgen-Zimmerman [11].

The article is organised as follows. In §2 we recall some prerequisite terminology from multi-sorted model theory. In §3 we specify to compactly generated triangulated categories by recalling the canonical multi-sorted language of Garkusha and Prest [8]. In §4 we gather results about products and coproducts of pp-definable subgroups in this context, following ideas of Huisgen-Zimmerman [11]. In §5 we highlight a connection between annihilator subobjects of finitely generated functors and pp-definable subgroups, where the presentation of the functor determines the sort of the subgroup. In §6 we begin combining the results developed in the previous sections with results of Krause [18]. In §7 we complete the proof of Theorem 1.1. In §8 we introduce endoperfect objects, and see applications of Theorem 1.1.

2. Multi-sorted languages, structures and homomorphisms.

There are various module-theoretic characterisations of purity in terms of positive-primitive formulas in the underlying one-sorted language of modules over a ring. Similarly, purity in compactly generated triangulated categories may be discussed in terms of formulas in a multi-sorted language. Although Definitions 2.1, 2.3, 2.5, 2.6 and 2.7 are well-known, we recall them for completeness. We closely follow [5, §2, §7] for consistency.

Definition 2.1.

[5, Definition 34] For a non-empty set $\mathtt{S}$ , an $\mathtt{S}$ -sorted predicate language $\mathfrak{L}$ is a tuple $\langle\mathrm{pred}_{\mathtt{S}},\,\mathrm{func}_{\mathtt{S}},\,\mathrm{ar}_{\mathtt{S}},\,\mathrm{sort}_{\mathtt{S}}\rangle$ where:

(1)

each $s\in\mathtt{S}$ is called a sort;
(2)

the symbol $\mathrm{pred}_{\mathtt{S}}$ denotes a non-empty set of sorted predicate symbols;
(3)

the symbol $\mathrm{func}_{\mathtt{S}}$ denotes a set of sorted function symbols;
(4)

the symbol $\mathrm{ar}_{\mathtt{S}}$ denotes an arity function $\mathrm{pred}_{\mathtt{S}}\sqcup\mathrm{func}_{\mathtt{S}}\to\mathbb{N}=\{0,1,2,\dots\}$ ;
(5)

the symbol $\mathrm{sort}_{\mathtt{S}}$ denotes a sort function, taking any $n$ -ary $R\in\mathrm{pred}_{\mathtt{S}}$ (respectively $F\in\mathrm{func}_{\mathtt{S}}$ ) to a sequence in $\mathtt{S}$ of length $n$ (respectively $n+1$ ).

When $n>0$ in condition (5) we often write $\mathrm{sort}_{\mathtt{S}}(R)=(s_{1},\dots,s_{n})$ (respectively $\mathrm{sort}_{\mathtt{S}}(F)=(s_{1},\dots,s_{n},s)$ ). Note that functions $F$ with $\mathrm{ar}_{\mathtt{S}}(F)=0$ have a sort.

For each sort $s$ we introduce a countable set $\mathcal{V}_{s}$ of variables of sort $s$ . The terms of $\mathfrak{L}$ each have their own sort, and are defined inductively by stipulating: any variable $x$ of sort $s$ will be considered a term of sort $s$ ; and for any $F\in\mathrm{func}_{\mathtt{S}}$ with $\mathrm{sort}_{\mathtt{S}}(F)=(s_{1},\dots,s_{n},s)$ and for any terms $t_{1},\dots,t_{n}$ of sort $s_{1},\dots,s_{n}$ respectively, $F(t_{1},\dots,t_{n})$ is considered a term of sort $s$ . Note that constant symbols, given by functions $F$ with $\mathrm{ar}_{\mathtt{S}}(F)=0$ , are also terms.

The atomic formulas with which $\mathfrak{L}$ is equipped are built from the equality $t=_{s}t^{\prime}$ between terms $t,t^{\prime}$ of common sort $s$ , together with the formulas $R(t_{1},\dots,t_{n})$ where $R\in\mathrm{pred}_{\mathtt{S}}$ , $\mathrm{sort}_{\mathtt{S}}(R)=(s_{1},\dots,s_{n})$ and where each $t_{i}$ is a term of sort $s_{i}$ . First-order formulas $\varphi$ in $\mathfrak{L}$ are built from: the variables of each sort; the atomic formulas; binary connectives $\wedge$ , $\vee$ , and $\implies$ ; negation $\neg$ ; and the quantifiers $\forall$ and $\exists$ .

A positive-primitive or pp formula $\varphi(x_{1},\dots,x_{n})$ (with $x_{i}$ free) has the form

\begin{array}[]{c}\exists\,w_{n+1},\dots,w_{m}\colon\bigwedge_{j=1}^{k}\psi_{j}(x_{1},\dots,x_{n},w_{n+1},\dots,w_{m}),\end{array}

where each $\psi_{j}$ is an atomic formula (see, for example, [10, p.50]).

One may build a theory for a multi-sorted language $\mathfrak{L}$ by specficying a set of axioms. For our purposes these axioms are those charaterising objects and morphisms in a fixed category. We explain this idea by means of examples.

Example 2.2.

[14, §6] Let $A$ be a unital ring. We recall how the language $\mathfrak{L}_{A}$ of $A$ -modules may be considered as a predicate language in the sense of Definition 2.1. In this case there is only one sort, which we ignore, and which uniquely determines the function $\mathrm{sort}_{A}$ . Let $\mathrm{pred}_{A}=\{0\}$ . Let $\mathrm{func}_{A}=\{+\}\cup\{a\times-\mid a\in A\}$ where $+$ is binary and $a\times-$ is unary.

In Definition 2.3 the notion of a structure is recalled. For the language $\mathfrak{L}_{A}$ we have that this notion, together with certain axioms, recovers the defining properties of left $A$ -modules. Later we consider homomorphisms.

Definition 2.3.

[5, Definition 35] Fix a set $\mathtt{S}\neq\emptyset$ and an $\mathtt{S}$ -sorted predicate language $\mathfrak{L}$ . An $\mathfrak{L}$ -structure is a tuple

\mathsf{M}=\langle\mathtt{S}(\mathsf{M}),(R(\mathsf{M})\mid R\in\mathrm{pred}_{\mathtt{S}}),(F(\mathsf{M})\mid F\in\mathrm{func}_{\mathtt{S}})\rangle

such that:

(1)

the symbol $\mathtt{S}(\mathsf{M})$ denotes a family of sets $\{s(\mathsf{M})\mid s\in\mathtt{S}\}$ ;
(2)

if $\mathrm{sort}_{\mathtt{S}}(R)=(s_{1},\dots,s_{n})$ then $R(\mathsf{M})$ is a subset of $s_{1}(\mathsf{M})\times\dots\times s_{n}(\mathsf{M})$ ; and
(3)

if $\mathrm{sort}_{\mathtt{S}}(F)=(s_{1},\dots,s_{n},s)$ then $F(\mathsf{M})$ is a map $s_{1}(\mathsf{M})\times\dots\times s_{n}(\mathsf{M})\to s(\mathsf{M})$ .

Denoting the cardinality of any set $\mathtt{X}$ by $|\mathtt{X}|$ , let $|\mathfrak{L}|=|\mathrm{pred}_{\mathtt{S}}\sqcup\mathrm{func}_{\mathtt{S}}|$ and, for $\mathsf{M}$ as above, let $|\mathsf{M}|$ be the sum of the cardinalities $|s(\mathsf{M})|$ as $s$ runs through $\mathtt{S}$ .

The so-called one-sorted language from Example 2.2 is trivial in the sense that there is only one possibility for the sort function. In this way, Example 2.4 is a non-trivial example of the multi-sorted languages we recalled in Definition 2.1.

Example 2.4.

[14, §9] Here we recall an example of an $\{\mathtt{r},\mathtt{m}\}$ -sorted predicate language which is in contrast to Example 2.2. The predicates in this language will be the unary symbols $0_{\mathtt{r}}$ and $1_{\mathtt{r}}$ of sort $\mathtt{r}$ , and $0_{\mathtt{m}}$ of sort $\mathtt{m}$ . The functions in this language will be the ternary symbols $+$ and $\times$ where $\mathrm{sort}_{\mathtt{r},\mathtt{m}}(+)=(\mathtt{m},\mathtt{m},\mathtt{m})$ and $\mathrm{sort}_{\mathtt{r},\mathtt{m}}(\times)=(\mathtt{r},\mathtt{m},\mathtt{m})$ . After specifying the appropriate axioms, structures ${}_{\mathsf{A}}\mathsf{M}$ are tuples $(A,M)$ where $A$ is a unital ring and $M$ is a left $A$ -module.

In this way one interprets the symbols $0_{\mathtt{r}}$ and $1_{\mathtt{r}}$ as the additive and multiplicative identities in $A$ . Similarly the symbol $0_{\mathtt{m}}$ is interpreted as the additive identity in $M$ . In Definition 2.5 the notion of a homomorphism between structures is recalled. In this sense, a homomorphism $(A,M)\to(B,N)$ is given by a pair $(f,l)$ where $f\colon A\to B$ is a homomorphism of rings and $l\colon M\to N$ is a homomorphism of left $A$ -modules with the action of $A$ on $N$ given by $f$ .

Definition 2.5.

[5, Definition 3] Fix a non-empty set $\mathtt{S}$ , an $\mathtt{S}$ -sorted predicate language $\mathfrak{L}$ and $\mathfrak{L}$ -structures $\mathsf{L}$ and $\mathsf{M}$ . By an $\mathfrak{L}$ -homomorphism $\mathsf{h}\colon\mathsf{L}\to\mathsf{M}$ we mean a family $\{\mathsf{h}_{s}\mid s\in\mathtt{S}\}$ of functions $\mathsf{h}_{s}\colon s(\mathsf{L})\to s(\mathsf{M})$ such that:

(1)

if $\mathrm{sort}_{\mathtt{S}}(R)=(s_{1},\dots,s_{n})$ then $R(\mathsf{M})$ is the set of $(\mathsf{h}_{s_{1}}(a_{1}),\dots,\mathsf{h}_{s_{n}}(a_{n}))$ such that $(a_{1},\dots,a_{n})\in R(\mathsf{L})$ ;
(2)

and if $\mathrm{sort}_{\mathtt{S}}(F)=(s_{1},\dots,s_{n},s)$ then for all $(a_{1},\dots,a_{n})\in s_{1}(\mathsf{L})\times\dots\times s_{n}(\mathsf{L})$ we have $\mathsf{h}_{s}(F(\mathsf{L})(a_{1},\dots,a_{n}))=F(\mathsf{M})(\mathsf{h}_{s_{1}}(a_{1}),\dots,\mathsf{h}_{s_{n}}(a_{n}))$ .

Note that, in the notation of Definition 2.5, [5, Theorem 17] says that a collection of functions $\mathsf{h}_{s}:s(\mathsf{L})\to s(\mathsf{M})$ defines an $\mathfrak{L}$ -homomorphism if and only if, whenever $\varphi(x_{1},\dots,x_{n})$ is an atomic formula with $\mathrm{sort}_{\mathtt{S}}(x_{i})=s_{i}$ and $(a_{1},\dots,a_{n})$ lies in $s_{1}(\mathsf{L})\times\dots\times s_{n}(\mathsf{L})$ , we have that

\mathsf{L}\models\varphi(a_{1},\dots,a_{n})\implies\mathsf{M}\models\varphi(\mathsf{h}_{s_{1}}(a_{1}),\dots,\mathsf{h}_{s_{n}}(a_{n})).

We are now ready to recall the idea of purity coming from model theory.

Definition 2.6.

[5, Definition 36] Fix a set $\mathtt{S}\neq\emptyset$ and an $\mathtt{S}$ -sorted predicate language $\mathfrak{L}$ . By an $\mathfrak{L}$ -embedding we mean an $\mathfrak{L}$ -homomorphism $\mathsf{h}\colon\mathsf{L}\to\mathsf{M}$ such that:

(1)

if $\varphi(x_{1},\dots,x_{n})$ is an atomic formula with $\mathrm{sort}_{\mathtt{S}}(x_{i})=s_{i}$ , then for all $(a_{1},\dots,a_{n})\in s_{1}(\mathsf{L})\times\dots\times s_{n}(\mathsf{L})$ , $\mathsf{L}\models\varphi(a_{1},\dots,a_{n})$ if and only if $\mathsf{M}\models\varphi(\mathsf{h}_{s_{1}}(a_{1}),\dots,\mathsf{h}_{s_{n}}(a_{n}))$ .

By an $\mathfrak{L}$ -pure embedding we mean an $\mathfrak{L}$ -homomorphism $\mathsf{h}\colon\mathsf{L}\to\mathsf{M}$ such that:

(2)

if $\varphi(x_{1},\dots,x_{n})$ is a pp formula with $\mathrm{sort}_{\mathtt{S}}(x_{i})=s_{i}$ , then for all $(a_{1},\dots,a_{n})\in s_{1}(\mathsf{L})\times\dots\times s_{n}(\mathsf{L})$ , if $\mathsf{M}\models\varphi(\mathsf{h}_{s_{1}}(a_{1}),\dots,\mathsf{h}_{s_{n}}(a_{n}))$ then $\mathsf{L}\models\varphi(a_{1},\dots,a_{n})$ .

Note that $\mathfrak{L}$ -pure embeddings are $\mathfrak{L}$ -embeddings. Note also that the statement of Definition 2.5(2) is the contrapositive of the definition in [10, p.50], so in this sense, over a ring $A$ and in the notation from Example 2.2, an injective left $A$ -module homomorphism is pure if and only if it is an $\mathfrak{L}_{A}$ -pure embedding.

Definition 2.7.

Fix a non-empty set $\mathtt{S}$ , an $\mathtt{S}$ -sorted predicate language $\mathfrak{L}$ and $\mathfrak{L}$ -structures $\mathsf{L}$ and $\mathsf{M}$ . We say $\mathsf{L}$ is an $\mathfrak{L}$ -substructure of $\mathsf{M}$ if $s(\mathsf{L})\subseteq s(\mathsf{M})$ for each $s\in\mathtt{S}$ and, labelling these inclusions $\mathsf{i}_{s}$ , the family $\{\mathsf{i}_{s}\mid s\in\mathtt{S}\}$ defines an $\mathfrak{L}$ -homomorphism $\mathsf{i}\colon\mathsf{L}\to\mathsf{M}$ . If, additionally, $\mathsf{i}\colon\mathsf{L}\to\mathsf{M}$ is an $\mathfrak{L}$ -pure embedding, we say $\mathsf{L}$ is an $\mathfrak{L}$ -pure substructure of $\mathsf{M}$ .

3. Purity in the canonical language of a triangulated category.

We now specify the setting of multi-sorted model theory outlined in §2. Throughout the sequel we consider a fixed compactly generated triangulated category; see Assumption 3.3. Before recalling Definition 3.2 we fix some notation.

Notation 3.1.

Let $\mathcal{A}$ be an additive category. Denote the hom-sets $\mathcal{A}(X,Y)$ and the identity maps $1_{X}$ . For any set $\mathtt{I}$ and any collection $\mathrm{B}=\{B_{i}\mid i\in\mathtt{I}\}$ of objects in $\mathcal{A}$ , if the categorical product $\prod_{i}B_{i}$ exists in $\mathcal{A}$ , we write $p_{j,\mathrm{B}}\colon\prod_{i}B_{i}\to B_{j}$ for the natural morphisms equipping it, in which case the universal property gives unique morphisms $v_{j,\mathrm{B}}\colon B_{j}\to\prod_{i}B_{i}$ such that $p_{j,\mathrm{B}}v_{j,\mathrm{B}}$ is the identity $1_{j}$ on $B_{j}$ for each $j$ . Similarly $u_{j,\mathrm{B}}\colon B_{j}\to\coprod_{i}B_{i}$ will denote the morphisms equipping the coproduct $\coprod_{i}B_{i}$ if it exists, in which case there exist unique morphisms $q_{j,\mathrm{B}}\colon\coprod_{i}B_{i}\to B_{j}$ such that $q_{j,\mathrm{B}}u_{j,\mathrm{B}}=1_{j}$ for each $j$ .

Fix an object $A$ in $\mathcal{A}$ and consider the covariant functor $\mathcal{A}(A,-)$ . Note that both the product and coproduct of the collection $\mathcal{A}(A,\mathrm{B})=\{\mathcal{A}(A,B_{i})\mid i\in\mathtt{I}\}$ exist in the category $\mathbf{Ab}$ of abelian groups. We identify $\coprod_{i\in\mathtt{I}}\mathcal{A}(A,B_{i})$ with the subgroup of $\prod_{i\in\mathtt{I}}\mathcal{A}(A,B_{i})$ consisting of tuples $(g_{i}\mid i\in\mathtt{I})$ such that $g_{i}=0$ for all but finitely many $i\in\mathtt{I}$ .

Consequently, if $\prod_{i}B_{i}$ exists in $\mathcal{A}$ then map $\lambda_{A,\mathrm{B}}\colon\mathcal{A}(A,\prod_{i}B_{i})\to\prod_{i}\mathcal{A}(A,B_{i})$ from the universal property is given by $f\mapsto(p_{i,\mathrm{B}}f\mid i\in\mathtt{I})$ for each $f\in\mathcal{A}(A,\prod_{i}B_{i})$ . Similarly if $\coprod_{i}B_{i}$ exists in $\mathcal{A}$ then map $\gamma_{A,\mathrm{B}}\colon\coprod_{i}\mathcal{A}(A,B_{i})\to\mathcal{A}(A,\coprod_{i}B_{i})$ from the universal property is given by $\gamma_{A,\mathrm{B}}(g_{i}\mid i\in\mathtt{I})=\sum_{i}u_{i,\mathrm{B}}g_{i}$ . In general each of the morphisms $\lambda_{A,\mathrm{B}}$ are isomorphisms.

Definition 3.2.

[19, Definition 1.1] Let $\mathcal{T}$ be a triangulated category with suspension functor $\Sigma$ , and assume all small coproducts in $\mathcal{T}$ exist. An object $X$ of $\mathcal{T}$ is said to be compact if, for any set $\mathtt{I}$ and collection $\mathrm{M}=\{M_{i}\mid i\in\mathtt{I}\}$ of objects in $\mathcal{T}$ the morphism $\gamma_{X,\mathrm{M}}$ is an isomorphism. Let $\mathcal{T}^{c}$ be the full triangulated subcategory of $\mathcal{T}$ consisting of compact objects.

Given a set $\mathcal{G}$ of compact objects in $\mathcal{T}$ we say that $\mathcal{T}$ is compactly generated by $\mathcal{G}$ if there are no non-zero objects $M$ in $\mathcal{T}$ satisfying $\mathcal{T}(X,M)=0$ for all $X\in\mathcal{G}$ (or, said another way, any non-zero object $M$ gives rise to a non-zero morphism $X\to M$ for some $X\in\mathcal{G}$ ). If $\mathcal{T}$ is compactly generated by $\mathcal{G}$ we call $\mathcal{G}$ a generating set provided $\Sigma X\in\mathcal{G}$ for all $X\in\mathcal{G}$ .

Assumption 3.3.

In the remainder of §3 fix a triangulated category $\mathcal{T}$ with suspension functor $\Sigma$ , and we assume:

•

that $\mathcal{T}$ has all small coproducts;
•

that $\mathcal{T}$ is compactly generated by a generating set $\mathcal{G}$ ; and
•

that the subcategory $\mathcal{T}^{c}$ of compact objects is skeletally small.

Definition 3.4 and Remark 3.6 closley follow [8, §3], in which a multi-sorted language associated to the category $\mathcal{T}$ is introduced.

Definition 3.4.

In what follows let $\mathcal{S}$ denote a fixed set of objects in $\mathcal{T}^{c}$ given by choosing exactly one representative of each isomorphism class. Such a set $\mathcal{S}$ exists because we are assuming that $\mathcal{T}^{c}$ is skeletally small.

[8, §3] The canonical language $\mathfrak{L}^{\mathcal{T}}$ of $\mathcal{T}$ is given by an $\mathcal{S}$ -sorted predicate language $\langle\mathrm{pred}_{\mathcal{S}},\,\mathrm{func}_{\mathcal{S}},\,\mathrm{ar}_{\mathcal{S}},\,\mathrm{sort}_{\mathcal{S}}\rangle$ , defined as follows. The set $\mathrm{pred}_{\mathcal{S}}$ consists of a symbol $\mathsf{0}_{G}$ with $\mathrm{sort}_{\mathcal{S}}(\mathsf{0}_{G})=G$ for each $G\in\mathcal{S}$ . The set $\mathrm{func}_{\mathcal{S}}$ consists of: a ternary symbol $+_{G}$ with $\mathrm{sort}_{\mathcal{S}}(+_{G})=(G,G,G)$ for each $G\in\mathcal{S}$ ; and a unary operation $-\circ a$ with $\mathrm{sort}_{\mathcal{S}}(-\circ a)=(H,G)$ for each morphism $a:G\to H$ with $G,H\in\mathcal{S}$ . Variables of sort $G\in\mathcal{S}$ will be denoted $v_{G}$ .

Notation 3.5.

Suppose $\mathcal{A}$ is any additive category. We write $\mathcal{A}\text{-}\mathbf{Mod}$ (respectively $\mathbf{Mod}\text{-}\mathcal{A}$ ) for the category of additive covariant (respectively contravariant) functors $\mathcal{A}\to\mathbf{Ab}$ where $\mathbf{Ab}$ is the category of abelian groups.

For any object $M$ of $\mathcal{T}$ we let $\mathcal{T}(-,M)|$ denote the object of $\mathbf{Mod}\text{-}\mathcal{T}^{c}$ defined by restriction of $\mathcal{T}(-,M)$ to compact objects. We write

\mathbf{Y}\colon\mathbf{Mod}\text{-}\mathcal{T}^{c}\to\mathbf{Ab}

to denote the restricted Yoneda functor. That is, $\mathbf{Y}$ takes an object $M$ to $\mathbf{Y}(M)=\mathcal{T}(-,M)|$ , and takes a morphism $h:L\to M$ to the natural transformation $\mathbf{Y}(h)\colon\mathcal{T}(-,L)|\to\mathcal{T}(-,M)|$ given by defining, for each compact object $X$ , the map $\mathbf{Y}(h)_{X}\colon\mathcal{T}(X,L)\to\mathcal{T}(X,M)$ by $g\mapsto hg$ .

Remark 3.6.

[8, §3] Consider the theory given from the set of axioms expressing the positive atomic diagram of the objects in $\mathcal{T}^{c}$ including the specification that all functions are additive. In this way, the category of models for the above theory is equivalent to the category $\mathbf{Mod}\text{-}\mathcal{T}^{c}$ where objects $M$ of $\mathcal{T}$ are regarded as structures $\mathsf{M}$ for this language.

That is, in the notation of Definition 2.3, we let $G(\mathsf{M})=\mathcal{T}(G,M)$ , we interpret the predicate symbol $0_{G}$ as the identitly element of $G(\mathsf{M})$ , we interpret $+_{G}$ as the additive group operation on $G(\mathsf{M})$ , and we interpret $-\circ a$ as the map $G(\mathsf{M})\to H(\mathsf{M})$ given by $f\mapsto fa$ .

Lemma 3.7.

Let $L$ and $M$ be objects in $\mathcal{T}$ with corresponding $\mathfrak{L}^{\mathcal{T}}$ -structures $\mathsf{L}$ and $\mathsf{M}$ . Then the choice of $\mathcal{S}$ made in Definition 3.4 defines a bijection between morphisms $\mathbf{Y}(L)\to\mathbf{Y}(M)$ (that is, natural transformations) in $\mathbf{Mod}\text{-}\mathcal{T}^{c}$ and $\mathfrak{L}^{\mathcal{T}}$ -homomorphisms $\mathsf{L}\to\mathsf{M}$ .

Proof.

Any object $X$ of $\mathcal{T}^{c}$ lies in the same isoclass as some unique $c(X)\in\mathcal{S}$ , in which case we choose an isomorphism $\phi_{X}\colon c(X)\to X$ . In this way, any object $N$ defines an isomorphism $-\circ\phi_{X,N}\colon\mathcal{T}(X,N)\to\mathcal{T}(c(X),N)$ by precomposition with $\phi_{X}$ . Recall that here the $\mathfrak{L}^{\mathcal{T}}$ -structure $\mathsf{N}$ is defined by setting $G(\mathsf{N})=\mathcal{T}(G,N)$ for each sort $G\in\mathcal{S}$ . In case $X\in\mathcal{S}$ we assume, without loss of generality, that $\phi_{X}=1_{X}$ . Define the required bijection as follows. Fix an $\mathfrak{L}^{\mathcal{T}}$ -homomorphism $\mathsf{h}\colon\mathsf{L}\to\mathsf{M}$ . For any object $X$ of $\mathcal{T}^{c}$ define the function $\mathscr{H}(\mathsf{h})_{X}\colon\mathcal{T}(X,L)\to\mathcal{T}(X,M)$ by $l\mapsto(\mathsf{h}_{c(X)}(l\phi_{X}))\phi^{-1}_{X}$ . Converlsey, fixing a natural transformation $\mathscr{H}\colon\mathbf{Y}(L)\to\mathbf{Y}(M)$ , let $\mathsf{h}(\mathscr{H})_{G}=\mathscr{H}_{G}$ for each $G\in\mathcal{S}$ .

It suffices to explain why these assignments swap between morphisms in $\mathbf{Mod}\text{-}\mathcal{T}^{c}$ and $\mathfrak{L}^{\mathcal{T}}$ -homomorphisms. To do so, we explain why the compatability conditions which define these morphisms are in correspondence. To this end, note firstly that the preservation of (the predicate symbol $0_{G}$ and the function symbols $+_{G}$ ) is equivalent to saying that each function $\mathscr{H}_{X}$ is a homomorphism of abelian groups. Letting $b\colon X\to Y$ be a morphism in $\mathcal{T}^{c}$ and $a=\phi_{Y}^{-1}b\phi_{X}$ , for any object $N$ of $\mathcal{T}$ the sorted function symbol $-\circ a$ is interpreted in $\mathsf{N}$ by the equation $(-\circ a)(\mathsf{N})=-\circ(\phi_{N,Y}^{-1})b\phi_{N,X}$ . Thus, by construction, saying that the function symbols $-\circ a$ are preserved is equivalent to saying that the collection of $\mathscr{H}_{X}$ (for $X$ compact) defines a natural transformation. ∎

In what follows we discuss the notion of purity in the context of compactly generated triangulated categories.

Definition 3.8.

[18, Definition 1.1] We say $h\colon L\to M$ in $\mathcal{T}$ is a pure monomorphism if $\mathbf{Y}(h)_{X}\colon\mathcal{T}(X,L)\to\mathcal{T}(X,M)$ is injective for each compact object $X$ .

Now we may begin to build results which mimic well-known ideas from the model theory of modules. To consistently compare and contrast our work with the module-theoretic setting, we use a book of Jensen and Lenzing [14]. In this spirit, Lemma 3.9 is analogous to [14, Theorem 6.4(i,ii)], and Lemma 4.2 is analogous to [14, Proposition 6.6]. Similar analogies are found throughout the sequel.

Lemma 3.9.

A morphism $h\colon L\to M$ is a pure monomorphism if and only if the image $\mathsf{h}\colon\mathsf{L}\to\mathsf{M}$ under the bijection in Lemma 3.7 is an $\mathfrak{L}^{\mathcal{T}}$ -pure embedding.

Proof.

By [8, Proposition 3.1] any pp-formula $\varphi(v_{G})$ is equivalent to a divisibility formula $\exists u_{H}\colon v_{G}=u_{H}a$ where $a\colon G\to H$ is morphism and $G,H\in\mathcal{S}$ . By Definition 2.6, $\mathsf{h}$ is an $\mathfrak{L}^{\mathcal{T}}$ -pure embedding if and only if, for any morphism $a\colon G\to H$ with $G,H\in\mathcal{S}$ and any pair $(f,g)\in G(\mathsf{L})\times H(\mathsf{L})$ , if $hg=hfa$ then $g=fa$ . Since any compact object is isomorphic to an object in $\mathcal{S}$ , this is equivalent to the condition which says that, for each compact object $X$ , the morphism $\mathcal{T}(X,L)\to\mathcal{T}(X,M)$ given by $g\mapsto hg$ is injective. ∎

4. Products, coproducts, coherent functors and pp-formulas.

Recall, from Definition 2.1, that pp-formulas in $\mathfrak{L}^{\mathcal{T}}$ are those lying in the closure of the set of equations under conjunction and existential quantification.

Definition 4.1.

[8, §2] Given $G\in\mathcal{S}$ and an object $M$ of $\mathcal{T}$ with $\mathfrak{L}^{\mathcal{T}}$ -structure $\mathsf{M}$ , a pp-definable subgroup of $M$ of sort $G$ is the set $\varphi(M)=\{f\in G(\mathsf{M})\mid\mathsf{M}\models\varphi(f)\}$ of solutions (in $\mathsf{M}$ ) to some pp-formula $\varphi(v_{G})$ in one free variable of sort $G\in\mathcal{S}$ .

For any morphism $b\colon X\to Y$ in $\mathcal{T}^{c}$ and any object $M$ in $\mathcal{T}$ recall the map $\mathbf{Y}(b)\colon\mathcal{T}(Y,M)\to\mathcal{T}(X,M)$ is defined by precomposition. In this case let

Mb=\mathrm{im}(\mathbf{Y}(b))=\{fb\in\mathcal{T}(X,M)\mid f\in\mathcal{T}(Y,M)\}.

If $G,H\in\mathcal{S}$ and $\phi_{X}\colon G\to X$ and $\phi_{Y}\colon H\to Y$ are isomorphisms in $\mathcal{T}$ (as in the proof of Lemma 3.7), then $fb\mapsto f\phi^{-1}_{Y}b\phi_{X}$ defines a isomorphism $Mb\to\varphi(M)$ in $\mathbf{Ab}$ where $\varphi(v_{G})$ is the pp-formula $(\exists u_{H}\colon v_{G}=u_{H}a)$ where $a=\phi^{-1}_{Y}b\phi_{X}$ .

Lemma 4.2.

Let $G\in\mathcal{S}$ and let $\varphi(v_{G})$ be a pp-formula in one free variable of sort $G$ . If $h\colon L\to M$ is a pure monomorphism then $\varphi(L)=\{g\in\mathcal{T}(G,L)\mid hg\in\varphi(M)\}$ .

Proof.

The claim follows from Lemma 3.9, together with Definition 2.6, which we recall defines $\mathfrak{L}^{\mathcal{T}}$ -pure embeddings as those which preserve solutions to the negations of all pp-formulas. ∎

We continue, slightly abusing terminology, by reffering to any set of the form $Mb$ (for some $b\in\mathcal{T}(X,Y)$ ) as a pp-definable subgroup of $M$ of sort $X$ . Lemma 4.3 is an analgoue of the corresponding result for module categories; see for example [20, Corollary 2.2(i)]. Note that we state it here for convenience, but that it is only used in §8, and not in the proof of Theorem 1.1.

Lemma 4.3.

For any object $M$ in $\mathcal{T}$ and any morphism $b\in\mathcal{T}(X,Y)$ the pp-definable subgroup $Mb$ is a left $\mathrm{End}_{\mathcal{T}}(M)$ -submodule of $\mathcal{T}(X,M)$ .

Proof.

By the associativity of the composition of morphisms in $\mathcal{T}$ the set $Mb$ is closed under postcomposition with endomorphisms of $M$ . ∎

Recall that a covariant functor $\mathscr{F}:\mathcal{T}\to\mathbf{Ab}$ is coherent if there is an exact sequence in $\mathcal{T}\text{-}\mathbf{Mod}$ of the form $\mathcal{T}(A,-)\to\mathcal{T}(B,-)\to\mathscr{F}\to 0$ . If $t\colon M\to N$ and $a:G\to H$ are morphisms in $\mathcal{T}$ with $G,H\in\mathcal{S}$ , then $tv\in Na$ for any $v\in Ma$ . So, for the pp-formula $\varphi(v_{G})=(\exists u_{H}\colon v_{G}=u_{H}a)$ in $\mathfrak{L}^{\mathcal{T}}$ the assignment of objects $M\mapsto\varphi(M)$ defines a functor.

Furthermore, by [8, Lemma 4.3] these functors are coherent, and any such coherent functor arises this way. We now recall that the categories we are considering have all small products.

Remark 4.4.

As a result of Assumption 3.3, by the Brown representability theorem we have that $\mathcal{T}$ has all small products. See [17, Lemma 1.5] for details.

Lemma 4.5 is analogous to [14, Proposition 6.7(i,ii)]. Recall Notation 3.1.

Lemma 4.5.

Let $G\in\mathcal{S}$ and let $\varphi(v_{G})$ be a pp-formula in one free variable of sort $G$ . For any set $I$ and any collection $\mathrm{M}=\{M_{i}\mid i\in\mathtt{I}\}$ of objects in $\mathcal{T}$ the restrictions of $\gamma_{G,\mathrm{M}}$ and $\lambda_{G,\mathrm{M}}$ define isomorphisms of abelian groups

\begin{array}[]{cc}\coprod_{i}\varphi(M_{i})\to\varphi(\coprod_{i}M_{i}),&\varphi(\prod_{i}M_{i})\to\prod_{i}\varphi(M_{i}).\end{array}

Proof.

By the existence of small products and coproducts in $\mathcal{T}$ and the functorality of $\varphi$ , the universal properties give morphisms $\delta\colon\coprod_{i}\varphi(M_{i})\to\varphi(\coprod_{i}M_{i})$ and $\mu\colon\varphi(\prod_{i}M_{i})\to\prod_{i}\varphi(M_{i})$ . By [8, Lemma 4.3] the functor $\varphi$ is coherent, so by the equivalence of statements (1) and (3) from [18, Theorem A] the morphisms $\delta$ and $\mu$ are isomorphisms. It is straightforward to check that $\delta$ and $\mu$ are the respectively restrictions of $\gamma_{G,\mathrm{M}}$ and $\lambda_{G,\mathrm{M}}$ from Notation 3.1. ∎

We now adapt some technical results from work of Huisgen-Zimmerman [11], in which a (now well-known) charaterisation of $\Sigma$ -pure-injective modules was given. Our adaptations, namely Lemmas 4.8 and 7.7, are used in the sequel.

Notation 4.6.

Fix collections $\mathrm{M}=\{M_{i}\mid i\in\mathbb{N}\}$ and $\mathrm{L}=\{L_{j}\mid j\in\mathtt{J}\}$ of objects in $\mathcal{T}$ and let $M=L$ where $M=\prod_{i}M_{i}$ and $L=\coprod_{j}L_{j}$ exist. By Notation 3.1, $p_{i,\mathrm{M}}\colon M\to M_{i}$ and $u_{j,\mathrm{L}}\colon L_{j}\to L$ denote the morphisms equipping the product and coproduct, and $v_{i,\mathrm{M}}\colon M_{i}\to M$ and $q_{j,\mathrm{L}}\colon L\to L_{j}$ denote the morphisms given by universal properties such that $p_{i,\mathrm{M}}v_{i,\mathrm{M}}=1_{M_{i}}$ and $q_{j,\mathrm{L}}u_{j,\mathrm{L}}=1_{L_{j}}$ for each $i$ and $j$ .

Corollary 4.7.

Consider Notation 4.6, let $a\colon G\to H$ be a morphism with $G,H\in\mathcal{S}$ , and let $\varphi(v_{G})=(\exists u_{H}\colon v_{G}=u_{H}a)$ . Then there is an isomorphism

\begin{array}[]{c}\kappa\langle\varphi\rangle\colon\prod_{i\in\mathbb{N}}\varphi(M_{i})\to\coprod_{j\in\mathtt{J}}\varphi(L_{j}),\,(f_{i}\mid i\in\mathbb{N})\mapsto(q_{j,\mathrm{L}}f\mid j\in\mathtt{J})\end{array}

where $f\colon G\to\prod_{i}M_{i}$ is given by the universal property, and whose inverse is

\begin{array}[]{c}\kappa^{-1}\langle\varphi\rangle\colon\coprod_{j\in\mathtt{J}}\varphi(L_{j})\to\prod_{i\in\mathbb{N}}\varphi(M_{i}),\,(g_{j}\mid j\in\mathtt{J})\mapsto(\sum_{j\in\mathtt{J}}p_{i,\mathrm{M}}u_{j,\mathrm{L}}g_{j}\mid i\in\mathbb{N}).\end{array}

Proof.

By Lemma 4.5 the restriction of $\gamma_{G,\mathrm{L}}$ and $\lambda_{G,\mathrm{M}}$ define isomorphisms

\begin{array}[]{cc}\delta\colon\coprod_{j}\varphi(L_{j})\to\varphi(\coprod_{j}L_{j}),\mu\colon\varphi(\prod_{i}M_{i})\to\prod_{i}\varphi(M_{i}).\end{array}

Letting $\kappa\langle\varphi\rangle=\delta^{-1}\mu^{-1}$ and $\kappa\langle\varphi\rangle^{-1}=\mu\delta$ , the proof is straightforward. ∎

The proof of Lemma 4.8 follows the proof of the cited result of Huisgen-Zimmerman.

Lemma 4.8.

[11, Lemma 4] Consider Notation 4.6 and let $\varphi_{1}(M)\supseteq\varphi_{2}(M)\supseteq\dots$ be a descending chain of pp-definable subgroups of $M$ of some sort $G\in\mathcal{S}$ . Let

\begin{array}[]{cc}\Psi(n)=\{\varphi_{n}(L_{j})\mid j\in\mathtt{J}\},&\Pi(n)=\{\prod_{i<n}\varphi_{n}(M_{i}),\prod_{i\geq n}\varphi_{n}(M_{i})\},\end{array}

for any $n\in\mathbb{N}$ , and for any $j$ consider the map $\rho_{n,j}=q_{j,\Psi(n)}\kappa\langle\varphi_{n}\rangle u_{\geq,\Pi(n)}$ given by

For some $r\in\mathbb{N}$ and some $\mathtt{J}^{\prime}\subseteq\mathtt{J}$ finite, $\mathrm{im}(\rho_{r,j})\subseteq\varphi_{n}(L_{j})$ for all $n\geq r$ and $j\notin\mathtt{J}^{\prime}$ .

Proof.

Considering Notation 3.1, for each $n\in\mathbb{N}$ , we have that $u_{\geq,\Pi(n)}$ is (the inclusion) given by sending $(x_{n+i}\mid i\in\mathbb{N})=(x_{n},x_{n+1},\dots)$ (where $x_{i}\in\varphi_{n}(M_{i})$ ) to the sequence $(0,\dots,0,x_{n},x_{n+1},\dots)$ , the initial $n$ terms of which are $0$ . Similarly, we have that $q_{j,\Psi(n)}$ is the restriction of $q_{j,\mathcal{T}(G,\mathrm{L})}$ for each $j$ and each $n$ .

We may assume $\mathtt{J}$ is non-empty since otherwise the statement is automatic. Assume for a contradiction that for any $r\in\mathbb{N}$ and any finite subset $\mathtt{J}^{\prime}$ of $\mathtt{J}$ , there exists $n\in\mathbb{N}$ with $n\geq r$ and there exists $j\in\mathtt{J}\setminus\mathtt{J}^{\prime}$ such that $\mathrm{im}(\rho_{r,j})\not\subseteq\varphi_{n}(L_{j})$ . Following [11, Lemma 4], we firstly claim that there exists: a strictly increasing sequence of integers $r(0)<r(1)<r(2)<r(3)<\dots$ ; a sequence $j(0),j(1),j(2),\dots$ of pairwise distinct elements of $\mathtt{J}$ (so, where $j(n)\notin\{j(0),\dots,j(n-1)\}$ for all $n>0$ ); and a sequence of $\mathbb{N}$ -tuples $\underline{m}_{0},\underline{m}_{1},\underline{m}_{2},\dots$ where

\begin{array}[]{ccc}\underline{m}_{d}\in\prod_{s\geq r(d)}\varphi_{r(d)}(M_{s}),&\rho_{r(d),j(d)}(\underline{m}_{d})\notin\varphi_{r(d+1)}(L_{j(d)}),&\rho_{r(d),j(d)}(\underline{m}_{t})=0,\end{array}

for all $d\in\mathbb{N}$ and all $t\in\mathbb{N}$ with $t<d$ . We proceed inductively. Choose an element $j(-1)\in\mathtt{J}$ . Let $\mathtt{J}^{\prime}_{0}=\{j(-1)\}$ and $r(0)=1$ . By our assumption that the conclusion is false, there exists an integer $r(1)>0$ and an element $j(0)\in\mathtt{J}\setminus\{j(-1)\}$ where $\mathrm{im}(\rho_{r(0),j(0)})\not\subseteq\varphi_{r(1)}(L_{j(0)})$ . Hence there exists $\underline{n}_{0}\in\prod_{i\geq r(0)}\varphi_{r(0)}(M_{i})$ where $\rho_{r(0),j(0)}(\underline{n}_{0})\notin\varphi_{r(1)}(L_{j(0)})$ , and so $r(1)>r(0)$ .

We now iterate this process, yielding sequences $(r(d)\mid d\in\mathbb{N})$ , $(j(d)\mid d\in\mathbb{N})$ and $(\underline{n}_{d}\mid d\in\mathbb{N})$ where $r(d)$ is a positive integer, $j(d)\in\mathtt{J}$ and $\underline{n}_{d}\in\prod_{i\geq r(d)}\varphi_{r(d)}(M_{i})$ and such that $r(d+1)>r(d)$ , $j(d+1)\in J\setminus\{j(-1),\dots,j(d)\}$ , and $\rho_{r(d),j(d)}(\underline{n}_{d})\notin\varphi_{r(d+1)}(L_{j(d)})$ . Now fix $t\in\mathbb{N}$ . Since we are considering coproducts of abelian groups, note that we have $\rho_{r(t),j}(\underline{n}_{t})=0$ for all but finitely many $j\in\mathtt{J}$ . For each $j$ define the map $l_{j}^{t}\colon G\to L_{j}$ in $\varphi_{r(t)}(L_{j})$ by $l_{j}^{t}=\rho_{r(t),j}(\underline{n}_{t})$ if $j=j(d)$ for some $d>t$ , and $l_{j}^{t}=0$ otherwise. Now let

\begin{array}[]{c}\tilde{\underline{m}}_{d}=u_{\geq,\Pi(r(d))}(\underline{n}_{d})-\kappa^{-1}\langle\varphi_{r(d)}\rangle(l^{d}_{j}\mid j\in\mathtt{J})\in\prod_{i\in\mathbb{N}}\varphi_{r(d)}(M_{i}).\end{array}

By construction we have $p_{i,\mathrm{M}}(\tilde{\underline{m}}_{d})=0$ for all $i<r(d)$ , and if $t<d$ then

\begin{array}[]{c}q_{j(d),\mathcal{T}(G,\mathrm{L})}(\kappa\langle\varphi_{r(d)}\rangle(\tilde{\underline{m}}_{t}))=q_{j(d),\mathcal{T}(G,\mathrm{L})}(\kappa\langle\varphi_{r(d)}\rangle(u_{\geq,\Pi(r(d))}(\underline{n}_{t})))-l^{t}_{j(d)}=0.\end{array}

Now, writing $\tilde{\underline{m}}_{d}=(m_{d,0},m_{d,1},\dots)$ where $m_{d,i}\in\varphi_{r(d)}(M_{i})$ for all $i\in\mathbb{N}$ , the above gives $m_{d,i}=0$ for all $i<r(d)$ , so we may define

\begin{array}[]{c}\underline{m}_{d}=(m_{d,r(d)+i}\mid i\in\mathbb{N})=(m_{d,r(d)},m_{d,r(d)+1},m_{d,r(d)+2}\dots)\in\prod_{s\geq r_{d}}\varphi_{r(d)}(M_{s}).\end{array}

So we have $\tilde{\underline{m}}_{d}=u_{\geq,\Pi(r(d))}(\underline{m}_{d})$ , and therefore

\begin{array}[]{c}\rho_{r(d),j(d)}(\underline{m}_{d})=q_{j(d),\mathcal{T}(G,\mathrm{L})}(\kappa\langle\varphi_{r(d)}\rangle(\tilde{\underline{m}}_{d}))=\rho_{r(d),j(d)}(\underline{n}_{d})-l^{d}_{j(d)}\notin\varphi_{r(d+1)}(L_{j(d)}),\end{array}

since by definition $l^{d}_{j(d)}=0$ . Our calculations above likewise show that if $t<d$ then we have $\rho_{r(d),j(d)}(\underline{m}_{t})=0$ . This verifies our initial claim. Now let $d$ vary. Since $r(d)<r(d+1)$ for all $d$ , for each $i\in\mathbb{N}$ observe that there are finitely many $d$ with $r(d)\leq i$ . Hence the sum

\begin{array}[]{c}\tilde{\underline{m}}=\sum_{i\in\mathbb{N}}\tilde{\underline{m}}_{i}=(\sum_{d\in\mathbb{N},r(d)\leq i}m_{d,i}\mid i\in\mathbb{N})\in\prod_{i\in\mathbb{N}}\mathcal{T}(G,M_{i})\end{array}

is well-defined. Now, for each $l\in\mathbb{N}$ , by combining everything so far with Corollary 4.7, we have

\begin{array}[]{c}q_{j(l),\mathcal{T}(G,\mathrm{L})}(\gamma^{-1}_{G,\mathrm{L}}(\lambda^{-1}_{G,\mathrm{M}}(\tilde{\underline{m}})))=\sum_{i\in\mathbb{N}}q_{j(l),\mathcal{T}(G,\mathrm{L})}(\gamma^{-1}_{G,\mathrm{L}}(\lambda^{-1}_{G,\mathrm{M}}(\tilde{\underline{m}}_{i})))\\ =\sum_{i\in\mathbb{N}}q_{j(l),\mathcal{T}(G,\mathrm{L})}(\kappa\langle\varphi_{r(i)}\rangle(\tilde{\underline{m}}_{i}))=\sum_{i\geq l}q_{j(l),\mathcal{T}(G,\mathrm{L})}(\kappa\langle\varphi_{r(i)}\rangle(\tilde{\underline{m}}_{i}))\\ =\sum_{i\geq l}\rho_{r(i),j(l)}(\underline{m}_{i})=\rho_{r(l),j(l)}(\underline{m}_{l})+\sum_{i>l}\rho_{r(i),j(l)}(\underline{m}_{i}).\end{array}

Now recall that $\rho_{r(l),j(l)}(\underline{m}_{l})\notin\varphi_{r(l+1)}(L_{j(l)})$ . Since $\varphi_{1}(M)\supseteq\varphi_{2}(M)\supseteq\dots$ is descending and $r(i+1)>r(i)$ for all $i$ , we have $\rho_{r(i),j(l)}(\underline{m}_{i})\in\varphi_{r(i)}(L_{j(l)})\subseteq\varphi_{r(l+1)}(L_{j(l)})$ whenever $i>l$ . Together with the above, this shows

q_{j(l),\mathcal{T}(G,\mathrm{L})}(\gamma^{-1}_{G,\mathrm{L}}(\lambda^{-1}_{G,\mathrm{M}}(\tilde{\underline{m}})))\neq 0

for all $l$ . Since $\gamma^{-1}_{G,\mathrm{L}}(\lambda^{-1}_{G,\mathrm{M}}(\tilde{\underline{m}}))$ lies in the coproduct $\coprod_{j}\mathcal{T}(G,L_{j})$ , and therefore must have had finite support over $j\in\mathtt{J}$ , we have a contradiction, since the set $\{j(l)\mid l\in\mathbb{N}\}$ is in bijection with $\mathbb{N}$ . ∎

5. Annihilator subobjects and pp-definable subgroups.

Recall that, in a category with all small coproducts, a set $\{\mathscr{G}_{\alpha}\mid\alpha\in\mathtt{X}\}$ of objects is called a set of generators provided, for each object $\mathscr{Q}$ , there is an epimorphism $\coprod_{\alpha}\mathscr{G}_{\alpha}\to\mathscr{Q}$ . In case $\mathtt{X}$ is a singleton we say the category has a generator. Recall an additive category $\mathcal{A}$ is Grothendieck provided: $\mathcal{A}$ is abelian; $\mathcal{A}$ has all small coproducts; $\mathcal{A}$ has a generator; and the direct limit of any short exact sequence in $\mathcal{A}$ is again exact.

Remark 5.1.

Let $\mathcal{A}$ be a Grothendieck category. Recall that an object $\mathscr{Q}$ of $\mathcal{A}$ is finitely presented provided the functor $\mathcal{A}(\mathscr{Q},-)\colon\mathcal{A}\to\mathbf{Ab}$ commutes with direct limits. Recall that an object $\mathscr{S}$ of $\mathcal{A}$ is finitely generated provided there is an exact sequence $\mathscr{R}\to\mathscr{Q}\to 0$ in $\mathcal{A}$ . The categories considered both in work of Garcia and Dung [7] and in work of Harada [9] were Grothendieck categories with a set of finitely generated generators.

Following Krause [15], a category $\mathcal{A}$ is said to be locally coherent provided: $\mathcal{A}$ is a Grothendieck category; $\mathcal{A}$ has a set $\{\mathscr{G}_{\alpha}\mid\alpha\in\mathtt{X}\}$ of generators such that each $\mathscr{G}_{\alpha}$ is finitely presented; and the full subcategory of $\mathcal{A}$ consisting of finitely presented objects is abelian. As noted at the top of [8, p.3], $\mathbf{Mod}\text{-}\mathcal{T}^{c}$ is locally coherent.

Thus, in Definition 5.2 and Lemmas 5.3 and 5.4, we specify various definitions and results from [7] and [9] to $\mathbf{Mod}\text{-}\mathcal{T}^{c}$ (which can be done by Remark 5.1). We now recall a notion introduced by Harada.

Definition 5.2.

[9, §1] Let $\mathcal{A}$ be a Grothendieck category with a set $\{\mathscr{G}_{\alpha}\mid\alpha\in\mathtt{X}\}$ of finitely generated generators. Let $\mathscr{Q}$ and $\mathscr{R}$ be objects in $\mathcal{A}$ . A subobject $\mathscr{P}$ of $\mathscr{Q}$ is said to be an $\mathscr{R}$ -annihilator subobject of $\mathscr{Q}$ provided $\mathscr{P}=\bigcap_{\mathfrak{f}\in\mathtt{K}}\mathrm{ker}(\mathfrak{f})$ where the intersection is taken over morphisms $\mathfrak{f}\colon\mathscr{Q}\to\mathscr{R}$ running through some $\mathtt{K}\subseteq\mathcal{A}(\mathscr{Q},\mathscr{R})$ .

Lemma 5.3 focuses on a particular context of Definition 5.2. That is, we specifiy to the locally coherent category $\mathbf{Mod}\text{-}\mathcal{T}^{c}$ and consider the image of objects under $\mathbf{Y}$ . Note Lemma 5.3 was written only to simplify the proof of Lemma 5.4.

Lemma 5.3.

Let $\mathfrak{q}\colon\mathbf{Y}(X)\to\mathscr{Q}$ be an epimorphism in $\mathbf{Mod}\text{-}\mathcal{T}^{c}$ where the object $X$ of $\mathcal{T}$ is compact. If $M$ and $Z$ are objects of $\mathcal{T}$ and $\mathcal{T}^{c}$ respectively, then for any $\mathbf{Y}(M)$ -annihilator subobject $\mathscr{P}=\bigcap_{\mathfrak{f}\in\mathtt{K}}\mathrm{ker}(\mathfrak{f})$ of $\mathscr{Q}$ (where $\mathtt{K}\subseteq\mathcal{A}(\mathscr{Q},\mathscr{R})$ ) we have

\mathscr{P}(Z)=\{\mathfrak{q}_{Z}(g)\mid g\in\mathcal{T}(Z,X)\text{ and }\mathfrak{f}_{X}(\mathfrak{q}_{X}(1_{X}))g=0\text{ for all }\mathfrak{f}\in\mathtt{K}\}.

Proof.

It suffices to assume $\mathtt{K}\neq\emptyset$ . Let $p\in\mathscr{Q}(Z)$ . Since $\mathfrak{q}_{Z}$ is onto, $p=\mathfrak{q}_{Z}(g)$ for some $g\in\mathcal{T}(Z,X)$ . Since $\mathfrak{q}$ and each $\mathfrak{f}$ are morphisms in $\mathbf{Mod}\text{-}\mathcal{T}^{c}$ the diagram of abelian groups given by

commutes. By the commutativity of the diagram we have $\mathfrak{q}_{Z}(g)=\mathscr{Q}(g)(\mathfrak{q}_{X}(1_{X}))$ and (hence) $\mathfrak{f}_{Z}(p)=\mathfrak{f}_{X}(\mathfrak{q}_{X}(1_{X}))g$ . Now suppose $p\in\bigcap_{\mathfrak{f}\in\mathtt{K}}\mathrm{ker}(\mathfrak{f}_{Z})$ . By the above this means $\mathfrak{f}_{X}(\mathfrak{q}_{X}(1_{X}))g=0$ for all $\mathfrak{f}$ . Hence $\mathscr{P}(Z)$ lies in the ride hand side of the required equality. The reverse inclusion is straightforward. ∎

Lemma 5.4 is based on a proof of a given by Huisgen-Zimmerman [12, Corollary 7] of a well-known characterisation of $\Sigma$ -injective modules due to Faith [6, Proposition 3]. We use Lemma 5.4 to simplify the proof of Lemma 6.6, a key result employed in the sequel.

Lemma 5.4.

Let $M$ and $X$ be objects in $\mathcal{T}$ and $\mathcal{T}^{c}$ respectively, and let $\mathcal{T}(-,X)\to\mathscr{Q}\to 0$ be a sequence in $\mathbf{Mod}\text{-}\mathcal{T}^{c}$ which is exact. Any strictly ascending chain of ( $\mathbf{Y}(M)$ -annihilator subobjects of $\mathscr{Q}$ ) gives a strictly descending chain of (pp-definable subgroups of $M$ of sort $X$ ).

Proof.

Suppose $\mathscr{P}_{1}\subsetneq\mathscr{P}_{2}\subsetneq\dots$ is a strictly ascending chain of $\mathbf{Y}(M)$ -annihilator subobjects of $\mathscr{Q}$ , say where, for each integer $n>0$ , we have $\mathscr{P}_{n}=\bigcap_{\mathfrak{f}\in\mathtt{K}[n]}\mathrm{ker}(\mathfrak{f})$ for some subset $\mathtt{K}[n]$ of morphisms in $\mathbf{Mod}\text{-}\mathcal{T}^{c}$ of the form $\mathfrak{f}\colon\mathscr{Q}\to\mathbf{Y}(M)$ .

We assume $\mathtt{K}[1]\supsetneq\mathtt{K}[2]\supsetneq\dots$ without loss of generality. For each $n$ there is an object $Z_{n}$ of $\mathcal{T}^{c}$ for which $\mathscr{P}_{n}(Z_{n})\subsetneq\mathscr{P}_{n+1}(Z_{n})$ , and we choose $h_{n}\in\mathscr{P}_{n+1}(Z_{n})\setminus\mathscr{P}_{n}(Z_{n})$ . Let $\mathfrak{q}\colon\mathcal{T}(X,-)\to\mathscr{Q}$ be the epimorphism in $\mathbf{Mod}\text{-}\mathcal{T}^{c}$ giving the exact sequence $\mathcal{T}(-,X)\to\mathscr{Q}\to 0$ . Since $\mathfrak{q}_{Z_{n}}$ is onto and $h_{n}\in\mathscr{Q}(Z_{n})$ we have $h_{n}=\mathfrak{q}_{Z_{n}}(g_{n})$ for some morphism $g_{n}\colon Z_{n}\to X$ . By Lemma 5.3, since $h_{n}\in\mathscr{P}_{n+1}(Z_{n})$ we have that $\mathfrak{f}_{X}(\mathfrak{q}_{X}(1_{X}))g_{n}=0$ for all $\mathfrak{f}\in\mathtt{K}[n+1]$ . Similarly, since $h_{n}\notin\mathscr{P}_{n}(Z_{n})$ there exists $\mathfrak{s}(n)\in\mathtt{K}[n]$ such that $\mathfrak{s}(n)_{X}(\mathfrak{q}_{X}(1_{X}))g_{n}\neq 0$ .

We now follow the proof of [8, Proposition 3.1]. Since $\mathcal{T}^{c}$ is triangulated any morphism $g:Z\to X$ in $\mathcal{T}^{c}$ yields a triangle in $\mathcal{T}^{c}$ and an exact sequence in $\mathbf{Ab}$

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given by applying the contravaraint functor $\mathcal{T}(-,W)\colon\mathcal{T}\to\mathbf{Ab}$ . In other words, $b$ is a pseudocokernel of $g$ , and so any morphism $t\colon X\to W$ in $\mathcal{T}^{c}$ with $tg=0$ must satisfy $t=sb$ for some $s\colon Y\to Z$ .

Let $t_{n}=\mathfrak{s}(n)_{X}(\mathfrak{q}_{X}(1_{X}))$ for each $n$ . Combining what we have so far, for each $n$ we have $\mathfrak{s}(n+1)\in\mathtt{K}[n+1]$ , so $t_{n+1}g_{n}=0$ , and so $t_{n+1}=s_{n}b_{n}$ for some morphism $s_{n}\colon Y_{n}\to M$ , and so $t_{n+1}\in Mb_{n}$ . On the other hand, if $t_{n+1}\in Mb_{n+1}$ then $t_{n+1}g_{n+1}=0$ which contradicts that $\mathfrak{s}(n+1)_{X}(\mathfrak{q}_{X}(1_{X}))g_{n+1}\neq 0$ , and so $t_{n+1}\notin Mb_{n+1}$ . This gives a strict descending chain

\begin{array}[]{c}Mb_{1}\supsetneq Mb_{1}\cap Mb_{2}\supsetneq Mb_{1}\cap Mb_{2}\cap Mb_{3}\supsetneq\dots\supsetneq\bigcap_{i=1}^{d}Mb_{i}\supsetneq\dots\end{array}

A direct application of [8, Proposition 3.1] shows that each finite intersection $\bigcap_{i=1}^{d}Mb_{i}$ has the form $Ma_{d}$ for some morphism in $\mathcal{T}^{c}$ of the form $a_{d}\colon X\to W_{d}$ . So the chain above is, as required, a strictly descending chain of pp-definable subgroups of $M$ of sort $X$ . ∎

Definition 5.5.

[9, §1] Let $\mathcal{A}$ be a Grothendieck category with a set of finitely generated generators. Fix an object $\mathscr{M}$ of $\mathcal{A}$ . We say that $\mathscr{M}$ is $\Sigma$ -injective if, for any set $\mathtt{I}$ , the coproduct $\mathscr{M}^{(\mathtt{I})}=\coprod_{i\in\mathtt{I}}\mathscr{M}$ is injective. We say that $\mathscr{M}$ is fp-injective if, whenever $0\to\mathscr{P}\to\mathscr{R}\to\mathscr{Q}\to 0$ is an exact sequence in $\mathcal{A}$ where $\mathscr{Q}$ is finitely presented, any morphism $\mathscr{P}\to\mathscr{M}$ extends to a morphism $\mathscr{R}\to\mathscr{M}$ ; see [7, §1].

For the proof of Corollary 5.8 we recall two results: Proposition 5.6, due to Garcia and Dung, characterises $\Sigma$ -injectivity in the fp-injective setting; and Lemma 5.7, due to Krause, shows that it is sufficient to consider the fp-injective setting.

Proposition 5.6.

[7, Proposition 1.3] Let $\mathscr{M}$ be an fp-injecitve in a Grothendieck category $\mathcal{A}$ which has a set $\mathtt{G}$ of finitely presented generators $\mathscr{G}$ . Then $\mathscr{M}$ is $\Sigma$ -injective if and only if, for each $\mathscr{G}\in\mathtt{G}$ , every ascending chain of $\mathscr{M}$ -annihilator subobjects of $\mathscr{G}$ must stabilise.

Lemma 5.7.

[17, Lemma 1.6] For any $M$ in $\mathcal{T}$ the image $\mathbf{Y}(M)$ is fp-injective.

Corollary 5.8.

Let $M$ be an object in $\mathcal{T}$ such that for any compact object $X$ of $\mathcal{T}$ we have that each descending chain $Ma_{1}\supseteq Ma_{2}\supseteq\dots$ of pp-definable subgroups of $M$ of sort $X$ must stabilise. Then the image $\mathbf{Y}(M)$ of $M$ in $\mathbf{Mod}\text{-}\mathcal{T}^{c}$ is $\Sigma$ -injective.

Proof.

We prove the contrapositive, so we assume $\mathbf{Y}(M)$ is not $\Sigma$ -injective. Recall, from Remark 5.1, that $\mathbf{Mod}\text{-}\mathcal{T}^{c}$ is locally coherent, and so it is a Grothendieck category with a set $\mathtt{G}$ of finitely presented generators.

Note $\mathscr{M}=\mathbf{Y}(M)$ is fp-injective by Lemma 5.7, and combining our initial assumption with Proposition 5.6 shows that, for some $\mathscr{G}\in\mathtt{G}$ , there exists a strictly ascending chain of $\mathscr{M}$ -annihilator subobjects of $\mathscr{G}$ . Since $\mathscr{G}$ is finitely presented, there is an exact sequence of the form $\mathcal{T}(-,Y)\to\mathcal{T}(-,X)\to\mathscr{G}\to 0$ in $\mathbf{Mod}\text{-}\mathcal{T}^{c}$ where $X$ and $Y$ lie in $\mathcal{T}^{c}$ .

By Lemma 5.4 the aforementioned ascending chain strict ascending chain gives rise to a strictly descending chain of pp-definable subgroups of $M$ of sort $X$ . ∎

6. $\Sigma$ -pure-injective objects and canonical morphisms.

Definition 6.1.

Recall Notation 3.1. Let $\mathtt{I}$ be a set and let $M$ be an object of $\mathcal{T}$ . By the universal properties of the product and coproduct of the collection $\mathrm{M}=\{M\mid i\in\mathtt{I}\}$ , there exists a unique summation morphism $\sigma_{\mathtt{I},\mathrm{M}}:\coprod_{i}M\to M$ and a unique canonical morphism $\iota_{\mathtt{I},\mathrm{M}}:\coprod_{i}M\to\prod_{i}M$ for which $\sigma_{\mathtt{I},\mathrm{M}}u_{i,\mathrm{M}}=1_{M}$ and $\iota_{\mathtt{I},\mathrm{M}}u_{i,\mathrm{M}}=v_{i,\mathrm{M}}$ for each $i$ .

Proposition 6.2.

Let $M$ be an object of $\mathcal{T}$ and let $\mathtt{I}$ be a set. Then the canonical morphism $\iota_{\mathtt{I},\mathrm{M}}$ is a pure monomorphism.

Proof.

Let $X$ be a object in $\mathcal{T}$ which is compact. In general: the morphism $\lambda_{X,\mathrm{M}}$ is an isomorphism; the canonical morphism $\iota_{\mathtt{I},\mathcal{T}(X,\mathrm{M})}$ is injective; and $\iota_{\mathtt{I},\mathcal{T}(X,\mathrm{M})}$ is the composition $\lambda_{X,\mathrm{M}}\mathcal{T}(X,\iota_{\mathtt{I},\mathrm{M}})\gamma_{X,\mathrm{M}}$ .

Since $X$ is compact the morphism $\gamma_{X,\mathrm{M}}$ is an isomorphism. This shows $\mathcal{T}(X,\iota_{\mathtt{I},\mathrm{M}})$ is injective if $X$ is compact, and so $\iota_{\mathtt{I},\mathrm{M}}$ is a pure monomorphism. ∎

Definition 6.3.

[17, Definition 1.1] An object $M$ of $\mathcal{T}$ is called pure-injective if each pure monomorphism $M\to N$ is a section, and $M$ is called $\Sigma$ -pure-injective if, for any set $\mathtt{I}$ , the coproduct $\coprod_{i\in\mathtt{I}}M=M^{(\mathtt{I})}$ is pure-injective.

At this point it is worth recalling some characterisations of purity due to Krause. Theorem 6.4 is analogous to [14, Theorem 7.1 (ii,v,vi)].

Theorem 6.4.

[18, Theorem 1.8, (1,3,5)] For an object $M$ of $\mathcal{T}$ the following statements are equivalent.

(1)

The object $M$ of $\mathcal{T}$ is pure-injective.
(2)

The object $\mathbf{Y}(M)$ of $\mathbf{Mod}\text{-}\mathcal{T}^{c}$ is injective.
(3)

For any set $\mathtt{I}$ the morphism $\sigma_{\mathtt{I},\mathrm{M}}$ factors through the morphism $\iota_{\mathtt{I},\mathrm{M}}$ .

Proposition 6.5 is analogous to parts (i) and (ii) in [14, Theorem 8.1].

Proposition 6.5.

An object $M$ of $\mathcal{T}$ is $\Sigma$ -pure-injective if and only if, for each set $\mathtt{I}$ , the canonical morphism $\iota_{\mathtt{I},\mathrm{M}}$ is a section.

Proof.

Assume that $M$ is $\Sigma$ -pure-injective and that $\mathtt{I}$ is a set. By assumption the domain of $\iota_{\mathtt{I},\mathrm{M}}$ is pure-injective. Since $\iota_{\mathtt{I},\mathrm{M}}$ is a pure monomorphism, this means it is a section. Supposing conversley that $\iota_{\mathtt{I},\mathrm{M}}$ is a section for each set $\mathtt{I}$ , it remains to show that $M$ is $\Sigma$ -pure-injective. Choose a set $\mathtt{T}$ and let $N=\coprod_{t\in\mathtt{T}}M$ . It suffices to prove $N$ is pure-injective. Let $\mathtt{S}$ be any set and consider the collection $\mathrm{N}=\{N\mid s\in\mathtt{S}\}$ . By Theorem 6.4 it suffices to find a map $\theta_{\mathtt{S},\mathrm{N}}\colon\prod_{s}N\to N$ such that $\sigma_{\mathtt{S},\mathrm{N}}=\theta_{\mathtt{S},\mathrm{N}}\iota_{\mathtt{S},\mathrm{N}}$ . Let $\mathrm{M}=\{M\mid t\in\mathtt{T}\}$ .

For each $(s,t)\in\mathtt{S}\times\mathtt{T}$ the morphisms $u_{s,\mathrm{N}}u_{t,\mathrm{M}}$ satisfy the universal property of the coproduct $\coprod_{s,t}M$ , and so we assume $u_{s,t,\mathrm{M}}=u_{s,\mathrm{N}}u_{t,\mathrm{M}}$ without loss of generality. Consider the morphisms $\varphi_{s,t,\mathrm{M}}=q_{t,\mathrm{M}}p_{s,\mathrm{N}}$ for each $(s,t)\in\mathtt{S}\times\mathtt{T}$ . Since $u_{s,t,\mathrm{M}}=u_{s,\mathrm{N}}u_{t,\mathrm{M}}$ we have $q_{s,t,\mathrm{M}}=q_{t,\mathrm{M}}q_{s,\mathrm{N}}$ by uniqueness. Consequently $\varphi_{s,t,\mathrm{M}}v_{s,\mathrm{N}}q_{s,\mathrm{N}}u_{s,t,\mathrm{M}}$ is the identity on $M$ . By the universal property of the product, there is a morphism $\Xi\colon\prod_{s}N\to\prod_{s,t}M$ such that $p_{s,t,\mathrm{M}}\Xi=\varphi_{s,t,\mathrm{M}}$ for each $(s,t)$ . It suffices to let $\theta_{\mathtt{S},\mathrm{N}}=\sigma_{\mathtt{S},\mathrm{N}}\pi_{\mathtt{S}\times\mathtt{T},\mathrm{M}}\Xi$ . By the uniqueness of the involved morphisms, it is straightforward to see that $\sigma_{\mathtt{S},\mathrm{N}}=\theta_{\mathtt{S},\mathrm{N}}\iota_{\mathtt{S},\mathrm{N}}$ .

∎

Lemma 6.6 is analogous to [14, Theorem 8.1(ii,iii)].

Lemma 6.6.

If $G\in\mathcal{S}$ and $M$ is a $\Sigma$ -pure-injective object in $\mathcal{T}$ then every descending chain of pp-definable subgroups of $M$ of sort $G$ stabilises.

Proof.

For a contradiction we assume the existence of a strictly descending chain $Ma_{0}\supsetneq Ma_{1}\supsetneq Ma_{2}\supsetneq\dots$ of (pp-definable subgroups of $M$ of sort $G$ ) for some $G\in\mathcal{S}$ . Hence there is a collection of compact objects $H_{n}\in\mathcal{S}$ such that $a_{n}\in\mathcal{T}(G,H_{n})$ for each $n\in\mathbb{N}$ . By our assumption we may choose elements $b_{n}\in\mathcal{T}(H_{n},M)$ such that $b_{n}a_{n}\notin Ma_{n+1}$ . By Proposition 6.5 the canonical morphism $\iota_{\mathbb{N},\mathrm{M}}:\coprod_{\mathbb{N}}M\to\prod_{\mathbb{N}}M$ is a section, and so there is some morphism $\pi_{\mathbb{N},\mathrm{M}}\colon\prod_{\mathbb{N}}M\to\coprod_{\mathbb{N}}M$ such that $\pi_{\mathbb{N},\mathrm{M}}\iota_{\mathbb{N},\mathrm{M}}$ is the identity on $\coprod_{\mathbb{N}}M$ . After applying the functor $\mathcal{T}(G,-)$ (from $\mathcal{T}$ to the category of abelian groups) this means $\mathcal{T}(G,\pi_{\mathbb{N},\mathrm{M}})\mathcal{T}(G,\iota_{\mathbb{N},\mathrm{M}})$ is the identity on $\mathcal{T}(G,\coprod_{\mathbb{N}}M)$ . Let $\underline{ba}=(b_{n}a_{n}\mid n\in\mathbb{N})\in\prod_{n}\mathcal{T}(G,M)$ . Fix $n\in\mathbb{N}$ and let $\varphi_{n}(v_{G})$ be the formula $(\exists u_{H_{n}}\colon v_{G}=u_{H_{n}}a_{n})$ . Let $\mathrm{M}=\{M\mid n\in\mathbb{N}\}$ . Recall $\lambda_{G,\mathrm{M}}$ is always an isomorphism, and since $G$ is compact, $\gamma_{G,\mathrm{M}}$ is an isomorphism. Define $\mu$ by

\begin{array}[]{c}\mu=(\gamma_{G,\mathrm{M}})^{-1}\mathcal{T}(G,\pi_{\mathbb{N},\mathrm{M}})(\lambda_{G,\mathrm{M}})^{-1}\colon\prod_{n\in\mathbb{N}}\mathcal{T}(G,M)\to\coprod_{n\in\mathbb{N}}\mathcal{T}(G,M).\end{array}

Let $\mu(\underline{ba})=(c_{n}\mid n\in\mathbb{N})$ . The contradiction we will find is that $c_{l}\neq 0$ for all $l\in\mathbb{N}$ , which contradicts that $\mu$ has codomain $\coprod_{n\in\mathbb{N}}\mathcal{T}(G,M)$ . Fix $l\in\mathbb{N}$ . Now let

\underline{ba}_{\leq l}=(b_{0}a_{0},\dots,b_{l}a_{l},0,0,\dots)\text{ and }\underline{ba}_{>l}=(0,\dots,0,b_{l+1}a_{l+1},b_{l+2}a_{l+2},\dots),

where the first $l$ entries of $\underline{ba}_{>l}$ are $0$ .

Note that $\underline{ba}_{\leq l}\in\coprod_{n}\mathcal{T}(G,M)$ . Furthermore, since the chain $Ma_{0}\supseteq Ma_{1}\supseteq\cdots$ is descending, we have $b_{n}a_{n}\in\varphi_{l}(M)$ for all $n>l$ and so $\underline{ba}_{>l}\in\prod_{n}\varphi_{l+1}(M)$ . By Lemma 4.2(ii), the restrictions of $(\lambda_{G,\mathrm{M}})^{-1}$ and $(\gamma_{G,\mathrm{M}})^{-1}$ respectively define isomorphisms $\prod_{n}\varphi_{l+1}(M)\to\varphi_{l+1}(\prod_{n}M)$ and $\varphi_{l+1}(\coprod_{n}M)\to\coprod_{n}\varphi_{l+1}(M)$ . Similarly $\mathcal{T}(G,\pi_{\mathbb{N},\mathrm{M}})$ restricts to define a morphism $\varphi_{l+1}(\prod_{n}M)\to\varphi_{l+1}(\coprod_{n}M)$ . Altogether we have that $\mu$ restricts to a morphism $\prod_{n}\varphi_{l+1}(M)\to\coprod_{n}\varphi_{l+1}(M)$ . Let $\mu(\underline{ba}_{>l})=(d_{n}\mid n\in\mathbb{N})$ , and so $d_{n}\in\varphi_{l+1}(M)$ for all $n$ .

Recall it suffices to show $c_{l}\neq 0$ where $\mu(\underline{ba})=(c_{n}\mid n\in\mathbb{N})$ . From the above,

\begin{array}[]{c}(c_{0},\dots,c_{l},c_{l+1},\dots)=\mu(\underline{ba})=\mu(\underline{ba}_{\leq l}+\underline{ba}_{>l})=\underline{ba}_{\leq l}+\mu(\underline{ba}_{>l})\\ =(b_{0}a_{0}+d_{0},\dots,b_{l}a_{l}+d_{l},b_{l+1}a_{l+1}+d_{l+1},\dots),\end{array}

and so $c_{l}\neq 0$ as otherwise $\varphi_{l+1}(M)\ni-d_{l}=b_{l}a_{l}\notin\varphi_{l+1}(M)$ . ∎

We now recall a result of Krause which is used heavily in the sequel.

Corollary 6.7.

[17, Corollary 1.10] The functor $\mathbf{Y}$ gives an equivalence between the full subcategory of $\mathcal{T}$ consisting of pure-injective objects and the full subcategory of $\mathbf{Mod}\text{-}\mathcal{T}^{c}$ consitsting of injective objects.

Note that, since $\mathbf{Y}$ is additive, by Corollary 6.7 a pure-injective object $M$ is indecomposable if and only if $\mathbf{Y}(M)$ is an indecomposable (injective) object.

Remark 6.8.

Fix a collection $\mathrm{M}=\{M_{i}\mid i\in\mathtt{I}\}$ of objects in $\mathcal{T}$ . Since small coproducts exist in $\mathcal{T}$ (by Assumption 3.3) and $\mathbf{Ab}$ , the morphisms $\gamma_{X,\mathrm{M}}$ combine to define a natural transformation $\coprod_{i}\mathcal{T}(-,M_{i})\to\mathcal{T}(-,\coprod_{i}M_{i})$ . By Definition 3.2 this transformation is in fact an isomorphism $\coprod_{i}\mathbf{Y}(M_{i})\simeq\mathbf{Y}(\coprod_{i}M_{i})$ in $\mathbf{Mod}\text{-}\mathcal{T}^{c}$ which shows that $\mathbf{Y}$ preserves small coproducts.

Corollary 6.9.

The functor $\mathbf{Y}$ gives an equivalence between the full subcategory of $\mathcal{T}$ consisting of $\Sigma$ -pure-injective objects and the full subcategory of $\mathbf{Mod}\text{-}\mathcal{T}^{c}$ consitsting of $\Sigma$ -injective objects.

Proof.

By definition, an object $M$ of $\mathcal{T}$ is $\Sigma$ -pure-injective if and only if, for every set $\mathtt{I}$ , the coproduct $M^{(\mathtt{I})}$ is pure-injective. By Corollary 6.7 and Remark 6.8, given any such $\mathtt{I}$ , $M^{(\mathtt{I})}$ is pure-injective if and only if $\mathbf{Y}(M^{(\mathtt{I})})\simeq(\mathbf{Y}(M))^{(\mathtt{I})}$ is injective.

This shows that $\mathbf{Y}$ induces a functor from the full subcategory of $\Sigma$ -pure-injectives in $\mathcal{T}$ and the full subcategory of $\mathbf{Mod}\text{-}\mathcal{T}^{c}$ consisting of $\Sigma$ -injectives. That this functor is full, faithful and dense, follows by Corollary 6.7 together with the fact that any $\Sigma$ -pure-injective object of $\mathcal{T}$ is pure-injective. ∎

Corollary 6.10 is analogous to [14, Corollary 8.2(i,ii)].

Corollary 6.10.

Let $M$ be a $\Sigma$ -pure-injective object of $\mathcal{T}$ and let $G\in\mathcal{S}$ .

(1)

For any set $\mathtt{I}$ , the objects $M^{(\mathtt{I})}$ and $M^{\mathtt{I}}$ are $\Sigma$ -pure-injective.
(2)

If $h\colon L\to M$ is a pure monomorphism in $\mathcal{T}$ then $L$ is $\Sigma$ -pure-injective and $h$ is a section.

Proof.

It is worth noting that we now have the equivalence of (1) and (5) of Theorem 1.1. That is, by Lemma 6.6 and Corollaries 5.8 and 6.9, an object $M$ is $\Sigma$ -pure-injective if and only if every descending chain of pp-definable subgroups of $M$ of each sort stabilises.

The proof of part (1) is now straightforward, recalling that, by Lemma 4.5, we have that $\varphi(M)^{(\mathtt{I})}\simeq\varphi(M^{(\mathtt{I})})$ and $\varphi(M^{\mathtt{I}})\simeq\varphi(M)^{\mathtt{I}}$ for any pp-formula $\varphi$ of sort $G$ . For (2), as above it suffices to recall that $\varphi(L)=\{g\in\mathcal{T}(G,L)\mid hg\in\varphi(M)\}$ for any pp-formula $\varphi$ of sort $G$ , by Lemma 4.2. ∎

To prove Lemma 6.12 we use Corollary 6.11, a result of Garcia and Dung.

Corollary 6.11.

[7, Corollary 1.6] Let $\mathcal{A}$ be a Grothendieck category with a set of finitely generated generators. Any $\Sigma$ -injective object of $\mathcal{A}$ is a coproduct of indecomposable objects.

Lemma 6.12 is analogous to the implication of (v) by (i) in [14, Theorem 8.1]. The case $\mathtt{I}=\{0\}$ shows that $\Sigma$ -pure-injectives decompose into indecomposables.

Lemma 6.12.

If $M$ is a $\Sigma$ -pure-injective object of $\mathcal{T}$ then for any set $\mathtt{I}$ the product $M^{\mathtt{I}}$ is a coproduct of indecomposable $\Sigma$ -pure-injectives.

Proof.

Recall $\mathbf{Mod}\text{-}\mathcal{T}^{c}$ is a Grothendieck category with a set of finitely generated generators. Let $K=M^{\mathtt{I}}$ so that, by Corollary 6.10(1), $K$ is $\Sigma$ -pure-injective, and so $\mathbf{Y}(K)$ is $\Sigma$ -injective by Corollary 6.9. By Corollary 6.11 this means $\mathbf{Y}(K)\simeq\coprod_{j\in\mathtt{J}}\mathscr{L}_{j}$ where $\mathscr{L}_{j}$ is an indecomposable object of $\mathbf{Mod}\text{-}\mathcal{T}^{c}$ for each $j$ .

For each $j$ , there is a section $u_{j}\colon\mathscr{L}_{j}\to\mathbf{Y}(K)$ , which means $\mathscr{L}_{j}$ is injective, and so by Corollary 6.7 we have $\mathscr{L}_{j}\simeq\mathbf{Y}(L_{j})$ for some pure-injective object $L_{j}$ . Since $\mathscr{L}_{j}$ is indecomposable, we have that $L_{j}$ is indecomposable. Again, applying Corollary 6.7 gives a section $h_{j}\colon L_{j}\to K$ in $\mathcal{T}$ with $\mathbf{Y}(h_{j})=u_{j}$ . Altogether this means $h_{j}$ is a pure monomorphism into a $\Sigma$ -pure-injective object, and so $L_{j}$ is $\Sigma$ -pure-injective by Corollary 6.10(2). By Remark 6.8 we have that $\mathbf{Y}(\coprod_{j}L_{j})\simeq\mathbf{Y}(K)$ is injective, and so $\coprod_{j}L_{j}\simeq K$ by Corollary 6.7. ∎

7. Completing the proof of the characterisation.

Before proving Theorem 1.1 we note a consequence of the results gathered so far. Recall, from Definitions 2.3 and 3.2, that the cardinality of the $\mathfrak{L}^{\mathcal{T}}$ -structure $\mathsf{M}$ underlying any object $M$ of $\mathcal{T}$ is defined and denoted $|\mathsf{M}|=|\bigsqcup_{G\in\mathcal{S}}\mathcal{T}(G,M)|$ where $\mathcal{S}$ is a fixed chosen set of isoclass representatives, one for each class. Corollary 7.1 is analogous to [14, Corollary 8.2(iii)].

Corollary 7.1.

There exists a cardinal $\kappa$ such that the $\mathfrak{L}^{\mathcal{T}}$ -structure underlying any indecomposable pure-injective object of $\mathcal{T}$ has cardinality at most $\kappa$ .

We delay the proof of Corollary 7.1 until after Corollary 7.3.

Remark 7.2.

We now note a non-trivial complication in our setting of compactly generated triangulated categories, which is absent in the module-theoretic situation. In the spirit of the results presented so far, it is natural to ask if one may adapt the proof of [14, Corollary 8.2(iii)] to prove Corollary 7.1. This seems straightforward at first glance, since there are multi-sorted versions of the downward Löwenheim-Skolem theorem; see for example [5, Theorem 37].

Recall that, in the language $\mathfrak{L}_{A}$ of modules over a fixed ring $A$ from Example 2.2, any $\mathfrak{L}_{A}$ -structure is an object in the category of $A$ -modules, and any elementary embedding of $\mathfrak{L}_{A}$ -structures is a pure embedding of $A$ -modules. The same correspondence between structures need not be true here. Although objects of $\mathbf{Mod}\text{-}\mathcal{T}^{c}$ correspond to structures over $\mathfrak{L}^{\mathcal{T}}$ by Remark 3.6, the former need not be given by objects of $\mathcal{T}$ as $\mathbf{Y}$ need not be essentially surjective. Fortunately, here we may instead use Corollary 7.3, a remark due to Krause, which also shortens the proof.

Corollary 7.3.

(See [17, Corollary 1.10]). There is a set $\mathtt{Sp}$ of isomorphism classes of objects in $\mathcal{T}$ which are pure-injective indecomposable.

Proof of Corollary 7.1.

It suffices to let $\kappa=|\bigsqcup\mathcal{T}(G,M)|$ where $G$ runs through $\mathcal{S}$ and $M$ runs through the set $\mathtt{Sp}$ from Corollary 7.3. ∎

We now proceed toward proving Theorem 1.1. For this we require Corollary 7.8, and to this end, in Theorem 7.4 we recall well-known results about decompositions of coproducts in abelian categories. For consistency we follow [20].

Theorem 7.4.

Let $\mathcal{A}$ be an abelian category, and let $\mathrm{L}=\{\mathscr{L}_{j}\mid j\in\mathtt{J}\}$ and $\mathrm{N}=\{\mathscr{N}_{k}\mid k\in\mathtt{K}\}$ be collections of objects in $\mathcal{A}$ . The following statements hold.

(1)

[20, p.82, Theorem 4.A7] Suppose the collections $\mathrm{L}$ and $\mathrm{N}$ consist of indecomposable objects with local endomorphism rings. If we have $\coprod_{j\in\mathtt{J}}\mathscr{L}_{j}\simeq\coprod_{k\in\mathtt{K}}\mathscr{N}_{k}$ then there is a bijection $\sigma\colon\mathtt{J}\to\mathtt{K}$ with $\mathscr{L}_{j}\simeq\mathscr{N}_{\sigma(j)}$ for all $j$ .
(2)

[20, p.82, Theorem 4.A11] Suppose that $\mathtt{J}=\mathtt{K}$ , and that for all $j$ , $\mathscr{L}_{j}$ is indecomposable and $\mathscr{L}_{j}$ is the injective hull of $\mathscr{N}_{j}$ . Suppose there is a section $\mathscr{P}\to\coprod_{j\in\mathtt{J}}\mathscr{L}_{j}$ . Then there is a subset $\mathtt{H}\subseteq\mathtt{J}$ with $\coprod_{j\in\mathtt{J}}\mathscr{L}_{j}\simeq\mathscr{P}\amalg\coprod_{j\in\mathtt{H}}\mathscr{L}_{j}$ .

To apply Theorem 7.4 we use the following observation of Garkusha and Prest.

Lemma 7.5.

[8, Lemma 2.2] A pure-injective object $M$ of $\mathcal{T}$ is indecomposable if and only if the endomorphism ring $\mathrm{End}_{\mathcal{T}}(M)$ is local.

Corollary 7.6.

Let $\mathrm{L}=\{L_{j}\mid j\in\mathtt{J}\}$ be a collection of indecomposable objects in $\mathcal{T}$ such that the coproduct $\coprod_{j\in\mathtt{J}}L_{j}$ is pure injective. If $P$ is a summand of $\coprod_{j\in\mathtt{J}}L_{j}$ then there exists $\mathtt{H}\subseteq\mathtt{J}$ such that $\coprod_{j\in\mathtt{J}}L_{j}\simeq P\amalg\coprod_{j\in\mathtt{H}}L_{j}$ .

If additionally $P\simeq\coprod_{k\in\mathtt{K}}N_{k}$ where each $N_{k}$ is indecomposable, then there is a bijection $\sigma\colon\mathtt{J}\to\mathtt{K}\sqcup\mathtt{H}$ with $L_{j}\simeq N_{\sigma(j)}$ for all $j\in\mathtt{J}$ .

Proof.

Let $\mathscr{L}_{j}=\mathbf{Y}(L_{j})$ for each $j$ . Let $L_{\mathtt{T}}=\coprod_{j\in\mathtt{T}}L_{j}$ and $\mathscr{L}_{\mathtt{T}}=\coprod_{j\in\mathtt{T}}\mathscr{L}_{j}$ for any subset $\mathtt{T}\subseteq\mathtt{J}$ . Since $\mathbf{Y}$ preserves small coproducts by Remark 6.8, and since each $L_{j}$ is a summand of $L_{\mathtt{J}}$ , each $\mathscr{L}_{j}$ is a summand of $\mathscr{L}_{\mathtt{J}}$ . By Theorem 6.4 and Remark 6.8 $\mathscr{L}_{\mathtt{J}}$ is an injective object. Thus each $\mathscr{L}_{j}$ is injective. Since each $L_{j}$ is indecomposable, each $\mathscr{L}_{j}$ is indecomposable. Similarly, each $N_{k}$ is pure-injective.

Let $\mathscr{P}=\mathbf{Y}(P)$ . Since $P$ is a direct summand of $L_{\mathtt{J}}$ by assumption, $\mathscr{P}$ is a direct summand of $\mathscr{L}_{\mathtt{J}}$ . Hence by Theorem 7.4(2) there exists a subset $\mathtt{H}\subseteq\mathtt{J}$ such that $\mathscr{L}_{\mathtt{J}}\simeq\mathscr{P}\amalg\mathscr{L}_{\mathtt{H}}$ . By Corollary 6.7, since we assume $L_{\mathtt{J}}$ is pure injective, $\mathscr{L}_{\mathtt{J}}$ is injective, and hence so too are $\mathscr{P}$ and $\mathscr{L}_{\mathtt{H}}$ . Again, by Corollary 6.7 this gives $L_{\mathtt{J}}\simeq P\amalg L_{\mathtt{H}}$ .

Now suppose also $P\simeq\coprod_{k\in\mathtt{K}}N_{k}$ where each $N_{k}$ is indecomposable and, as above, neccesarilly pure-injective. By Lemma 7.5 each of the objects $L_{j}$ and $N_{k}$ has a local endomorphism ring. By Corollary 6.7, for any pure-injective object $Z$ of $\mathcal{T}$ there is a ring isomorphism

\mathrm{End}_{\mathcal{T}}(Z)\simeq\mathrm{End}_{\mathbf{Mod}\text{-}\mathcal{T}^{c}}(\mathbf{Y}(Z)).

Since $\mathscr{L}_{\mathtt{J}}\simeq\mathscr{P}\amalg\mathscr{L}_{\mathtt{H}}$ , the second claim follows, as above, by Theorem 7.4(1), using again that $\mathbf{Y}$ preserves coproducts by Remark 6.8. ∎

Lemma 7.7 is analogous to the cited result of Huisgen-Zimmerman.

Lemma 7.7.

[11, Lemma 5] Recall and consider Notation 4.6, where $M=\prod_{i\in\mathbb{N}}M_{i}=\coprod_{j\in\mathtt{J}}L_{j}$ . Let $X$ be compact and let $\varphi_{1}(M)\supseteq\varphi_{2}(M)\supseteq\dots$ be a descending chain of pp-definable subgroups of sort $X$ . Suppose additionally that each object $L_{j}$ is both pure-injective and indecomposable.

Then there exists $r\in\mathbb{N}$ such that, for each collection $\mathrm{N}=\{N_{i}\mid i\in\mathbb{N}\}$ where $N_{i}$ is an indecomposable summand of $M_{i}$ , we have $\varphi_{r}(N_{i})=\varphi_{n}(N_{i})$ for all $n\in\mathbb{N}$ with $n\geq r$ and all but finitely many $i\in\mathbb{N}$ .

Proof.

Recall the morphisms $\rho_{n,j}=q_{j,\Psi(n)}\kappa\langle\varphi_{n}\rangle u_{\geq,\Pi(n)}$ defined by the composition

Here the isomorphism $\kappa\langle\varphi_{n}\rangle$ is defined in Corollary 4.7, and the maps $u_{\geq,\Pi(n)}$ and $q_{j,\Psi(n)}$ are the canonical inclusion and projection respectively. By Lemma 4.8 there exists $r\in\mathbb{N}$ and a finite subset $\mathtt{J}^{\prime}$ of $\mathtt{J}$ such that we have the containment $\mathrm{im}(\rho_{r,j})\subseteq\varphi_{n}(L_{j})$ for all $n\in\mathbb{N}$ with $n\geq r$ and all $j\in\mathtt{J}\setminus\mathtt{J}^{\prime}$ .

Choose arbitrary $m\in\mathbb{N}$ with $m>|\mathtt{J}^{\prime}|$ , and choose arbitrary $i_{1},\dots,i_{m}\in\mathbb{N}$ with $i_{p}\geq r$ for each $p$ . Let $P=N_{i_{1}}\amalg\dots\amalg N_{i_{m}}$ so that $P$ is a summand of $M=\coprod_{j}L_{j}$ . By Corollary 7.6, since $P$ is a coproduct of $m$ indecomposable pure-injective objects, we have $M\simeq P\amalg\coprod_{j\in\mathtt{K}}L_{j}$ and a bijection $\sigma\colon\mathtt{J}\to\mathtt{K}\sqcup\{i_{1},\dots,i_{m}\}$ with $L_{j}\simeq N_{\sigma(j)}$ for all $j\in\mathtt{J}$ .

Since $m>|\mathtt{J}^{\prime}|$ there exists $p=1,\dots,m$ such that $\sigma^{-1}(i_{p})\notin\mathtt{J}^{\prime}$ . Let $j=\sigma^{-1}(i_{p})$ . It is straightforward to check that post composition with the isomorphism $L_{j}\simeq N_{\sigma(j)}$ defines an isomorphism $\varphi_{n}(L_{j})\simeq\varphi_{n}(N_{\sigma(j)})$ for each $n\in\mathbb{N}$ . Since $N_{\sigma(j)}$ is a summand of $M_{\sigma(j)}$ and $i_{p}\geq r$ the group $\varphi_{n}(N_{\sigma(j)})$ embedds into $\prod_{i\geq r}\varphi_{r}(M_{i})$ . Considering the image under $\rho_{r,j}$ the result follows by Lemma 4.8 as in the proof of [11, Lemma 5]. ∎

Corollary 7.8.

Let $M$ be a pure-injective object of $\mathcal{T}$ such that $M^{I}$ is a coproduct of indecomposable pure-injectives for any set $I$ . Then $M$ is $\Sigma$ -pure-injective.

Proof.

Let $K=M^{\mathbb{N}}$ and $M_{i}=M$ for each $i\in\mathbb{N}$ , so that $K=\prod_{i\in\mathbb{N}}M_{i}$ . By hypothesis we have $K=\coprod_{j\in\mathtt{J}}L_{j}$ where each $L_{j}$ is an indecomposable pure-injective object of $\mathcal{T}$ . By Lemma 7.7 there exists $r\in\mathbb{N}$ such that, for each collection $\mathrm{N}=\{N_{i}\mid i\in\mathbb{N}\}$ where $N_{i}$ is an indecomposable summand of $M_{i}$ , we have $\varphi_{r}(N_{i})=\varphi_{n}(N_{i})$ for all $n\in\mathbb{N}$ with $n\geq r$ and all but finitely many $i\in\mathbb{N}$ . Fixing $j\in\mathtt{J}$ , for the collection given by $N_{i}=L_{j}$ for all $i$ , we have $\varphi_{r}(L_{j})=\varphi_{n}(L_{j})$ for all $n\in\mathbb{N}$ , and $\varphi_{n}(K)\simeq\coprod_{j}\varphi_{n}(L_{j})$ for all $n$ by Lemma 4.5. ∎

Theorem 7.9 goes back to a characterisation due to Faith [6, Proposition 3].

Theorem 7.9.

[9, Theorem 1] (see also [7, Lemma 1.1]). Let $\mathscr{M}$ be an injective object in a Grothendieck category $\mathcal{A}$ which has a set $\mathtt{G}$ of finitely generated generators. Then the following statements are equivalent.

(1)

$\mathscr{M}$ is $\Sigma$ -injective.
(2)

The countable coproduct $\mathscr{M}^{(\mathbb{N})}$ is injective.
(3)

For each $\mathscr{G}\in\mathtt{G}$ , any ascending chain of $\mathscr{M}$ -annihilator subobjects of $\mathscr{G}$ eventually stabilises.

Finally, we may now complete the proof of our main result.

of Theorem 1.1.

Taking $I=\mathbb{N}$ shows (1) implies (2). That (2) implies (3) follows from Remark 6.8 and Theorem 7.9. That (3) implies (1) follows from Remark 5.1, Proposition 5.6 and Lemma 5.7. The equivalence of (1) and (4) follows from Proposition 6.5. That (1) implies (5) follows from Lemma 6.6, and the converse follows from Corollaries 5.8 and 6.9. The equivalence of (1) and (6) follows from Lemma 6.12 and Corollary 7.8. ∎

8. Applications to endoperfection.

To consider direct applications of Theorem 1.1, recall Lemma 4.3, which says that pp-definable subgroups of an object $M$ of a fixed sort $X$ define $\mathrm{End}_{\mathcal{T}}(M)$ -submodules of $\mathcal{T}(X,M)$ . We begin by introducing the definition of an endoperfect object; see Definition 8.1. This concept is motivated by the idea of a perfect module as in Björk [3], and generalises the finite endolength objects defined by Krause [16], recalled in Definition 8.6.

Definition 8.1.

An object $M$ of $\mathcal{T}$ is said to be endoperfect if for all compact objects $X$ of $\mathcal{T}$ the left $\mathrm{End}_{\mathcal{T}}(M)$ -module $\mathcal{T}(X,M)$ has the descending chain condition on cyclic submodules. That is, $M$ is endoperfect provided $\mathcal{T}(X,M)$ is a perfect $\mathrm{End}_{\mathcal{T}}(M)$ -module in the sense of [3, §1] for each compact object $X$ .

As the terminology suggests, any object with a perfect endomorphism is endoperfect by the famous characterisation of perfect rings due to Bass [1, Theorem P]. We now relate this to purity by adapting the notion of endonoerthian modules to our categorical setting. This was motivated by work of Huisgen-Zimmermann and Saorín [13], in which conditions for endomorphism rings are related to $\Sigma$ -pure-injective modules.

Definition 8.2.

An object $M$ of $\mathcal{T}$ is called endonoetherian (respectively, endoartinian) if $\mathcal{T}(X,M)$ is noetherian (respectively, artinian) over $\mathrm{End}_{\mathcal{T}}(M)$ for each compact object $X$ .

By using a result of Björk [3] we now see a direct application of Theorem 1.1.

Corollary 8.3.

Any endoperfect endonoetherian object of $\mathcal{T}$ is $\Sigma$ -pure-injective, and hence decomposes into a coproduct of indecomposable objects with local endomorphism rings.

Proof.

Let $X$ be a compact object of $\mathcal{T}$ and $\varphi_{1}(M)\supseteq\varphi_{2}(M)\supseteq\dots$ be a descending chain of pp-definable subgroups of $M$ of sort $X$ . By Lemma 4.3 this defines a descending chain of $\mathrm{End}_{\mathcal{T}}(M)$ -submodules of $\mathcal{T}(X,M)$ . Since $M$ is endonoetherian each submodule (of the form $\varphi_{n}(M)$ ) is finitely generated. Since $M$ is endoperfect the module $\mathcal{T}(X,M)$ has the descending chain condition on cyclic submodules. This means $\mathcal{T}(X,M)$ is perfect (over $\mathrm{End}_{\mathcal{T}}(M)$ ) in the sense of Björk [3, §1]. By [3, Theorem 2] this means that the descending chain of finitely generated submodules $\varphi_{1}(M)\supseteq\varphi_{2}(M)\supseteq\dots$ of $\mathcal{T}(X,M)$ must terminate.

By the equivalence of (1) and (5) in Theorem 1.1 this means $M$ is $\Sigma$ -pure-injective. That $M$ decomposes into indecomposables with local endomorphism rings follows by the equivalence of (1) and (6) in Theorem 1.1 (setting $\mathtt{I}=\{0\}$ ). ∎

Lemma 8.4 was motivated by a result due to Crawley-Boevey [4, §4, Lemma].

Lemma 8.4.

If $M$ is a pure-injective object of $\mathcal{T}$ and $f\colon X\to M$ is a morphism in $\mathcal{T}$ with $X$ compact, then the cyclic $\mathrm{End}_{\mathcal{T}}(M)$ -submodule $\mathrm{End}_{\mathcal{T}}(M)f$ of $\mathcal{T}(X,M)$ is an intersection of pp-definable subgroups of $M$ of sort $X$ .

Proof.

We follow [4, §1.7]. Suppose $\mathscr{K}$ is a finitely generated subobject of $\mathrm{ker}(\mathbf{Y}(f))$ , so that we have an exact sequence in $\mathbf{Mod}\text{-}\mathcal{T}^{c}$ of the form $\mathbf{Y}(Z)\to\mathscr{K}\to 0$ with $Z$ compact. By construction the quotient $\mathbf{Y}(X)/\mathscr{K}$ is finitely presented, so there is a morphism $k\colon Z\to X$ such that $\mathbf{Y}(k)$ is the composition $\mathbf{Y}(Z)\to\mathscr{K}\subseteq\mathbf{Y}(X)$ . Similarly it is straightforward to check that any morphism $\mathbf{Y}(X)/\mathscr{K}\to\mathbf{Y}(M)$ is given by a morphism $h\colon X\to M$ in $\mathcal{T}$ such that $hk=0$ .

By [8, Proposition 3.1] this shows that the set of morphisms $\mathbf{Y}(X)/\mathscr{K}\to\mathbf{Y}(M)$ is the set of $\mathbf{Y}(h)$ where $h$ lies in a pp-definable subgroup of $M$ of sort $X$ . We now follow [4, §4, Lemma]. Since $\mathbf{Mod}\text{-}\mathcal{T}^{c}$ is a locally finitely generated Grothendieck category we have that $\mathrm{ker}(\mathbf{Y}(f))$ is the sum $\sum\mathscr{K}$ over the finitely generated subobjects $\mathscr{K}$ of $\mathrm{ker}(\mathbf{Y}(f))$ . Hence the set of morphisms $\mathbf{Y}(X)/\mathrm{ker}(\mathbf{Y}(f))\to\mathbf{Y}(M)$ is the intersection over each $\mathscr{K}$ of the set of morphisms $\mathbf{Y}(X)/\mathscr{K}\to\mathbf{Y}(M)$ . Altogether this shows the set of $\mathbf{Y}(X)/\mathrm{ker}(\mathbf{Y}(f))\to\mathbf{Y}(M)$ in $\mathbf{Mod}\text{-}\mathcal{T}^{c}$ is given by the set of $\mathbf{Y}(h)$ where $h$ lies in an intersection of pp-definable subgroup of $M$ of sort $X$ .

The pure-injectivity of $M$ in $\mathcal{T}$ is equivalent, by Corollary 6.7, to the injectivity of its image $\mathbf{Y}(M)$ . Thus, considering the canonical embedding $\mathbf{Y}(X)/\mathrm{ker}(\mathbf{Y}(f))\to\mathbf{Y}(M)$ given by $\mathbf{Y}(f)$ , any morphism $\mathbf{Y}(X)/\mathrm{ker}(\mathbf{Y}(f))\to\mathbf{Y}(M)$ factors through an endomorphism of $\mathbf{Y}(M)$ . Altogether this shows that the cyclic module $\mathrm{End}_{\mathcal{T}}(M)f$ is the intersection $\bigcap_{\mathscr{K}}\varphi_{\mathscr{K}}(M)$ where each pp-formula $\varphi_{\mathscr{K}}$ is given by $\varphi_{\mathscr{K}}(v_{G})=\exists u_{Z}\colon v_{G}=u_{Z}k$ in the above notation. ∎

Hence we may now provide partial a converse to Corollary 8.3.

Corollary 8.5.

Any $\Sigma$ -pure-injective object $M$ of $\mathcal{T}$ is endoperfect.

Proof.

By the equivalence of (1) and (5) in Theorem 1.1 we have that for each compact object $X$ any descending chain of pp-definable subgroups of $M$ of sort $X$ must stabilise. By Lemma 8.4 any cyclic $\mathrm{End}_{\mathcal{T}}(M)$ -submodule of $\mathcal{T}(X,M)$ is an interesction, and hence a finite intersection, of such subgroups. Thus any cyclic $\mathrm{End}_{\mathcal{T}}(M)$ -submodule of $\mathcal{T}(X,M)$ is a pp-definable subgroup of $M$ of sort $X$ . So, by definition, and by the equivalence of (1) and (5) in Theorem 1.1, any descending chain of cyclic $\mathrm{End}_{\mathcal{T}}(M)$ -submodules stabilises. ∎

The author would be interested in finding, if it exists, an endoperfect object in a compactly generated triangulated category which is not $\Sigma$ -pure-injective. Of course, by Corollary 8.3 such an example would not be endonoetherian.

Definition 8.6.

[16, Definition 1.1] An object $M$ of $\mathcal{T}$ is called endofinite if $\mathcal{T}(X,M)$ has finite length over $\mathrm{End}_{\mathcal{T}}(M)$ for all compact objects $X$ .

To conclude, we recover a result of Krause. Note endofinite objects are endorartinian, endonoetherian, hence endoperfect and $\Sigma$ -pure-injective by Corollary 8.3.

Theorem 8.7.

[16, Theorem 1.2(1)] Every endofinite object of $\mathcal{T}$ decomposes into a coproduct of indecomposable objects with local endomorphism rings.

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Characterisations of Σ\Sigma-pure-injectivity in triangulated categories and applications to endoperfect objects.

Abstract.

2020 Mathematics Subject Classification:

1. Introduction.

Theorem 1.1.

2. Multi-sorted languages, structures and homomorphisms.

Definition 2.1.

Example 2.2.

Definition 2.3.

Example 2.4.

Definition 2.5.

Definition 2.6.

Definition 2.7.

3. Purity in the canonical language of a triangulated category.

Notation 3.1.

Definition 3.2.

Assumption 3.3.

Definition 3.4.

Notation 3.5.

Remark 3.6.

Lemma 3.7.

Proof.

Definition 3.8.

Lemma 3.9.

Proof.

4. Products, coproducts, coherent functors and pp-formulas.

Definition 4.1.

Lemma 4.2.

Proof.

Lemma 4.3.

Proof.

Remark 4.4.

Lemma 4.5.

Proof.

Notation 4.6.

Corollary 4.7.

Proof.

Lemma 4.8.

Proof.

5. Annihilator subobjects and pp-definable subgroups.

Remark 5.1.

Definition 5.2.

Lemma 5.3.

Proof.

Lemma 5.4.

Proof.

Definition 5.5.

Proposition 5.6.

Lemma 5.7.

Corollary 5.8.

Proof.

6. Σ\Sigma-pure-injective objects and canonical morphisms.

Definition 6.1.

Proposition 6.2.

Proof.

Definition 6.3.

Theorem 6.4.

Proposition 6.5.

Proof.

Lemma 6.6.

Proof.

Corollary 6.7.

Remark 6.8.

Corollary 6.9.

Proof.

Corollary 6.10.

Proof.

Corollary 6.11.

Lemma 6.12.

Proof.

7. Completing the proof of the characterisation.

Corollary 7.1.

Remark 7.2.

Corollary 7.3.

Proof of Corollary 7.1.

Theorem 7.4.

Lemma 7.5.

Corollary 7.6.

Proof.

Lemma 7.7.

Characterisations of $\Sigma$ -pure-injectivity in triangulated categories and applications to endoperfect objects.

6. $\Sigma$ -pure-injective objects and canonical morphisms.