# Generalized periodicity theorems

Leonid Positselski  Institute of Mathematics, Czech Academy of Sciences
Žitná 25, 115 67 Praha 1, Czech Republic<EMAIL_ADDRESS>
###### Abstract.

Let $R$ be a ring and $\mathsf{S}$ be a class of strongly finitely presented
($\mathrm{FP}_{\infty}$) $R$-modules closed under extensions, direct summands,
and syzygies. Let $(\mathsf{A},\mathsf{B})$ be the (hereditary complete)
cotorsion pair generated by $\mathsf{S}$ in
${\operatorname{\mathsf{Mod--}}}R$, and let $(\mathsf{C},\mathsf{D})$ be the
(also hereditary complete) cotorsion pair in which
$\mathsf{C}=\varinjlim\mathsf{A}=\varinjlim\mathsf{S}$. We show that any
$\mathsf{A}$-periodic module in $\mathsf{C}$ belongs to $\mathsf{A}$, and any
$\mathsf{D}$-periodic module in $\mathsf{B}$ belongs to $\mathsf{D}$. Further
generalizations of both results are obtained, so that we get a common
generalization of the flat/projective and fp-projective periodicity theorems,
as well as a common generalization of the fp-injective/injective and cotorsion
periodicity theorems. Both are applicable to modules over an arbitrary ring,
and in fact, to Grothendieck categories.

###### Contents

  1. 1 Generalized Fp-injective/Injective and Cotorsion Periodicity
  2. 2 Generalized Flat/Projective and Fp-projective Periodicity I
  3. 3 Direct Limit Closures of Classes of Finitely Presentables
  4. 4 Exact Categories of Grothendieck Type
  5. 5 Pure Exact Structure and Deconstructibility
  6. 6 Classes of $\kappa$-Presentables and Their $\kappa$-Direct Limit Closures
  7. 7 Generalized Flat/Projective and Fp-projective Periodicity II

## Introduction

### 0.0.

_Periodicity theorems_ in homological algebra apply to the following setup.
Let $R$ be an associative ring and

($*$) $0\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muM\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5muL\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5muM\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu0$

be a short exact sequence of (right) $R$-modules with the leftmost term
isomorphic to the rightmost one. Then it is known that

  1. (1)

if the $R$-module $M$ is flat and the $R$-module $L$ is projective, then the
$R$-module $M$ is projective (Benson and Goodearl 2000 [7], rediscovered by
Neeman in 2008 [26]);

  2. (2)

if the exact sequence ($*$ ‣ 0.0) is pure and the $R$-module $L$ is pure-
projective, then the $R$-module $M$ is pure-projective (Simson 2002 [40]);

  3. (3)

if the exact sequence ($*$ ‣ 0.0) is pure and the $R$-module $L$ is pure-
injective, then the $R$-module $M$ is pure-injective (Št’ovíček 2014 [44]);

  4. (4)

in particular, if the $R$-module $M$ is fp-injective and the $R$-module $L$ is
injective, then the $R$-module $M$ is injective;

  5. (5)

if the $R$-module $L$ is cotorsion, then the $R$-module $M$ is cotorsion
(Bazzoni, Cortés-Izurdiaga, and Estrada 2017 [4]);

  6. (6)

if the ring $R$ is right coherent and the right $R$-module $L$ is fp-
projective, then the $R$-module $M$ is fp-projective (Šaroch and Št’ovíček
2018 [37]);

  7. (7)

over any ring $R$, if the $R$-module $L$ is fp-projective, then the $R$-module
$M$ is weakly fp-projective (Bazzoni, Hrbek, and the present author 2022 [5]).

Periodicity phenomena are linked to behavior of the modules of cocycles in
acyclic complexes. This means that the assertions (1–7) can be restated as
follows:

  1. (1${}^{\text{c}}$)

in any acyclic complex of projective modules with flat modules of cocycles,
the modules of cocycles are actually projective (so the complex is
contactible);

  2. (2${}^{\text{c}}$)

in any pure acyclic complex of pure-projective modules, the modules of
cocycles are pure-projective (so the complex is contractible);

  3. (3${}^{\text{c}}$)

in any pure acyclic complex of pure-injective modules, the modules of cocycles
are pure-injective (so the complex is contractible);

  4. (4${}^{\text{c}}$)

in any acyclic complex of injective modules with fp-injective modules of
cocycles, the modules of cocycles are actually injective (so the complex is
contractible);

  5. (5${}^{\text{c}}$)

in any acyclic complex of cotorsion modules, the modules of cocycles are
cotorsion;

  6. (6${}^{\text{c}}$)

in any acyclic complex of fp-projective right modules over a right coherent
ring, the modules of cocycles are fp-projective;

  7. (7${}^{\text{c}}$)

in any acyclic complex of fp-projective modules (over any ring), the modules
of cocycles are weakly fp-projective.

We refer to the introduction to the preprint [5] for a more detailed
discussion of the periodicity theorems (1–7) and
(1${}^{\text{c}}$–7${}^{\text{c}}$).

### 0.1.

The aim of this paper is to obtain a common generalization of (1) and (6–7),
and also a common generalization of (4) and (5), in the context of a chosen
class of modules or objects in a Grothendieck category. Let us start with
presenting the most symmetric and nicely looking formulation of a special case
of our main results, and then proceed to further generalizations.

Let $R$ be a ring. An $R$-module is said to be _strongly finitely presented_
if it has an (infinite) resolution by finitely generated projective
$R$-modules. In the terminology of the book [22], such modules are called
“$\mathrm{FP}_{\infty}$-modules”.

Let $\mathsf{S}$ be a class (up to an isomorphism, of course, a set) of
strongly finitely presented (right) $R$-modules. Assume that the free
$R$-module $R$ belongs to $\mathsf{S}$, and that the class of modules
$\mathsf{S}$ is closed under direct summands, extensions, and kernels of
epimorphisms in ${\operatorname{\mathsf{Mod--}}}R$. In particular, for any
module $S\in\mathsf{S}$ there exists a (finitely generated) projective
$R$-module $P$ together with an $R$-module epimorphism $P\longrightarrow S$
whose kernels also belongs to $\mathsf{S}$. The latter property is expressed
by saying that “the class of modules $\mathsf{S}$ is closed under syzygies”.

Denote by $\mathsf{B}=\mathsf{S}^{\perp_{1}}$ the class of all $R$-modules $B$
such that $\operatorname{Ext}^{1}_{R}(S,B)=0$ for all $S\in\mathsf{S}$.
Furthermore, denote by $\mathsf{A}={}^{\perp_{1}}\mathsf{B}$ the class of all
$R$-modules $A$ such that $\operatorname{Ext}^{1}_{R}(A,B)=0$ for all
$B\in\mathsf{B}$. The pair of classes of modules $(\mathsf{A},\mathsf{B})$ is
called _the cotorsion pair generated by $\mathsf{S}$ in
${\operatorname{\mathsf{Mod--}}}R$_.

Let $\mathsf{C}=\varinjlim\mathsf{S}$ denote the class of all $R$-modules that
can be obtained as direct limits of diagrams of modules from $\mathsf{S}$,
indexed by directed posets. Since $\mathsf{S}$ is a class of finitely
presented modules, one can see that $\varinjlim\mathsf{S}$ coincides with the
direct limit closure or $\mathsf{S}$ in ${\operatorname{\mathsf{Mod--}}}R$
[25, 15, 24].

Furthermore, since $\mathsf{S}$ is a class of strongly finitely presented
modules ($\mathrm{FP}_{2}$ is sufficient), the class $\mathsf{C}$ is closed
under extensions in ${\operatorname{\mathsf{Mod--}}}R$ [3]. Taking into
account the description of $\mathsf{A}$ as the class of all direct summands of
transfinitely iterated extensions of modules from $\mathsf{S}$ [22, Corollary
6.14], one concludes that $\mathsf{A}\subset\mathsf{C}$. Hence
$\mathsf{C}=\varinjlim\mathsf{A}$ is the class of all direct limits of modules
from $\mathsf{A}$.

Denote by $\mathsf{D}=\mathsf{C}^{\perp_{1}}$ the class of all $R$-modules $D$
such that $\operatorname{Ext}^{1}_{R}(C,D)=0$ for all $C\in\mathsf{C}$. Then
one has $\mathsf{A}\subset\mathsf{C}$ and $\mathsf{B}\supset\mathsf{D}$.

Part (a) of the following theorem is one of the main results of this paper,
while part (b) follows rather easily from a result of Bazzoni, Cortés-
Izurdiaga, and Estrada [4, Theorem 4.7] together with a result of Angeleri
Hügel and Trlifaj [3, Corollary 2.4].

######  Theorem 0.

Let $R$ be a ring and $\mathsf{S}$ be a class of strongly finitely presented
$R$-modules, containing the free $R$-module $R$ and closed under direct
summands, extensions, and kernels of epimorphisms. Put
$\mathsf{B}=\mathsf{S}^{\perp_{1}}$, $\mathsf{A}={}^{\perp_{1}}\mathsf{B}$,
$\mathsf{C}=\varinjlim\mathsf{S}$, and $\mathsf{D}=\mathsf{C}^{\perp_{1}}$.
Then the following assertions hold:

(a) For any short exact sequence ($*$ ‣ 0.0) with $L\in\mathsf{A}$ and
$M\in\mathsf{C}$, one has $M\in\mathsf{A}$. In other words, in any acyclic
complex of modules from $\mathsf{A}$ with the modules of cocycles belonging to
$\mathsf{C}$, the modules of cocycles actually belong to $\mathsf{A}$.

(b) For any short exact sequence ($*$ ‣ 0.0) with $L\in\mathsf{D}$ and
$M\in\mathsf{B}$, one has $M\in\mathsf{D}$. In other words, in any acyclic
complex of modules from $\mathsf{D}$ with the modules of cocycles belonging to
$\mathsf{B}$, the modules of cocycles actually belong to $\mathsf{D}$.

Theorem 0(a) is a common generalization of items (1) or (1${}^{\text{c}}$) and
(6) or (6${}^{\text{c}}$) on the list of Section 0.0. Taking
$\mathsf{S}=\\{R\\}$ to be the class consisting of the free $R$-module $R$
only, one obtains the flat/projective periodicity theorem of Benson–Goodearl
[7, Theorem 2.5] and Neeman [26, Remark 2.15] as a particular case of Theorem
0(a). Assuming the ring $R$ to be right coherent and taking $\mathsf{S}$ to be
the class of all finitely presented right $R$-modules, one obtains the fp-
projective periodicity theorem of Šaroch and St’ovíček [37, Example 4.3] as a
particular case of Theorem 0(a).

Theorem 0(b) is a common generalization of items (4) or (4${}^{\text{c}}$)
(for coherent rings) and (5) or (5${}^{\text{c}}$) on the list of Section 0.0.
Assuming the ring $R$ to be right coherent and taking $\mathsf{S}$ to be the
class of all finitely presented right $R$-modules, one obtains the fp-
injective/injective periodicity theorem, essentially due to St’ovíček [44,
Corollary 5.5] (see also [4, Theorem 1.2(1) or 5.1(1)]), as a particular case
of Theorem 0(b). Taking $\mathsf{S}=\\{R\\}$ to be the class consisting of the
free $R$-module $R$ only, one obtains the cotorsion periodicity theorem of
Bazzoni, Cortés-Izurdiaga, and Estrada [4, Theorem 1.2(2) or 5.1(2)] as a
particular case of Theorem 0(b).

### 0.2.

Both parts (a) and (b) of Theorem 0 admit far-reaching generalizations in
several directions simultaneously (allowing, in particular, to drop the
coherence assumptions on the ring $R$ in the preceding two paragraphs). Let us
state these results.

We consider a Grothendieck abelian category $\mathsf{K}$. For any class of
objects $\mathsf{S}\subset\mathsf{K}$, let
$\varinjlim\mathsf{S}\subset\mathsf{K}$ denote the class of all direct limits
in $\mathsf{K}$ of diagrams of objects from $\mathsf{S}$ indexed by directed
posets. Furthermore, given a regular cardinal $\kappa$, we let
$\varinjlim^{(\kappa)}\mathsf{S}\subset\mathsf{K}$ denote the class of all
direct limits in $\mathsf{K}$ of diagrams of objects from $\mathsf{S}$ indexed
by _$\kappa$ -directed_ posets. Here a poset $X$ is said to be
$\kappa$-directed if any its subset of cardinality less than $\kappa$ has an
upper bound in $X$.

Part (iii) of following theorem is the main result of this paper, formulated
in full generality and strength. It is a generalization of Theorem 0(a). Parts
(i) and (ii) are supplementary comments on part (iii), providing some
sufficient conditions for validity of the main assumption in (iii).

######  Theorem A.

Let $\mathsf{K}$ be a Grothendieck abelian category, and let $\kappa$ be a
regular cardinal such that $\mathsf{K}$ is a locally $\kappa$-presentable
category. Let $\mathsf{S}\subset\mathsf{K}$ be a class of (some)
$\kappa$-presentable objects closed under transfinitely iterated extensions
indexed by ordinals smaller than $\kappa$. Put
$\mathsf{C}=\varinjlim^{(\kappa)}\mathsf{S}\subset\mathsf{K}$. Then

(i) If $\kappa=\aleph_{0}$ is the countable cardinal and the class
$\mathsf{S}$ consists of (some) objects of type $\mathrm{FP}_{2}$, then the
class $\mathsf{C}$ is closed under extensions in $\mathsf{K}$.

(ii) If $\kappa=\aleph_{0}$ and the class $\mathsf{C}$ is closed under
extensions in $\mathsf{K}$, then the class $\mathsf{C}$ is deconstructible in
$\mathsf{K}$.

(iii) For any regular cardinal $\kappa$, assume that the class $\mathsf{C}$ is
deconstructible in $\mathsf{K}$. Denote by
$\mathsf{A}=\mathsf{Fil}(\mathsf{S})^{\oplus}$ the class of all direct
summands of transfinitely iterated extensions of objects from $\mathsf{S}$ in
$\mathsf{K}$. Let $\mathsf{B}^{\prime}=\mathsf{S}^{\perp_{\geq
1}}\cap\mathsf{C}$ be the class of all objects $B\in\mathsf{C}$ such that
$\operatorname{Ext}_{\mathsf{K}}^{n}(S,B)=0$ for all $S\in\mathsf{S}$ and
$n\geq 1$. Let
$\mathsf{A}^{\prime}=\mathsf{C}\cap{}^{\perp_{1}}\mathsf{B}^{\prime}$ be the
class of all objects $A\in\mathsf{C}$ such that
$\operatorname{Ext}_{\mathsf{K}}^{1}(A,B)=0$ for all $B\in\mathsf{B}^{\prime}$
(so $\mathsf{A}\subset\mathsf{A}^{\prime}\subset\mathsf{C}$). Then, in any
acyclic complex of objects from $\mathsf{A}$ with the objects of cocycles
belonging to $\mathsf{C}$, the objects of cocycles actually belong to
$\mathsf{A}^{\prime}$.

Let us empasize that deconstructibility in Theorem A is understood in the
strong sense of the word. So, a class of objects $\mathsf{C}$ is
_deconstructible_ if it is closed under transfinitely iterated extensions and
contains a subset $\mathsf{T}\subset\mathsf{C}$ such that all objects from
$\mathsf{C}$ are transfinitely iterated extensions of objects from
$\mathsf{T}$.

Taking $\kappa=\aleph_{0}$ and assuming $\mathsf{S}$ to be a class of strongly
finitely presentable ($\mathrm{FP}_{\infty}$) objects closed under extensions
in $\mathsf{K}$ makes the assertions of Theorem A(i–ii) applicable, so the
assumption of Theorem A(iii) is satisfied. This makes Theorem 0(a) a
particular case of Theorem A (for
$\mathsf{K}={\operatorname{\mathsf{Mod--}}}R$).

Taking $\mathsf{S}$ to be the class of all $\kappa$-presentable objects in
$\mathsf{K}$ makes the assumption of Theorem A(iii) satisfied as well, since
$\mathsf{C}=\mathsf{K}$ in this case. So one obtains the theorem of Bazzoni,
Hrbek and the present author ([5, Theorem 4.1] for $\kappa=\aleph_{0}$, listed
above as item (7) or (7${}^{\text{c}}$) in the case of
$\mathsf{K}={\operatorname{\mathsf{Mod--}}}R$; or [5, Remark 4.11] for other
$\kappa$) as a particular case of Theorem A(iii).

### 0.3.

The next theorem is a generalization of Theorem 0(b). Part (i), which is the
main claim, is rather easily deduced from a result of St’ovíček and the
present author [33, Theorem 6.1] (which, in turn, is a generalization of [4,
Theorem 4.7]). Part (ii), which is a supplementary comment on part (i)
(explaining what (i) means under some additional assumptions), turns out to be
more involved.

######  Theorem B.

Let $\mathsf{K}$ be a Grothendieck category and $\mathsf{S}\subset\mathsf{K}$
be a class of objects. Let $\mathsf{T}\subset\mathsf{K}$ be any class of
objects of finite projective dimension in $\mathsf{K}$ such that the union
$\mathsf{S}\cup\mathsf{T}$ contains a set of generators for $\mathsf{K}$.
Denote by $\mathsf{C}\subset\mathsf{K}$ the closure of
$\mathsf{S}\cup\mathsf{T}$ under coproducts, direct limits, extensions, and
kernels of epimorphisms in $\mathsf{K}$. Then

(i) Let $\mathsf{B}=\mathsf{S}^{\perp_{1}}$ be the class of all objects
$B\in\mathsf{K}$ such that $\operatorname{Ext}^{1}_{\mathsf{K}}(S,B)=0$ for
all $S\in\mathsf{S}$, and let $\mathsf{D}=\mathsf{C}^{\perp_{1}}$ be the class
of all objects $D\in\mathsf{K}$ such that
$\operatorname{Ext}^{1}_{\mathsf{K}}(C,D)=0$ for all $C\in\mathsf{C}$. Then,
for any acyclic complex of objects from $\mathsf{D}$ with the objects of
cocycles belonging to $\mathsf{B}$, the objects of cocycles actually belong to
$\mathsf{D}$.

(ii) If the class $\mathsf{S}\cup\mathsf{T}$ consists of (some) finitely
presentable objects and is closed under extensions and kernels of epimorphisms
in $\mathsf{K}$, then $\mathsf{C}=\varinjlim(\mathsf{S}\cup\mathsf{T})$ is the
class of all direct limits of diagrams of objects from
$\mathsf{S}\cup\mathsf{T}$, indexed by directed posets.

One can see that in the context of Theorem B(ii) the class
$\mathsf{S}\cup\mathsf{T}$ has to consist of strongly finitely presentable
($\mathrm{FP}_{\infty}$) objects. Taking $\mathsf{T}=\varnothing$ makes
Theorem 0(b) a particular case of Theorem B(i–ii) (for
$\mathsf{K}={\operatorname{\mathsf{Mod--}}}R$).

Taking $\mathsf{S}$ to be the class of all finitely presentable objects in a
locally finitely presentable abelian category $\mathsf{K}$ and
$\mathsf{T}=\varnothing$, one obtains the assertion (4) or (4${}^{\text{c}}$)
on the list of Section 0.0, essentially due to St’ovíček [44, Corollary 5.5],
as a particular case of Theorem B(i) (for
$\mathsf{K}={\operatorname{\mathsf{Mod--}}}R$).

### 0.4.

The proofs of Theorems 0(b) and B(i) are presented in Section 1. Theorem 0(a)
is proved in Section 2. The proofs of Theorems A(i) and B(ii) are given in
Section 3. Theorem A(ii) is proved in Section 5. The possibilities and
difficulties of extending Theorem A(i–ii) to higher cardinals $\kappa$ are
discussed in Section 6. The proof of the main result, Theorem A(iii), is
presented in Section 7.

One comment on the style of the exposition may be in order. This paper is
written with the intent to be at least partially understandable to readers not
necessarily feeling at ease with advanced category-theoretic concepts. In
order not to intimidate a reader mostly interested in module-theoretic rather
than category-theoretic applications, category-theoretic terminology is
introduced slowly and gradually as the paper progresses from the less general
results such as Theorem 0 to the more general ones such as Theorems B and A.

### Acknowledgement

I want to thank Michal Hrbek, Silvana Bazzoni, Jan Šaroch, and Jan Trlifaj for
helpful discussions and comments. Long conversations with Jan Št’ovíček were
particularly illuminating, and to him goes my special gratitude. The author is
supported by the GAČR project 20-13778S and research plan RVO: 67985840.

##  1\. Generalized Fp-injective/Injective and Cotorsion Periodicity

In this section we prove Theorems 0(b) and B(i). This is not difficult, given
the preceding results in [4, Theorem 4.7] and [33, Theorem 6.1]. The former
theorem needs to be used together with [3, Corollary 2.4], and the latter one
together with [33, Lemmas 4.2, 4.3, and 6.4].

Let us formally introduce some notation and terminology which was already used
throughout the introduction. Given an abelian (or exact [10]) category
$\mathsf{K}$ and a class of objects $\mathsf{A}\subset\mathsf{K}$, one denotes
by $\mathsf{A}^{\perp_{1}}\subset\mathsf{K}$ the class of all objects
$X\in\mathsf{K}$ such that $\operatorname{Ext}^{1}_{\mathsf{K}}(A,X)=0$ for
all $A\in\mathsf{A}$. Dually, for a class of objects
$\mathsf{B}\subset\mathsf{K}$, the notation
${}^{\perp_{1}}\mathsf{B}\subset\mathsf{K}$ stands for the class of all
objects $Y\in\mathsf{K}$ such that
$\operatorname{Ext}^{1}_{\mathsf{K}}(Y,B)=0$ for all $B\in\mathsf{B}$.
Similarly, $\mathsf{A}^{\perp_{\geq 1}}\subset\mathsf{K}$ is the class of all
objects $X\in\mathsf{K}$ such that
$\operatorname{Ext}^{n}_{\mathsf{K}}(A,X)=0$ for all $A\in\mathsf{A}$ and
$n\geq 1$. Dually, ${}^{\perp_{\geq 1}}\mathsf{B}\subset\mathsf{K}$ is the
class of all objects $Y\in\mathsf{K}$ such that
$\operatorname{Ext}^{n}_{\mathsf{K}}(Y,B)=0$ for all $B\in\mathsf{B}$ and
$n\geq 1$.

A class of objects $\mathsf{A}\subset\mathsf{K}$ is said to be _generating_
(or _a class of generators_) if every object of $\mathsf{K}$ is a quotient
object of a coproduct of objects from $\mathsf{A}$. A class of objects
$\mathsf{B}\subset\mathsf{K}$ is said to be _cogenerating_ (or _a class of
cogenerators_) if every object of $\mathsf{K}$ is a subobject of a product of
objects from $\mathsf{B}$.

The previous definitions, as well as generally all category-theoretic
definitions in this paper, are transferred from abelian to exact categories in
the obvious way: all the mentions of “subobjects”, “quotients”,
“monomorphisms”, “epimorphisms”, “exact sequences”, etc., are understood to
mean admissible monomorphisms, admissible epimorphisms, admissible exact
sequences, etc.

A pair of classes of objects $(\mathsf{A},\mathsf{B})$ in $\mathsf{K}$ is said
to be a _cotorsion pair_ if $\mathsf{A}={}^{\perp_{1}}\mathsf{B}$ and
$\mathsf{B}=\mathsf{A}^{\perp_{1}}$. Notice that, for any cotorsion pair
$(\mathsf{A},\mathsf{B})$ in $\mathsf{K}$, the class
$\mathsf{A}\subset\mathsf{K}$ is closed under coproducts (i. e., those
coproducts that exist in $\mathsf{K}$), and the class
$\mathsf{B}\subset\mathsf{K}$ is closed under products (in the same sense)
[12, Corollary 8.3], [14, Corollary A.2].

For any class of objects $\mathsf{S}\subset\mathsf{K}$, the pair of classes
$\mathsf{B}=\mathsf{S}^{\perp_{1}}$ and $\mathsf{A}={}^{\perp_{1}}\mathsf{B}$
is a cotorsion pair in $\mathsf{K}$. The cotorsion pair
$(\mathsf{A},\mathsf{B})$ obtained in this way is said to be _generated_ by
the class $\mathsf{S}$. Dually, for any class of objects
$\mathsf{T}\subset\mathsf{K}$, the pair of classes
$\mathsf{A}={}^{\perp_{1}}\mathsf{T}$ and $\mathsf{B}=\mathsf{A}^{\perp_{1}}$
is also a cotorsion pair in $\mathsf{K}$. The latter cotorsion pair
$(\mathsf{A},\mathsf{B})$ is said to be _cogenerated_ by the class
$\mathsf{T}$.

Let $(\mathsf{A},\mathsf{B})$ be a cotorsion pair in $\mathsf{K}$ such that
the class $\mathsf{A}$ is generating and the class $\mathsf{B}$ is
cogenerating in $\mathsf{K}$. So every object of $\mathsf{K}$ is a quotient
object of an object from $\mathsf{A}$ and a subobject of an object from
$\mathsf{B}$. These conditions are satisfied automatically for any cotorsion
pair in an abelian category $\mathsf{K}$ with enough projective and injective
objects (because all projective objects belong to $\mathsf{A}$ and all
injective objects belong to $\mathsf{B}$). In particular, this applies to the
module categories $\mathsf{K}={\operatorname{\mathsf{Mod--}}}R$.

In the assumptions of the previous paragraph, the following conditions are
equivalent [19, Theorem 1.2.10], [22, Lemma 5.24], [5, Section 1], [33, Lemma
4.1]:

  1. (1)

the class $\mathsf{A}$ is closed under kernels of epimorphisms in
$\mathsf{K}$;

  2. (2)

the class $\mathsf{B}$ is closed under cokernels of monomorphisms in
$\mathsf{K}$;

  3. (3)

$\operatorname{Ext}_{\mathsf{K}}^{2}(A,B)=0$ for all $A\in\mathsf{A}$ and
$B\in\mathsf{B}$;

  4. (4)

$\operatorname{Ext}_{\mathsf{K}}^{n}(A,B)=0$ for all $A\in\mathsf{A}$,
$B\in\mathsf{B}$, and $n\geq 1$.

A cotorsion pair $(\mathsf{A},\mathsf{B})$ satisfying conditions (1–4) is said
to be _hereditary_.

Given a class of objects $\mathsf{L}\subset\mathsf{K}$, an object
$M\in\mathsf{K}$ is said to be _$\mathsf{L}$ -periodic_ if there exists a
short exact sequence $0\longrightarrow M\longrightarrow L\longrightarrow
M\longrightarrow 0$ ($*$ ‣ 0.0) in $\mathsf{K}$ with $L\in\mathsf{L}$. We
recall that the notation $\varinjlim\mathsf{L}\subset\mathsf{K}$ stands for
the class of all direct limits in $\mathsf{K}$ of diagrams of objects from
$\mathsf{L}$ (indexed by directed posets).

A short exact sequence of right $R$-modules $0\longrightarrow K\longrightarrow
L\longrightarrow M\longrightarrow 0$ is said to be _pure_ if it remains exact
after taking the tensor product with any left $R$-module. Equivalently, a
short exact sequence $0\longrightarrow K\longrightarrow L\longrightarrow
M\longrightarrow 0$ is pure if and only if it remains exact after the functor
$\operatorname{Hom}_{R}(S,{-})$ from any finitely presented right $R$-module
$S$ is applied [22, Definition 2.6 and Lemma 2.19]. If this is the case, the
object $K$ is said to be a _pure subobject_ of $L$, while the object $M$ is
called a _pure epimorphic image_ (or a _pure quotient_) of $M$. The _pure
exact structure_ on ${\operatorname{\mathsf{Mod--}}}R$ is formed by the class
of all pure exact sequences. The projective objects of the exact category
${\operatorname{\mathsf{Mod--}}}R$ with the pure exact structure are called
_pure-projective_ $R$-modules, and the injective objects are called _pure-
injective_.

An $R$-module $S$ is said to be $\mathrm{FP}_{n}$ (where $n\geq 0$ is an
integer) if it admits a fragment of projective resolution
$P_{n}\longrightarrow P_{n-1}\longrightarrow\dotsb\longrightarrow
P_{0}\longrightarrow S\longrightarrow 0$ with finitely generated projective
modules $P_{i}$. So a module is $\mathrm{FP}_{0}$ if and only if it is
finitely generated, and it is $\mathrm{FP}_{1}$ if and only if it is finitely
presented. A module $S$ is said to be $\mathrm{FP}_{\infty}$ if it admits a
resolution by finitely generated projective modules; equivalently, this means
that $S$ is $\mathrm{FP}_{n}$ for all $n\geq 0$. Modules of type
$\mathrm{FP}_{\infty}$ are otherwise known as _strongly finitely presented_.

A class of modules $\mathsf{S}$ is said to be _closed under syzygies_ if for
every module $S\in\mathsf{S}$ there exists a short exact sequence
$0\longrightarrow K\longrightarrow P\longrightarrow S\longrightarrow 0$ with a
projective module $P$ and $K\in\mathsf{S}$. For any other short exact sequence
$0\longrightarrow K^{\prime}\longrightarrow P^{\prime}\longrightarrow
S\longrightarrow 0$ with a projective module $P^{\prime}$, it then follows
that $K^{\prime}\oplus P\simeq K\oplus P^{\prime}$, which often implies that
$K^{\prime}\in\mathsf{S}$ as well. Dually, a class of modules $\mathsf{T}$ is
_closed under cosyzygies_ if for every module $T\in\mathsf{T}$ there exists a
short exact sequence $0\longrightarrow T\longrightarrow J\longrightarrow
L\longrightarrow 0$ with an injective module $J$ and $L\in\mathsf{T}$. For any
other short exact sequence $0\longrightarrow T\longrightarrow
J^{\prime}\longrightarrow L^{\prime}\longrightarrow 0$ with an injective
module $J^{\prime}$, it then follows that $L^{\prime}\oplus J\simeq L\oplus
J^{\prime}$, which often implies that $L^{\prime}\in\mathsf{T}$, too.

###### Proof of Theorem 0(b) from Section 0.1.

Let us first prove the first assertion, then deduce the second one. In order
to apply [4, Theorem 4.7], we need to show that $(\mathsf{C},\mathsf{D})$ is a
hereditary cotorsion pair in ${\operatorname{\mathsf{Mod--}}}R$. First of all,
$(\mathsf{C},\mathsf{D})$ is indeed a cotorsion pair by [3, Corollary 2.4]
(see also [22, Corollary 8.42]). Alternatively, Corollary 5.4 below provides a
more general result.

To show that the cotorsion pair $(\mathsf{C},\mathsf{D})$ is hereditary, one
can argue as follows. The cotorsion pair $(\mathsf{A},\mathsf{B})$ generated
by $\mathsf{S}$ in ${\operatorname{\mathsf{Mod--}}}R$ is hereditary, since the
class $\mathsf{S}$ is closed under syzygies [22, Corollary 5.25(a)], [5, Lemma
1.3], [33, Lemma 4.1]. Consequently, the class $\mathsf{B}$ is closed under
the cokernels of monomorphisms, and in particular, under cosyzygies in
${\operatorname{\mathsf{Mod--}}}R$. By [3, Corollary 2.4] or [22, Corollary
8.42], the cotorsion pair $(\mathsf{C},\mathsf{D})$ is cogenerated by the
class of all pure-injective modules belonging to $\mathsf{B}$. The class of
all pure-injective modules is closed under cosyzygies by [22, Lemma 6.20], so
the class of all pure-injective modules belonging to $\mathsf{B}$ is also
closed under cosyzygies. Applying [22, Corollary 5.25(b)], we conclude that
the cotorsion pair $(\mathsf{C},\mathsf{D})$ is hereditary. Alternatively, one
can use Proposition 3.5 below, which is a more general result.

We also need to know that the class $\mathsf{C}$ is closed under pure
epimorphic images. This is [25, Proposition 2.1], [15, Section 4.1], [3,
Theorem 2.3], or [22, Theorem 8.40]. By the latter two references, we also
have $\mathsf{A}\subset\mathsf{C}$, hence $\mathsf{B}\supset\mathsf{D}$.

Therefore, the result of [4, Theorem 4.7] is applicable to the cotorsion pair
$(\mathsf{C},\mathsf{D})$, and it tells that the class
${}^{\perp_{1}}\\{M\\}\cap\mathsf{C}$ is closed under direct limits in
${\operatorname{\mathsf{Mod--}}}R$ for any $\mathsf{D}$-periodic module $M$.
Now, if $M\in\mathsf{B}$, then the class ${}^{\perp_{1}}\\{M\\}\cap\mathsf{C}$
contains $\mathsf{A}$. Thus
$\mathsf{C}=\varinjlim\mathsf{A}\subset{}^{\perp_{1}}\\{M\\}$ and
$M\in\mathsf{D}$.

To deduce the second assertion of Theorem 0(b) from the first one, suppose
given an acyclic complex $D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ of
modules from $\mathsf{D}$ with the modules of cocycles belonging to
$\mathsf{B}$. All one needs to do is to chop up the complex
$D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ into short exact sequence
pieces and take the infinite product of the pieces. The resulting short exact
sequence has the form ($*$ ‣ 0.0) with $L\in\mathsf{D}$ and $M\in\mathsf{B}$,
and the first assertion of the theorem can be applied to it, providing the
desired conclusion. This argument uses the observations that countable
products are exact in ${\operatorname{\mathsf{Mod--}}}R$, the classes
$\mathsf{D}$ and $\mathsf{B}$ are closed under countable products, and the
class $\mathsf{D}$ is closed under direct summands in
${\operatorname{\mathsf{Mod--}}}R$ (cf. [11, proof of Proposition 7.6] or [18,
Proposition 2]). ∎

Let $\mathsf{K}$ be an abelian category. A class of objects
$\mathsf{C}\subset\mathsf{K}$ is said to be _self-generating_ [5, Section 1],
[33, Section 4] if for any epimorphism $K\longrightarrow C$ in $\mathsf{K}$
with $C\in\mathsf{C}$ there exists a morphism $C^{\prime}\longrightarrow K$ in
$\mathsf{K}$ with $C^{\prime}\in\mathsf{C}$ such that the composition
$C^{\prime}\longrightarrow K\longrightarrow C$ is en epimorphism in
$\mathsf{K}$. A class of objects $\mathsf{C}$ is said to be _self-resolving_
[33, Section 6] if it is self-generating and closed under extensions and
kernels of epimorphisms.

Before proving Theorem B(i) as it is stated, let us explicitly formulate and
prove the following periodicity assertion.

######  Theorem 1.1.

Let $\mathsf{K}$ be a Grothendieck category and $\mathsf{S}\subset\mathsf{K}$
be a class of objects. Let $\mathsf{T}\subset\mathsf{K}$ be any class of
objects of finite projective dimension in $\mathsf{K}$ such that the union
$\mathsf{S}\cup\mathsf{T}$ contains a set of generators for $\mathsf{K}$.
Denote by $\mathsf{C}\subset\mathsf{K}$ the closure of
$\mathsf{S}\cup\mathsf{T}$ under coproducts, direct limits, extensions, and
kernels of epimorphisms in $\mathsf{K}$.

Put $\mathsf{B}=\mathsf{S}^{\perp_{1}}$ and
$\mathsf{D}=\mathsf{C}^{\perp_{1}}\subset\mathsf{K}$. Then, for any short
exact sequence ($*$ ‣ 0.0) as in Section 0.0 with objects $L\in\mathsf{D}$ and
$M\in\mathsf{B}$, one has $M\in\mathsf{D}$. In other words, any
$\mathsf{D}$-periodic object belonging to $\mathsf{B}$ actually belongs to
$\mathsf{D}$.

###### Proof.

The class of objects $\mathsf{C}$ contains a set of generators for
$\mathsf{K}$ and is closed under coproducts; hence, in particular, it is self-
generating. The class $\mathsf{C}$ is also closed under extensions and kernels
of epimorphisms in $\mathsf{K}$; so it is self-resolving. Finally, the class
$\mathsf{C}$ is closed under direct limits in $\mathsf{K}$, and the direct
limits are exact in $\mathsf{K}$. Thus the assumptions of [33, Theorem 6.1]
are satisfied for the class $\mathsf{C}\subset\mathsf{K}$, which tells that,
for any $\mathsf{D}$-periodic object $M\in\mathsf{K}$, the class
${}^{\perp_{1}}\\{M\\}\cap\mathsf{C}$ is closed under direct limits in
$\mathsf{K}$.

By [5, Lemma 1.3] or [33, Lemma 4.1], we have
$\operatorname{Ext}^{n}_{\mathsf{K}}(C,D)=0$ for all objects $C\in\mathsf{C}$,
$D\in\mathsf{D}$, and integers $n\geq 1$. By [33, Lemma 4.2], it follows that
the class ${}^{\perp_{1}}\\{M\\}\cap\mathsf{C}$ contains all objects of the
class $\mathsf{C}$ which have finite projective dimension in $\mathsf{K}$.
Thus $\mathsf{T}\subset{}^{\perp_{1}}\\{M\\}\cap\mathsf{C}$. If
$M\in\mathsf{B}$, then we also have
$\mathsf{S}\subset{}^{\perp_{1}}\\{M\\}\cap\mathsf{C}$.

On the other hand, [33, Lemma 4.3] tells that the class
${}^{\perp_{1}}\\{M\\}\cap\mathsf{C}$ is closed under extensions and kernels
of admissible epimorphisms in the exact category $\mathsf{C}$ (with the exact
category structure inherited from the abelian exact structure of
$\mathsf{K}$). Since the class $\mathsf{C}$ is closed under extensions and
kernels of epimorphisms in $\mathsf{K}$, it follows that the class
${}^{\perp_{1}}\\{M\\}\cap\mathsf{C}$ is closed under extensions and kernels
of epimorphisms in $\mathsf{K}$. Finally, the class
${}^{\perp_{1}}\\{M\\}\cap\mathsf{C}$ is closed under coproducts in
$\mathsf{K}$, since it is closed under finite direct sums and direct limits.

We have shown that the class ${}^{\perp_{1}}\\{M\\}\cap\mathsf{C}$ contains
$\mathsf{S}\cup\mathsf{T}$ and is closed under extensions, kernels of
epimorphisms, coproducts, and direct limits in $\mathsf{K}$. Hence we can
conclude that ${}^{\perp_{1}}\\{M\\}\cap\mathsf{C}=\mathsf{C}$, so
$\mathsf{C}\subset{}^{\perp_{1}}\\{M\\}$ and $M\in\mathsf{D}$. ∎

###### Proof of Theorem B(i) from from Section 0.3.

Let $D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ be an acyclic complex
in $\mathsf{K}$ with the terms $D^{i}\in\mathsf{D}$ and the objects of
cocycles $B^{i}\in\mathsf{B}$. So we have short exact sequences
$0\longrightarrow B^{i}\longrightarrow D^{i}\longrightarrow
B^{i+1}\longrightarrow 0$ in $\mathsf{K}$. Taking the product of these short
exact sequences over $i\in\mathbb{Z}$, we obtain a sequence

(1) $0\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\prod\nolimits_{i\in\mathbb{Z}}B^{i}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\prod\nolimits_{i\in\mathbb{Z}}D^{i}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\prod\nolimits_{i\in\mathbb{Z}}B^{i}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu0.$

In order to show that (1) is exact, we apply [33, Lemma 6.4]. By assumption,
the class $\mathsf{S}\cup\mathsf{T}$ contains a set of generators of the
Grothendieck category $\mathsf{K}$. So there exists a family of objects
$(G_{\xi})_{\xi\in\Xi}$ in $\mathsf{S}\cup\mathsf{T}$ together with an
epimorphism
$G=\coprod_{\xi\in\Xi}G_{\xi}\longrightarrow\prod_{i\in\mathbb{Z}}B^{i}$ in
$\mathsf{K}$. It remains to show that
$\operatorname{Ext}^{1}_{\mathsf{K}}(G,B^{i})=0$ for every $i\in\mathbb{Z}$.

By [12, Corollary 8.3] or [14, Corollary A.2], it suffices to check that
$\operatorname{Ext}^{1}_{\mathsf{K}}(G_{\xi},B^{i})=0$ for every
$i\in\mathbb{Z}$ and $\xi\in\Xi$. There are two cases. If
$G_{\xi}\in\mathsf{S}$, then it remains to recall that
$B^{i}\in\mathsf{B}=\mathsf{S}^{\perp_{1}}$. If $G_{\xi}\in\mathsf{T}$, then
$G_{\xi}\in\mathsf{C}$ and the projective dimension of $G_{\xi}$ in
$\mathsf{K}$ is finite. From the short exact sequences $0\longrightarrow
B^{j}\longrightarrow D^{j}\longrightarrow B^{j+1}\longrightarrow 0$ we get
$\operatorname{Ext}^{1}_{\mathsf{K}}(G_{\xi},B^{i})\simeq\operatorname{Ext}^{2}_{\mathsf{K}}(G_{\xi},B^{i-1})\simeq\operatorname{Ext}^{3}_{\mathsf{K}}(G_{\xi},B^{i-2})\simeq\dotsb=0$,
since $\operatorname{Ext}^{n}_{\mathsf{K}}(G_{\xi},D^{j})=0$ for all
$j\in\mathbb{Z}$ and $n\geq 1$ as explained above (cf. [33, proof of
Proposition 6.5]). So [33, Lemma 6.4] tells that the short sequence (1) is
exact.

Applying [12, Corollary 8.3] or [14, Corollary A.2] again, we see that both
the classes $\mathsf{B}$ and $\mathsf{D}$ are closed under infinite products
in $\mathsf{K}$. Hence $\prod_{i\in\mathbb{Z}}B^{i}\in\mathsf{B}$ and
$\prod_{i\in\mathbb{Z}}D^{i}\in\mathsf{D}$. So $\prod_{i\in\mathbb{Z}}B^{i}$
is a $\mathsf{D}$-periodic object in $\mathsf{B}$. By Theorem 1.1, it follows
that $\prod_{i\in\mathbb{Z}}B^{i}\in\mathsf{D}$. Finally, the class
$\mathsf{D}$ is closed under direct summands in $\mathsf{K}$, hence
$B^{i}\in\mathsf{D}$ for all $i\in\mathbb{Z}$. ∎

##  2\. Generalized Flat/Projective and Fp-projective Periodicity I

The aim of this section is to prove Theorem 0(a). It is restated below as
Theorem 2.9(a) and Corollary 2.11. The argument follows the ideas of the proof
of [5, Theorems 0.7–0.8 or Corollaries 4.7–4.9]. The result is module-
theoretic, but the proof has a category-theoretic flavor in that the approach
of [5] needs to be applied _within the class $\mathsf{C}$ viewed as an exact
subcategory $\mathsf{C}\subset{\operatorname{\mathsf{Mod--}}}R$_.

Let $\mathsf{K}$ be an exact category (in Quillen’s sense). We suggest the
survey paper [10] as a general reference source on exact categories. The
definition of a (_hereditary_) _cotorsion pair_ $(\mathsf{A},\mathsf{B})$ in
$\mathsf{K}$ was already given in the beginning of Section 1. The intersection
of the two classes $\mathsf{A}\cap\mathsf{B}\subset\mathsf{K}$ is called the
_kernel_ of a cotorsion pair $(\mathsf{A},\mathsf{B})$. Let us define the
important concept of a _complete_ cotorsion pair.

A cotorsion pair $(\mathsf{A},\mathsf{B})$ in $\mathsf{K}$ is said to be
_complete_ if for every object $K\in\mathsf{K}$ there exist (admissible) short
exact sequences in $\mathsf{K}$ of the form

(2) $\displaystyle 0\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5muB^{\prime}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5muA\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5muK\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu0$ (3)
$\displaystyle 0\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muK\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5muB\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5muA^{\prime}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu0$

with $A$, $A^{\prime}\in\mathsf{A}$ and $B$, $B^{\prime}\in\mathsf{B}$. The
sequence (2) is called a _special precover sequence_. The sequence (3) is
called a _special preenvelope sequence_. Collectively, the sequences (2–3) are
referred to as the _approximation sequences_.

Let $\mathsf{E}\subset\mathsf{K}$ be a full subcategory closed under
extensions. Then we endow $\mathsf{E}$ with the exact category structure
_inherited from_ the exact category structure of $\mathsf{K}$. The short exact
sequences in the inherited exact structure on $\mathsf{E}$ are the short exact
sequences in $\mathsf{K}$ with the terms belonging to $\mathsf{E}$.

######  Lemma 2.1.

Let $(\mathsf{C},\mathsf{D})$ be a complete cotorsion pair in an exact
category $\mathsf{K}$. Then the exact category $\mathsf{C}$ (with the exact
structure inherited from $\mathsf{K}$) has enough injective objects. The class
of all injective objects in $\mathsf{C}$ is precisely the kernel
$\mathsf{C}\cap\mathsf{D}$ of the cotorsion pair $(\mathsf{C},\mathsf{D})$.
Dually, the exact category $\mathsf{D}$ has enough projective objects, and the
kernel $\mathsf{C}\cap\mathsf{D}$ is precisely the class of all projectives in
$\mathsf{D}$.

###### Proof.

The proof is left to the reader. ∎

Let $\mathsf{K}$ be an exact category and $\mathsf{E}\subset\mathsf{K}$ be a
full subcategory closed under extensions, endowed with the inherited exact
category structure. Let $(\mathsf{A},\mathsf{B})$ be a complete cotorsion pair
in $\mathsf{K}$. We will say that the cotorsion pair $(\mathsf{A},\mathsf{B})$
_restricts to_ (_a complete cotorsion pair in_) the exact subcategory
$\mathsf{E}$ if the pair of classes ($\mathsf{E}\cap\mathsf{A}$,
$\mathsf{E}\cap\mathsf{B}$) is a complete cotorsion pair in $\mathsf{E}$.

######  Lemma 2.2.

Let $(\mathsf{A},\mathsf{B})$ be a complete cotorsion pair in an exact
category $\mathsf{K}$, and let $\mathsf{E}\subset\mathsf{K}$ be a full
subcategory closed under extensions and kernels of admissible epimorphisms.
Assume that $\mathsf{A}\subset\mathsf{E}$. Then

(a) the cotorsion pair $(\mathsf{A},\mathsf{B})$ restricts to $\mathsf{E}$, so
$(\mathsf{A}$, $\mathsf{E}\cap\mathsf{B})$ is a complete cotorsion pair in
$\mathsf{E}$;

(b) if the cotorsion pair $(\mathsf{A},\mathsf{B})$ is hereditary in
$\mathsf{K}$, then the restricted cotorsion pair $(\mathsf{A}$,
$\mathsf{E}\cap\mathsf{B})$ is hereditary in $\mathsf{E}$.

###### Proof.

This is fairly standard and easy to prove. The details can be found, e. g., in
[29, Lemmas 1.5(a) and 1.6]. ∎

Given an additive category $\mathsf{K}$, we denote by $\mathbf{C}(\mathsf{K})$
the additive category of complexes in $\mathsf{K}$ (with the usual morphisms
of complexes) and by $\mathbf{H}(\mathsf{K})$ the triangulated homotopy
category of complexes in $\mathsf{K}$. So the morphisms in
$\mathbf{H}(\mathsf{K})$ are the cochain homotopy classes of morphisms in
$\mathbf{C}(\mathsf{K})$. When $\mathsf{K}$ is an exact category, the category
$\mathbf{C}(\mathsf{K})$ is endowed with the exact category structure in which
a short sequence of complexes is exact if and only if it is exact at every
degree. We denote by
$K^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longmapsto
K^{\text{\smaller\smaller$\scriptstyle\bullet$}}[n]$ the functor of grading
shift on the complexes; so
$K^{\text{\smaller\smaller$\scriptstyle\bullet$}}[n]^{i}=K^{n+i}$ for all $n$,
$i\in\mathbb{Z}$.

######  Lemma 2.3.

Let $\mathsf{K}$ be an exact category, and let
$A^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ and
$B^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ be two complexes in
$\mathsf{K}$. Assume that $\operatorname{Ext}^{1}_{\mathsf{K}}(A^{n},B^{n})=0$
for every $n\in\mathbb{Z}$. Then there is a natural isomorphism of abelian
groups

$\operatorname{Ext}^{1}_{\mathbf{C}(\mathsf{K})}(A^{\text{\smaller\smaller$\scriptstyle\bullet$}},B^{\text{\smaller\smaller$\scriptstyle\bullet$}})\simeq\operatorname{Hom}_{\mathbf{H}(\mathsf{K})}(A^{\text{\smaller\smaller$\scriptstyle\bullet$}},B^{\text{\smaller\smaller$\scriptstyle\bullet$}}[1]).$

###### Proof.

This is also standard and well-known. More generally, for any two complexes
$A^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ and
$B^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ in $\mathsf{K}$, the
subgroup of _termwise split_ extensions $0\longrightarrow
B^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
C^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
A^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow 0$ in
$\operatorname{Ext}^{1}_{\mathsf{K}}(A^{\text{\smaller\smaller$\scriptstyle\bullet$}},B^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is naturally isomorphic to the group of morphisms
$A^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
B^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ in the homotopy category
$\mathbf{H}(\mathsf{K})$. We refer to [5, Lemma 1.6] for the details. ∎

At this point, let us specialize our discussion to Grothendieck abelian
categories $\mathsf{K}$. Let $F\in\mathsf{K}$ be an object and $\alpha$ be an
ordinal. A family of subobjects $(F_{\beta}\subset F)_{0\leq\beta\leq\alpha}$
is said to be an _$\alpha$ -indexed filtration_ on $F$ if the following
conditions are satisfied:

  * •

$F_{0}=0$ and $F_{\alpha}=F$;

  * •

$F_{\gamma}\subset F_{\beta}$ for all $0\leq\gamma\leq\beta\leq\alpha$;

  * •

$F_{\beta}=\bigcup_{\gamma<\beta}F_{\gamma}$ for all limit ordinals
$\beta\leq\alpha$.

An object $F\in\mathsf{K}$ endowed with an ordinal-indexed filtration
$(F_{\beta})_{0\leq\beta\leq\alpha}$ is said to be _filtered by_ the quotient
objects $S_{\beta}=F_{\beta+1}/F_{\beta}$, $0\leq\beta<\alpha$. In an
alternative terminology, the object $F$ is called a _transfinitely iterated
extension_ (_in the sense of the direct limit_) of the objects
$(S_{\beta})_{0\leq\beta<\alpha}$.

Given a class of objects $\mathsf{S}\subset\mathsf{K}$, the class of all
objects in $\mathsf{K}$ filtered by (objects isomorphic to) objects from
$\mathsf{S}$ is denoted by $\mathsf{Fil}(\mathsf{S})\subset\mathsf{K}$. A
class of objects $\mathsf{F}\subset\mathsf{K}$ is said to be _deconstructible_
if there exists a _set_ of objects $\mathsf{S}\subset\mathsf{K}$ such that
$\mathsf{F}=\mathsf{Fil}(\mathsf{S})$. It is easy to see that any
deconstructible class (in the sense of this definition) is closed under
transfinitely iterated extensions.

The following result is known as the _Eklof lemma_ [17, Lemma 1], [22, Lemma
6.2].

######  Lemma 2.4.

For any class of objects $\mathsf{B}\subset\mathsf{K}$, the class
${}^{\perp_{1}}\mathsf{B}$ is closed under transfinitely iterated exensions.
In other words,
$\mathsf{Fil}({}^{\perp_{1}}\mathsf{B})={}^{\perp_{1}}\mathsf{B}$.

###### Proof.

This assertion, properly understood (as per the definitions in Section 4
below), holds in any exact category $\mathsf{K}$. See the references in [5,
Lemma 1.1]. The general formulation can be also found in [33, Lemma 4.4.]. ∎

The next theorem is goes back to Eklof and Trlifaj [17, Theorems 2 and 10],
[22, Theorem 6.11 and Corollary 6.14]. For any class of objects
$\mathsf{F}\subset\mathsf{K}$, we denote by
$\mathsf{F}^{\oplus}\subset\mathsf{K}$ the class of all direct summands of
objects from $\mathsf{F}$ in $\mathsf{K}$.

######  Theorem 2.5.

Let $\mathsf{K}$ be a Grothendieck category and $(\mathsf{A},\mathsf{B})$ be
the cotorsion pair generated by a _set_ of objects
$\mathsf{S}\subset\mathsf{K}$. Then

(a) If the class $\mathsf{A}$ is generating in $\mathsf{K}$, then the
cotorsion pair $(\mathsf{A},\mathsf{B})$ is complete.

(b) If the class $\mathsf{Fil}(\mathsf{S})$ is generating in $\mathsf{K}$,
then $\mathsf{A}=\mathsf{Fil}(\mathsf{S})^{\oplus}$.

###### Proof.

This result, properly stated, holds in any locally presentable abelian
category $\mathsf{K}$. See [5, Theorem 1.2] for a discussion with references,
and Theorem 4.3 below for a version for efficient exact categories. ∎

We refer to the book [1, Definition 1.9 and Theorem 1.11] for the definition
of a _locally finitely presentable_ category. Any locally finitely presentable
abelian category is Grothendieck [1, Proposition 1.59]. We will have a
detailed discussion of such categories below in Section 3, where several
further references are suggested. The abelian category of modules
${\operatorname{\mathsf{Mod--}}}R$ is locally finitely presentable for any
ring $R$.

The following result of Št’ovíček was already used in a similar way in the
paper [5], where it is stated as [5, Lemma 3.4].

######  Proposition 2.6.

Let $\mathsf{K}$ be a locally finitely presentable abelian category and
$\mathsf{S}\subset\mathsf{K}$ be a class of finitely presentable objects
closed under extensions in $\mathsf{K}$. Let
$A^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ be a complex in
$\mathsf{K}$ whose terms are $\mathsf{S}$-filtered objects. Then the complex
$A^{\text{\smaller\smaller$\scriptstyle\bullet$}}$, viewed as an object of the
abelian category of complexes $\mathbf{C}(\mathsf{K})$, is filtered by bounded
below complexes of objects from $\mathsf{S}$.

###### Proof.

This is the particular case of [42, (proof of) Proposition 4.3] for the
countable cardinal $\kappa=\aleph_{0}$. The argument is based on the Hill
lemma ([42, Theorem 2.1] or [22, Theorem 7.10]). ∎

In addition to the abelian exact structure on the module category
$\mathsf{K}={\operatorname{\mathsf{Mod--}}}R$, we are interested in the pure
exact structure. The definition of the pure exact structure on
${\operatorname{\mathsf{Mod--}}}R$ was already given in Section 1. A complex
in ${\operatorname{\mathsf{Mod--}}}R$ is said to be _pure acyclic_ (or _pure
exact_) if it is acyclic in the pure exact structure, i. e., can be obtained
by splicing pure short exact sequences. The following result due to Neeman
[26] and Št’ovíček [44] is a stronger version of the pure pure-projective
periodicity theorem (item (2) or (2${}^{\text{c}}$) on the list of Section
0.0).

######  Theorem 2.7.

Let $R$ be an associative ring. Let
$P^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ be a complex of pure-
projective $R$-modules, and let
$X^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ be a pure acyclic complex
of $R$-modules. Then any morphism of complexes of $R$-modules
$P^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
X^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is homotopic to zero.

###### Proof.

This was first stated in [44, Theorem 5.4] based on [26, Theorem 8.6]. We
refer to the paper [4, Theorem 1.1] for a generalization, and to [5, Section
0.2 and proof of Theorem 4.3] for a discussion with some details. ∎

Now let $\mathsf{S}$ be a class of finitely presented $R$-modules closed under
finite direct sums and containing the free $R$-module $R$. Put
$\mathsf{C}=\varinjlim\mathsf{S}\subset{\operatorname{\mathsf{Mod--}}}R$.

######  Lemma 2.8.

The full subcategory $\mathsf{C}=\varinjlim\mathsf{S}$ is closed under pure
extensions (as well as pure submodules and pure epimorphic images) in
${\operatorname{\mathsf{Mod--}}}R$. In the exact category structure on
$\mathsf{C}$ inherited from the pure exact structure on
${\operatorname{\mathsf{Mod--}}}R$, a short sequence $0\longrightarrow
K\longrightarrow L\longrightarrow M\longrightarrow 0$ is exact if and only if
the short sequence of abelian groups
$0\longrightarrow\operatorname{Hom}_{R}(S,K)\longrightarrow\operatorname{Hom}_{R}(S,L)\longrightarrow\operatorname{Hom}_{R}(S,M)\longrightarrow
0$ is exact for every module $S\in\mathsf{S}$.

###### Proof.

The first assertion is the result of [25, Proposition 2.2]. The second
assertion claims that a short sequence $0\longrightarrow K\longrightarrow
L\longrightarrow M\longrightarrow 0$ with $K$, $L$, $M\in\mathsf{C}$ is pure
exact in ${\operatorname{\mathsf{Mod--}}}R$ if and only if the functor
$\operatorname{Hom}_{R}(S,{-})$ takes it to a short exact sequence for every
$S\in\mathsf{S}$. The point is that any morphism $T\longrightarrow M$ from a
finitely presented $R$-module $T$ into the module
$M\in\mathsf{C}=\varinjlim\mathsf{S}$ factorizes through some module
$S\in\mathsf{S}$. So if every morphism $S\longrightarrow M$ lift to a morphism
$S\longrightarrow L$, then also every morphism $T\longrightarrow M$ lifts to a
morphism $T\longrightarrow L$. ∎

Now we can formulate and prove the main results of the section (though we will
need yet another lemma in between).

######  Theorem 2.9.

Let $R$ be a ring and $\mathsf{S}$ be a class of finitely presented
$R$-modules, containing the free $R$-module $R$ and closed under extensions
and the kernels of epimorphisms. Put $\mathsf{B}=\mathsf{S}^{\perp_{1}}$,
$\mathsf{A}={}^{\perp_{1}}\mathsf{B}$, $\mathsf{C}=\varinjlim\mathsf{S}$, and
$\mathsf{D}=\mathsf{C}^{\perp_{1}}\subset{\operatorname{\mathsf{Mod--}}}R$.
Then

(a) in any acyclic complex of modules from $\mathsf{A}$ with the modules of
cocycles belonging to $\mathsf{C}$, the modules of cocycles actually belong to
$\mathsf{A}$;

(b) let $A^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ be a complex in
${\operatorname{\mathsf{Mod--}}}R$ whose terms belong to $\mathsf{A}$, and let
$X^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ be an acyclic complex in
${\operatorname{\mathsf{Mod--}}}R$ whose terms belong to
$\mathsf{B}\cap\mathsf{C}$ and the modules of cocycles also belong to
$\mathsf{B}\cap\mathsf{C}$. Then any morphism of complexes of modules
$A^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
X^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is homotopic to zero.

The following lemma tells that modules from the class
$\mathsf{B}\cap\mathsf{C}$ are “absolutely pure within the exact category
$\mathsf{C}$”.

######  Lemma 2.10.

In the notation of Theorem 0 or Theorem 2.9, let $0\longrightarrow
B\longrightarrow L\longrightarrow C\longrightarrow 0$ be a short exact
sequence in ${\operatorname{\mathsf{Mod--}}}R$ with the terms $B$, $L$,
$C\in\mathsf{C}$. Assume that the module $B$ belongs to the class
$\mathsf{B}\cap\mathsf{C}$. Then the short exact sequence $0\longrightarrow
B\longrightarrow L\longrightarrow C\longrightarrow 0$ is pure in
${\operatorname{\mathsf{Mod--}}}R$.

###### Proof.

It is only important that $C\in\mathsf{C}$ and $B\in\mathsf{B}$. By Lemma 2.8,
it suffices to check that any morphism $S\longrightarrow C$ with
$S\in\mathsf{S}$ lifts to a morphism $S\longrightarrow L$. This holds because
$B\in\mathsf{B}=\mathsf{S}^{\perp_{1}}\subset{\operatorname{\mathsf{Mod--}}}R$.
∎

###### Proof of Theorem 2.9(b).

The argument follows the ideas of the proof of [5, Theorem 4.2], with suitable
modifications. By the Eklof–Trlifaj theorem (Theorem 2.5(b)), we have
$\mathsf{A}=\mathsf{Fil}(\mathsf{S})^{\oplus}$. Without loss of generality we
can assume that the terms of the complex
$A^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ belong to
$\mathsf{Fil}(\mathsf{S})$. Then, by Proposition 2.6 (applied in the case of
the module category $\mathsf{K}={\operatorname{\mathsf{Mod--}}}R$), the
complex $A^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is filtered by
(bounded below) complexes with the terms belonging to $\mathsf{S}$.

By Lemma 2.3, for any complex
$A^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ with the terms in
$\mathsf{A}$ and any complex
$B^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ with the terms in
$\mathsf{B}$ we have an isomorphism of abelian groups

$\operatorname{Ext}^{1}_{\mathbf{C}({\operatorname{\mathsf{Mod--}}}R)}(A^{\text{\smaller\smaller$\scriptstyle\bullet$}},B^{\text{\smaller\smaller$\scriptstyle\bullet$}}[-1])\simeq\operatorname{Hom}_{\mathbf{H}({\operatorname{\mathsf{Mod--}}}R)}(A^{\text{\smaller\smaller$\scriptstyle\bullet$}},B^{\text{\smaller\smaller$\scriptstyle\bullet$}}).$

So, instead of showing that
$\operatorname{Hom}_{\mathbf{H}({\operatorname{\mathsf{Mod--}}}R)}(A^{\text{\smaller\smaller$\scriptstyle\bullet$}},X^{\text{\smaller\smaller$\scriptstyle\bullet$}})=0$
as desired in the theorem, it suffices to prove that
$\operatorname{Ext}^{1}_{\mathbf{C}({\operatorname{\mathsf{Mod--}}}R)}(A^{\text{\smaller\smaller$\scriptstyle\bullet$}},X^{\text{\smaller\smaller$\scriptstyle\bullet$}}[-1])=0$.
In view of the Eklof lemma (Lemma 2.4) applied in the abelian category
$\mathsf{K}=\mathbf{C}({\operatorname{\mathsf{Mod--}}}R)$, the question
reduces to showing that
$\operatorname{Ext}^{1}_{\mathbf{C}({\operatorname{\mathsf{Mod--}}}R)}(S^{\text{\smaller\smaller$\scriptstyle\bullet$}},X^{\text{\smaller\smaller$\scriptstyle\bullet$}}[-1])=0$
for any complex $S^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ with the
terms belonging to $\mathsf{S}$ and any complex
$X^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ as in the theorem. Using
Lemma 2.3 again, we conclude that it suffices to show that any morphism of
complexes $S^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
X^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is homotopic to zero.

Finally, we observe that all finitely presented $R$-modules are pure-
projective (by the definitions), while any acyclic complex of modules with the
modules of cocycles in $\mathsf{B}\cap\mathsf{C}$ is pure acyclic (by Lemma
2.10). Thus any morphism of complexes
$S^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
X^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is homotopic to zero by the
Neeman–Št’ovíček theorem (Theorem 2.7). ∎

###### Proof of Theorem 2.9(a).

It is clear that in the assumptions of the theorem all the modules from
$\mathsf{S}$ have to be strongly finitely presented ($\mathrm{FP}_{\infty}$).
Thus [3, Theorem 2.3 and Corollary 2.4] or [22, Theorem 8.40, Corollary 8.42,
and Theorem 6.19] are applicable, telling that $(\mathsf{C},\mathsf{D})$ is a
complete cotorsion pair in ${\operatorname{\mathsf{Mod--}}}R$. The cotorsion
pair $(\mathsf{A},\mathsf{B})$ is complete in
${\operatorname{\mathsf{Mod--}}}R$ by the Eklof–Trlifaj theorem (Theorem
2.5(a)).

Both the cotorsion pairs $(\mathsf{A},\mathsf{B})$ and
$(\mathsf{C},\mathsf{D})$ are hereditary, as it was explained in the proof of
Theorem 0(b) in Section 1. Applying Lemma 2.2 to the abelian category
$\mathsf{K}={\operatorname{\mathsf{Mod--}}}R$ and the full subcategory
$\mathsf{E}=\mathsf{C}$, we conclude that ($\mathsf{A}$,
$\mathsf{B}\cap\mathsf{C}$) is a hereditary complete cotorsion pair in the
exact category $\mathsf{C}$. Lemma 2.1 tells that there are enough injective
objects in the exact category $\mathsf{C}$, and the class of such injective
objects coincides with the intersection $\mathsf{C}\cap\mathsf{D}$.

Let $A^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ be an acyclic complex
of modules from $\mathsf{A}$. Then one can easily see that the modules of
cocycles of $A^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ belong to
$\mathsf{A}$ if and only if the complex of abelian groups
$\operatorname{Hom}_{R}(A^{\text{\smaller\smaller$\scriptstyle\bullet$}},B)$
is acyclic for any module $B\in\mathsf{B}$. This holds because
$(\mathsf{A},\mathsf{B})$ is a cotorsion pair in
${\operatorname{\mathsf{Mod--}}}R$ (so $\mathsf{A}={}^{\perp_{1}}\mathsf{B}$).
We will use a version of this observation made within the exact category
$\mathsf{C}$.

So let $A^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ be an acyclic
complex of modules from $\mathsf{A}$ with the modules of cocycles belonging to
$\mathsf{C}$. Then we observe that the modules of cocycles of
$A^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ belong to $\mathsf{A}$ if
and only if the complex
$\operatorname{Hom}_{R}(A^{\text{\smaller\smaller$\scriptstyle\bullet$}},B)$
is acyclic for any module $B\in\mathsf{B}\cap\mathsf{C}$. This holds because
($\mathsf{A}$, $\mathsf{B}\cap\mathsf{C}$) is a cotorsion pair in
$\mathsf{C}$, so
$\mathsf{A}=\mathsf{C}\cap{}^{\perp_{1}}(\mathsf{B}\cap\mathsf{C})$.

Now let $D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ be an injective
resolution of the object $B$ in the exact category $\mathsf{C}$. So we have
$B\in\mathsf{B}\cap\mathsf{C}$ by assumption, and $0\longrightarrow
B\longrightarrow D^{0}\longrightarrow D^{1}\longrightarrow
D^{2}\longrightarrow\dotsb$ is an acyclic complex in
${\operatorname{\mathsf{Mod--}}}R$ with the modules
$D^{n}\in\mathsf{C}\cap\mathsf{D}$ and the modules of cocycles belonging to
$\mathsf{C}$. We observe that the modules of cocycles of the complex
$D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ actually belong to
$\mathsf{B}\cap\mathsf{C}$, because $\mathsf{D}\subset\mathsf{B}$ and the
class $\mathsf{B}$ is closed under cokernels of monomorphisms. Essentially,
this is a restatement of the claim that the cotorsion pair
$(\mathsf{A},\mathsf{B})$ is hereditary in ${\operatorname{\mathsf{Mod--}}}R$,
or more specifically, that the cotorsion pair ($\mathsf{A}$,
$\mathsf{B}\cap\mathsf{C}$) is hereditary in $\mathsf{C}$.

Denote by $X^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ the acyclic
complex $(B\to D^{\text{\smaller\smaller$\scriptstyle\bullet$}})$. Then the
complex of abelian groups
$\operatorname{Hom}_{R}(A^{\text{\smaller\smaller$\scriptstyle\bullet$}},X^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is acyclic by Theorem 2.9(b) (which we have proved above). This holds because
$A^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is a complex with the terms
in $\mathsf{A}$, while $X^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is
an acyclic complex with the terms in $\mathsf{B}\cap\mathsf{C}$ and the
modules of cocycles in $\mathsf{B}\cap\mathsf{C}$.

On the other hand, the complex of abelian groups
$\operatorname{Hom}_{R}(A^{\text{\smaller\smaller$\scriptstyle\bullet$}},D^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is acyclic as well. This holds quite generally for any acyclic complex
$A^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ and any bounded below
complex of injective objects
$D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ in any exact category
$\mathsf{C}$. Notice that in the situation at hand the complex of modules
$A^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is acyclic in the exact
category $\mathsf{C}$, as its modules of cocycles belong to $\mathsf{C}$ by
assumption.

Since both the complexes
$\operatorname{Hom}_{R}(A^{\text{\smaller\smaller$\scriptstyle\bullet$}},X^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
and
$\operatorname{Hom}_{R}(A^{\text{\smaller\smaller$\scriptstyle\bullet$}},D^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
are acyclic, and the complex
$X^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ has the form
$X^{\text{\smaller\smaller$\scriptstyle\bullet$}}=(B\to
D^{\text{\smaller\smaller$\scriptstyle\bullet$}})$, we can finally conclude
that the complex of abelian groups
$\operatorname{Hom}_{R}(A^{\text{\smaller\smaller$\scriptstyle\bullet$}},B)$
is acyclic. ∎

######  Corollary 2.11.

Let $R$ be a ring and $\mathsf{S}$ be a class of finitely presented
$R$-modules, containing the free $R$-module $R$ and closed under extensions
and the kernels of epimorphisms. Put
$\mathsf{A}={}^{\perp_{1}}(\mathsf{C}^{\perp_{1}})$ and
$\mathsf{C}=\varinjlim\mathsf{S}$. Then, for any short exact sequence ($*$ ‣
0.0) as in Section 0.0 with modules $L\in\mathsf{A}$ and $M\in\mathsf{C}$, one
has $M\in\mathsf{A}$. In other words, any $\mathsf{A}$-periodic module
belonging to $\mathsf{C}$ actually belongs to $\mathsf{A}$.

###### Proof.

This is a corollary of Theorem 2.9(a), provable by splicing up a doubly
unbounded sequence of short exact sequences ($*$ ‣ 0.0) and applying Theorem
2.9(a) to the resulting doubly unbounded complex. (Cf. [18, Proposition 1].) ∎

###### Proof of Theorem 0(a) from Section 0.1.

The first assertion of Theorem 0(a) is provided by Corollary 2.11, and the
second one by Theorem 2.9(a). ∎

##  3\. Direct Limit Closures of Classes of Finitely Presentables

The aim of this section is to prove Theorems A(i) and B(ii). They are restated
below as Propositions 3.3 and 3.5.

In this section we work with _locally finitely presentable_ abelian
categories. We suggest the book [1] as a general reference source on
nonadditive locally (finitely) presentable and (finitely) accessible
categories.

The definitions of a finitely presentable object and and a locally finitely
presentable category can be found in [1, Definitions 1.1 and 1.9, and Theorem
1.11] (it is helpful to keep in mind that in abelian categories the notions of
a generator and a strong generator coincide). All locally finitely presentable
abelian categories have exact direct limit functors, so they are Grothendieck
[1, Proposition 1.59]. The abelian category of modules over an arbitrary ring
$\mathsf{K}={\operatorname{\mathsf{Mod--}}}R$ is an important example of a
locally finitely presentable abelian category.

Finitely accessible categories [1, Definition 2.1 and Remark 2.2(1)] form a
wider class than the locally finitely presentable ones. The theory of finitely
accessible additive categories goes back to the paper [25, Section 2] (where
they were not defined yet). Subsequently they were studied in the papers [15,
24] under the name of “locally finitely presented” additive categories.

We suggest [32, Sections 8.1–8.2] as an additional reference source on locally
finitely presentable abelian categories. Our proof of Theorem A(i)
(Proposition 3.3 below) is a slight generalization of the argument in [32,
Proposition 8.4].

_Locally finitely generated_ categories also form a wider class than the
locally finitely presentable ones. We refer to [1, Section 1.E] for a general
discussion of locally generated (nonadditive) categories and to [31, Corollary
9.6] for a very general form of the assertion that any locally finitely
generated abelian category is Grothendieck. A good reference source on locally
finitely generated Grothendieck categories and finitely generated/finitely
presentable objects in them is [41, §V.3].

The following definitions are very general. Let $\mathsf{K}$ be a category
with direct limits. An object $S\in\mathsf{K}$ is said to be _finitely
presentable_ if the functor
$\operatorname{Hom}_{\mathsf{K}}(P,{-})\colon\mathsf{K}\longrightarrow\mathsf{Sets}$
preserves direct limits. An object $S$ is said to be _finitely generated_ if
the same functor preserves the direct limits of diagrams of monomorphisms.

An abelian category $\mathsf{K}$ with set-indexed coproducts is said to be
_locally finitely generated_ if it has a set of generators consisting of
finitely generated objects. In particular, the category $\mathsf{K}$ is
_locally finitely presentable_ if it has a set of generators consisting of
finitely presentable objects.

Given a locally finitely presentable abelian category $\mathsf{K}$, we denote
by $\mathsf{K}_{\mathsf{fp}}\subset\mathsf{K}$ the full subcategory of
finitely presentable objects in $\mathsf{K}$. The full subcategory
$\mathsf{K}_{\mathsf{fp}}$ is closed under cokernels [1, Proposition 1.3] and
extensions [32, Lemma 8.1] in $\mathsf{K}$. Similarly, the full subcategory of
finitely generated objects in a locally finitely generated abelian category
$\mathsf{K}$ is closed under extensions and quotients [41, Lemma V.3.1 and
Proposition V.3.2].

######  Proposition 3.1.

Let $\mathsf{K}$ be a locally finitely presentable abelian category and
$\mathsf{S}\subset\mathsf{K}_{\mathsf{fp}}$ be a class of finitely presentable
objects closed under finite direct sums. Then the class of objects
$\varinjlim\mathsf{S}\subset\mathsf{K}$ is closed under coproducts and direct
limits in $\mathsf{K}$. An object $L\in\mathsf{K}$ belongs to
$\varinjlim\mathsf{S}$ if and only if, for any object
$T\in\mathsf{K}_{\mathsf{fp}}$, any morphism $T\longrightarrow L$ in
$\mathsf{K}$ factorizes through an object from $\mathsf{S}$.

###### Proof.

This is [25, Proposition 2.1], [15, Section 4.1], or [24, Proposition 5.11]. ∎

######  Corollary 3.2.

Let $\mathsf{K}$ be a locally finitely presentable abelian category and
$\mathsf{S}\subset\mathsf{K}_{\mathsf{fp}}$ be a class of finitely presentable
objects closed under finite direct sums. Let $(H_{i})_{i\in I}$ be a direct
system of objects $H_{i}\in\varinjlim\mathsf{S}$, indexed by a directed poset
$I$. Then the kernel of the natural epimorphism

(4) $\coprod\nolimits_{i\in I}H_{i}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\varinjlim\nolimits_{i\in I}H_{i}$

belongs to $\varinjlim\mathsf{S}$.

###### Proof.

One can argue from purity considerations, observing that the epimorphism (4)
is pure (in the sense of the definition from [15, Section 3], [44, Section 4]
reproduced below in Section 5) and the class $\varinjlim\mathsf{S}$ is closed
under pure subobjects by the category-theoretic version of [25, Proposition
2.2]. Alternatively, one can notice that the kernel of (4) is a direct limit
of coproducts of copies of the objects $H_{i}$, following [6, proof of
Proposition 4.1]; then it remains to refer to Proposition 3.1. ∎

The following definitions appeared in the papers [21, 9]. Let $\mathsf{K}$ be
a Grothendieck category and $n\geq 1$ be an integer. An object
$S\in\mathsf{K}$ is said to be _of type $\mathrm{FP}_{n}$_ if the functors
$\operatorname{Ext}_{\mathsf{K}}^{i}(S,{-})\colon\mathsf{K}\longrightarrow\mathsf{Ab}$
preserve direct limits for all $0\leq i\leq n-1$. So the objects of type
$\mathrm{FP}_{1}$ are, by the definition, the finitely presentable ones, while
the objects of type $\mathrm{FP}_{n}$ for $n\geq 2$ form more narrow classes.

An object $S\in\mathsf{K}$ is said to be _of type $\mathrm{FP}_{0}$_ if it is
finitely generated. An object $S$ is said to be _of type
$\mathrm{FP}_{\infty}$_ if it is of type $\mathrm{FP}_{n}$ for every $n\geq
0$, that is, in other words, the functors
$\operatorname{Ext}_{\mathsf{K}}^{i}(S,{-})\colon\mathsf{K}\longrightarrow\mathsf{Ab}$
preserve direct limits for all $i\geq 0$.

We use the term _strongly finitely presentable object_ as a synonym for “type
$\mathrm{FP}_{\infty}$”. In the case of the module category
$\mathsf{K}={\operatorname{\mathsf{Mod--}}}R$, these definitions are
equivalent to the ones from Section 1 (see [9, Corollary 2.14]).

Closure properties of the classes of objects of type $\mathrm{FP}_{n}$ and
$\mathrm{FP}_{\infty}$ in locally finitely presentable abelian categories
$\mathsf{K}$ are listed in [21, Corollary 3.3] and [9, Proposition 2.8]. In
particular, [9, Proposition 2.8(1)] tells that the class of all objects of
type $\mathrm{FP}_{n}$ is closed under extensions in $\mathsf{K}$.

######  Proposition 3.3.

Let $\mathsf{K}$ be a locally finitely presentable abelian category and
$\mathsf{S}$ be a class of (some) objects of type $\mathrm{FP}_{2}$ closed
under extensions in $\mathsf{K}$. Then the class of objects
$\varinjlim\mathsf{S}$ is also closed under extensions in $\mathsf{K}$.

###### Proof.

We follow the proof of [32, Proposition 8.4]. Given an abelian category
$\mathsf{K}$ and two classes of objects $\mathsf{X}$,
$\mathsf{Y}\subset\mathsf{K}$, denote by $\mathsf{X}*\mathsf{Y}$ the class of
all objects $Z\in\mathsf{K}$ for which there exists a short exact sequence
$0\longrightarrow X\longrightarrow Z\longrightarrow Y\longrightarrow 0$ in
$\mathsf{K}$ with $X\in\mathsf{X}$ and $Y\in\mathsf{Y}$. In the situation at
hand, we need to prove that
$\varinjlim\mathsf{S}*\varinjlim\mathsf{S}\subset\varinjlim\mathsf{S}$. For
this purpose, we claim that the three inclusions

(5)
$\varinjlim\mathsf{T}*\varinjlim\mathsf{T}\subset\varinjlim(\varinjlim\mathsf{T}*\mathsf{T})\subset\varinjlim\varinjlim(\mathsf{T}*\mathsf{T})\subset\varinjlim(\mathsf{T}*\mathsf{T})$

hold for any class of $\mathrm{FP}_{2}$ objects $\mathsf{T}$ closed under
finite direct sums in $\mathsf{K}$.

More generally, it is explained in [32, first part of the proof of Proposition
8.4] that the inclusion
$\mathsf{X}*\varinjlim\mathsf{Y}\subset\varinjlim(\mathsf{X}*\mathsf{Y})$
holds for any two classes of objects $\mathsf{X}$ and $\mathsf{Y}$ in a
Grothendieck category $\mathsf{K}$. This takes care of the leftmost inclusion
in (5).

Furthermore, any $\mathrm{FP}_{2}$ object is, by definition, finitely
presentable; and the class of all finitely presentable objects is closed under
extensions in $\mathsf{K}$ by [32, Lemma 8.1]. So we certainly have
$\mathsf{T}*\mathsf{T}\subset\mathsf{K}_{\mathsf{fp}}$. It is clear that the
class $\mathsf{T}*\mathsf{T}$ is closed under finite direct sums whenever the
class $\mathsf{T}$ is. For any class of finitely presentable objects
$\mathsf{S}$ closed under finite direct sums in $\mathsf{K}$, we have
$\varinjlim\varinjlim\mathsf{S}=\varinjlim\mathsf{S}$ by Proposition 3.1. This
explains the rightmost inclusion in (5).

Finally, the middle inclusion in (5) is provable similarly to the proof in
[32]. Let $\mathsf{K}$ be a Grothendieck abelian category,
$\mathsf{X}\subset\mathsf{K}$ be a class of objects, and
$\mathsf{T}\subset\mathsf{K}$ be a class of objects such that the functor
$\operatorname{Ext}^{1}_{\mathsf{K}}(T,{-})\colon\mathsf{K}\longrightarrow\mathsf{Ab}$
preserves direct limits for all objects $T\in\mathsf{T}$. We claim that the
inclusion
$(\varinjlim\mathsf{X})*\mathsf{T}\subset\varinjlim(\mathsf{X}*\mathsf{T})$
holds. The argument from [32, second part of the proof of Proposition 8.4]
applies. ∎

######  Lemma 3.4.

Let $\mathsf{K}$ be a Grothendieck category and $\mathsf{S}\subset\mathsf{K}$
be a class of finitely generated objects containing a set of generators of
$\mathsf{K}$ and closed under finite direct sums and kernels of epimorphisms.
Then all the objects from $\mathsf{S}$ are strongly finitely presentable (type
$\mathrm{FP}_{\infty}$).

###### Proof.

First of all, the category $\mathsf{K}$ is locally finitely generated, since
it has a set of finitely generated generators by assumption. In this context,
it is explained in [41, Proposition V.3.4] that an object $S\in\mathsf{K}$ is
finitely presentable if and only if the kernel of any epimorphism onto $S$
from any finitely generated object $T\in\mathsf{K}$ is finitely generated.
Following the argument in [41], based on [41, Lemma V.3.3], one can see that
it suffices to let $T$ range over the quotient objects of finite direct sums
of objects from a chosen set $\mathsf{G}$ of finitely generated generators of
$\mathsf{K}$. Furthermore, the passage to the quotients holds automatically,
and so it suffices to let $T$ range over the finite direct sums of objects
from $\mathsf{G}$. In the situation at hand, choosing
$\mathsf{G}\subset\mathsf{S}$, we conclude that all the objects in
$\mathsf{S}$ are finitely presentable.

The rest of the proof is similar to [32, proof of Lemma 8.3]. For any two
objects $X$ and $Y$ in any abelian category $\mathsf{K}$, the abelian group
$\operatorname{Ext}^{n}_{\mathsf{K}}(X,Y)$ can be computed as the filtered
direct limit of cohomology groups
$H^{n}\operatorname{Hom}_{\mathsf{K}}(R_{\text{\smaller\smaller$\scriptstyle\bullet$}},Y)$,
taken over the (large) filtered category of exact complexes
$\dotsb\longrightarrow R_{2}\longrightarrow R_{1}\longrightarrow
R_{0}\longrightarrow X\longrightarrow 0$ in $\mathsf{K}$. Here the morphisms
in the category of such arbitrary resolutions
$R_{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow X$ are the
usual morphisms of complexes acting by the identity maps on the object $X$ and
viewed up to cochain homotopy.

In the situation at hand, for any object $S\in\mathsf{S}$, the full
subcategory of resolutions
$T_{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow S$ consisting
of objects $T_{i}\in\mathsf{S}$ is cofinal in the category of all resolutions
$R_{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow S$ with
$R_{i}\in\mathsf{K}$. So one can compute the group
$\operatorname{Ext}^{n}_{\mathsf{K}}(S,Y)$ as the filtered direct limit of
$H^{n}\operatorname{Hom}_{\mathsf{K}}(T_{\text{\smaller\smaller$\scriptstyle\bullet$}},Y)$
taken over all the resolutions
$T_{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow S$ with
$T_{i}\in\mathsf{S}$. Now, since all the objects of $\mathsf{S}$ are finitely
presentable, the functor
$\operatorname{Hom}_{\mathsf{K}}(T_{\text{\smaller\smaller$\scriptstyle\bullet$}},{-})$
takes direct limits in $\mathsf{K}$ to direct limits of complexes of abelian
groups. It remains to recall that the functors of cohomology of a complex of
abelian groups preserve direct limits, and direct limits commute with direct
limits. ∎

######  Proposition 3.5.

Let $\mathsf{K}$ be a Grothendieck category and $\mathsf{S}\subset\mathsf{K}$
be a class of finitely generated objects containing a set of generators of
$\mathsf{K}$ and closed under extensions and kernels of epimorphisms. Then the
class of objects $\varinjlim\mathsf{S}\subset\mathsf{K}$ is closed under
coproducts, direct limits, extensions, and kernels of epimorphisms.

###### Proof.

By Lemma 3.4, all objects in $\mathsf{S}$ are finitely presentable, and in
fact even strongly finitely presentable. As the class $\mathsf{S}$ contains a
set of generators for $\mathsf{K}$, it follows that the category $\mathsf{K}$
is locally finitely presentable. So Proposition 3.1 is applicable, telling
that the class $\varinjlim\mathsf{S}$ is closed under coproducts and direct
limits in $\mathsf{K}$. Furthermore, Proposition 3.3 tells that the class
$\varinjlim\mathsf{S}$ is closed under extensions. It remains to prove its
closedness under kernels of epimorphisms.

Let $C\longrightarrow D$ be an epimorphism in $\mathsf{K}$ between two objects
$C$, $D\in\varinjlim\mathsf{S}$. Then there exists a direct system
$(S_{i})_{i\in I}$, indexed by a directed poset $I$, such that
$S_{i}\in\mathsf{S}$ for all $i\in I$ and $D=\varinjlim_{i\in I}S_{i}$. For
every $i\in I$, consider the pullback diagram

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0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
31.93619pt\raise-37.75664pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{D}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{gathered}$

Here $C_{i}$ is the pullback of the given epimorphism $C\longrightarrow D$ and
the natural morphism to the direct limit $S_{i}\longrightarrow D$. As the
index $i\in I$ varies, the upper lines of (6) form a direct system of
(epi)morphisms in $\mathsf{K}$, whose direct limit is the epimorphism
$C\longrightarrow D$ in the lower line of the diagram.

Choose a set of generators $\mathsf{G}\subset\mathsf{S}$ of the category
$\mathsf{K}$, and put $H=\coprod_{G\in\mathsf{G}}G$. For every index $i\in I$,
denote by $\Xi_{i}$ the underlying set of the image of the abelian group map
$\operatorname{Hom}_{\mathsf{K}}(H,C_{i})\longrightarrow\operatorname{Hom}_{\mathsf{K}}(H,S_{i})$
induced by the morphism $C_{i}\longrightarrow S_{i}$. Then, for every pair of
indices $i<j\in I$, the transition morphism $S_{i}\longrightarrow S_{j}$
induces a map of sets $\Xi_{i}\longrightarrow\Xi_{j}$. So we obtain a direct
system of sets $(\Xi_{i})_{i\in I}$.

For any object $K\in\mathsf{K}$ and any set $\Xi$, let us denote by
$K^{(\Xi)}$ the coproduct of $\Xi$ copies of $K$ in $\mathsf{K}$. Notice that
the assigment $(K,\Xi)\longmapsto K^{(\Xi)}$ is a covariant functor
$\mathsf{K}\times\mathsf{Sets}\longrightarrow\mathsf{K}$ (i. e., a covariant
functor of both the arguments $K\in\mathsf{K}$ and $\Xi\in\mathsf{Sets}$).

For every index $i\in I$ we have a natural morphism $h_{i}\colon
H^{(\Xi_{i})}\longrightarrow S_{i}$ in $\mathsf{K}$. Since $H$ is a generator
of the category $\mathsf{K}$, and the morphism $C_{i}\longrightarrow S_{i}$ is
an epimorphism, the morphism $h_{i}$ is an epimorphism in $\mathsf{K}$ as
well. As the index $i$ varies, the morphisms $h_{i}$ form a direct system
$(h_{i})_{i\in I}$ in the category of morphisms in $\mathsf{K}$.

Let us show that the kernel $L_{i}$ of the morphism $h_{i}$ belongs to
$\varinjlim\mathsf{S}$. The object $H^{(\Xi_{i})}$ is the coproduct of copies
of all the objects $G\in\mathsf{G}$, each of them taken with the multiplicity
$\Xi_{i}$. Since the object $S_{i}$ is finitely generated, there exists a
finite subcoproduct in this coproduct mapping epimorphically onto $S_{i}$. So
we have a direct sum decomposition $H^{(\Xi_{i})}=H^{\prime}_{i}\oplus
H^{\prime\prime}_{i}$, where $H^{\prime}_{i}$ is a finite direct sum of
objects from $\mathsf{G}$ and the restriction of $h_{i}$ onto $H^{\prime}_{i}$
is an epimorphism $h^{\prime}_{i}\colon H^{\prime}_{i}\longrightarrow S_{i}$.
Denote by $K_{i}$ the kernel of $h^{\prime}_{i}$. We have constructed a
pushout diagram

---
$\textstyle{K_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{\prime}_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{L_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S_{i}}$$\textstyle{H^{\prime\prime}_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{\prime\prime}_{i}}$

Now $H^{\prime}_{i}\in\mathsf{S}$, since the class $\mathsf{S}$ is closed
under finite direct sums and $\mathsf{G}\subset\mathsf{S}$. Hence
$K_{i}\in\mathsf{S}$, as the class $\mathsf{S}$ is closed under kernels of
epimorphisms. On the other hand,
$H^{\prime\prime}_{i}\in\varinjlim\mathsf{S}$, since the class
$\varinjlim\mathsf{S}$ is closed under coproducts. As we already know that the
class $\varinjlim\mathsf{S}$ is closed under extensions, we can conclude from
the short exact sequence $0\longrightarrow K_{i}\longrightarrow
L_{i}\longrightarrow H^{\prime\prime}_{i}\longrightarrow 0$ that
$L_{i}\in\varinjlim\mathsf{S}$.

Passing to the direct limit of $h_{i}$ over $i\in I$, we see that the kernel
of the epimorphism

$\varinjlim\nolimits_{i\in I}H^{(\Xi_{i})}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\varinjlim\nolimits_{i\in I}S_{i}=D$

belongs to $\varinjlim\varinjlim\mathsf{S}=\varinjlim\mathsf{S}$. We already
know from Corollary 3.2 that the kernel of the epimorphism (4) (for
$H_{i}=H^{(\Xi_{i})}$) belongs to $\varinjlim\mathsf{S}$. Since the class
$\varinjlim\mathsf{S}$ is closed under extensions, it follows that the kernel
$M$ of the composition of epimorphisms

$H=\coprod\nolimits_{i\in I}H^{(\Xi_{i})}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\varinjlim\nolimits_{i\in I}H^{(\Xi_{i})}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5muD$

belongs to $\varinjlim\mathsf{S}$.

The final observation is that the epimorphism $H=\coprod_{i\in
I}H^{(\Xi_{i})}\longrightarrow D$ factorizes through the epimorphism
$C\longrightarrow D$, essentially due to the construction of the sets
$\Xi_{i}$ in the beginning of this proof. Now we consider the pullback diagram

$\textstyle{N\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{N\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{D}$

where $X$ is the pullback of the pair of epimorphisms $C\longrightarrow D$ and
$H\longrightarrow D$, while $N$ is the kernel of the morphism
$C\longrightarrow D$. Since the epimorphism $H\longrightarrow D$ factorizes
through the epimorphism $C\longrightarrow D$, the short exact sequence
$0\longrightarrow N\longrightarrow X\longrightarrow H\longrightarrow 0$
splits. We have $M\in\varinjlim\mathsf{S}$ and $C\in\varinjlim\mathsf{S}$, so
it follows from the short exact sequence $0\longrightarrow M\longrightarrow
X\longrightarrow C\longrightarrow 0$ that $X\in\varinjlim\mathsf{S}$. It
remains to notice that the class $\varinjlim\mathsf{S}$ is closed under direct
summands (since it is closed under direct limits) in $\mathsf{K}$. So
$N\in\varinjlim\mathsf{S}$ as $N$ is a direct summand of $X$. ∎

We conclude the section by presenting formal proofs of Theorems A(i) and
B(ii).

###### Proof of Theorem A(i) from from Section 0.2.

This is precisely the assertion of Proposition 3.3. ∎

###### Proof of Theorem B(ii) from from Section 0.3.

Applying Proposition 3.5 to the class
$\mathsf{S}\cup\mathsf{T}\subset\mathsf{K}$, we see that the class
$\varinjlim(\mathsf{S}\cup\mathsf{T})$ is closed under coproducts, direct
limits, extensions, and kernels of epimorphisms in $\mathsf{K}$. So
$\varinjlim(\mathsf{S}\cup\mathsf{T})$ is precisely the class $\mathsf{C}$ as
defined in the formulation of Theorem B. ∎

##  4\. Exact Categories of Grothendieck Type

In this section we recall some basic concepts of the theory of _efficient
exact categories_ and _exact categories of Grothendieck type_ , as developed
by Saorín and Št’ovíček [35, 43]. The exposition in Št’ovíček’s paper [43]
is particularly convenient as a reference source for our purposes.

Let $\mathsf{E}$ be a category. By a _well-ordered chain_ (of morphisms) in
$\mathsf{E}$ one means a direct system $(f_{\beta,\gamma}\colon E_{\gamma}\to
E_{\beta})_{0\leq\gamma<\beta<\alpha}$ in $\mathsf{E}$ indexed by an ordinal
$\alpha$. A well-ordered chain $(E_{\beta})_{0\leq\beta<\alpha}$ is said to be
_smooth_ if $E_{\beta}=\varinjlim_{\gamma<\beta}E_{\gamma}$ for all limit
ordinals $0<\beta<\alpha$. If the direct limit
$E_{\alpha}=\varinjlim_{\beta<\alpha}E_{\alpha}$ exists in $\mathsf{E}$, then
the natural morphism $E_{0}\longrightarrow E_{\alpha}$ is said to be the
_composition_ of the smooth chain $(E_{\beta})_{0\leq\beta<\alpha}$. The
morphism $E_{0}\longrightarrow E_{\alpha}$ is also called the _transfinite
composition_ of the morphisms $E_{\beta}\longrightarrow E_{\beta+1}$, where
$0\leq\beta<\alpha$.

Let $\mathsf{E}$ be a category, $\mathsf{D}$ be a class of morphisms in
$\mathsf{E}$, and $\kappa$ be a regular cardinal. An object $X\in\mathsf{E}$
is said to be _$\kappa$ -small relative to $\mathsf{D}$_ if, for any smooth
chain $(E_{\beta})_{0\leq\beta<\alpha}$ indexed by an ordinal $\alpha$ of
cofinality $\geq\kappa$ with the morphisms $E_{\beta}\longrightarrow
E_{\beta+1}$ belonging to $\mathsf{D}$ for all $0\leq\beta<\alpha$ and the
direct limit $E_{\alpha}=\varinjlim_{\beta<\alpha}E_{\alpha}$, the induced map
of sets

$\varinjlim\nolimits_{\beta<\alpha}\operatorname{Hom}_{\mathsf{E}}(X,E_{\beta})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{Hom}_{\mathsf{E}}(X,E_{\alpha})$

is a bijection. An object $X\in\mathsf{E}$ is called _small relative to
$\mathsf{D}$_ if it is $\kappa$-small relative to $\mathsf{D}$ for some
regular cardinal $\kappa$.

The following definition is taken from [43, Definition 3.4]. The definition in
[35, Proposition 2.6] is slightly more general (the result of [35, Proposition
2.7(2)] or [43, Proposition 5.3(2)] provides the comparison). An exact
category $\mathsf{E}$ is called _efficient_ if the following conditions hold:

  1. (Ef0)

$\mathsf{E}$ is weakly idempotent complete, i. e., any pair of morphisms
$p\colon X\longrightarrow Y$ and $i\colon Y\longrightarrow X$ in $\mathsf{E}$
with $pi=\operatorname{id}_{Y}$ arises from a direct sum decomposition
$X=Y\oplus Z$;

  2. (Ef1)

all transfinite compositions of admissible monomorphisms exist in
$\mathsf{E}$, and the class of admissible monomorphisms is closed under
transfinite compositions;

  3. (Ef2)

every object of $\mathsf{E}$ is small relative to the class of all admissible
monomorphisms;

  4. (Ef3)

the exact category $\mathsf{E}$ has a generator, i. e., there is an object
$G\in\mathsf{E}$ such that every object $E\in\mathsf{E}$ is the codomain of an
admissible epimorphism $G^{(I)}\longrightarrow E$ from a coproduct $G^{(I)}$
of some set $I$ of copies of the object $G$.

The definition of a _filtration_ or _transfinitely iterated extension_ was
already given in Section 2 in the context of Grothendieck abelian categories.
It is generalized to exact categories in the following way. A smooth chain
$(E_{\beta})_{0\leq\beta<\alpha}$ is said to be an _$\alpha$ -indexed
filtration_ (of the object $E_{\alpha}=\varinjlim_{\beta<\alpha}E_{\alpha}$)
if $E_{0}=0$ and, for every ordinal $0\leq\beta<\alpha$, the morphism
$E_{\beta}\longrightarrow E_{\beta+1}$ is an admissible monomorphism in
$\mathsf{E}$.

If this is the case, the object $E_{\alpha}$ is said to be _filtered by_ the
cokernels $S_{\beta}$ of the admissible monomorphisms
$E_{\beta}\longrightarrow E_{\beta+1}$. Alternatively, the object $E_{\alpha}$
is called a _transfinitely iterated extension_ of the objects
$(S_{\beta})_{0\leq\beta<\alpha}$. Given a class of objects
$\mathsf{S}\subset\mathsf{E}$, the class of all objects in $\mathsf{E}$
filtered by (objects isomorphic to) objects from $\mathsf{S}$ is denoted by
$\mathsf{Fil}(\mathsf{S})\subset\mathsf{E}$.

The next definition is taken from [43, Definition 3.11]. An _exact category of
Grothendieck type_ is an efficient exact category $\mathsf{E}$ satisfying the
additional axiom

  1. (GT4)

the category $\mathsf{E}$ is deconstructible in itself, i. e., there exists a
_set_ of objects $\mathsf{S}\subset\mathsf{E}$ such that
$\mathsf{E}=\mathsf{Fil}(\mathsf{S})$.

The following result is important for our purposes.

######  Theorem 4.1.

Any exact category of Grothendieck type has enough injective objects.

###### Proof.

This is [43, Corollary 5.9]. ∎

The next lemma is an exact category version of [5, Lemma 1.4].

######  Lemma 4.2.

Let $\mathsf{E}$ be an exact category and $\mathsf{T}\subset\mathsf{E}$ be a
class of objects. Put $\mathsf{B}=\mathsf{T}^{\perp_{\geq 1}}$, and assume
that every object of $\mathsf{E}$ is an admissible subobject of an object from
$\mathsf{B}$ (in particular, this holds if there are enough injective objects
in $\mathsf{E}$). Then

(a) ${}^{\perp_{1}}\mathsf{B}={}^{\perp_{\geq 1}}\mathsf{B}\subset\mathsf{E}$;

(b) if the class $\mathsf{A}={}^{\perp_{1}}\mathsf{B}={}^{\perp_{\geq
1}}\mathsf{B}$ is generating in $\mathsf{E}$, then $(\mathsf{A},\mathsf{B})$
is a hereditary cotorsion pair in $\mathsf{E}$ (as defined in Section 1).

###### Proof.

The argument from [5, Lemma 1.4] applies. All the injective objects of
$\mathsf{E}$ always belong to $\mathsf{B}$; so if there are enough such
injective objects, then every object of $\mathsf{E}$ is an admissible
subobject of an object from $\mathsf{B}$. In part (b), it is helpful to keep
in mind that the class ${}^{\perp_{1}}\mathsf{B}$ is closed under coproducts
in $\mathsf{E}$ for any class $\mathsf{B}\subset\mathsf{E}$ [12, Corollary
8.3], [14, Corollary A.2]. So the conditions that any object of $\mathsf{E}$
is an admissible epimorphic image of an object from $\mathsf{A}$ and that it
is an admissible epimorphic image of a coproduct of objects from $\mathsf{A}$
are equivalent. ∎

The following version of the Eklof–Trlifaj theorem for efficient exact
categories was obtained in the papers [35, 43].

######  Theorem 4.3.

Let $\mathsf{E}$ be an efficient exact category and $(\mathsf{A},\mathsf{B})$
be the cotorsion pair generated by a set of objects
$\mathsf{S}\subset\mathsf{E}$. Then

(a) If the class $\mathsf{A}$ is generating in $\mathsf{E}$, then the
cotorsion pair $(\mathsf{A},\mathsf{B})$ is complete.

(b) If the class $\mathsf{Fil}(\mathsf{S})$ is generating in $\mathsf{E}$,
then $\mathsf{A}=\mathsf{Fil}(\mathsf{S})^{\oplus}$.

###### Proof.

This is [35, Corollary 2.15] or [43, Theorem 5.16]. ∎

The next proposition is the efficient exact category version of [5,
Proposition 1.5].

######  Proposition 4.4.

Let $\mathsf{E}$ be an efficient exact category and
$\mathsf{T}\subset\mathsf{E}$ be a set of objects. Put
$\mathsf{B}=\mathsf{T}^{\perp_{\geq 1}}$, and assume that the class
$\mathsf{B}$ is cogenerating in $\mathsf{E}$ (in partular, this holds if
$\mathsf{E}$ is an exact category of Grothendieck type). Put
$\mathsf{A}={}^{\perp_{1}}\mathsf{B}={}^{\perp_{\geq 1}}\mathsf{B}$, as per
Lemma 4.2, and assume that the class $\mathsf{A}$ is generating in
$\mathsf{E}$. Then $(\mathsf{A},\mathsf{B})$ is a hereditary complete
cotorsion pair in $\mathsf{E}$ generated by a certain set of objects
$\mathsf{S}$.

###### Proof.

If $\mathsf{E}$ is of Grothendieck type, then there are enough injective
objects in $\mathsf{E}$ by Theorem 4.1. In view of Lemma 4.2 and Theorem 4.3,
we only need to construct a set of objects $\mathsf{S}\subset\mathsf{E}$ such
that $\mathsf{S}^{\perp_{1}}=\mathsf{T}^{\perp_{\geq 1}}$. Clearly, we have
$\mathsf{T}\subset\mathsf{A}$. Arguing by induction similarly to [5, proof of
Proposition 1.5], it suffices to show that for every object $S\in\mathsf{A}$
and an integer $n\geq 2$ there exists a set of objects
$\mathsf{S}^{\prime}\subset\mathsf{A}$ such that for any given
$X\in\mathsf{E}$ one has $\operatorname{Ext}^{n}_{\mathsf{E}}(S,X)=0$ whenever
$\operatorname{Ext}^{n-1}_{\mathsf{E}}(S^{\prime},X)=0$ for all
$S^{\prime}\in\mathsf{S}^{\prime}$.

Let $\mathsf{J}_{S}$ be a set of admissible monomorphisms provided by [43,
Proposition 5.3] for the object $S\in\mathsf{E}$. For every short exact
sequence $0\longrightarrow E\overset{j}{\longrightarrow}H\longrightarrow
S\longrightarrow 0$ in $\mathsf{E}$ with $j\in\mathsf{J}_{S}$, choose an
admissible epimorphism $A\longrightarrow H$ onto $H$ from an object
$A\in\mathsf{A}$ (cf. the proof of Lemma 4.2), and set $S^{\prime}$ to be the
kernel of the composition $A\longrightarrow H\longrightarrow S$. Then one has
$S^{\prime}\in\mathsf{A}$, since $\mathsf{A}$ is closed under the kernels of
admissible epimorphisms (as the cotorsion pair $(\mathsf{A},\mathsf{B}$) in
$\mathsf{E}$ is hereditary).

Let $\mathsf{S}^{\prime}$ be the set of all objects $S^{\prime}$ obtained in
this way. For any Yoneda extension class
$\xi\in\operatorname{Ext}^{n}_{\mathsf{E}}(S,X)$, there exists a short exact
sequence $0\longrightarrow Z\longrightarrow Y\longrightarrow S\longrightarrow
0$ in $\mathsf{E}$ such that the class $\xi$ is the composition of the class
$\operatorname{Ext}^{1}$ represented by this short exact sequence with some
Yoneda extension class $\eta\in\operatorname{Ext}_{\mathsf{E}}^{n-1}(Z,X)$. By
construction, any short exact sequence $0\longrightarrow Z\longrightarrow
Y\longrightarrow S\longrightarrow 0$ in $\mathsf{E}$ is a pushout of a short
exact sequence $0\longrightarrow E\overset{j}{\longrightarrow}H\longrightarrow
S\longrightarrow 0$ with $j\in\mathsf{J}_{S}$, which in turn is a pushout of a
short exact sequence $0\longrightarrow S^{\prime}\longrightarrow
A\longrightarrow S\longrightarrow 0$ with $A\in\mathsf{A}$ and
$S^{\prime}\in\mathsf{S}^{\prime}$. It follows easily that
$\operatorname{Ext}_{\mathsf{E}}^{n-1}(S^{\prime},X)=0$ for all
$S^{\prime}\in\mathsf{S}^{\prime}$ implies
$\operatorname{Ext}_{\mathsf{E}}^{n}(S,X)=0$. ∎

The next theorem of Št’ovíček plays a key role in our proof of Theorem
A(iii).

######  Theorem 4.5.

Let $\mathsf{K}$ be a Grothendieck abelian category and
$\mathsf{E}\subset\mathsf{K}$ be a deconstructible class of objects (as
defined in Section 2). Assume additionally that the full subcategory
$\mathsf{E}$ is closed under direct summands in $\mathsf{K}$. Then the
category $\mathsf{E}$, endowed with the exact category structure inherited
from the abelian exact structure of $\mathsf{K}$, is an exact category of
Grothendieck type.

###### Proof.

This is [43, Theorem 3.16]. ∎

######  Lemma 4.6.

Let $\mathsf{K}$ be a efficient exact category and
$\mathsf{E}\subset\mathsf{K}$ be a full subcategory closed under transfinitely
iterated extensions, endowed with the inherited exact structure. Let
$\mathsf{S}\subset\mathsf{E}$ be a class of objects. Then the notation
$\mathsf{Fil}(\mathsf{S})$ is unambiguous: the class of all
$\mathsf{S}$-filtered objects in $\mathsf{E}$ coincides with the class of all
$\mathsf{S}$-filtered objects in $\mathsf{K}$.

###### Proof.

This is a part of [45, Lemma 1.11] or [43, Lemma 3.18]. ∎

##  5\. Pure Exact Structure and Deconstructibility

The aim of this section is to prove Theorem A(ii). We restate it now as the
following Proposition 5.1.

Let $\mathsf{K}$ be a Grothendieck category. We will say that a class of
objects $\mathsf{F}\subset\mathsf{K}$ is _weakly deconstructible_ if there
exists a set of objects $\mathsf{T}\subset\mathsf{F}$ such that
$\mathsf{F}\subset\mathsf{Fil}(\mathsf{T})$. Clearly, a class of objects is
deconstructible if and only if it is weakly deconstructible _and_ closed under
transfinitely iterated extensions.

######  Proposition 5.1.

Let $\mathsf{K}$ be a locally finitely presentable abelian category and
$\mathsf{S}\subset\mathsf{K}$ be a class of finitely presentable objects
closed under finite direct sums. Then the class of objects
$\varinjlim\mathsf{S}\subset\mathsf{K}$ is weakly deconstructible.

The definition of the pure exact structure on the module category
${\operatorname{\mathsf{Mod--}}}R$ was already given in Section 1. It is
extended to locally finitely presentable abelian categories $\mathsf{K}$ as
follows [15, Section 3], [44, Section 4].

A short exact sequence $0\longrightarrow K\longrightarrow L\longrightarrow
M\longrightarrow 0$ in $\mathsf{K}$ is said to be _pure_ if the functor
$\operatorname{Hom}_{\mathsf{K}}(T,{-})\colon\mathsf{K}\longrightarrow\mathsf{Ab}$
takes it to a short exact sequence of abelian groups for any finitely
presentable object $T\in\mathsf{K}$. The _pure exact structure_ on
$\mathsf{K}$ is defined by the collection of all pure short exact sequences.
With the pure exact structure in mind, one can speak about _pure subobjects_ ,
_pure quotients_ , _pure monomorphisms_ , _pure epimorphisms_ , _pure-
projective objects_ , and _pure acyclic complexes_ in $\mathsf{K}$.

For any small preadditive category $\mathcal{S}$ (i. e., a small category
enriched in abelian groups), we denote by
${\operatorname{\mathsf{Mod--}}}\mathcal{S}=\mathsf{Funct}_{\mathsf{ad}}(\mathcal{S}^{\mathsf{op}},\mathsf{Ab})$
the category of contravariant additive functors from $\mathcal{S}$ to
$\mathsf{Ab}$, and by
$\mathcal{S}{\operatorname{\mathsf{--Mod}}}=\mathsf{Funct}_{\mathsf{ad}}(\mathcal{S},\mathsf{Ab})$
the category of covariant additive functors
$\mathcal{S}\longrightarrow\mathsf{Ab}$. A small preadditive category
$\mathcal{S}$ can be viewed as a “ring with many objects” or “a nonunital ring
with enough idempotents”; then the objects of
${\operatorname{\mathsf{Mod--}}}\mathcal{S}$ and
$\mathcal{S}{\operatorname{\mathsf{--Mod}}}$ are interpreted as right and left
$\mathcal{S}$-modules.

The abelian category ${\operatorname{\mathsf{Mod--}}}\mathcal{S}$ is locally
finitely presentable and has enough projective objects. Representable functors
play the role of free modules with one generator in
${\operatorname{\mathsf{Mod--}}}\mathcal{S}$, and the projective objects are
the direct summands of coproducts of representables. There is a naturally
defined tensor product functor
$\otimes_{\mathcal{S}}\colon{\operatorname{\mathsf{Mod--}}}\mathcal{S}\times\mathcal{S}{\operatorname{\mathsf{--Mod}}}\longrightarrow\mathsf{Ab}$,
and its derived functor $\operatorname{Tor}^{\mathcal{S}}_{*}$ can be
constructed as usual. Hence one can speak of _flat_ right and left
$\mathcal{S}$-modules. We denote the full subcategory of flat modules by
${\operatorname{\mathsf{Mod_{fl}--}}}\mathcal{S}\subset{\operatorname{\mathsf{Mod--}}}\mathcal{S}$
(cf. the discussion in [5, Section 2]).

Part (b) of the next proposition is a surprisingly nontrivial generalization
of Lemma 2.8.

######  Proposition 5.2.

Let $\mathsf{K}$ be a locally finitely presentable abelian category and
$\mathsf{S}\subset\mathsf{K}$ be a class of finitely presentable objects
closed under finite direct sums. Then

(a) The full subcategory $\mathsf{C}=\varinjlim\mathsf{S}\subset\mathsf{K}$ is
closed under pure extensions (as well as pure subobjects and pure quotients)
in $\mathsf{K}$, so it inherits an exact category structure from the pure
exact structure on $\mathsf{K}$.

(b) In the inherited exact category structure on $\mathsf{C}$, a short
sequence $0\longrightarrow K\longrightarrow L\longrightarrow M\longrightarrow
0$ is exact if and only if the short sequence of abelian groups
$0\longrightarrow\operatorname{Hom}_{\mathsf{K}}(S,K)\longrightarrow\operatorname{Hom}_{\mathsf{K}}(S,L)\longrightarrow\operatorname{Hom}_{\mathsf{K}}(S,M)\longrightarrow
0$ is exact for every object $S\in\mathsf{S}$.

(c) Denote by $\mathcal{S}$ a small category equivalent to the full
subcategory $\mathsf{S}\subset\mathsf{K}$. Then there is a natural equivalence
between the category $\mathsf{C}$ and the category of flat right
$\mathcal{S}$-modules,
$\mathsf{C}\simeq{\operatorname{\mathsf{Mod_{fl}--}}}\mathcal{S}$. Under this
equivalence, the exact structure on $\mathsf{C}$ inherited from the pure exact
structure on $\mathsf{K}$ corresponds to the exact structure on
${\operatorname{\mathsf{Mod_{fl}--}}}\mathcal{S}$ inherited from the abelian
exact structure on ${\operatorname{\mathsf{Mod--}}}\mathcal{S}$.

###### Proof.

Part (a) is a straightforward generalization of [25, Propostion 2.2] which was
already mentioned in the proof of Corollary 3.2.

Part (c): the first assertion can be be obtained by combining [15, Theorems
1.4(2) and 4.1] (cf. [44, Proposition 4.2] and [5, Lemma 2.2]). The functor
$\mathsf{C}\longrightarrow{\operatorname{\mathsf{Mod_{fl}--}}}\mathcal{S}$
assigns to an object $C\in\mathsf{C}\subset\mathsf{K}$ the contravariant
functor
$\operatorname{Hom}_{\mathsf{K}}({-},C)\colon\mathsf{K}^{\mathsf{op}}\longrightarrow\mathsf{Ab}$
restricted to the full subcategory
$\mathsf{S}\subset\mathsf{C}\subset\mathsf{K}$. This functor identifies the
full subcategory $\mathsf{S}\subset\mathsf{C}$ with the full subcategory of
representable functors in ${\operatorname{\mathsf{Mod--}}}\mathcal{S}$, and
preserves direct limits (as the objects from $\mathsf{S}$ are finitely
presentable). For an arbitrary preadditive category $\mathcal{S}$, the
representable functors play the role of free modules with one generator in
${\operatorname{\mathsf{Mod--}}}\mathcal{S}$; when $\mathcal{S}$ is an
additive category, as in the situation at hand, these are the same things as
the finitely generated free modules. It remains to recall that the flat
modules are the direct limits of finitely generated free ones (also over a
ring with many objects [27, Theorem 3.2]).

To prove the second assertion, it suffices to say that the equivalence
$\mathsf{C}\simeq{\operatorname{\mathsf{Mod_{fl}--}}}\mathcal{S}$, viewed as a
functor
$\mathsf{C}\longrightarrow{\operatorname{\mathsf{Mod_{fl}--}}}\mathcal{S}$, is
exact (with respect to the respective inherited exact structures) by the
definition of the pure exact structure on $\mathsf{K}$. On the other hand, the
inverse functor
${\operatorname{\mathsf{Mod_{fl}--}}}\mathcal{S}\longrightarrow\mathsf{C}$ is
exact because every short exact sequence of flat modules is a direct limit of
split short exact sequences. (Notice also that the full subcategory
$\mathsf{C}\subset\mathsf{K}$ is preserved by the direct limits by Proposition
3.1, and direct limits of pure short exact sequences are pure exact in
$\mathsf{K}$.)

Part (b): with the argument from the proof of Lemma 2.8 in mind, the following
still needs to be proved. Let $K\longrightarrow L\longrightarrow M$ be a
composable pair of morphisms in $\mathsf{C}$. Suppose that the sequence of
abelian groups
$0\longrightarrow\operatorname{Hom}_{R}(S,K)\longrightarrow\operatorname{Hom}_{R}(S,L)\longrightarrow\operatorname{Hom}_{R}(S,M)\longrightarrow
0$ is exact for every object $S\in\mathsf{S}$. One has to show that the
sequence $0\longrightarrow K\longrightarrow L\longrightarrow M\longrightarrow
0$ is exact in $\mathsf{K}$. When the class $\mathsf{S}$ contains a set of
generators of the abelian category $\mathsf{K}$, this is obvious. In the
general case, this is a restatement of the claim that the equivalence functor
${\operatorname{\mathsf{Mod_{fl}--}}}\mathcal{S}\longrightarrow\mathsf{C}$ is
exact, which we have proved already in part (c). ∎

With the assertions of the proposition in mind, we will simply call the exact
category structure on $\mathsf{C}=\varinjlim\mathsf{S}$ described in
Proposition 5.2 “the pure exact structure”.

###### Proof of Proposition 5.1.

We have to show that there is a set of objects $\mathsf{T}\subset\mathsf{C}$
such that all the objects of $\mathsf{C}$ are filtered by $\mathsf{T}$ in
$\mathsf{K}$. Let us prove a stronger assertion instead: there exists a set of
objects $\mathsf{T}\subset\mathsf{C}$ such that all the objects of
$\mathsf{C}$ are _pure filtered_ by $\mathsf{T}$, that is, filtered by objects
from $\mathsf{T}$ in the pure exact structure on $\mathsf{C}$. Clearly, any
filtration in the pure exact structure on $\mathsf{C}$ is also a filtration in
the pure exact structure on $\mathsf{K}$, and consequently it is a filtration
in the abelian exact structure on $\mathsf{K}$ (as all pure short exact
sequences in $\mathsf{K}$ are exact).

Now we use the equivalence of exact categories
$\mathsf{C}\simeq{\operatorname{\mathsf{Mod_{fl}--}}}\mathcal{S}$ from
Proposition 5.2(c). In view of this equivalence, it remains to observe that
the class of all flat $\mathcal{S}$-modules is deconstructible (in itself
viewed as an exact category, or equivalently, in the abelian category of
modules ${\operatorname{\mathsf{Mod--}}}\mathcal{S}$, cf. Lemma 4.6). This is
the result of [8, Lemma 1] (see also [22, Lemma 6.23]). ∎

######  Remark 5.3.

The assertion of Proposition 5.1 admits a far-reaching generalization: all the
finite presentability conditions can be dropped. For any Grothendieck abelian
category $\mathsf{K}$ and any _set_ of objects $\mathsf{S}\subset\mathsf{K}$
closed under finite direct sums, the class
$\varinjlim\mathsf{S}\subset\mathsf{K}$ is weakly deconstructible. This is a
Grothendieck category multiobject generalization of [30, Corollary 3.4],
provable by an argument similar to the one in [30] and extending the proofs of
Propositions 5.1–5.2 in the following way.

For consistency of notation, let us denote again by $\mathcal{S}$ the full
additive subcategory in $\mathsf{K}$ corresponding to the class $\mathsf{S}$.
Then there is no longer an category equivalence as in Proposition 5.2(c), but
there is still a right exact, direct limit-preserving functor
$\widetilde{\Theta}\colon{\operatorname{\mathsf{Mod--}}}\mathcal{S}\longrightarrow\mathsf{K}$
left adjoint to the restricted Yoneda functor
$K\longmapsto\operatorname{Hom}_{\mathsf{K}}({-},K)|_{\mathsf{S}}$. In the
spirit of the argument in [30], one can interpret $\widetilde{\Theta}$ as a
_tensor product_ functor. Specifically, this is a restriction of the category-
theoretic tensor product operation

$\otimes_{\mathcal{S}}\colon\mathsf{Funct}_{\mathsf{ad}}(\mathcal{S}^{\mathsf{op}},\mathsf{Ab})\times\mathsf{Funct}_{\mathsf{ad}}(\mathcal{S},\mathsf{K})\longrightarrow\mathsf{K}$

(see [27, Section 1] for the definition). The functor $\widetilde{\Theta}$ is
constructed by tensoring the usual right $\mathcal{S}$-modules with one
specific left $\mathcal{S}$-module given by the covariant identity inclusion
functor
$\operatorname{Id}\colon\mathcal{S}=\mathsf{S}\longrightarrow\mathsf{K}$; so

$\widetilde{\Theta}(\mathcal{M})=\mathcal{M}\otimes_{\mathcal{S}}\operatorname{Id}$

for all
$\mathcal{M}\in{\operatorname{\mathsf{Mod--}}}\mathcal{S}=\mathsf{Funct}_{\mathsf{ad}}(\mathcal{S}^{\mathsf{op}},\mathsf{Ab})$.
Denoting by
$\Theta\colon{\operatorname{\mathsf{Mod_{fl}--}}}\mathcal{S}\longrightarrow\mathsf{K}$
the restriction of $\widetilde{\Theta}$ to the full subcategory of flat
modules
${\operatorname{\mathsf{Mod_{fl}--}}}\mathcal{S}\subset{\operatorname{\mathsf{Mod--}}}\mathcal{S}$,
one observes that $\Theta$ is an exact functor (since all the exact sequences
of flat modules are direct limits of split ones). The functor $\Theta$ is
_not_ fully faithful, but the full subcategory $\mathsf{C}\subset\mathsf{K}$
is the essential image of $\Theta$ (essentially for the reasons explained in
[30]).

Denoting by
$\mathcal{T}\subset{\operatorname{\mathsf{Mod_{fl}--}}}\mathcal{S}$ a set of
flat modules such that
${\operatorname{\mathsf{Mod_{fl}--}}}\mathcal{S}=\mathsf{Fil}(\mathcal{T})$,
one concludes that $\mathsf{T}=\Theta(\mathcal{T})\subset\mathsf{K}$ is a set
of objects such that $\mathsf{T}\subset\mathsf{C}$ and
$\mathsf{C}\subset\mathsf{Fil}(\mathsf{T})$, since the functor $\Theta$
preserves transfinitely iterated extensions.

###### Proof of Theorem A(ii) from Section 0.2.

By assumption, the class $\mathsf{C}=\varinjlim\mathsf{S}$ is closed under
extensions in $\mathsf{K}$. Since $\mathsf{C}$ is also closed under direct
limits in $\mathsf{K}$ by Proposition 3.1, it follows that $\mathsf{C}$ is
closed under transfinitely iterated extensions in $\mathsf{K}$. Since, on the
other hand, the class $\mathsf{C}$ is weakly deconstructible in $\mathsf{K}$
by Proposition 5.1, we can conclude that $\mathsf{C}$ is deconstructible under
our assumptions. ∎

######  Corollary 5.4.

Let $\mathsf{K}$ be a locally finitely presentable abelian category, and let
$\mathsf{S}\subset\mathsf{K}$ be a class of objects of type $\mathrm{FP}_{2}$
(as defined in Section 3) containing a set of generators of the abelian
category $\mathsf{K}$. Put $\mathsf{C}=\varinjlim\mathsf{S}\subset\mathsf{K}$.
Then there is a complete cotorsion pair $(\mathsf{C},\mathsf{D})$ in
$\mathsf{K}$.

###### Proof.

By Theorem A(i–ii), or in other words, by Propositions 3.1, 3.3, and 5.1, the
class $\mathsf{C}$ is deconstructible in $\mathsf{K}$; so
$\mathsf{C}=\mathsf{Fil}(\mathsf{T})$ for a set of objects
$\mathsf{T}\subset\mathsf{K}$. Furthermore, by assumption, the class
$\mathsf{C}$ contains a set of generators of $\mathsf{K}$. Let
$(\mathsf{C}^{\prime},\mathsf{D})$ be the cotorsion pair in $\mathsf{K}$
generated by $\mathsf{T}$. Applying Theorem 2.5, we conclude that
$(\mathsf{C}^{\prime},\mathsf{D})$ is a complete cotorsion pair and
$\mathsf{C}^{\prime}=\mathsf{C}$. ∎

##  6\. Classes of $\kappa$-Presentables and Their $\kappa$-Direct Limit
Closures

In this section we discuss generalizations of some results from Sections 2, 3,
and 5 from the countable cardinal $\aleph_{0}$ to arbitrary regular cardinals
$\kappa$. In particular, we present a version of Theorem A(i) for regular
cardinals $\kappa$, stated below as Proposition 6.4, and discuss the
difficulties involved with an attempt to extend Theorem A(ii) to higher
cardinals (see Remark 6.8).

We refer to the book [1, Definitions 1.13 and 1.17, and Theorem 1.20] for the
definitions of a _$\kappa$ -presentable object_ and a _locally
$\kappa$-presentable_ category (for a regular cardinal $\kappa$). The functors
of $\kappa$-direct limit (i. e., direct limits indexed by $\kappa$-directed
posets) are exact in any locally $\kappa$-presentable category [1, Proposition
1.59]. Up to an isomorphism, in a locally $\kappa$-presentable category there
is only a set of $\kappa$-presentable objects [1, Remark 1.19]. Any
Grothendieck abelian category is locally presentable, i. e., locally
presentable for _some_ regular cardinal $\kappa$ [42, Lemma A.1].

The following proposition is a generalization of Proposition 3.1 to higher
cardinals. We state it for nonabelian and even nonadditive categories, as the
abelian category case is no easier than the general one.

######  Proposition 6.1.

Let $\mathsf{K}$ be a locally $\kappa$-presentable category and
$\mathsf{S}\subset\mathsf{K}$ be a class of $\kappa$-presentable objects. Then
the class $\varinjlim^{(\kappa)}\mathsf{S}$ of all $\kappa$-direct limits of
objects from $\mathsf{S}$ in $\mathsf{K}$ (i. e., direct limits indexed by
$\kappa$-directed posets) is closed under $\kappa$-direct limits in
$\mathsf{K}$. An object $L\in\mathsf{K}$ belongs to
$\varinjlim^{(\kappa)}\mathsf{S}$ if and only if, for any $\kappa$-presentable
object $T$ in $\mathsf{K}$, any morphism $T\longrightarrow L$ in $\mathsf{K}$
factorizes through an object from $\mathsf{S}$. If the class $\mathsf{S}$ is
closed under $\kappa$-small coproducts (i. e., coproducts indexed by sets of
cardinality $<\kappa$), then the class $\varinjlim^{(\kappa)}\mathsf{S}$ is
closed under all coproducts in $\mathsf{K}$.

###### Proof.

Two assertions need to be explained: the “if” implication and the closedness
under coproducts. The “only if” implication is obvious; and the closedness
under $\kappa$-direct limits follows from the “if and only if”.

Concerning the “if” implication, the argument is similar to the one in [1,
Proposition 1.22]. It is convenient to use [1, Theorem 1.5 and Remark 1.21] to
the effect that it suffices to construct a $\kappa$-filtered category $D$ and
a $D$-indexed diagram $(S_{d})_{d\in D}$ of objects $S_{d}\in\mathsf{S}$ in
$\mathsf{K}$ such that $L=\varinjlim_{d\in D}S_{d}$. For this purpose, let $D$
be the essentially small category of all pairs $d=(S_{d},f_{d})$, where
$S_{d}\in\mathsf{S}$ and $f_{d}\colon S_{d}\longrightarrow L$ is an arbitrary
morphism. Morphisms in the category $D$ are defined in the obvious way, and
the construction of the diagram $D\longrightarrow\mathsf{K}$ is also obvious.

Concerning the coproducts, let $(I_{\xi})_{\xi\in\Xi}$ be a family of
$\kappa$-directed posets, indexed by a set $\Xi$; and let $(K_{i,\xi})_{i\in
I}$ be a diagram in $\mathsf{K}$, indexed by the poset $I_{\xi}$ and given for
every $\xi\in\Xi$. Then the coproduct of $\kappa$-direct limits
$\coprod_{\xi\in\Xi}\varinjlim_{i\in I_{\xi}}K_{i,\xi}$ can be expressed as
the following $\kappa$-direct limit of $\kappa$-small coproducts. Denote by
$J$ the set of all pairs $j=(\Upsilon,t)=(\Upsilon_{j},t_{j})$, where
$\Upsilon\subset\Xi$ is a subset of cardinality smaller than $\kappa$ and
$t\colon\Upsilon\longrightarrow\coprod_{\upsilon\in\Upsilon}I_{\upsilon}$ is a
function assigning to every element $\upsilon\in\Upsilon$ an element
$t(\upsilon)\in I_{\upsilon}$. Given two elements $j$ and $k\in J$, we say
that $j\leq k$ if $\Upsilon_{j}\subset\Upsilon_{k}$ and, for every
$\upsilon\in\Upsilon_{j}$, the inequality $t_{j}(\upsilon)\leq
t_{k}(\upsilon)$ holds in $I_{\upsilon}$. Then $J$ is a $\kappa$-directed
poset; and it is easy to construct a natural $J$-indexed diagram in
$\mathsf{K}$, with the object
$K_{j}=\coprod_{\upsilon\in\Upsilon_{j}}K_{t_{j}(\upsilon),\upsilon}$ sitting
at the vertex $j\in J$, such that $\varinjlim_{j\in
j}K_{j}=\coprod_{\xi\in\Xi}\varinjlim_{i\in I_{\xi}}K_{i,\xi}$. ∎

Let $\mathsf{K}$ be a locally $\kappa$-presentable abelian category. A short
exact sequence $0\longrightarrow K\longrightarrow L\longrightarrow
M\longrightarrow 0$ in $\mathsf{K}$ is said to be _$\kappa$ -pure_ if the
functor
$\operatorname{Hom}_{\mathsf{K}}(T,{-})\colon\mathsf{K}\longrightarrow\mathsf{Ab}$
takes it to a short exact sequence of abelian groups for any
$\kappa$-presentable object $T\in\mathsf{K}$. Any $\kappa$-direct limit of
$\kappa$-pure short exact sequences is a $\kappa$-pure short exact sequence.
The _$\kappa$ -pure exact structure_ on $\mathsf{K}$ is defined by the
collection of all $\kappa$-pure short exact sequences. So one can speak about
_$\kappa$ -pure subobjects_, _$\kappa$ -pure quotients_, _$\kappa$ -pure
acyclic complexes_, _$\kappa$ -pure-projective objects_, etc. (similary to
Section 5). We refer to the paper [2] for some details.

######  Proposition 6.2.

Let $\mathsf{K}$ be a locally $\kappa$-presentable abelian category and
$\mathsf{S}\subset\mathsf{K}$ be a class of $\kappa$-presentable objects
closed under $\kappa$-small coproducts in $\mathsf{K}$. Then the class of
objects $\varinjlim^{(\kappa)}\mathsf{S}$ is closed under $\kappa$-pure
subobjects, $\kappa$-pure quotients, and $\kappa$-pure extensions in
$\mathsf{K}$.

###### Proof.

This is the $\kappa$-version of [25, Proposition 2.2], provable in the similar
way. It is helpful to keep in mind that the class of $\kappa$-presentable
objects is closed under colimits of diagrams with less than $\kappa$ vertices
and arrows [1, Proposition 1.16]. ∎

The next corollary is a generalization of Corollary 3.2.

######  Corollary 6.3.

Let $\mathsf{K}$ be a locally $\kappa$-presentable abelian category and
$\mathsf{S}\subset\mathsf{K}$ be a class of $\kappa$-presentable objects
closed under $\kappa$-small coproducts. Let $(H_{i})_{i\in I}$ be a
$\kappa$-direct system of objects $H_{i}\in\varinjlim^{(\kappa)}\mathsf{S}$,
indexed by a $\kappa$-directed poset $I$. Then the kernel of the natural
epimorphism $\coprod_{i\in I}H_{i}\longrightarrow\varinjlim_{i\in I}H_{i}$ (4)
belongs to $\varinjlim^{(\kappa)}\mathsf{S}$.

###### Proof.

Both the proofs of Corollary 3.2 can be readily adopted to the situation at
hand: one can use either Proposition 6.2, or the construction from [6, proof
of Proposition 4.1] together with Proposition 6.1. ∎

Let $\mathsf{K}$ be an abelian category with exact functors of $\kappa$-direct
limits, and let $n\geq 1$ be an integer. We will say that an object
$S\in\mathsf{K}$ is _of type $\kappa\text{-}\mathrm{P}_{n}$_ if the functors
$\operatorname{Ext}_{\mathsf{K}}^{i}(S,{-})\colon\mathsf{K}\longrightarrow\mathsf{Ab}$
preserve $\kappa$-direct limits for $0\leq i\leq n-1$. So the objects of type
$\kappa\text{-}\mathrm{P}_{1}$ are, by the definition, the
$\kappa$-presentable ones.

One can further define types $\kappa\text{-}\mathrm{P}_{0}$ and
$\kappa\text{-}\mathrm{P}_{\infty}$, similarly to the discussion in Section 3,
but we will not need these definitions. The following proposition refers to
objects of type $\kappa\text{-}\mathrm{P}_{2}$, which form a subclass of the
class of $\kappa$-presentable objects.

######  Proposition 6.4.

Let $\mathsf{K}$ be a locally $\kappa$-presentable abelian category and
$\mathsf{S}$ be a class of (some) objects of type
$\kappa\text{-}\mathrm{P}_{2}$ closed under extensions in $\mathsf{K}$. Then
the class of objects $\varinjlim^{(\kappa)}\mathsf{S}$ is also closed under
extensions in $\mathsf{K}$.

###### Proof.

This is a straightforward generalization of Proposition 3.3, provable in the
similar way. The class of all $\kappa$-presentable objects in $\mathsf{K}$ is
closed under extensions by [42, Lemma A.4] (the running assumption in [42]
that the category is Grothendieck is not needed for this lemma).

The following observations play the key role. Let $\mathsf{K}$ be an abelian
category with exact functors of $\kappa$-direct limits. Then

  1. (i)

for any two classes of objects $\mathsf{X}$ and $\mathsf{Y}$ in $\mathsf{K}$,
one has
$\mathsf{X}*\varinjlim^{(\kappa)}\mathsf{Y}\subset\varinjlim^{(\kappa)}(\mathsf{X}*\mathsf{Y})$;

  2. (ii)

for any class of objects $\mathsf{X}\subset\mathsf{K}$ and any class of
objects $\mathsf{T}\subset\mathsf{K}$ such that the functor
$\operatorname{Ext}_{\mathsf{K}}^{1}(T,{-})\colon\mathsf{K}\longrightarrow\mathsf{Ab}$
preserves $\kappa$-direct limits for all $T\in\mathsf{T}$, one has
$(\varinjlim^{(\kappa)}\mathsf{X})*\mathsf{T}\subset\varinjlim^{(\kappa)}(\mathsf{X}*\mathsf{T})$.

Once again, the arguments from [33, proof of Proposition 8.4] apply. ∎

Let $R$ be an associative ring. One important particular case of the
construction of the class $\varinjlim^{(\kappa)}\mathsf{S}$ occurs when
$\mathsf{K}={\operatorname{\mathsf{Mod--}}}R$ is the module category and
$\mathsf{S}$ is the class of all free (or projective) $R$-modules with less
than $\kappa$-generators. An $R$-module is said to be _$\kappa$ -flat_ if it
can be presented as a $\kappa$-direct limit of projective $R$-modules, or
equivalently, as a $\kappa$-direct limit of free $R$-modules with less than
$\kappa$ generators [28, Theorem 6.1]. The class of all $\kappa$-flat
$R$-modules is closed under extensions and the kernels of epimorphisms in
${\operatorname{\mathsf{Mod--}}}R$; and any short exact sequence of
$\kappa$-flat $R$-modules is a $\kappa$-direct limit of split short exact
sequences of $\kappa$-flat $R$-modules [28, Lemma 6.2].

More generally, let $\mathcal{S}$ be a small preadditive category. Then a
right $\mathcal{S}$-module (i. e., an additive functor
$\mathcal{S}^{\mathsf{op}}\longrightarrow\mathsf{Ab}$) is said to be _$\kappa$
-flat_ if it can be presented as a $\kappa$-direct limit of projective
$\mathcal{S}$-modules, or equivalently, as a $\kappa$-direct limit of
coproducts of less than $\kappa$ representable functors. We will denote the
full subcategory of $\kappa$-flat modules by
${\operatorname{\mathsf{Mod_{\kappa-
fl}--}}}\mathcal{S}\subset{\operatorname{\mathsf{Mod--}}}\mathcal{S}$. The
same results from [28, Section 6] remain valid in this context; so the exact
category structure inherited by ${\operatorname{\mathsf{Mod_{\kappa-
fl}--}}}\mathcal{S}$ from the abelian exact structure of
${\operatorname{\mathsf{Mod--}}}\mathcal{S}$ coincides with the exact
structure on ${\operatorname{\mathsf{Mod_{\kappa-fl}--}}}\mathcal{S}$
inherited from the $\kappa$-pure exact structure on
${\operatorname{\mathsf{Mod--}}}\mathcal{S}$.

The following proposition is the $\kappa$-version of Proposition 5.2.

######  Proposition 6.5.

Let $\mathsf{K}$ be a locally $\kappa$-presentable abelian category and
$\mathsf{S}\subset\mathsf{K}$ be a class of $\kappa$-presentable objects
closed under $\kappa$-small coproducts. Then

(a) The full subcategory
$\mathsf{C}=\varinjlim^{(\kappa)}\mathsf{S}\subset\mathsf{K}$ inherits an
exact category structure from the $\kappa$-pure exact structure on
$\mathsf{K}$.

(b) In the inherited exact category structure on $\mathsf{C}$, a short
sequence is exact if and only if the functor
$\operatorname{Hom}_{\mathsf{K}}(S,{-})$ takes it to a short exact sequnce of
abelian groups for every object $S\in\mathsf{S}$.

(c) Denote by $\mathcal{S}$ a small category equivalent to the full
subcategory $\mathsf{S}\subset\mathsf{K}$. Then there is a natural equivalence
between the category $\mathsf{C}$ and the category of $\kappa$-flat right
$\mathcal{S}$-modules, $\mathsf{C}\simeq{\operatorname{\mathsf{Mod_{\kappa-
fl}--}}}\mathcal{S}$. Under this equivalence, the exact structure on
$\mathsf{C}$ inherited from the $\kappa$-pure exact structure on $\mathsf{K}$
corresponds to the exact structure on ${\operatorname{\mathsf{Mod_{\kappa-
fl}--}}}\mathcal{S}$ discussed above.

###### Proof.

Part (a) holds by Proposition 6.2. The proof of parts (b–c) is similar to the
proof of Proposition 5.2(b–c). In part (b), first of all notice the fact that
any morphism from a $\kappa$-presentable object in $\mathsf{K}$ to an object
from $\mathsf{C}$ factorizes through an object from $\mathsf{S}$. In part (c),
it is important that $\kappa$-direct limits of $\kappa$-pure short exact
sequences are $\kappa$-pure exact in $\mathsf{K}$. ∎

The next theorem is a generalization of Theorem 2.7 suggested in [5, Remark
4.11].

######  Theorem 6.6.

Let $\mathsf{K}$ be a locally $\kappa$-presentable abelian category. Let
$P^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ be a complex of
$\kappa$-pure-projective objects, and let
$X^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ be a $\kappa$-pure acyclic
complex in $\mathsf{K}$. Then any morphism of complexes
$P^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
X^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is homotopic to zero.

###### Proof.

Let $\mathsf{S}$ be the class of all $\kappa$-presentable objects in
$\mathsf{K}$. Applying Proposition 6.5(c), we conclude that the exact category
$\mathsf{K}$ with the $\kappa$-pure exact structure is equivalent to the exact
category ${\operatorname{\mathsf{Mod_{\kappa-fl}--}}}\mathcal{S}$ of
$\kappa$-flat right $\mathcal{S}$-modules.

Notice that all the projective $\mathcal{S}$-modules are $\kappa$-flat, and
the kernel of any surjective morphism from a projective $\mathcal{S}$-module
to a $\kappa$-flat one is $\kappa$-flat by [28, Lemma 6.2(a)]. It follows that
there are enough projective objects in the exact category
${\operatorname{\mathsf{Mod_{\kappa-fl}--}}}\mathcal{S}$, and these projective
objects are precisely the projective $\mathcal{S}$-modules.

Hence the equivalence of exact categories
$\mathsf{K}\simeq{\operatorname{\mathsf{Mod_{\kappa-fl}--}}}\mathcal{S}$ takes
the $\kappa$-pure-projective objects of $\mathsf{K}$ to the projective
$\mathcal{S}$-modules. It also takes $\kappa$-pure acyclic complexes to
acyclic complexes in the exact category ${\operatorname{\mathsf{Mod_{\kappa-
fl}--}}}\mathcal{S}$, which means acyclic complexes of $\kappa$-flat
$\mathcal{S}$-modules with $\kappa$-flat modules of cocycles. As all
$\kappa$-flat modules are flat, it remains to apply [26, Theorem
8.6(iii)$\Rightarrow$(i)] or [5, Theorem 4.4]. ∎

The following assertion extends Proposition 2.6 to arbitrary regular cardinals
$\kappa$.

######  Proposition 6.7.

Let $\mathsf{K}$ be a locally $\kappa$-presentable Grothendieck category. Let
$\mathsf{S}\subset\mathsf{K}$ be a class of $\kappa$-presentable objects
closed under transfinitely iterated extensions of families of objects of
cardinality $<\kappa$ (i. e., indexed by ordinals $\alpha<\kappa$). Let
$A^{\text{\smaller\smaller$\scriptstyle\bullet$}}\in\mathbf{C}(\mathsf{Fil}(\mathsf{S}))$
be a complex in $\mathsf{K}$ whose terms are $\mathsf{S}$-filtered objects.
Then the complex $A^{\text{\smaller\smaller$\scriptstyle\bullet$}}$, viewed as
an object of the abelian category of complexes $\mathbf{C}(\mathsf{K})$, is
filtered by bounded below complexes whose terms belong to $\mathsf{S}$.

###### Proof.

This is still [42, (proof of) Proposition 4.3]. Once again, the argument is
based on the Hill lemma [42, Theorem 2.1]. ∎

######  Remark 6.8.

Let $\mathsf{K}={\operatorname{\mathsf{Mod--}}}R$ be the module category and
$\mathsf{S}\subset{\operatorname{\mathsf{Mod--}}}R$ be a class of objects of
type $\kappa\text{-}\mathrm{P}_{2}$ (or $\kappa\text{-}\mathrm{P}_{\infty}$)
closed under transfinitely iterated extensions indexed by ordinals smaller
than $\kappa$. Then Propositions 6.1 and 6.4 tell that the class of modules
$\varinjlim^{(\kappa)}\mathsf{S}$ is closed under extensions, coproducts, and
$\kappa$-direct limits in ${\operatorname{\mathsf{Mod--}}}R$. But is it closed
under transfinitely iterated extensions? For $\kappa=\aleph_{0}$ we said, in
the proof of Theorem A(ii) in Section 5, that transfinitely iterated
extensions are built up from extensions and direct limits. But this requires
all direct limits (of chains of monomorphisms) and _not_ only $\kappa$-direct
limits.

On the other hand, does the analogue of Proposition 5.1 hold for $\kappa$? In
other words, is the class $\varinjlim^{(\kappa)}\mathsf{S}$ weakly
deconstructible? Arguing similarly to the proof of Proposition 5.1 and using
Proposition 6.5, it would be sufficient to know that the class of
$\kappa$-flat $\mathcal{S}$-modules is weakly deconstructible. But is this
true?

Furthermore, the class $\mathsf{C}=\varinjlim^{(\kappa)}\mathsf{S}$ is a
_Kaplansky class_ in the sense of [20, 38]: for any regular cardinal $\lambda$
there exists a regular cardinal $\mu$ such that for any object
$C\in\mathsf{C}$ and any $\lambda$-presentable subobject $X\subset C$ there
exists a $\mu$-presentable subobject $K\subset C$ such that $X\subset K$ and
both the objects $K$ and $C/K$ belong to $\mathsf{C}$. This is provable using
Proposition 6.2 and a suitable version of purification procedure (cf. [8,
first paragraph of the proof of Theorem 5]). Still the class $\mathsf{C}$ is
_not_ closed under direct limits in general, but only under $\kappa$-direct
limits; so [23, Lemma 6.9] or [38, Lemma 2.5(2)] cannot be used in order to
deduce deconstructibility of $\mathsf{C}$ (cf. [22, Sections 10.1–10.2]).

Let us point out some partial answers to the questions above that are
available in the literature. To begin with, we observe that the answers to the