# Almost free modules, perfect decomposition and Enochs’s conjecture

Manuel Cortés-Izurdiaga  Departamento de Matemática Aplicada, Universidad de
Málaga, 29071, Málaga, Spain<EMAIL_ADDRESS>and  Alejandro Poveda  Harvard
University, Department of Mathematics and Center of Mathematical Sciences and
Applications, Cambridge (MA), 02138, USA<EMAIL_ADDRESS>
###### Abstract.

Given a module $X$ and a regular cardinal $\kappa$ we study various notions of
$(\kappa,\mathrm{Add}(X))$-freeness and
$(\kappa,\mathrm{Add}(X))$-separability. Bearing on appropriate set-theoretic
assumptions, we construct a non-trivial $\kappa^{+}$-generated,
$(\kappa^{+},\mathrm{Add}(X))$-free and
$(\kappa^{+},\mathrm{Add}(X))$-separable module. Our construction allows
$\kappa$ to be singular thus extending [GCIT10, Theorem 4.7]. Bearing on
similar set-theoretic assumptions, we characterize when every module $X$ has a
perfect decomposition. As a subproduct we show that Enoch’s conjecture for
classes $\mathrm{Add}(X)$ is consistent with ZFC – a fact first proved by
Šaroch [Šar23].

###### Key words and phrases:

Enoch’s conjecture, Mittag-Leffler modules, almost free, $\mathrm{Add}(X)$

###### 2020 Mathematics Subject Classification:

Primary 16E50, 16U40

The first author is partially supported by the Spanish Government under grants
PID2020-113552GB-I00 which include FEDER funds of the EU., and by Junta de
Andalucía under grant P20-00770. The second author acknowledges support from
the Department of Mathematics at Harvard University as well as from the
Harvard Center of Mathematical Sciences and Applications.

##  1\. Introduction

An infinite abelian group $G$ is called _almost free_ whenever every subgroup
of strictly smaller cardinality is free yet $G$ itself fails to be free. A
natural question in infinite abelian group theory is which infinite cardinals
accommodate almost free groups [Fuc70]. Almost free groups where introduced
back in the late 30’s by Kurosch [Kur39] who employed the terminology _locally
free groups_. Kurosch himself provided the first construction of an almost
free group of cardinality $\aleph_{0}$. A decade later, Higman [Hig51]
produced an almost free group of cardinality $\aleph_{1}$. Ever since, major
algebrist and set theorist have devoted stubborn efforts to produce a rich and
fine-structured theory of almost freeness [She75, Ekl75, EM02, Šar23]. The
existence of almost free groups is an instance of a more general phenomenon
called _compactness_ [MS94]. In turn, the study of compactness is of paramount
importance in infinitary combinatorics. This explains the narrow historical
ties between the study of almost freeness, stationary reflection and large
cardinal among others.

Given $R$ a unital (non-necessarily commutative) ring one may inquire whether
this supports almost free (left $R$-)modules. The first thing to be made
precise is what is meant for a module to be almost free. Indeed, if $R$ fails
to be hereditary then there might be free modules none of whose non-zero
submodules are free – thus the natural definition is faulty. This issue is
addressed in Eklof-Mekler’s book [EM02, Ch. IV] where a module $M$ is said to
be _$\kappa$ -free_ provided it carries a directed system of
${<}\kappa$-generated free submodules that is _big_ in certain set-theoretic
sense. Thus a _module is almost free_ whenever it is $\kappa$-generated and
$\kappa$-free in the above-described sense. In [EM02, Ch. IV and VII] the
authors offer an extensive study of almost freeness. Weakening of almost
freeness were more recently considered by Herbera and Trlifaj in their study
of Mittag-Leffler modules [HT12].

In this paper we investigate a notion of $\kappa$-freeness (namely, _
$(\kappa,\mathrm{Add}(X))$-freeness_) relative to classes of the form
$\mathrm{Add}(X)$. For a module $X$ the class $\mathrm{Add}(X)$ consists
(modulo isomorphism) of all modules that are direct summands of $X^{(\mu)}$
for some cardinal $\mu$. Classes of this form subsume (among others) the
projective and pure-projective modules. Thus
$(\kappa,\mathrm{Add}(X))$-freeness yields a natural extension of Eklof-Mekler
$\kappa$-freeness. Notions of freeness akin to ours were previously considered
by Herbera-Trlifaj [HT12] in the study of Mittag-Leffler modules and more
recently by Šaroch [Šar23] in connection to the so-called _Enoch’s
conjecture_.

This paper shall also study the notion of _almost separability_ relative to
$\mathrm{Add}(X)$. Recall that a module $M$ is called _$\kappa$ -separable_
whenever every submodule $N$ of cardinality ${<}\kappa$ is contained in a
${<}\kappa$-generated free direct summand of $M$. Assuming that $X$ is
${<}\kappa$-generated, a module $M$ will be called _$(\kappa,\mathrm{Add}(X))$
-separable_ provided the module $N$ can be taken from $\mathrm{Add}(X).$ This
notion of separability was first studied by the first author, Guil and
Torrecillas in [GCIT10] in connection to when certain categories of modules
are closed under colimits of directed systems.

In the manuscript we discuss the interplay between
$(\kappa,\mathrm{Add}(X))$-freeness and
$(\kappa,\mathrm{Add}(X))$-separability with two major themes of research in
module theory.

The first topic regards the existence of perfect decompositions of a module
$X$; to wit, when every module in $\mathrm{Add}(X)$ has a decomposition
complementing direct summands. Assuming the Generalized Continuum Hypothesis,
[GCIT10, Corollary 5.13] shows that if $X$ has a perfect decomposition then
there is a non-trivial111A $(\kappa,\mathrm{Add}(X))$-separable module is said
to be trivial if it belongs to the class $\mathrm{Add}(X)$.
$(\kappa^{+},\mathrm{Add}(X))$-separable module, being $\kappa$ the
cardinality of a generating set of $X$. In this paper, we improve the
construction of [GCIT10] by showing that non-trivial
$(\kappa^{+},\mathrm{Add}(X))$-separable modules exist even for singular
cardinals $\kappa$ (the construction in [GCIT10] only allows regular cardinals
$\kappa$). Second, we characterize, modulo a set-theoretic hypothesis
consistent with ZFC, when every module $X$ has a perfect decomposition.
Granting this hypothesis, $X$ does not have a perfect decomposition if and
only if there is a non-trivial $(\kappa^{+},\mathrm{Add}(X))$-separable module
(Theorem 5.2).

The second topic under consideration regards the so-called Enoch’s Conjecture.
For a class of modules $\mathcal{X}$ the conjecture asserts that $\mathcal{X}$
is closed under direct limits provided it is covering. Just recently J. Šaroch
[Šar23] has proved (modulo an extra set-theoretic hypothesis) the consistency
of the conjecture for classes of the form $\mathrm{Add}(X)$ [Šar23]. Saroch’s
argument is based on the construction of a certain
$(\kappa,\mathrm{Add}(X))$-free module. In what we are concerned, here we
establish the precise relationship between the existence of a non-trivial
$(\kappa,\mathrm{Add}(X))$-separable module and the existence of
$\mathrm{Add}(X)$-covers (see Lemma 5.3). As a bi-product we provide an
alternative proof of Enoch’s conjecture for $\mathrm{Add}(X)$ using separable
modules. Like Šaroch’s, our argument also requires an extra set-theoretic
assumption.

The following is a succinct account of the paper’s contents. In Section 2 we
provide all the pertinent module and set-theoretic prelimminaries and
notations. Section 3 analyzes various notions of almost freeness relative to
the class $\mathrm{Add}(X)$ and connects them with the main device used in the
paper – $\mathrm{Add}(X)$-filtrations. A key role will be played by Lemma 3.3
which will allow us to elucidate whether an almost free module is trivial
(i.e., whether it belongs to $\mathrm{Add}(X)$). In Section 4 we will shift
our attention to $(\kappa,\mathrm{Add}(X))$-separable modules. Here we prove
that under suitable set-theoretic assumptions there are $\kappa^{+}$-generated
non-trivial $(\kappa^{+},\mathrm{Add}(X))$-separable (and
$(\kappa^{+},\mathrm{Add}(X))$-free) modules, even when $\kappa$ is a singular
cardinal. Finally, Section 5 is devoted to applications. Here we show: first,
under a consistent set-theoretic assumption, the existence of a non-trivial
$(\kappa,\mathrm{Add}(X))$-free/separable module is equivalent to $X$ not
having a perfect decomposition; second, under the same assumptions, Enoch’s
Conjecture holds for classes of the form $\mathrm{Add}(X)$. This provides an
alternative proof to Šaroch’s [Šar23]; third, the class of all left
$R$-modules with $\mathcal{Q}$-Mittag-Leffler dimension less than or equal to
$n$ ([CI16a]) satisfies Enoch’s conjecture, thus extending [BYT0, Theorem 2.6]
to the non-flat setting.

##  2\. Preliminaries

The cardinality of a set $A$ will be denoted by $|A|$ and its power set by
$\mathcal{P}(A)$. The restriction of a map $f:A\rightarrow B$ to a subset
$A^{\prime}$ of $A$ will be denoted $f\upharpoonright A^{\prime}$. As
customary, infinite cardinals will be denoted by Greek letters, such as
$\kappa,\lambda,\delta$, etc. The cofinality of an infinite cardinal $\kappa$
will be denoted by $\operatorname{cf}(\kappa)$. Recall that a cardinal
$\kappa$ is _regular_ if $\kappa=\operatorname{cf}(\kappa)$; otherwise
$\kappa$ is called singular. Let $\beta$ be a limit ordinal with uncountable
cofinality. A set $C\subseteq\beta$ is called _a club_ if it is closed and
unbounded with respect to the order topology of $\beta$. A set
$S\subseteq\beta$ is called _stationary_ if $S\cap C\neq\emptyset$ for all
clubs $C\subseteq\beta$. The set of all stationary subsets of $\beta$ will be
denoted by $\mathrm{NS}_{\beta}^{+}$. A typical stationary set is
$E^{\kappa}_{\lambda}:=\\{\alpha<\kappa\mid\mathrm{cf}(\alpha)=\lambda\\}$
whenever $\lambda<\kappa$ are both regular cardinals. A stationary set
$S\subseteq\beta$ _reflects_ if there is $\alpha<\beta$ with
$\operatorname{cf}(\alpha)>\omega$ such that $S\cap\alpha$ is stationary in
$\alpha$. We will say that $E$ reflects at $\alpha$. $I[\kappa^{+}]$ will
denote _Shelah’s Approchability ideal_ on $\kappa^{+}$ (cf. [She23, Eis09]).

Through the manuscript $R$ will stand for a (non-necessarily commutative) ring
with unit. By a _module_ we mean a left $R$-module – i.e., a member of
$R$-$\mathrm{Mod}$.

Let $\kappa$ be an infinite (non-necessarily regular) cardinal. A module $M$
is called _$\kappa$ -generated_ if it has a set of generators $G\subseteq M$
with size $\kappa$; to wit, every $x\in M$ can be written as
$\sum_{i=0}^{n}r_{i}y_{i}$ for $r_{i}\in R$, $y_{i}\in G$ and $n<\omega.$
Similarly, $M$ will be called _${ <}\kappa$-generated_ if it admits a set of
generators of cardinality ${<}\kappa.$ We will denote by $\lambda_{M}$ the
minimum of the set

$\\{\lambda\mid\text{There is a set of generators for $M$ of cardinality
$\lambda$}\\}.$

Thus $M$ will be ${<}\kappa$-generated for an infinite cardinal $\kappa$ if
and only if $\lambda_{M}<\kappa$. Through the paper we will use $``M$ is
${<}\kappa$-generated” and $``\lambda_{X}<\kappa$” interchangeably as often
times the latter phrasing becomes more convenient than the former.

The module $M$ is $\kappa$-presented (resp. $<\kappa$-presented) if it is
$\kappa$-generated (resp. $<\kappa$-generated) and has a free presentation
with $\kappa$-generated kernel.

Filtrations of modules will also play a prominent role in this work:

######  Definition 2.1.

Let $M$ be a module and $\kappa$ a regular cardinal. A _$\kappa$ -filtration_
of $M$ is a sequence $\langle M_{\alpha}\mid\alpha<\kappa\rangle$ consisting
of ${<}\kappa$-generated submodules of $M$ such that

  1. (1)

$M_{\alpha}\subseteq M_{\beta}$ for all $\alpha<\beta$,

  2. (2)

$M_{\alpha}=\bigcup_{\beta<\alpha}M_{\beta}$ whenever $\beta$ is a limit
ordinal,

  3. (3)

$M=\bigcup_{\alpha<\kappa}M_{\alpha}.$

The class $\mathrm{Add}(X)$, where $X$ is a fixed module, will have a promient
role in this paper. Recall that if $\mathcal{X}$ is a class of modules,
$\mathrm{Sum}(\mathrm{Add}(X))$ is the class of all direct sums of modules
from $\mathcal{X}$ and

$\mathrm{Add}(\mathcal{X})=\\{N\in\text{$R$-$\mathrm{Mod}$}\mid
N\lesssim_{\oplus}X\,\text{for some }X\in\textrm{Sum}(\mathcal{X})\\},$

where $N\lesssim_{\oplus}X$ is a shorthand for $``N$ is isomorphic to a direct
summand of $X$”. Clearly, $\mathrm{Add}(\mathcal{X})$ is closed both under
direct summands and direct sums. If $\mathcal{X}$ consists of one single
module, we will write $\mathrm{Add}(X)$ and $\textrm{Sum}(X)$ instead of
$\mathrm{Add}(\\{X\\})$ and $\textrm{Sum}(\\{X\\})$.

Often times an infinite cardinal $\kappa$ will be fixed and we will consider
the classes $\textrm{Sum}_{\kappa}(\mathcal{X})$ consisting of all direct sums
of less than $\kappa$ modules belonging to $\mathcal{X}$ and
$\mathrm{Add}_{\kappa}(\mathcal{X})$, of all modules isomorphic to a direct
summand of a module from $\textrm{Sum}_{\kappa}(\mathcal{X})$. Notice that, if
$\kappa$ is regular and $\lambda_{X}<\kappa$, the modules in
$\mathrm{Add}_{\kappa}(X)$ are the $<\kappa$-generated modules in
$\mathrm{Add}(X)$. For instance, $\mathrm{Add}(R)$ is the class of all
projective modules. Similarly, if $M$ is the direct sum of all finitely
presented modules up to isomorphism, $\mathrm{Add}(M)$ is the class of all
pure-projective modules. More generally, if $\kappa$ is an uncountable regular
cardinal and $M$ is the direct sum of all $<\kappa$-presented modules up to
isomorphism, then $\mathrm{Add}(M)$ consists of all $\kappa$-pure-projective
modules, i.e., those modules $N$ which are projective with respect to all
$\kappa$-pure exact sequences [GCIT10, Proposition 5.1].

A classical result we shall bear on is Walker’s theorem [AF92, Theorem 26.1] –
this asserts that every direct summand of a module which is a direct sum of
$\kappa$-generated modules is itself a direct sum of $\kappa$-generated
modules. In particular, if $X$ is a direct sum of $\kappa$-generated modules
then so is every module in $\mathrm{Add}(X)$. Walker’s result is a
generalization of Kaplansky’s theorem concerning projective modules. In turn,
Walker’s result can be easily obtained for $\kappa$-presented modules:

######  Lemma 2.2.

Let $\kappa$ be a regular cardinal and $M$ be a module which is a direct sum
of $\kappa$-presented modules. Then every direct summand of $M$ is a direct
sum of $\kappa$-presented modules.

###### Proof.

Let $K$ be a direct summand of $M$. By Walker’s Theorem, $K$ is a direct sum
of $\kappa$-generated modules. Note that every $\kappa$-generated direct
summand $K^{\prime}$ of $K$ is $\kappa$-presented: If $M=\bigoplus_{i\in
I}M_{i}$ for a family $\\{M_{i}\mid i\in I\\}$ of $\kappa$-presented
submodules of $M$, $K^{\prime}\leq\bigoplus_{j\in J}M_{j}$ for some subset $J$
of $I$ with cardinality $\kappa$. Then $K^{\prime}$ is a direct summand of the
$\kappa$-presented module $\bigoplus_{j\in J}M_{j}$ and it is itself
$\kappa$-presented. ∎

The next (truly useful) fact will be used repeatedly through the paper.

######  Fact 2.3 ([CI17, Lemma 1.1]).

Suppose that $\langle M_{\alpha}\mid\alpha<\kappa\rangle$ is a
$\kappa$-filtration of a module $M$ and that for each $\alpha<\kappa$ there is
$N_{\alpha}\leq M$ such that $M_{\alpha+1}=M_{\alpha}\oplus N_{\alpha}$. Then,
$M=\bigoplus_{\alpha<\kappa}N_{\alpha}$.

A direct system of modules $\mathcal{M}$ where the underlying poset is well-
ordered is called a _well ordered system_. A class $\mathcal{X}$ of modules is
called _closed under direct limits_ (resp. _closed under well-ordered limits_)
if for any direct (resp. well-ordered) system $\mathcal{M}$ consisting of
modules in $\mathcal{X}$, $\varinjlim\mathcal{M}\in\mathcal{X}.$

######  Lemma 2.4.

Let $\mathcal{X}$ be a class of modules. The following assertions are
equivalent:

  1. (1)

$\mathcal{X}$ is closed under direct limits.

  2. (2)

$\mathcal{X}$ is closed under well ordered direct limits.

  3. (3)

For any infinite cardinal $\kappa$, $\mathcal{X}$ is closed under well ordered
direct limits of cardinality greater than or equal to $\kappa$.

  4. (4)

There exists an infinite cardinal $\kappa$ such that $\mathcal{X}$ is closed
under well ordered direct limits of cardinality greater than or equal to
$\kappa$.

###### Proof.

(1) $\Leftrightarrow$ (2). Well known.

(2) $\Rightarrow$ (3) $\Rightarrow$ (4). Clear.

(4) $\Rightarrow$ (2). Take a direct system of modules belonging to
$\mathcal{X}$,
$(X_{\delta},u_{\delta\varepsilon}\mid\delta\leq\varepsilon<\lambda)$, for
some cardinal $\lambda<\kappa$ and denote by
$(X,u_{\delta}\mid\delta<\lambda)$ its direct limit. We may assume that
$\lambda$ is regular since, otherwise, $\lambda$ has a cofinal subset of
cardinality some regular cardinal.

Let $\mu\geq\kappa$ be a cardinal with $\operatorname{cf}(\mu)=\lambda$. Take
a strictly increasing map $\gamma:\lambda\rightarrow\mu$ such that
$\\{\gamma(\delta)\mid\delta<\lambda\\}$ is unbounded in $\mu$. Let
$\sigma:\mu\rightarrow\lambda$ be the map defined by
$\sigma(\alpha)=\sup\\{\delta<\lambda\mid\gamma(\delta)\leq\alpha\\}$ for any
$\alpha<\mu$. Set $Y_{\alpha}=X_{\sigma(\alpha)}$ and
$v_{\alpha\beta}=u_{\sigma(\alpha)\sigma(\beta)}$ for each
$\alpha\leq\beta<\lambda$ so that we get a well ordered direct system
$(Y_{\alpha},v_{\alpha\beta}\mid\alpha\leq\beta<\mu)$. Since $\sigma$ is
increasing, its image $C$ is unbounded in $\lambda$ and the direct limit of
$(Y_{\alpha},v_{\alpha\beta}\mid\alpha\leq\beta<\mu)$ is precisely
$(X,u_{\delta}\mid\delta\in C)$. By (4), $X$ belongs to $\mathcal{X}$. ∎

In the last section of this paper we will be concerned with _Enoch’s
conjecture_. The two backbone notions underpinning this conjecture are
_precovering_ and _covering_ classes of modules.

######  Definition 2.5.

Let $\mathcal{X}$ be a class of modules. We say that $\mathcal{X}$ is:

  1. (1)

_Precovering_ : If every module $M$ posseses an _$\mathcal{X}$ -precover_;
namely, a morphism $f\colon X\rightarrow M$ with $X\in\mathcal{X}$ such that
$\mathrm{Hom}(Y,f)$ is surjective for all $Y\in\mathcal{X}.$

  2. (2)

_Covering_ : If every module $M$ posseses an _$\mathcal{X}$ -cover_; namely,
an $\mathcal{X}$-precover $f\colon X\rightarrow M$ such that every
$g\in\mathrm{End}(X)$ is an automorphism if $f\circ g=f$.

The key classical result bridging these concepts is Enoch’s theorem [GT06,
Theorem 5.31] asserting that any precovering class closed under direct limits
is covering. The converse of this assertion is the so-called _Enoch’s
conjecture_. Namely,

######  Conjecture (Enochs).

Every covering class of modules is closed under direct limits.

##  3\. $(\kappa,\mathrm{Add}(X))$-free modules

Related to an infinite regular cardinal $\kappa$, there are several notions of
$\kappa$-free modules which have been extensively studied in the category of
abelian groups [EM02]: $\kappa$-free in the weak sense, $\kappa$-free,
strongly $\kappa$-free and $\kappa$-separable. In this paper we are interested
in these notions but relative to the class $\mathrm{Add}(X)$ for a fixed
module $X$. We begin with the almost free modules in the weak sense which were
introduced in [EM02, p. 89]. We consider here the slightly more general
approach of [CI16b, Definition 2.2] which is based on [HT12, Definition 2.5]:

######  Definition 3.1 ($(\kappa,\mathcal{X})$-freeness in the weak sense).

Let $\kappa$ be an uncountable regular cardinal and $\mathcal{X}$, a class of
modules. We say that a module $M$ is $(\kappa,\mathcal{X})$-free in the weak
sense if there is a direct system $\mathcal{S}$ of submodules of $M$ belonging
to $\mathcal{X}$ such that any subset of $M$ of cardinality smaller than
$\kappa$ is contained in an element of $\mathcal{S}$. We say that $M$ is
trivial if $M$ belongs to $\mathcal{X}$.

Notice that in this definition the modules in $\mathcal{S}$ need not be
$<\kappa$-generated as in the definition by Eklof and Mekler. But if we take
$\mathcal{X}$ the class of all $<\kappa$-generated free modules, then the
$(\kappa,\mathcal{X})$-free modules in the weak sense are the $\kappa$-free
modules in the weak sense of Eklof and Mekler.

The relationship between weak almost freeness and filtrations is the
following:

######  Proposition 3.2.

Let $X$ be a module and $\kappa$ an uncountable regular cardinal. Let $M$ be
any module.

  1. (1)

If $M$ has a filtration, $(M_{\alpha}\mid\alpha<\kappa)$, with
$M_{\alpha+1}\in\mathrm{Add}(X)$ for each $\alpha<\kappa$, then $M$ is
$(\kappa,\mathrm{Add}(X))$-free in the weak sense.

  2. (2)

If $\kappa>\lambda_{X}$, $M$ is $\kappa$-generated and is
$(\kappa,\mathrm{Add}(X))$-free in the weak sense, then it has a
$\kappa$-filtration, $(M_{\alpha}\mid\alpha<\kappa)$, with
$M_{\alpha+1}\in\mathrm{Add}_{\kappa}(X)$ for each $\alpha<\kappa$. In
particular, $M$ is $(\kappa,\mathrm{Add}_{\kappa}(X))$-free in the weak sense.

###### Proof.

(1) Just notice that $\\{M_{\alpha+1}\mid\alpha<\kappa\\}$ is a direct system
satisfying the conditions in Definition 3.1.

(2) Let $\\{m_{\alpha}\mid\alpha<\kappa\\}$ be a generating system of $M$ and
$\mathcal{S}$ a direct system of sumodules of $M$ witnessing the
$(\kappa,\mathrm{Add}(X))$-freeness of $M$. We can construct the modules
$M_{\alpha}$ in the $\kappa$-filtration satisfying $m_{\alpha}\in
M_{\alpha+1}$ recursively on $\alpha$: Set $M_{0}=0$ and, for $\alpha<\kappa$
limit, $M_{\alpha}=\bigcup_{\gamma<\alpha}M_{\gamma}$. For the case
$\alpha+1$, take a submodule $N$ of $M$ belonging to $\mathcal{S}$ and
containing $M_{\alpha}\cup\\{x_{\alpha}\\}$. Since $N$ is a direct sum of
$\lambda_{X}$-generated modules belonging to $\mathrm{Add}(X)$ by Walker’s
lemma, $\lambda_{X}<\kappa$, and $\kappa$ is regular, we can find a
$<\kappa$-generated direct summand $M_{\alpha+1}$ of $N$ containing
$M_{\alpha}\cup\\{x_{\alpha}\\}$. This concludes the construction. ∎

One of the main topics discussed in this paper is when an almost free module
relative to $\mathrm{Add}(X)$ is trivial. As the next lemma evidences,
stationary sets play an important role when it comes to this issue:

######  Lemma 3.3.

Let $X$ a module and $\kappa$, an uncountable regular cardinal with
$\lambda_{X}<\kappa$. Let $M$ be a $\kappa$-generated and
$(\kappa,\mathrm{Add}(X))$-free module in the weak sense. Then the following
assertions are equivalent:

  1. (1)

$M\in\mathrm{Add}(X)$.

  2. (2)

$M$ is a direct sum of $<\kappa$-generated modules.

  3. (3)

For any $\kappa$-filtration of $M$, $\langle
M_{\alpha}\mid\alpha<\kappa\rangle$, with $M_{\alpha+1}\in\mathrm{Add}(X)$ for
every $\alpha<\kappa$, the set

$E=\\{\alpha<\kappa\mid M_{\alpha}\nleq_{\oplus}M\\}$

is not stationary in $\kappa$.

  4. (4)

There is a $\kappa$-filtration of $M$, $\langle
M_{\alpha}\mid\alpha<\kappa\rangle$, with $M_{\alpha+1}\in\mathrm{Add}(X)$ for
every $\alpha<\kappa$ such that the set

$E=\\{\alpha<\kappa\mid M_{\alpha}\nleq_{\oplus}M\\}$

is not stationary in $\kappa$.

  5. (5)

For any $\kappa$-filtration of $M$, $\langle
M_{\alpha}\mid\alpha<\kappa\rangle$, with $M_{\alpha+1}\in\mathrm{Add}(X)$ for
every $\alpha<\kappa$, the set

$E^{\prime}=\\{\alpha<\kappa\mid\\{\beta>\alpha\mid
M_{\alpha}\nleq_{\oplus}M_{\beta}\\}\textrm{ is stationary in }\kappa\\}$

is not stationary in $\kappa$.

  6. (6)

There is a $\kappa$-filtration of $M$, $\langle
M_{\alpha}\mid\alpha<\kappa\rangle$, with $M_{\alpha+1}\in\mathrm{Add}(X)$ for
every $\alpha<\kappa$ such that the set

$E^{\prime}=\\{\alpha<\kappa\mid\\{\beta>\alpha\mid
M_{\alpha}\nleq_{\oplus}M_{\beta}\\}\textrm{ is stationary in }\kappa\\}$

is not stationary in $\kappa$.

###### Proof.

(1) $\Rightarrow$ (2). If $M\in\mathrm{Add}(X)$, then $M$ is a direct sum of
$\lambda_{X}$-generated modules by Walker’s lemma [AF92, Theorem 26.1].

(2) $\Rightarrow$ (3). Let $\langle M_{\alpha}\mid\alpha<\kappa\rangle$ be a
$\kappa$-filtration of $M$ with $M_{\alpha}\in\mathrm{Add}(X)$ for every
$\alpha<\kappa$. Write $M=\bigoplus_{\alpha<\kappa}X_{\alpha}$ for suitable
$X_{\alpha}\in\mathrm{Add}_{\kappa}(X)$. Setting
$N_{\alpha}=\bigoplus_{\gamma<\alpha}X_{\gamma}$ we get a $\kappa$-filtration
of $M$ with $N_{\alpha}\leq_{\oplus}M$ for each $\alpha<\kappa$. By [EM02,
Lemma IV.1.4], there is club $C$ such that $M_{\alpha}=N_{\alpha}$ for each
$\alpha<\kappa$. Then $C\cap E=\emptyset$ and $E$ is not stationary.

(3) $\Rightarrow$ (4). Trivial.

(4) $\Rightarrow$ (1). Let $C\subseteq\kappa$ be a club disjoint from $E$. Let
$C(\cdot)\colon\kappa\rightarrow C$ be an order-preserving continious
bijection and set $N_{\alpha}:=M_{C(\alpha)}$. Clearly, $\langle
N_{\alpha}\mid\alpha<\kappa\rangle$ is another $\kappa$-filtration of $M$
consisting of members of $\mathrm{Add}(X)$ which in addition satisfies that
$N_{\alpha}=M_{C(\alpha)}\leq_{\oplus}M_{C(\alpha)+1}=N_{\alpha+1}$. Hence,
$N_{\alpha+1}=N_{\alpha}\oplus N_{\alpha+1}^{\prime}$ for all $\alpha<\kappa$.
By Fact 2.3, $\textstyle M=\bigoplus_{\alpha<\kappa}N^{\prime}_{\alpha+1}$
which means that $M\in\mathrm{Add}(X)$.

(4) $\Rightarrow$ (5). Follows from the fact that $E\subseteq E^{\prime}$.

(5) $\Rightarrow$ (6). Trivial.

(6) $\Rightarrow$ (1). The same proof of [EM02, Proposition IV.1.7] applies. ∎

Let us consider $\kappa$-freeness of modules relative to a class [CI16b,
Definition 2.2]:

######  Definition 3.4 ($(\kappa,\mathcal{X})$-freeness).

Let $\kappa$ be an uncountable regular cardinal and $\mathcal{X}$, a class of
modules. A module $M$ is called _$(\kappa,\mathcal{X})$ -free_ if it has a
$(\kappa,\mathcal{X})$-dense system of submodules, i.e., a direct system
$\mathcal{S}\subseteq\mathcal{P}(M)$ consisting of submodules of $M$ such
that:

  1. (1)

Every module in $\mathcal{S}$ belongs to the class $\mathcal{X}$.

  2. (2)

$\mathcal{S}$ is closed under unions of well-ordered chains of length
${<}\kappa$.

  3. (3)

Every subset of $M$ of cardinality ${<}\kappa$ is contained in an element of
$\mathcal{S}$.

A $(\kappa,\mathcal{X})$-free module $M$ is called _trivial_ if
$M\in\mathcal{X}.$

Clearly, every $(\kappa,\mathcal{X})$-free module is
$(\kappa,\mathcal{X})$-free in the weak sense. If $\mathcal{X}$ is the class
of all $<\kappa$-generated free modules, then the $(\kappa,\mathcal{X})$-free
modules are the $\kappa$-free modules by Eklof and Mekler [EM02, Definition
IV.1.1]. If $\mathcal{X}$ is the class of all countably generated pure-
projective modules, then the $(\aleph_{1},\mathcal{X})$-free modules are the
Mittag-Leffler modules [HT12, Corollary 2.7]. If $\mathcal{X}$ is the class of
all countably generated projective modules, then the
$(\aleph_{1},\mathcal{X})$-free modules are the flat Mittag-Leffler modules
[HT12, Corollary 2.10].

######  Remark 3.5.

It is possible to tweak the above definition to encompass the case where
$\kappa$ is a singular cardinal (such as $\aleph_{\omega}$). In that case one
stipulates $M$ to be $(\kappa,\mathrm{Add}(X))$-free if it is
$(\lambda,\mathrm{Add}(X))$-free for every regular cardinal $\lambda<\kappa$.
By Shelah’s Singular Compactness Theorem [EM02, Theorem IV.3.7], every
$(\kappa,\mathrm{Add}(X))$-free module is trivial (i.e., it belongs to
$\mathrm{Add}(X)$) when $\kappa$ is a singular cardinal.

Similarly to the classical notion of $\kappa$-freeness of [EM02, IV.1.1],
$(\kappa,\mathrm{Add}_{\kappa}(X))$-freeness can be as well characterized by
the existence of certain $\kappa$-filtrations:

######  Proposition 3.6.

Let $\kappa$ be uncountable regular and $M$ and $X$ be modules.

  1. (1)

If $M$ has a filtration, $\langle M_{\alpha}\mid\alpha<\kappa\rangle$, with
$M_{\alpha}\in\mathrm{Add}(X)$ for each $\alpha<\kappa$, then $M$ is
$(\kappa,\mathrm{Add}(X))$-free.

  2. (2)

If $\kappa>\lambda_{X}$, $M$ is $\kappa$-generated and
$(\kappa,\mathrm{Add}_{\kappa}(X))$-free, then $M$ has a filtration, $\langle
M_{\alpha}\mid\alpha<\kappa\rangle$, with
$M_{\alpha}\in\mathrm{Add}_{\kappa}(X)$ for each $\alpha<\kappa$.

###### Proof.

(1) Just notice that $\\{M_{\alpha}\mid\alpha<\kappa\\}$ is a $\kappa$-dense
system consisting of modules belonging to $\mathrm{Add}(X)$.

(2) If $\mathcal{S}\subseteq\mathcal{P}(M)$ is a $(\kappa,\mathcal{S})$-dense
system of $M$, then we can reason as in Proposition 3.2 to obtain a
$\kappa$-filtration of $M$, $\langle M_{\alpha}\mid\alpha<\kappa\rangle$, with
$M_{\alpha+1}\in\mathcal{S}$. But, since $\mathcal{S}$ is closed under well-
ordered unions of length smaller than $\kappa$,
$M_{\alpha}\in\mathrm{Add}_{\kappa}(X)$ for every limit ordinal
$\alpha<\kappa$ as well. ∎

Another interesting instance of almost freeness is the following due to Eklof
and Mekler [EM02, Definition IV.1.8]:

######  Definition 3.7 (Strongly $(\kappa,\mathcal{X})$-freeness).

Let $\kappa$ be an uncountable regular cardinal and $M$, a module. We say that
$M$ is strongly $(\kappa,\mathcal{X})$-free if there is a set
$\mathcal{S}\subseteq\mathcal{X}$ of submodules of $M$ belonging to
$\mathcal{X}$ and containing $0$ such that for any $S\in\mathcal{S}$ and
$Y\subseteq M$ with $|Y|<\kappa$, there exists $S^{\prime}\in\mathcal{S}$
containing $Y\cup S$ and such that $S$ is a direct summand of $S^{\prime}$.

A strongly $(\kappa,\mathcal{X})$-free module $M$ is called _trivial_ if
$M\in\mathcal{X}.$

######  Remark 3.8.

Suppose that $M$ is strongly $(\kappa,\mathcal{X})$-free and that
$\mathcal{S}$ witnesses this. If $S,S^{\prime}\in\mathcal{S}$ and $S\leq
S^{\prime}$ then $S\leq_{\oplus}S^{\prime}$: By the definition of strongly
free we can find $S^{\prime\prime}\in\mathcal{S}$ with $S^{\prime}\leq
S^{\prime\prime}$ and $S$ being a direct summand of $S^{\prime\prime}$. In
particular, $S$ is a direct summand of $S^{\prime}$.

As in the classical context of _strong $\kappa$-freeness_ [EM02, IV.1], a
strongly $(\kappa,\mathrm{Add}(X))$-free module may not be
$(\kappa,\mathrm{Add}(X))$-free (see [Trl95, Theorem 8] for the case $X=R$).
Next we stablish the characterization of $\kappa$-generated strongly
$(\kappa,\mathrm{Add}(X))$-free modules via filtrations. In particular, they
are weakly $(\kappa,\mathrm{Add}(X))$-free.

######  Lemma 3.9.

Let $\kappa$ be an uncountable cardinal and $X$ a module with
$\lambda_{X}<\kappa$. The following assertions are equivalent for the
$\kappa$-generated module $M$:

  1. (1)

$M$ is strongly $(\kappa,\mathrm{Add}(X))$-free;

  2. (2)

$M$ is strongly $(\kappa,\mathrm{Add}_{\kappa}(X))$-free;

  3. (3)

$M$ admits a $\kappa$-filtration $\langle M_{\alpha}\mid\alpha<\kappa\rangle$
such that for $\alpha<\kappa$, $M_{\alpha+1}\in\mathrm{Add}_{\kappa}(X)$ and
$M_{\alpha+1}\leq_{\oplus}M_{\beta}$ for all $\alpha<\beta<\kappa.$

Thus, strongly $(\kappa,\mathrm{Add}(X))$-free modules are weakly
$(\kappa,\mathrm{Add}(X))$-free modules.

###### Proof.

(1) $\Rightarrow$ (2). Suppose that $M$ is strongly
$(\kappa,\mathrm{Add}(X))$-free and take the set $\mathcal{S}$ of submodules
of $M$ given by Definition 3.7. Let $\mathcal{S}^{\prime}$ be the set
consisting of all $<\kappa$-generated submodules of $M$ which are a direct
summand of some member of $\mathcal{S}$. Then
$\mathcal{S}^{\prime}\subseteq\mathrm{Add}_{\kappa}(X)$ and given any
$Y\subseteq M$ with $|Y|<\kappa$ and $S^{\prime}\in\mathcal{S}$, there exists
$S\in\mathcal{S}$ such that $S^{\prime}\leq_{\oplus}S$. Using that $M$ is
strongly $(\kappa,\mathrm{Add}(X))$-free, we can find $T\in\mathcal{S}$ with
$Y\cup S\leq T$ and $S\leq_{\oplus}T$. Since $T$ is a direct sum of
$\lambda_{X}$-generated modules by Walker’s lemma, there exists a
$<\kappa$-generated direct summand $T^{\prime}$ of $T$ containing $Y\cup
S^{\prime}$. Then $T^{\prime}\in\mathcal{S}^{\prime}$ and, as
$S^{\prime}\leq_{\oplus}T$, $S^{\prime}$ is a direct summand of $T^{\prime}$
as well. This means that $\mathcal{S}^{\prime}$ satisfies the conditions of
Definition 3.7, and that $M$ is strongly
$(\kappa,\mathrm{Add}_{\kappa}(X))$-free.

(2) $\Rightarrow$ (3). Suppose that $M$ is strongly
$(\kappa,\mathrm{Add}_{\kappa}(X))$-free and take $\mathcal{S}$ the set of
submodules of $M$ given in Definition 3.7. Repeat the argument provided in the
proof of $(1)\Rightarrow(2)$ in Proposition 3.6. This yields a
$\kappa$-filtration $\langle M_{\alpha}\mid\alpha<\kappa\rangle$ where
$M_{\alpha+1}\in\mathcal{S}$ for all $\alpha<\kappa$. By Remark 3.8,
$M_{\alpha+1}\leq_{\oplus}M_{\beta+1}$ for $\alpha<\beta<\kappa$. In
particular, $M_{\alpha+1}$ is a direct summmand of $M_{\beta}$ as well. ∎

In order to see if a strongly $(\kappa,\mathrm{Add}(X))$-free module is
trivial, we can look at another subset of $\kappa$ different from the one in
Lemma 3.3

######  Proposition 3.10.

Let $\kappa$ be an uncountable regular cardinal and $X$ a module with
$\lambda_{X}<\kappa$. The following are equivalent for a strongly
$(\kappa,\mathrm{Add}(X))$-free and $\kappa$-generated module $M$:

  1. (1)

$M$ is trivial;

  2. (2)

there exists a $\kappa$-filtration of $M$, $\langle
M_{\alpha}\mid\alpha<\kappa\rangle$, with
$M_{\alpha+1}\in\mathrm{Add}_{\kappa}(X)$ and
$M_{\alpha+1}\leq_{\oplus}M_{\beta}$ for each $\alpha<\beta<\kappa$, such that
the set

$E^{\prime\prime}=\\{\alpha<\kappa\mid
M_{\alpha}\nleq_{\oplus}M_{\alpha+1}\\}$

is not stationary in $\kappa$;

  3. (3)

for every $\kappa$-filtration of $M$, $\langle
M_{\alpha}\mid\alpha<\kappa\rangle$, with
$M_{\alpha+1}\in\mathrm{Add}_{\kappa}(X)$ and
$M_{\alpha+1}\leq_{\oplus}M_{\beta}$ for each $\alpha<\beta<\kappa$, that the
set

$E^{\prime\prime}=\\{\alpha<\kappa\mid
M_{\alpha}\nleq_{\oplus}M_{\alpha+1}\\}$

is not stationary in $\kappa$.

###### Proof.

Simply notice that every $\kappa$-filtration $\langle
M_{\alpha}\mid\alpha<\kappa\rangle$ of $M$ as in Lemma 3.9 we have the
equality

$E^{\prime\prime}=\\{\alpha<\kappa\mid\\{\beta>\alpha\mid
M_{\alpha}\nleq_{\oplus}M_{\beta}\\}\textrm{ is stationary in }\kappa\\}$

Then the result follows from Lemma 3.3. ∎

##  4\. On $(\kappa,\mathrm{Add}(X))$-separable modules

Continuing with our study of the class of (strongly)
$(\kappa,\mathrm{Add}(X))$-free modules in this section we shall be
preoccupied with the notion of $(\kappa,\mathrm{Add}(X))$-separability:

######  Definition 4.1.

Let $\kappa$ be an infinite regular cardinal and $\mathcal{X}$, a class of
modules. A module $M$ is called _$(\kappa,\mathcal{X})$ -separable_ if for
each $Y\subseteq M$ with $|Y|<\kappa$ there is $N\leq_{\oplus}M$ with
$N\in\mathcal{X}$ and $Y\subseteq N$.

A $(\kappa,\mathcal{X})$-separable module $M$ will be called _trivial_
whenever $M\in\mathcal{X}$.

In this paper we are interested in $(\kappa,\mathrm{Add}(X))$ and
$(\kappa,\mathrm{Add}_{\kappa}(X))$-separable modules for some fixed module
$X$. This seemingly strengthening of $(\kappa,\mathrm{Add}(X))$-separability
is equivalent to the former whenever $X$ is a ${<}\kappa$-generated module.
Proof of this will be provided in Lemma 4.5 below.

######  Example 4.2.

To demystify the concept of $(\kappa,\mathrm{Add}_{\kappa}(X))$-separability
let us consider a concrete example – the case where $\kappa=\aleph_{0}$ and
$X$ is the ring, $R$. In this scenario a module $M$ is
$(\aleph_{0},\mathrm{Add}_{\aleph_{0}}(R))$-separable whenever every finite
set $F\subseteq M$ is contained in a finitely-generated projective submodule
$N\leq_{\oplus}M.$ Namely, $M$ is a separable module in the traditional sense.

We should like to begin clarifying the connection between
$(\kappa,\mathrm{Add}(X))$-separability and strongly
$(\kappa,\mathrm{Add}(X))$-freeness. Namely,

######  Lemma 4.3.

Let $\kappa$ be an uncountable regular cardinal and $X$, a module with
$\lambda_{X}<\kappa$. If $M$ is $(\kappa,\mathrm{Add}(X))$-separable then it
is strongly $(\kappa,\mathrm{Add}_{\kappa}(X))$-free.

###### Proof.

Let $\mathcal{S}$ be the set of all $<\kappa$-generated direct summands of $M$
that belong to $\mathrm{Add}(X)$. Given $S\in\mathcal{S}$ and $Y\subseteq M$
with $|Y|<\kappa$, we can find, by the
$(\kappa,\mathrm{Add}(X))$-separability, $S^{\prime}\leq_{\oplus}M$ in
$\mathrm{Add}(X)$ such that $Y\cup S\subseteq S^{\prime}$. Since $S^{\prime}$
is a direct sum of $<\kappa$-generated modules by Walker’s lemma, we can find
a $<\kappa$-generated direct summand $S^{\prime\prime}$ of $S^{\prime}$
containing $Y\cup S$. Then, $S^{\prime\prime}$ is a direct summand of $M$, so
that it belongs to $\mathcal{S}$, and $S$, being a direct summand of $M$, is a
direct summand of $S^{\prime\prime}$ as well. ∎

######  Remark 4.4.

Notice that $(\kappa,\mathrm{Add}(X))$-separable modules need not be
$(\kappa,\mathrm{Add}(X))$-free, since strongly
$(\kappa,\mathrm{Add}(X))$-free modules need not be
$(\kappa,\mathrm{Add}(X))$-free by [Trl95, Theorem 8].

Now we give the characterization of separable modules in terms of filtrations:

######  Lemma 4.5.

Let $\kappa$ be an infinite regular cardinal. The following are equivalent for
a $\kappa$-generated module $M$; namely,

  1. (1)

$M$ is $(\kappa,\mathrm{Add}_{\kappa}(X))$-separable

  2. (2)

$M$ is $(\kappa,\mathrm{Add}(X))$-separable;

  3. (3)

$M$ admits a $\kappa$-filtration, $\langle M_{\alpha}\mid\alpha<\kappa\rangle$
with $M_{\alpha+1}\in\mathrm{Add}_{\kappa}(M)$ and
$M_{\alpha+1}\leq_{\oplus}M$ for each $\alpha<\kappa$.

###### Proof.

(1) $\Rightarrow$ (2): Is obvious.

(2) $\Rightarrow$ (3): The proof of Lemma 4.3 implies that there exists a set
$\mathcal{S}$ of direct summands of $M$ witnessing the strong
$(\kappa,\mathrm{Add}_{\kappa}(X))$-freeness of $M$. Using the proof of (2)
$\Rightarrow$ (3) of Lemma 3.9, we can find a $\kappa$-filtration of $M$,
$\langle M_{\alpha}\mid\alpha<\kappa\rangle$, with
$M_{\alpha+1}\in\mathcal{S}$ for each $\alpha<\kappa$. In particular,
$M_{\alpha+1}\leq_{\oplus}M$ and $M_{\alpha+1}\in\mathrm{Add}_{\kappa}(X)$ for
each $\alpha<\kappa$.

(3) $\Rightarrow$ (1): Trivial. ∎

In the next section we will employ Lemma 4.5 to produce a
$\kappa^{+}$-generated non-trivial $(\kappa^{+},\mathrm{Add}(X))$-separable
module for an infinite (non-necessarily regular) $\kappa$ – this will yield an
extension of [GCIT10, Theorem 4.7]. In particular, this will provide an
example of a non-trivial strongly $(\kappa^{+},\mathrm{Add}(X))$-free module
(see Lemma 4.3).

###  4.1. A construction of a non-trivial $(\kappa,\mathrm{Add}(X))$-separable
module

The main construction of this section is based on the notions of a template
and of a tree-like ladder system, both introduced next:

######  Definition 4.6.

Let $\lambda$ be an infinite regular cardinal and $X$, a module. A triple
$\langle N,L,{\mathcal{N}}\rangle$ is a _$(\lambda,\mathrm{Add}(X))$
-template_ if $N\subseteq L$ are modules, $\mathcal{N}$ is a continuous
filtration of $N$, $\langle N_{\alpha}\mid\alpha<\lambda\rangle$, satisfying
$N_{\alpha}\leq_{\oplus}L$, $L\in\mathrm{Add}(X)$ and $N\nleq_{\oplus}L.$

######  Definition 4.7 (Tree-like ladder system).

Let $\kappa$ be an uncountable regular cardinal and $S\subseteq\kappa$, a
stationary set consisting of limit ordinals of some fixed cofinality
$\lambda<\kappa$. A sequence $\langle c_{\eta}\mid\eta\in S\rangle$ is called
a _ladder system_ if $c_{\eta}\colon\lambda\rightarrow\eta$ is a strictly
increasing function and the range of $c_{\eta}$ is unbounded in $\eta.$ A
ladder system $\langle c_{\eta}\mid\eta\in S\rangle$ is said to be _tree-like_
if for each $\eta,\zeta\in S$, $\delta,\delta^{\prime}<\lambda$

$c_{\eta}(\delta)=c_{\zeta}(\delta^{\prime})\;\Rightarrow\;\delta=\delta^{\prime}\;\text{and}\;c_{\eta}\restriction\delta=c_{\zeta}\restriction\delta.$

######  Remark 4.8.

Given a tree-like ladder system $\langle c_{\eta}\mid\eta\in S\rangle$ one can
define another tree-like ladder system by stipulating
$c^{*}_{\eta}(\xi)=c_{\eta}(\xi)+1$ for any $\xi<\lambda$. As a consequence,
we may assume that $\mathrm{Im}(c_{\eta})\cap S=\emptyset$ and that
$\operatorname{Im}(c_{\eta})\cap\\{s+1\mid s\in S\\}=\emptyset$ for all
$\eta\in S$. We will assume that all of our ladder systems satisfy this.

Every stationary subset $S$ (consisting of ordinals of some fixed cofinality)
of every uncountable cardinal $\kappa$ carries a ladder system. If
$\kappa=\aleph_{1}$, this ladder system can be constructed with the tree-like
ladder property (see [EM02, Exercise XII.17]). If $\kappa>\aleph_{1}$ and $S$
belongs to the approchability ideal of $\kappa$, then it is possible to find a
club $C$ such that $E\cap C$ has a tree-like ladder system [EM02, Lemma
VI.5.13]. Other tree-like ladder systems have been constructed using aditional
set theoretical hypothesis, see [Ekl00, Theorem 9] and [GCIT10, Proposition
2.1].

######  Definition 4.9.

Given cardinals $\lambda<\mu$ with $\lambda$ regular denote by
$(\star)_{\mu,\lambda}$ the conjunction of the next two sentences:

  1. (1)

There is a non-reflecting stationary set $S\subseteq E^{\mu^{+}}_{\lambda}$.

  2. (2)

There is a tree-like ladder system $\langle c_{\eta}\mid\eta\in S\rangle$.

######  Remark 4.10.

If $\mu$ is a singular cardinal and $\lambda<\mu$ is regular, the existence of
a stationary set $S\subseteq E^{\mu^{+}}_{\lambda}$ in the approachability
ideal $I[\mu^{+}]$ turns to be a ZFC theorem – this is due to Shelah (see
[Eis09, Theorem 3.18])). Combining this with [EM02, Lemma 5.13] one can
construct (in ZFC) a tree-like ladder system supported on an approachable
stationary $S\subseteq E^{\mu^{+}}_{\lambda}$. However, it is consistent with
ZFC that every stationary subset of $\mu^{+}$ reflects. This holds in
Magidor’s model for stationary reflection at $\aleph_{\omega+1}$ [Mag82]. For
details see [Eis09, Corollary 3.41].

Under appropriate set-theoretic assumptions upon $\mu$ one can prove that
$(\star)_{\mu,\lambda}$ is consistent with the ZFC axioms. For instance:

######  Lemma 4.11.

If $\square_{\mu}$ holds then $(\star)_{\mu,\lambda}$ holds for all
$\lambda<\mu$ regular.

In particular, if $0^{\sharp}$ does not exist then $(\star)_{\mu,\lambda}$
holds for all singular cardinals $\mu$ and $\lambda<\mu$ regular.

###### Proof.

Since $\square_{\mu}$ holds we can let a non-reflecting stationary set
$S\subseteq E^{\mu^{+}}_{\lambda}$ [CFM01, Theorem 2.1]. In addition $S$ is
approachable (i.e., $S\in I[\mu^{+}]$) because so is $\mu^{+}$ and
$I[\mu^{+}]$ is an ideal. Now [EM02, Lemma 5.13] yields a club
$C\subseteq\mu^{+}$ and a tree-like ladder system $\langle c_{\eta}\mid\eta\in
S^{*}\rangle$ where $S^{*}:=S\cap C$. By [Eis09, Theorem 2.4], $S^{*}$ is a
non-reflecting stationary subset of $E^{\mu^{+}}_{\lambda}$ and $\langle
c_{\eta}\mid\eta\in S^{*}\rangle$ is the sought ladder system. The last
assertions follows from the fact that if $0^{\sharp}$ does not exists then
$\square_{\mu}$ holds for every singular cardinal $\mu$. ∎

We remind our readers that $``\lambda_{X}<\kappa$” was a shorthand for $``X$
is ${<}\kappa$-generated”. The main theorem of the section reads as follows:

######  Theorem 4.12.

Let $\lambda=\operatorname{cf}(\lambda)<\kappa$ be cardinals witnessing
$(\star)_{\kappa,\lambda}$. Let $X$ be a module with $\lambda_{X}<\kappa$, and
suppose that there is a $(\lambda,\mathrm{Add}(X))$-template, $\langle
N,L,\mathcal{N}\rangle$. Then there is a $\kappa^{+}$-generated non-trivial
$(\kappa^{+},\mathrm{Add}(X))$-free and
$(\kappa^{+},\mathrm{Add}(X))$-separable module.

Let $\langle c_{\eta}\mid\eta\in S\rangle$ be a tree-like ladder system
witnessing $(\star)_{\kappa,\lambda}$. The module $M$ witnessing the thesis of
Theorem 4.12 will be obtained as a union of a $\kappa^{+}$-filtration,
$\langle M_{\alpha}\mid\alpha<\kappa^{+}\rangle$, such that
$M_{\alpha+1}\in\mathrm{Add}(X)$ for each $\alpha<\kappa^{+}$ and

$S\subseteq\\{\alpha<\kappa^{+}\mid M_{\alpha}\nleq_{\oplus}M_{\alpha+1}\\}.$

This will entail the non-triviality of $M$ (i.e., $M\notin\mathrm{Add}(X)$) by
virtue of Proposition 3.10 and Lemma 4.5.

The idea is to construct each $M_{\alpha+1}$ by taking direct sums of
isomorphic copies of members $N_{\nu}$ of our fixed template. Before entering
into further considerations let us agree upon some notations.

######  Notation 4.13.

For each $\nu<\lambda$ and $\gamma<\kappa^{+}$ let us fix $N_{\gamma,\nu}$ an
isomorphic copy of $N_{\nu}$. Also, since $N_{\nu}\leq_{\oplus}N_{\nu+1}$
there is $N^{\prime}_{\nu}$ such that $N_{\nu+1}=N_{\nu}\oplus
N^{\prime}_{\nu}$. Thus, for each $\gamma<\kappa^{+}$, we may let isomorphic
copies $N^{*}_{\gamma,\nu}\simeq N_{\nu}$ and $N^{\prime}_{\gamma,\nu}\simeq
N^{\prime}_{\nu}$ such that

$N_{\gamma,\nu+1}=N^{*}_{\gamma,\nu}\oplus N^{\prime}_{\gamma,\nu}.$

Let us fix an isomorphism $s^{\nu}_{\gamma}\colon N^{\prime}_{\nu}\rightarrow
N^{\prime}_{\gamma,\nu}$.

Let us proceed with the construction. First, set $M_{0}:=0$. Assuming that
$\langle M_{\beta}\mid\beta<\alpha\rangle$ has been defined we construct
$M_{\alpha}$ as follows. If $\alpha$ is a limit ordinal we simply set
$M_{\alpha}:=\bigcup_{\beta<\alpha}M_{\beta}$. If $\alpha=\alpha_{*}+1$ we
distinguish among two cases:

Case $\alpha_{*}\notin S$: Let
$M^{\prime}_{\alpha_{*}}=\oplus_{\nu<\lambda}N_{\alpha_{*},\nu}$ and define

$M_{\alpha}:=M_{\alpha_{*}}\oplus M^{\prime}_{\alpha_{*}}.$

Case $\alpha_{*}\in S$: In this case $M_{\alpha}$ is defined as the pushout
between the inclusion $i\colon N\rightarrow L$ and a morphism
$\iota_{\alpha_{*}}\colon N\rightarrow M_{\alpha_{*}}$ that we are yet to
define. In turn, $\iota_{\alpha_{*}}$ will be defined as the limit of a direct
system of morphisms

$\langle\iota^{\nu}_{\alpha_{*}}\colon N_{\nu}\rightarrow
M_{\alpha_{*}},\;\nu<\lambda\rangle.$

First, $\iota^{0}_{\alpha_{*}}$ is declared to be the zero isomorphism. If we
have constructed the direct system
$\langle\iota^{\sigma}_{\alpha_{*}}\mid\sigma<\nu\rangle$ for some
$\nu<\lambda$, then one takes
$\iota^{\nu}_{\alpha_{*}}:=\varinjlim_{\sigma<\nu}\iota^{\sigma}_{\alpha_{*}}$
whenever $\nu$ is a limit ordinal; otherwise, if $\nu=\sigma_{*}+1$ one takes
$\iota^{\nu}_{\alpha_{*}}:=\iota^{\sigma_{*}}_{\alpha_{*}}\oplus
s^{\sigma_{*}}_{c_{\alpha_{*}}(\sigma_{*})}$. We note that
$\iota^{\nu}_{\alpha_{*}}$ is well-defined. In the limit case this is evident
and in the successor case it follows from

  * •

$s^{\sigma_{*}}_{c_{\alpha_{*}}(\sigma_{*})}\colon
N^{\prime}_{\sigma_{*}}\rightarrow
N^{\prime}_{c_{\alpha_{*}}(\sigma_{*}),\sigma_{*}}$

  * •

and
$N^{\prime}_{c_{\alpha_{*}}(\sigma_{*}),\sigma_{*}}\leq_{\oplus}M^{\prime}_{c_{\alpha_{*}}(\sigma_{*})}\leq
M_{\alpha}$ (as $c_{\alpha_{*}}(\sigma_{*})<\alpha_{*}$).

Finally, put
$\iota_{\alpha_{*}}:=\varinjlim_{\nu<\lambda}\iota^{\nu}_{\alpha_{*}}$. Let
$M_{\alpha_{*}+1}$ be the outcome of the pushout

(1)
${N}$${L}$${M_{\alpha_{*}}}$${M_{\alpha_{*}+1}.}$$\scriptstyle{i}$$\scriptstyle{\iota_{\alpha_{*}}}$$\scriptstyle{{\theta_{\alpha_{*}}}}$$\scriptstyle{i_{\alpha_{*}}}$

The above completes our construction of the sequence $\langle
M_{\alpha}\mid\alpha<\kappa^{+}\rangle$. Let us now prove that this is indeed
a $\kappa^{+}$-filtration satisfying Lemma 4.5.

First, we see that $M_{\alpha}$ is a direct summand of $M$ when $\alpha$ does
not belong to $S$. We will use the following technical fact:

######  Lemma 4.14.

Let $\eta\in S$ and $\delta<\kappa^{+}$ with $\delta<\eta$. Set
$\tau_{\eta,\delta}=\sup\\{\sigma+1\mid c_{\eta}(\sigma)\leq\delta\\}$. Then,
for any $\nu<\tau_{\eta,\delta}$, $c_{\eta}(\nu)\leq\delta$.

###### Proof.

If $c_{\eta}(\nu)>\delta$, then $\nu$ is an upper bound of the set
$\\{\sigma+1\mid c_{\eta}(\sigma)\leq\delta\\}$ so that
$\tau_{\eta,\delta}\leq\nu$. Consequently, if $\nu<\tau_{\eta,\delta}$ then it
is satisfied that $c_{\eta}(\nu)\leq\delta$. ∎

######  Lemma 4.15.

For each $\alpha\notin S$, $M_{\alpha}\leq_{\oplus}M$.

In particular, $M_{\alpha+1}\leq_{\oplus}M$ for all $\alpha<\kappa^{+}.$

###### Proof.

Fix $\alpha\notin S$. One has to consider two cases; namely, either $\alpha$
is successor or it is limit. The latter case will follow from the former in
that $M_{\alpha}\leq_{\oplus}M_{\alpha+1}\leq_{\oplus}M$. Consequently we
focus on analyzing the case where $\alpha$ is of the form $\delta+1$.

First, for each $\nu<\lambda$ (as $N_{\nu}\leq_{\oplus}L$) we fix
$\pi_{\nu}\colon L\rightarrow L$ an idempotent endomorphism such that
$\mathrm{Im}(\pi_{\nu})=N_{\nu}$. For each $\eta\in S$ above $\delta+1$ let
$\tau_{\eta}$ the ordinal $\tau_{\eta,\delta}$ defined in Lemma 4.14.

To show that $M_{\delta+1}$ is a direct summand of $M$ we construct a
projection

$p\colon M\rightarrow M_{\delta+1}.$

Namely, $p$ will be a homomorphism such that $p\restriction
M_{\delta+1}=1_{M_{\delta+1}}.$ In turn, $p$ is defined as the direct limit of
a direct system of homomorphisms

$\langle p_{\beta}\colon M_{\beta}\rightarrow
M_{\delta+1}\mid\beta<\kappa^{+}\rangle$

satisfying:

  1. (P1)

If $\beta\leq\delta+1$, then $p_{\beta}\upharpoonright
M_{\delta+1}=1_{M_{\delta+1}}$, and

  2. (P2)

if $\beta=c_{\eta}(\sigma)>\delta+1$ for some $\eta>\delta$ belonging to $S$,
then $p_{\beta+1}\upharpoonright
N^{\prime}_{\beta,\sigma}=\iota_{\eta}\circ\pi_{\tau_{\eta}}\circ(s_{\beta}^{\sigma})^{-1}$.

We construct the said system by induction on $\beta<\kappa^{+}$. For
$\beta\leq\delta+1$ take simply $p_{\beta}$ be the inclusion map. Suppose that
$\langle p_{\beta^{\prime}}\mid\beta^{\prime}<\beta\rangle$ has been defined.
If $\beta$ happens to be a limit ordinal we take $p_{\beta}$ the direct limit
of the previous $p_{\beta^{\prime}}$’s. Otherwise, $\beta$ takes the form
$\mu+1$ and we have to do something clever. We now distinguish three cases:

Case $\mu\notin S$ and $\mu\notin\bigcup_{\eta\in S}\mathrm{Im}(c_{\eta})$: In
this case we let $p_{\mu+1}:=p_{\mu}\oplus 0.$

Case $\mu\notin S$ and $\mu\in\bigcup_{\eta\in S}\mathrm{Im}(c_{\eta})$: In
this case $M_{\mu+1}$ decomposes as

$M_{\mu}\oplus
M^{\prime}_{\mu}=M_{\mu}\oplus(\bigoplus_{\nu\neq\sigma+1}N_{\mu,\nu})\oplus
N^{*}_{\mu,\sigma}\oplus N^{\prime}_{\mu,\sigma}$

where $\sigma$ is such that $\mu=c_{\eta}(\sigma)$ for some $\eta\in S$.
Consider the morphism
$q_{\mu}=\iota_{\eta}\circ\pi_{\tau_{\eta}}\circ(s_{\beta}^{\sigma})^{-1}$
defined on $N^{\prime}_{\mu,\sigma}$ whose image, by Lemma 4.14, is contained
in $M_{\delta+1}$. Thus, $p_{\mu+1}:=(p_{\mu}\oplus 0\oplus 0\oplus q_{\mu})$
defines a homomorphism between $M_{\mu+1}$ and $M_{\delta+1}$ which trivially
satisfies (P2) above.

We are left to show that this homomorphism does not depend upon the choice of
$\eta\in S$ – here is where the tree-likeness of $\langle c_{\eta}\mid\eta\in
S\rangle$ will come into play. Suppose that
$c_{\eta}(\sigma)=\mu=c_{\xi}(\rho)$. By tree-likeness, $\rho=\sigma$ (in
particular, $s^{\sigma}_{\mu}=s^{\xi}_{\mu}$) and

$c_{\eta}\restriction\sigma+1=c_{\xi}\restriction\rho+1.$

Now, since $c_{\eta}(\sigma)=c_{\xi}(\rho)>\delta$ it follows that
$\tau_{\eta}=\tau_{\rho}$ and thus $\pi_{\tau_{\eta}}=\pi_{\tau_{\rho}}$.
Finally, observe that $\iota_{\eta}\restriction
N_{\tau_{\eta}}=\iota_{\xi}\restriction N_{\tau_{\xi}}$ as both homomorphism
were constructed using the same $s^{\sigma}_{\beta}$’s in that
$c_{\eta}\restriction\sigma=c_{\xi}\restriction\sigma$.

Case $\mu\in S$: If we consider the morphism $\theta_{\mu}\pi_{\mu}$ from $L$
to $M_{\mu+1}$, whose image is inside $M_{\delta+1}$ by Lemma 4.14 we get, by
(P2), the following commutative diagram

${N}$${L}$${M_{\mu}}$${M_{\delta+1}.}$$\scriptstyle{i}$$\scriptstyle{\iota_{\mu}}$$\scriptstyle{\theta_{\mu}\pi_{\tau_{\mu}}}$$\scriptstyle{p_{\mu}}$

Since $M_{\mu+1}$ is the pushout of $\iota_{\mu}$ and $i$, the universality of
this construction yields a homomorphism $p_{\mu+1}\colon M_{\mu+1}\rightarrow
M_{\delta+1}$ such that $p_{\mu+1}\circ\ i_{\mu}=p_{\mu}$ and
$p_{\mu+1}\theta_{\mu}=\theta_{\mu}\pi_{\tau_{\mu}}.$ Clearly, $p_{\mu+1}$ is
as desired. ∎

The next step is to prove that $M$ is not trivial, equivalently, by
Proposition 3.10 and Lemma 4.3, that $\\{\alpha<\kappa^{+}\mid
M_{\alpha}\nleq_{\oplus}M_{\alpha+1}\\}$ is stationary. We need the following
result about $\iota_{\mu}$.

######  Lemma 4.16.

The morphism $\iota_{\mu}$ is a split monomorphism for every $\mu\in S$.

###### Proof.

First, observe that

$\textstyle\mathrm{Im}(\iota_{\mu})=\bigoplus_{\nu<\lambda}N^{\prime}_{c_{\mu}(\nu),\nu}$

and that
$N^{\prime}_{c_{\mu}(\nu),\nu}\leq_{\oplus}N_{c_{\mu}(\nu),\nu+1}\leq_{\oplus}M^{\prime}_{c_{\mu}(\nu)}.$
Thus,

$\textstyle\mathrm{Im}(\iota_{\mu})\leq_{\oplus}\bigoplus_{\nu<\lambda}M^{\prime}_{c_{\mu}(\nu)}.$

Let us show that $\bigoplus_{\nu<\lambda}M^{\prime}_{c_{\mu}(\nu)}$ is a
direct summand of $M_{\mu}$. To show this let

$C:=\mathrm{Im}(c_{\mu})\cup\\{\beta+1\mid\beta\in\mathrm{Im}(c_{\mu})\\}\cup\\{\sup\\{c_{\mu}(\nu)\mid\nu<\sigma\\}\mid\sigma<\lambda\text{
limit}\\}.$

######  Claim 4.16.1.

$C$ is a club in $\mu$ with $C\cap S=\emptyset$.

###### Proof of claim.

Clearly, $C\cap S=\emptyset$ because if $\alpha\in C$ is successor, then
$\alpha\notin S$ by Remark 4.8, and if $\alpha$ is limit, then its cofinality
is smaller than $\lambda$, and the ordinals in $S$ have cofinality equal to
$\lambda$.

Since $C$ is unbounded in $\mu$, it remains to see that it is closed. Let
$A\subseteq C$ be a subset of $C$ with $\sup A<\mu$. If $A$ has a cofinal
subset $A^{\prime}$ consisting of ordinals belonging to
$\mathrm{Im}(c_{\mu})\cup\\{\beta+1\mid\beta\in\mathrm{Im}(c_{\mu})\\}$, then
$\sup A=\sup
A^{\prime}\in\\{\sup\\{c_{\mu}(\nu)\mid\nu<\sigma\\}\mid\sigma<\lambda\text{
limit}\\}\subseteq C$. Otherwise, $A$ has a cofinal subset $A^{\prime}$
consisting of ordinals belonging to
$\\{\sup\\{c_{\mu}(\nu)\mid\nu<\sigma\\}\mid\sigma<\lambda\text{ limit}\\}$.
Take $\beta<\lambda$ such that
$A^{\prime}=\\{\alpha_{\gamma}\mid\gamma<\beta\\}$ and, for each
$\gamma<\beta$, take $\sigma_{\gamma}<\lambda$ satisfying

$\alpha_{\gamma}=\sup\\{c_{\mu}(\nu)\mid\nu<\sigma_{\gamma}\\}.$

Now let $\sigma=\sup\\{\sigma_{\gamma}\mid\gamma<\beta\\}$ and notice that,
since $\sup A^{\prime}<\mu$, $\sigma<\lambda$. Then it is easy to see that

$\sup A=\sup A^{\prime}=\sup\\{c_{\mu}(\nu)\mid\nu<\sigma\\}$

and, consequently, $\sup A\in C$ also. This finishes the proof of the claim. ∎

Let $C:\lambda\rightarrow C$ be a continuous strictly increasing map and
define de filtration $\\{Q_{\sigma}\mid\sigma<\lambda\\}$ of $M_{\mu}$ as
declaring $Q_{\sigma}=M_{C(\sigma)}$. Since $C\cap S\neq\emptyset$,
$Q_{\sigma}$ is a direct summand of $M_{\mu}$ for every $\sigma<\lambda$ by
Lemma 4.15. Now, for $\sigma<\lambda$, notice the following:

  * •

If $C(\sigma)=c_{\mu}(\nu)$ for some $\nu<\lambda$, then

$Q_{\sigma+1}=M_{c_{\mu}(\nu)+1}=Q_{\sigma}\oplus M^{\prime}_{c_{\mu}(\nu)}.$

  * •

If $C(\sigma)\neq c_{\mu}(\nu)$ for every $\nu<\lambda$, there exists a
submodule $B_{\sigma}$ of $M$ such that $Q_{\sigma+1}=Q_{\sigma}\oplus
B_{\sigma}$.

By Fact 2.3,

$M=\left(\bigoplus_{\sigma<\lambda}M^{\prime}_{c_{\mu}(\sigma)}\right)\bigoplus\left(\bigoplus_{\begin{subarray}{c}\sigma<\lambda\\\
C(\sigma)\notin\operatorname{Im}(c_{\mu})\end{subarray}}B_{\sigma}\right),$

which concludes the proof. ∎

######  Lemma 4.17.

$S\subseteq\\{\alpha<\kappa^{+}\mid M_{\alpha}\nleq_{\oplus}M_{\alpha+1}\\}$.

###### Proof.

Towards a contradiction suppose that
$M_{\alpha_{*}}\leq_{\oplus}M_{\alpha_{*}+1}$ for some ${\alpha_{*}}\in S$.
Then $i_{\alpha_{*}}$ is a split monomorphism so that
$\theta_{\alpha_{*}}i=i_{\alpha_{*}}\iota_{\alpha_{*}}$ is a split
monomorphism by the previous lemma. Then $i$ is a split monomorphism as well,
which is a contradiction since $\langle N,L,\mathcal{N}\rangle$ is a template.
∎

The last part of the proof is the fact that every module in the filtration
belongs to $\mathrm{Add}(X)$. Here, we use that $S$ does not reflect.

######  Lemma 4.18.

$M_{\alpha}\in\mathrm{Add}(X)$ for all $\alpha\notin S$.

In particular, $M_{\alpha+1}\in\mathrm{Add}(X)$ for all $\alpha<\kappa^{+}$.

###### Proof.

Let $\alpha\notin S$ and suppose by induction that
$M_{\beta}\in\mathrm{Add}(X)$ for all $\beta<\alpha$ not in $S$. If
$\alpha=\beta_{*}+1$ then we distinguish two cases: either $\beta_{*}\notin S$
or $\beta_{*}\in S$. In the former case,
$M_{\alpha}:=M_{\beta_{*}}\oplus\bigoplus_{\nu<\lambda}N_{\beta_{*},\nu}$, a
direct sum of members of $\mathrm{Add}(X)$ and thus a member of
$\mathrm{Add}(X).$ Alternatively, $\beta_{*}\in S$ and $M_{\alpha}$ is the
outcome of the pushout diagram

${N}$${L}$${M_{\beta_{*}}}$${M_{\alpha}.}$$\scriptstyle{i}$$\scriptstyle{\iota_{\beta_{*}}}$$\scriptstyle{\theta_{\beta_{*}}}$$\scriptstyle{i_{\beta_{*}}}$

By Lemma 4.16, $\iota_{\beta_{*}}$ is a split monomorphism hence so is
$\theta_{\beta_{*}}$. Using that
$M_{\alpha}\cong\operatorname{Coker}(\theta_{\beta_{*}})\oplus L$ and that
$\operatorname{Coker}(\theta_{\beta_{*}})=\operatorname{Coker}(\iota_{\beta_{*}})\in\mathrm{Add}(X)$
by induction hypothesis, we get that $M_{\alpha}$ belongs to $\mathrm{Add}(X)$
as well.

Finally, suppose that $\alpha$ is a limit ordinal. If
$\operatorname{cf}(\alpha)=\omega$ we fix an increasing cofinal sequence
$\langle\beta_{n}+1\mid n<\omega\rangle$ converging to $\alpha$. Since
$M_{\beta_{n+1}+1}=M_{\beta_{n}+1}\oplus M^{\prime}_{\beta_{n+1}+1}$ (see
Lemma 4.15) Fact 2.3 yields
$M_{\alpha}=\bigoplus_{n<\omega}M_{\beta_{n+1}+1}^{\prime}$ and as a result
$M_{\alpha}\in\mathrm{Add}(X).$ In case
$\operatorname{cf}(\alpha)\geq\omega_{1}$ we use the fact that $S\cap\alpha$
is non-stationary to find a club $C\subseteq\alpha$ avoiding $S$ and argue as
before with $\langle M_{\beta}\mid\beta\in C\rangle$. ∎

######  Lemma 4.19.

$M_{\alpha}$ is ${\leq}\kappa$-generated for all $\alpha<\kappa^{+}$.
Actually:

  1. (1)

If $\alpha<\max\\{\lambda,\lambda_{X}\\}$, then $M_{\alpha}$ is
$\leq\max\\{\lambda,\lambda_{X}\\}$-generated.

  2. (2)

If $\max\\{\lambda,\lambda_{X}\\}\leq\alpha<\kappa^{+}$, then $M_{\alpha}$ is
$|\alpha|$-generated.

###### Proof.

Follows by induction on $\beta<\kappa^{+}$ using that
$\lambda,\lambda_{X}<\kappa$ and that:

  * •

$M_{\beta}=M_{\alpha}\oplus\left(\bigoplus_{\nu<\lambda}N_{\nu}\right)$ if
$\beta=\alpha+1$ for $\alpha+1\notin S$;

  * •

$M_{\beta}\cong\operatorname{Coker}(\iota_{\alpha})\oplus L$ if
$\beta=\alpha+1$ for some $\alpha\in S$, and

  * •

$M_{\beta}=\bigcup_{\alpha<\kappa}M_{\alpha}$ if $\beta$ is limit.∎

Combining the above with Proposition 3.10 and Lemma 4.3 we conclude that
$M:=\bigcup_{\alpha<\kappa^{+}}M_{\alpha}$ is a $\kappa^{+}$-generated
$(\kappa^{+},\mathrm{Add}(X))$-free and
$(\kappa^{+},\mathrm{Add}(X))$-separable module that is non-trivial, as
sought.

A natural inquiry is whether the assumption in Theorem 4.12 is necessary:

######  Question 4.20.

Suppose that $\kappa$ is cardinal and that there is a $\kappa^{+}$-generated
non-trivial $(\kappa^{+},\mathrm{Add}(X))$-separable module. Must
$(\star)_{\kappa,\lambda}$ hold for some
$\lambda=\operatorname{cf}(\lambda)<\kappa$?

Related to this question, we can prove the following.

######  Proposition 4.21.

Let $\kappa$ be an infinite regular cardinal and $X$, a module with
$\lambda_{X}<\kappa$. Suppose that there is a non-trivial, $\kappa$-generated
$(\kappa,\mathrm{Add}_{\kappa}(X))$-free and strongly
$(\kappa,\mathrm{Add}_{\kappa}(X))$-free module that admits a
$\kappa$-filtration $\langle M_{\alpha}\mid\alpha<\kappa\rangle$ satisfying,
for every $\alpha<\kappa$:

  1. (1)

$M_{\alpha}\in\mathrm{Add}(X)$,

  2. (2)

$M_{\alpha+1}$ is a direct summand of $M_{\beta}$ for every $\beta>\alpha+1$,

  3. (3)

$M_{\alpha}$ is $\leq\lambda_{X}$-generated if $\alpha<\lambda_{X}$ and
$\leq|\alpha|$-generated if $\alpha\geq\lambda_{X}$.

Then there exists a stationary set $S\subseteq\kappa$ such that, for any
regular cardinal $\mu$ with $\lambda_{X}<\mu<\kappa$, $S$ does not reflect at
$\mu$.

###### Proof.

Since $M\notin\mathrm{Add}(X)$, the set

$S=\\{\alpha<\kappa\mid M_{\alpha}\nleq_{\oplus}M_{\alpha+1}\\}$

is, by virtue of Lemma 3.3, stationary in $\kappa$. Given any regular cardinal
$\mu$ with $\lambda_{X}<\mu<\kappa$ notice that $S\cap\mu$ is non-stationary.
Indeed, $\langle M_{\alpha}\mid\alpha<\mu\rangle$ serves as $\mu$-filtration
of $M_{\mu}$ consisting of modules belonging to $\mathrm{Add}_{\mu}(X)$, which
is (by construction) a $\mu$-generated trivial $(\mu,\mathrm{Add}(X))$-free
module. Thus, by Lemma 3.3, $S\cap\mu\in\mathrm{NS}_{\mu}$. ∎

The next is an outright consequence of the previous theorem which, in
particular, establishes the compactness result [EM02, Theorem IV.3.2] in our
setting of almost free modules relative to the class $\mathrm{Add}(X)$.

######  Corollary 4.22.

Let $X$ be a module and assume the following:

> $(\star)$ There exists a regular cardinal $\kappa>\lambda_{X}$ such that
> every stationary set $S\subseteq\kappa$ reflects at a regular cardinal $\mu$
> satisfying $\lambda_{X}<\mu<\kappa$.

Then every $\kappa$-generated $(\kappa,\mathrm{Add}(X))$-free and strongly
$(\kappa,\mathrm{Add}(X))$-free module with a filtration satisfying (1), (2)
and (3) of the preceding proposition is trivial.

In particular, this is true if $\kappa$ is a weakly compact cardinal.

###### Proof.

Follows from the preceding proposition. The last assertion is a consequence of
the fact that if $E$ is a stationary subset of a weakly compact cardinal,
there exists a stationary set $T$ of regular cardinals such that $S$ reflects
at $\alpha$ for each $\alpha\in T$, see [EM02, Lemma IV.3.1]. ∎

##  5\. Perfect decompositions and Enoch’s conjecture

For a module $M$ a local direct summand is a submodule $K\leq M$ which can be
expressed as $\bigoplus_{i\in I}K_{i}$ for a family of sumodules $\\{K_{i}\mid
i\in I\\}$ satisfying that for each finite set $J\subseteq I$,
$\bigoplus_{j\in J}K_{j}$ is a direct summand of $M$. A natural _compactness-
type_ 222Compactness is the abstract phenomenon by which the local properties
of a mathematical structure determine the global properties of the structure.
More specifically, if $\mathfrak{A}$ is a structure of cardinality $\kappa$
and every substructure $\mathfrak{B}$ of size ${<}\kappa$ witness a property
$\varphi$ then so does $\mathfrak{A}$ itself. question is whether every local
direct summand of a module $M$ is in fact a direct summand of it. The next
notion emerges from this speculation:

######  Definition 5.1.

A module $X$ is said to have a perfect decomposition if for each
$M\in\mathrm{Add}(X)$ every local direct summand of $M$ is a direct summand of
it.

The relationship between having a perfect decomposition and the existence of
non-trivial $\mathrm{Add}(X)$-separable modules was established in [GCIT10].
As a consequence of [GCIT10, Corollary 5.13], and assuming the generalized
continuum hypothesis, if $X$ has a perfect decomposition, then every
$\kappa$-generated and $(\kappa,\mathrm{Add}(X))$-separable module is trivial
for every uncountable cardinal $\kappa$ satisfying that $X$ is
$<\kappa$-presented (notice that if $X$ has a perfect decomposition, then $X$
satisfies i) of [GCIT10, Corollary 3.13] by [HS06, Theorem 1.4]). In this
paper, using Theorem 4.12, we prove the coverse of this result, that is, we
demonstrate that non-trivial almost free modules relative to $\mathrm{Add}(X)$
exist precisely when $X$ does not have a perfect decomposition. Indeed, there
is a natural expectation for this; on one hand, if $X$ does have a perfect
decomposition then $\mathrm{Add}(X)$ “is compact” in regards to direct
summands; on the other hand, if there is a non-trivial almost free module
relative to $\mathrm{Add}(X)$ then “compactness fails”.

In order to get the result, we assume that there exist a proper class of
cardinals $\kappa$ satisfying $(\star)_{\kappa,\lambda}$ for every regular
$\lambda<\kappa$. As a consequence of Lemma 4.11, this assumption is
relatively consistent with ZFC.

######  Theorem 5.2.

Assume that there exists a proper class of cardinals $\kappa$ satisfying
$(\star)_{\kappa,\lambda}$ for each regular $\lambda<\kappa$. The following
are equivalent for a module $X$:

  1. (1)

$X$ has a perfect decomposition.

  2. (2)

For every uncountable regular cardinal $\kappa>\lambda_{X}$ every
$\kappa$-generated and $(\kappa,\mathrm{Add}(X))$-free module is trivial.

  3. (3)

For every uncountable regular cardinal $\kappa>\lambda_{X}$ every
$\kappa$-generated and strongly $(\kappa,\mathrm{Add}(X))$-free module is
trivial.

  4. (4)

For every infinite regular cardinal $\kappa>\lambda_{X}$ every
$\kappa$-generated and $(\kappa,\mathrm{Add}(X))$-separable module is trivial.

  5. (5)

For every uncountable regular cardinal $\kappa>\lambda_{X}$ every
$\kappa$-generated, $(\kappa,\mathrm{Add}(X))$-free and
$(\kappa,\mathrm{Add}(X))$-separable module is trivial.

###### Proof.

(1) $\Rightarrow$ (2): Follows from Proposition 3.6 and [HS06, Theorem 1.4].

(1) $\Rightarrow$ (3): Follows from Lemma 4.5 and [HS06, Theorem 1.4].

(3) $\Rightarrow$ (4): Follows from Lemma 4.3.

(2) $\Rightarrow$ (5) and (4) $\Rightarrow$ (5) are trivial.

(5) $\Rightarrow$ (1): Suppose that $X$ does not have a perfect decomposition
and let a module $L\in\mathrm{Add}(X)$ admitting a local direct summand
$N:=\bigoplus_{i\in I}\bar{N}_{i}$ that is not a direct summand of it.
Moreover, this is chosen so that $I$ has the minimum cardinality witnessing
this. Set $|I|=\lambda$. It is not hard to show that $\lambda$ is regular.
Writing $I=\bigcup_{\alpha<\lambda}I_{\alpha}$ for a family of sets
$\\{I_{\alpha}\mid\alpha<\lambda\\}$ with cardinality smaller than $\lambda$
and setting $N_{\beta}:=\sum_{i\in I_{\alpha}}\bar{N}_{\alpha}$ we get a
continuous chain of direct summands of $L$ with union $N$. In particular,
$\langle N,L,\mathcal{N}\rangle$ is a $\lambda$-template, where
$\mathcal{N}=\langle N_{\alpha}\mid\alpha<\lambda\rangle$.

Now let $\kappa$ be a cardinal greater than $\max\\{\lambda_{X},\lambda\\}$
and satifying $(\star)_{\kappa,\lambda}$. We can apply Theorem 4.12 to get a
$\kappa^{+}$-generated, $(\kappa^{+},\mathrm{Add}(X))$-free and
$(\kappa^{+},\mathrm{Add}(X))$-separable module that is not trivial. Then (5)
above is false. ∎

Changing gears next we establish the consistency of ZFC with Enoch’s
conjecture for classes of the form $\mathrm{Add}(X)$ (see p.2). This result
was first proved by J. Šaroch in [Šar23, Theorem 2.2] under the existence of a
proper class of cardinals $\kappa$ satisfying that every stationary subset $E$
of $\kappa$ admits a non-reflecting stationary subset. In our case we will
employ the set-theoretic assumption used in Theorem 5.2.

We will use the following result, which is based on [BPŠ22, Corollary 5.3]
which states that a module $M$ belongs to $\mathcal{X}$ provided it has a
$\mathcal{X}$-precover with locally split kernel and an $\mathcal{X}$-cover
(here $\mathcal{X}$ is a class of modules closed under direct summands).
Recall that a morphism $m:K\rightarrow L$ is called locally split if for every
$k\in K$ there exists $h:L\rightarrow K$ with $hm(k)=k$.

######  Lemma 5.3.

Let $\kappa$ be an infinite regular cardinal, $X$ a module and $M$, a
$\kappa$-generated and $(\kappa,\mathrm{Add}(X))$-separable module with
$\kappa>\lambda_{X}$. Then the following assertions are equivalent:

  1. (1)

$M\in\mathrm{Add}(X)$;

  2. (2)

$M$ has an $\mathrm{Add}(X)$-cover.

Thus, if $\mathrm{Add}(X)$ is covering then every $\kappa$-generated and
$(\kappa,\mathrm{Add}(X))$-separable module is trivial for all regular
cardinals $\kappa>\lambda_{X}$.

###### Proof.

Let $\langle M_{\alpha}\mid\alpha<\kappa\rangle$ be a $\kappa$-filtration of
$M$ given by Lemma 4.5 and

$\textstyle\varphi\colon\bigoplus_{\alpha<\kappa}M_{\alpha}\rightarrow M$

be the homomorphism induced by the inclusion maps $i_{\alpha}\colon
M_{\alpha}\rightarrow M$. Since $\varphi$ is the morphism associated to a
totally ordered direct limit, it is locally split by [PA00, Lemma 2.1]. Since
$X$ is ${<}\kappa$-generated and $M$ is $(\kappa,\mathrm{Add}(X))$-separable,
$\varphi$ is an $\\{X\\}$-precover, hence an $\mathrm{Add}(X)$-precover. By
[BPS, Corollary 5.3], $M\in\mathrm{Add}(X).$ ∎

######  Theorem 5.4.

Assume that there exists a proper class of cardinals $\kappa$ satisfying
$(\star)_{\kappa,\lambda}$ for each regular $\lambda<\kappa$. The following
are equivalent for a module $X$:

  1. (1)

$X$ has a perfect decomposition;

  2. (2)

$\mathrm{Add}(X)$ is closed under direct limits;

  3. (3)

$\mathrm{Add}(X)$ is a covering class;

Therefore, under our set-theoretic assumption Enoch’s conjecture holds.

###### Proof.

(1) $\Rightarrow$ (2). By [HS06, Theorem 2.1] and Lemma 2.4.

(2) $\Rightarrow$ (3). The class $\mathrm{Add}(X)$ is always precovering. If
it is closed under direct limits, then it must be covering (see e.g., [Xu06,
Theorem 2.2.12]).

(3) $\Rightarrow$ (1) Follows from the preceding lemma and Theorem 5.2. ∎

Our forthcoming theorems will not bear on the instrumental Theorem 4.12 yet
will still draw further connections between almost free modules and Enoch’s
conjecture. Part of the subsequent discussions bear on the following concept:

######  Definition 5.5.

Let $\kappa$ be an infinite regular cardinal and $\mathcal{X}$ a class of
modules. We say that $\mathcal{X}$ is closed under $\kappa$-free modules if
every $(\kappa,\mathcal{X})$-free module belongs to $\mathcal{X}$.

A natural direction to asses the independence of Enoch’s conjecture from ZFC
is to find (perhaps, subject to extra set-theoretic hypothesis) a module $X$
such that the following hold: (1) $\mathrm{Add}(X)$ is not closed under direct
limits; (2) every $(\kappa,\mathrm{Add}(X))$-separable module is trivial for
every $\kappa>\lambda_{X}$. In view of Lemma 5.3, $\mathrm{Add}(X)$ would be a
candidate for a class not satisfying Enoch’s conjecture. The following result,
established in ZFC, says that for $\kappa=\aleph_{1}$ this is not possible:

######  Theorem 5.6.

Suppose that $\mathcal{X}$ is a class of modules closed under
$\aleph_{1}$-free modules, direct sums and direct summands. Then, if
$\mathcal{X}$ is precovering it is closed under direct limits. In particular,
Enoch’s conjecture holds for those classes.

###### Proof.

By Lemma 2.4 it suffices to show that $\mathcal{X}$ is closed under direct
limits of totally ordered systems of modules in $\mathcal{X}$. Since every
totally ordered set admits a cofinal subset that is linearly ordered we may
and do assume that our system of modules $\mathcal{F}$ is indexed by an
ordinal $\lambda$; thus
$\mathcal{F}=(F_{\alpha},\iota_{\alpha,\beta})_{\alpha\leq\beta<\lambda}$ with
$F_{\alpha}\in\mathcal{X}$ for every $\alpha<\kappa$. By passing to a cofinal
subset of $\lambda$ we may assume that $\lambda$ is a regular cardinal. Set
$F:=\varinjlim\mathcal{F}$.

Using the argument of [CI17, Lemma 2.3], we can find an infinite cardinal
$\kappa$ greater than $\max\\{\lambda,|F|,|R|\\}$ satisfying that
$\kappa^{\mu}=\kappa$ for each cardinal $\mu<\kappa$ and
$\kappa^{\lambda}=2^{\kappa}$. Following [CI17, Section 2], one constructs the
$(\aleph_{1},\mathrm{Sum}\\{F_{\alpha}\mid\alpha<\lambda\\})$-free module $L$
associated to $\mathcal{F}$ and $\kappa$. Since $\mathcal{X}$ is closed both
under direct sums and $\aleph_{1}$-free modules it follows that
$L\in\mathcal{X}.$ Also, by [CI17, Lemma 2.1] this module comes together with
a short exact sequence

${0}$${D}$${L}$${F^{(2^{\kappa})}}$${0.}$$\scriptstyle{\subseteq}$

where $|D|=\kappa$ by the election of $\kappa$. Since $\mathcal{X}$ is a
precovering class there is a short exact sequence

${0}$${M}$${A}$${F}$${0}$$\scriptstyle{m}$$\scriptstyle{f}$

with $A\in\mathcal{X}$ and $f$ an $X$-precover of $F$.

######  Claim 5.6.1.

$f$ splits.

###### Proof of claim.

This argument is due to Šaroch (see [Šar18, Lemma 3.2]) and we include it for
the reader’s convenience only. Apply $\mathrm{Hom}_{R}(-,m)$ to the previously
displayed short exact sequence to obtain the following diagram with exact
arrows

${\mathrm{Hom}_{R}(D,M)}$${\mathrm{Ext}^{1}_{R}(F,M)^{2^{\kappa}}}$${\mathrm{Ext}^{1}_{R}(L,M)}$${\mathrm{Hom}_{R}(D,A)}$${\mathrm{Ext}^{1}_{R}(F,A)^{2^{\kappa}}}$${\mathrm{Ext}^{1}_{R}(L,A).}$$\scriptstyle{\delta}$$\scriptstyle{\mathrm{Hom}_{R}(D,m)}$$\scriptstyle{\mathrm{Ext}^{1}_{R}(F,m)^{2^{\kappa}}}$$\scriptstyle{\mathrm{Ext}^{1}_{R}(L,m)}$$\scriptstyle{\overline{k}}$

Since $L\in\mathcal{X}$ and $f$ is an $\mathcal{X}$-precover,
$\mathrm{Ext}^{1}_{R}(L,m)$ is monic, hence
$\mathrm{Ker}(\mathrm{Ext}^{1}_{R}(F,m))^{2^{\kappa}}\subseteq\mathrm{Im}(\delta)$.
Nonetheless, $2^{\kappa}\geq|\mathrm{Hom}_{R}(D,M)|\geq|\mathrm{Im}(\delta)|$
and if $\mathrm{Ext}^{1}_{R}(F,m)$ is not monic, then
$\operatorname{Ker}(\mathrm{Ext}^{1}_{R}(F,m))^{2^{\kappa}})\geq
2^{2^{\kappa}}$, so that $\operatorname{Ext}^{1}_{R}(F,m)$ has to be monic.
This latter is equivalent to $\mathrm{Hom}_{R}(F,f)$ to be onto which implies,
in particular, that $f$ splits. ∎

Thus $F\leq_{\oplus}A\in\mathcal{X}$ and, since $\mathcal{X}$ is closed under
direct summands $F\in\mathcal{X}$. ∎

We are going to apply this result to some classes of relatively Mittag-Leffler
modules. For a class $\mathcal{Q}$ of right $R$-modules, a left $R$-module $M$
is called _$\mathcal{Q}$ -Mittag-Leffler_ if for each family of modules
$\\{Q_{i}\mid i\in I\\}$ belonging to $\mathcal{Q}$ the natural homomorphism

$\psi\colon M\otimes_{R}\prod_{i\in I}Q_{i}\rightarrow\prod_{i\in
I}M\otimes_{R}Q_{i}$

is monic. A left $R$-module $M$ is called _Mittag-Leffler_ whenever
$\mathcal{Q}=\mathrm{Mod}$-$R$.

Mittag-Leffler modules were introduced by Raynaud and Gruson in [RG71]. The
key property of the class of $\mathcal{Q}$-Mittag-Leffler modules is that it
is closed under $\aleph_{1}$-free modules [HT12, Theorem 2.6]. Actually, the
class of all modules with $\mathcal{Q}$-Mittag-Leffler dimension less than or
equal to $n$ is closed under $\aleph_{1}$-free modules [CI16a, Corollary 3.4].
Recall that a module $M$ has $\mathcal{Q}$-Mittag-Leffler dimension less than
or equal to $n\geq 1$ if $M$ has a projective resolution whose
$(n-1)st$-syzygy is $\mathcal{Q}$-Mittag-Leffler (and we say that $M$ has
$\mathcal{Q}$-Mittag-Leffler dimension $0$ if it is $\mathcal{Q}$-Mittag-
Leffler). We say that the left global $\mathcal{Q}$-Mittag-Leffler dimension
of $R$ is less than or equal to $n$ if every module has $\mathcal{Q}$-Mittag-
Leffler dimension less than or equal to $n$.

As an immediate consequence of Theorem 5.6 we get the following extension of
[BYT0, Theorem 2.6] to the non-flat setting:

######  Corollary 5.7.

Let $\mathcal{Q}$ be a class of right $R$-modules, $n$ a natural number. If
$\mathcal{M}$ is the class of all left $R$-modules with $\mathcal{Q}$-Mittag-
Leffler dimension less than or equal to $n$ then, $\mathcal{M}$ is precovering
if and only if the left global $\mathcal{Q}$-Mittag-Leffler of $R$ is less
than or equal to $n$.

In particular, Enoch’s conjecture holds for $\mathcal{M}$.

###### Proof.

Suppose that $\mathcal{M}$ is precovering. The class $\mathcal{M}$ is closed
under direct sums, direct summands and $\aleph_{1}$-free modules by [CI16a,
Corollary 3.4]. By the previous theorem, $\mathcal{M}$ is closed under direct
limits. But this implies that every module belongs to $\mathcal{M}$, since
every module is a direct limit of finitely presented modules and finitely
presented modules are $\mathcal{Q}$-Mittag-Leffler. ∎

Given $n$ a natural number, we say that the ring $R$ is left weak $n$-coherent
[CI16a, p. 4566] if the product of every family of flat right $R$-modules has
flat dimension less than or equal to $n$ ($R$ is left weak $0$-coherent if and
only if it is left cohererent). We get:

######  Corollary 5.8.

Let $\mathcal{F}$ be the class of all flat right $R$-modules, $n$ a natural
number and $\mathcal{M}_{n}$ the class of all left $R$-modules with
$\mathcal{F}$-Mittag-Leffler dimension less than or equal to $n$. Then:

  1. (1)

If $n=0$, $\mathcal{M}_{0}$ is precovering if and only if $R$ is left
noetherian.

  2. (2)

If $n=1$, $\mathcal{M}_{1}$ is precovering if and only if $R$ is left
coherent.

  3. (3)

If $n>1$, $\mathcal{M}_{n}$ is precovering if and only if $R$ is left weak
$n$-coherent.

###### Proof.

(2) and (3) follows from [CI16a, Theorem 4.2]. For $n=0$, the previous
corollary says that every left $R$-module is $\mathcal{F}$-Mittag-Leffler.
This is equivalent to $R$ being left noetherian by [AHH08, Example 5.6]. ∎

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