Tor-pairs: products and approximations

TL;DR

This paper extends the theory of modules with finite flat dimension to modules with finite b5-b5-projective dimension within a Tor-pair context, linking this to relative Mittag-Leffler dimensions and approximation properties.

Contribution

It characterizes when products of modules in b5 are of finite b5-projective dimension, relating it to relative b5-Mittag-Leffler dimensions and approximation theory.

Findings

Characterization of finite b5-projective dimension in Tor-pairs

Connections between b5-Mittag-Leffler dimension and module approximations

Simplified proofs of deconstructible classes being precovering and preenveloping

Abstract

Recently the author has studied rings for which products of flat modules have finite flat dimension. In this paper we extend the theory to characterize when products of modules in $\mathcal T$ have finite $\mathcal T$-projective dimension, where $\mathcal T$ is the left hand class of a Tor-pair $(\mathcal T,\mathcal S)$, relating this property with the relative $\mathcal T$-Mittag-Leffler dimension of modules in $\mathcal S$. We apply these results to study the existence of approximations by modules in $\mathcal T$. In order to do this, we give short proofs of the well known results that a deconstructible class is precovering and that a deconstructible class closed under products is preenveloping.

Equations

$$0$$

$${\mathrm{Tor}}_1^R(T,S)$$

$$0$$

$$\cdots$$

$${\mathrm{pd}}_{\mathcal{X}}(M)=\min \left(\{m<\omega :M\in {\mathcal{X}}_m\}\cup \{\omega \}\right)$$

$${\mathrm{pd}}_{\mathcal{X}}(\mathcal{Y})=\sup \{{\mathrm{pd}}_{\mathcal{X}}(Y):Y\in \mathcal{Y}\}$$

$${\mathrm{pd}}_{\mathcal{F}}(M)=\min \{n\le \omega :{\mathrm{Ext}}_R^{n+1}(M,C)=0\forall C\in \mathcal{C}\}$$

$${\mathrm{pd}}_{\mathcal{T}}(M)=\min \{n\le \omega :{\mathrm{Tor}}_{n+1}^R(M,S)=0\forall S\in \mathcal{S}\}.$$

$$0$$

$${\mathrm{Tor}}_{n+1}^R(B,S)$$

$${\mathrm{pd}}_{\mathcal{T}}\left(\mathrm{Prod}(\mathcal{T})\right)=n\iff {\mathrm{pd}}_{\mathrm{ML}(\mathcal{T})}(\mathcal{S})=n+1$$

$$\cdots$$

$$0$$

$$\left(\prod _{i\in I}{T}_i\right){\otimes }_R{K}_n$$

$$0$$

$$\left(\prod _{i\in I}{X}_i\right){\otimes }_{R}K$$

$$0$$

$${\mathrm{pd}}_{\mathcal{T}}(\mathrm{Prod}({\mathcal{T}}_m))\le {\mathrm{pd}}_{\mathcal{T}}\left(\mathrm{Prod}({\mathcal{T}}_{m+1})\right)\le {\mathrm{pd}}_{\mathcal{T}}\left(\mathrm{Prod}(\mathcal{T})\right)+m+1$$

$$0$$

$$t_{\mathcal{X}}(M)=\sum _{\begin{array}{c}f\in \mathrm{Hom}(X,M)\\ X\in \mathcal{X}\end{array}}\mathrm{Im}f$$

$$0$$

$$X\in \mathcal{X},S\le X,|S|\le \lambda \Rightarrow \exists Y\le X\text{ with }S\le Y,Y\in \mathcal{X}\text{ and }|Y|\le {\kappa }_{\lambda }$$

Full text

Introduction

Let $R$ be an associative ring with unit. A Tor-pair over $R$ is a pair of classes $(\mathcal{T},\mathcal{S})$ of right and left $R$-modules respectively, which are mutually $\operatorname{Tor}$-orthogonal (see Section 1 for details). The main objective of this work is to study two problems about Tor-pairs over $R$: when $\mathcal{T}$ is closed under products and when $\mathcal{T}$ provides for approximations.

The study of when $\mathcal{T}$ is closed under products is related with right Gorenstein regular rings. The ring $R$ is said to be right Gorenstein regular [ECIT14, Definition 2.1] if the category of right $R$-modules is a Gorenstein cateogry in the sense of [EEGR08, Definition 2.18]. These rings may be considered as the natural one-sided extension of classical Iwanaga-Gorenstein rings to non-noetherian rings (recall that the ring $R$ is Iwanaga-Gorenstein if it is two sided noetherian with finite left and right self-injective dimension).

In [BR07, Corollary VII.2.6] it is proved that the ring $R$ is right Gorenstein regular if and only if the class of all right $R$-modules with finite projective dimension coincides with the class of all right modules with finite injective dimension. If we look at the class $\mathrm{Proj}_{\omega}$ of all modules with finite projective dimension, this condition has two consequences: the right projective finitistic dimension of $R$ is finite (that is, $\mathrm{Proj}_{\omega}=\mathrm{Proj}_{n}$ for some natural number $n$, where $\mathrm{Proj}_{n}$ denotes the class of all modules with projective dimension less than or equal to $n$); and the class $\mathrm{Proj}_{\omega}$ is closed under products. As in the classical case of products of projective modules studied in [Cha60], this last property implies that products of modules with finite flat dimension have finite flat dimension. Consequently, the first step in order to understand right Gorenstein regular rings is to study rings with this property. This study is developed in [CI16].

In the first part of this paper we extend the theory of [CI16] to characterize, for a fixed Tor-pair $(\mathcal{T},\mathcal{S})$, when products of modules in $\mathcal{T}$ have finite $\mathcal{T}$-projective dimension (see Definition 1.5 for the definition of relative dimensions). As in the case of the flat modules, this property is related with the $\operatorname{ML}(\mathcal{T})$-projective dimension of modules in $\mathcal{S}$, see Theorem 2.1 (where $\operatorname{ML}(\mathcal{T})$ is the class of all Mittag-Leffler modules with respect to $\mathcal{T}$, see Definition 1.10).

In the second part of the paper we are interested in approximations by modules in $\mathcal{T}$ and in $\mathcal{T}_{n}$ (modules with $\mathcal{T}$-projective dimension less than or equal to $n$). The relationship of these approximations with the first part of the paper comes from the fact that if a class of right $R$-modules is preenveloping then it is closed under products [HJ08, Propostion 1.2]. So that, a natural question arises: if $\mathcal{T}_{n}$ is closed under products, when is it preenveloping?

One tool in order to construct approximations of modules is that of deconstruction of classes, because a deconstructible class is always precovering, [SŠ11, Theorem 2.14] and [Eno12, Theorem 5.5], and a deconstructible class closed under direct products is preenveloping [SŠ11, Theorem 4.19]). The procedure of deconstruction of a class $\mathcal{X}$ consists on finding a set $\mathcal{S}$ such that each module in $\mathcal{X}$ is $\mathcal{S}$-filtered, which means that for each $X\in\mathcal{X}$ there exists a continuous chain of submodules of $X$, $\{X_{\alpha}:\alpha<\kappa\}$ (where $\kappa$ is a cardinal), whose union is $X$ and such that $\frac{X_{\alpha+1}}{X_{\alpha}}\in\mathcal{S}$.

In Section 3 we give easy proofs of [SŠ11, Theorem 2.14] and [Eno12, Theorem 5.5] (in Theorem 3.3) and of [SŠ11, Theorem 4.19] (in Theorem 3.10), and we prove that $\mathcal{T}_{m}$ is deconstructible for each natural number $m$, so that it is always precovering and it is preenveloping precisely when it is closed under products (see Corollary 3.11).

Throught the paper $R$ will be an associative ring with unit. We shall denote by $\mathrm{Mod}\textrm{-}R$ and $R\textrm{-}\mathrm{Mod}$ the categories of all right $R$-modules and left $R$-modules respectively. Given a class $\mathcal{X}$ of right $R$-modules, we shall denote by $\mathrm{Prod}(\mathcal{X})$ the class consisting of all modules isomorphic to a direct products of modules in $\mathcal{X}$. The classes of flat and projective modules will be denoted by $\mathrm{Flat}_{R}$ and $\mathrm{Proj}_{R}$ respectively. If there is no possible confussion, we shall omit the subscript $R$. The cardinal of a set $X$ will be denoted by $|X|$.

1. Tor-pairs, relative dimensions and relative Mittag-Leffler modules

Given a class $\mathcal{X}$ of right (resp. left) $R$-modules we shall denote by $\mathcal{X}^{\top}$ (resp ${{}^{\top}}\mathcal{X}$) the class of all left (resp. right) $R$-modules $M$ satisfying $\operatorname{Tor}_{1}^{R}(X,M)=0$ (resp $\operatorname{Tor}_{1}^{R}(M,X)=0$) for each $X\in\mathcal{X}$. Recall that a Tor-pair is a pair of classes $(\mathcal{T},\mathcal{S})$ such that $\mathcal{T}={{}^{\top}}\mathcal{S}$ and $\mathcal{S}=\mathcal{T}^{\top}$. Given a class $\mathcal{X}$ of right $R$-modules (resp. left $R$-modules), the pair $({{}^{\top}}(\mathcal{X}^{\top}),\mathcal{X}^{\top})$ (resp. $({{}^{\top}}\mathcal{X},({{}^{\top}}\mathcal{X})^{\top})$) is a Tor-pair, which is called the Tor-pair generated by $\mathcal{X}$.

Given a class $\mathcal{X}$ of left modules, a short exact sequence of right modules

[TABLE]

is called $\mathcal{X}$-pure if the sequence

[TABLE]

is exact for each $X\in\mathcal{X}$. In such case, $f$ is called an $\mathcal{X}$-pure monomorphism and $g$ an $\mathcal{X}$-pure epimorphism. Note that each pure exact sequence is $\mathcal{X}$-pure exact.

Proposition 1.1**.**

Let $(\mathcal{T},\mathcal{S})$ be a Tor-pair. Then $\mathcal{T}$ is closed under direct limits, pure submodules and $\mathcal{S}$-pure quotients.

Proof.

$\mathcal{T}$ is closed under direct limits since the $\operatorname{Tor}$ functor commutes with direct limits. $\mathcal{T}$ is closed under pure submodules by [AHH08, Proposition 9.12]. In order to see that it is closed under $\mathcal{S}$-pure quotients, take $f:T\rightarrow T^{\prime}$ a pure epimorphism with $T\in\mathcal{T}$ and denote by $\iota$ the inclusion of $\operatorname{Ker}f$ into $T$. Given $S\in\mathcal{S}$ and applying $-\otimes_{R}S$ we get the exact sequence

[TABLE]

Since $T\in\mathcal{T}$, the first term is zero and, since $\ker f$ is a $\mathcal{S}$-pure submodule of $T$, $\iota\otimes_{R}S$ is monic. Then $\operatorname{Tor}_{1}^{R}(T^{\prime},S)=0$ and, as $S$ is arbitrary, $T^{\prime}$ belongs to $\mathcal{T}$. ∎

A class $\mathcal{X}$ of right $R$-modules is called resolving if it contains all projective modules and is closed under extensions and kernels of epimorphisms. A cotorsion pair $(\mathcal{F},\mathcal{C})$ is hereditary if $\mathcal{F}$ is resolving. Similarly, we shall call a Tor-pair $(\mathcal{T},\mathcal{S})$ hereditary if $\mathcal{T}$ is resolving. The following result is the Tor-pair version of the well known characterizations of hereditary cotorsion pairs [GaR99, Theorem 1.2.10].

Proposition 1.2**.**

Let $(\mathcal{T},\mathcal{S})$ be a Tor-pair. Then:

1. (1)

The Tor pair is hereditary. 2. (2)

$\mathcal{S}$* is resolving.* 3. (3)

$Tor^{R}_{n}(T,S)=0$* for each $T\in\mathcal{T}$, $S\in\mathcal{S}$ and nonzero natural number $n$.*

Proof.

(1) $\Rightarrow$ (3). From (1) follows that all syzygies of any module in $\mathcal{T}$ belong to $\mathcal{T}$. Then (3) is a consequence of [Rot09, Corollary 6.23].

(3) $\Rightarrow$ (1). Given a short exact sequence

[TABLE]

with $T_{1},T_{2}\in\mathcal{T}$, the induced long exact sequence when tensoring with any $S\in\mathcal{S}$ gives an isomorphism $\operatorname{Tor}_{2}^{R}(T_{2},S)\cong\operatorname{Tor}_{1}^{R}(K,S)$. Then (3) gives that $\operatorname{Tor}_{1}^{R}(K,S)=0$ and $K\in\mathcal{S}$.

(3) $\Leftrightarrow$ (2). Follows from the previous proof, since (3) is left-right symmetric. ∎

Proposition 1.3**.**

Let $(\mathcal{T},\mathcal{S})$ be a hereditary Tor-pair. Then $\mathcal{T}$ is closed under $\mathcal{S}$-pure submodules.

Proof.

We can argue as in [AHH08, Proposition 9.12]. Let $T$ be a module in $\mathcal{T}$ and $K$ a $\mathcal{S}$-pure submodule of $T$. Let $S$ be any module in $\mathcal{S}$ and take

[TABLE]

a projective presentation of $S$. We can construct the following commutative diagram with exact rows:

[TABLE]

Now $g$ is monic as the inclusion $K\rightarrow T$ is $\mathcal{S}$-pure and $S^{\prime}\in\mathcal{S}$ by Proposition 1.2. Then $f$ is monic and $\operatorname{Tor}_{1}^{R}(K,S)=0$ since $\operatorname{Tor}_{1}^{R}(T,S)=0$. Because $S$ is arbitrary, we conclude that $K\in\mathcal{T}$. ∎

Examples 1.4*.*

1. (1)

The pair of classes $(\mathrm{Flat},R\textrm{-}\mathrm{Mod})$ is a hereditary Tor-pair. 2. (2)

Recall that a left $R$-module $C$ is cyclically presented provided that $C\cong\frac{R}{Rx}$ for some $x\in R$. A right module $X$ satisfying $\operatorname{Tor}_{1}^{R}(X,C)=0$ for each cyclically presented left module $C$ is called torsion-free. We shall denote by $\operatorname{TFree}$ the class consisting of all torsion-free right modules. Then $(\operatorname{TFree},\operatorname{TFree}^{\top})$ is a Tor-pair.

We shall use the homological notation for projective resolutions so that, for a given a right $R$-module $M$, a projective resolution of $M$ will be denoted

[TABLE]

Then the $nth$-syzygy of $M$ will be $\operatorname{Ker}d_{n}$ for each natural number $n$.

Definition 1.5**.**

Let $\mathcal{X}$ be a class of left $R$-modules containing all projective modules.

1. (1)

Given a nonzero natural number $n$ and a left $R$-module $M$, we shall say that $M$ has projective dimension relative to $\mathcal{X}$ (or $\mathcal{X}$-projective dimension) less than or equal to $n$ (written $\operatorname{pd}_{\mathcal{X}}(M)\leq n$) if there exists a projective resolution of $M$ such that its $(n-1)st$ syzygy belongs to $\mathcal{X}$. We shall denote by $\mathcal{X}_{n}$ the class of all modules with $\mathcal{X}$-projective dimension less than or equal to $n$ (if $n=0$, $\mathcal{X}_{0}$ will be $\mathcal{X}$). Moreover, we shall denote $\mathcal{X}_{\omega}=\bigcup_{n<\omega}\mathcal{X}_{n}$. 2. (2)

Given a left $R$-module $M$ the $\mathcal{X}$-projective dimension of $M$ is

[TABLE]

Note that if $\mathcal{X}$ is closed under direct summands and finite direct sums, the $\mathcal{X}$-projective dimension does not depend on the chosen projective resolution, since, for each natural number $n$, any two $n$-sysygies of a module are projectively equivalent by [Rot09, Proposition 8.5].

Definition 1.6**.**

Let $\mathcal{X}$ and $\mathcal{Y}$ be a class of left modules such that $\mathcal{X}$ contains all projective modules. The $\mathcal{X}$-projective dimension of $\mathcal{Y}$ is

[TABLE]

Note that $\operatorname{pd}_{\mathrm{Proj}}(\mathrm{Mod}\textrm{-}R)$ is the right global dimension of the ring and $\operatorname{pd}_{\mathrm{Proj}}(\mathrm{Proj}_{\omega})$ is the right finitistic projective dimension of the ring, i. e., the supremmun of the projective dimensions of all modules with finite projective dimension. For a general class of right modules $\mathcal{X}$ containing all projective modules, $\operatorname{pd}_{\mathcal{X}}(\mathcal{X}_{\omega})$ is called in [BME17] the left finitistic $\mathcal{X}$-projective dimension of $R$ and is denoted there by $\operatorname{FPD}_{\mathcal{X}}(R)$.

As it is proved in [CI16, Lemma 3.9] using an argument from [BR07, Corollary VII.2.6], when $\mathcal{Y}$ is closed under countable direct sums or countable direct products, we only have to see that each module in $\mathcal{Y}$ has finite $\mathcal{X}$-projective dimension in order to get that $\operatorname{pd}_{\mathcal{X}}(\mathcal{Y})$ is finite.

Lemma 1.7**.**

[CI16, Lemma 3.9]** Let $\mathcal{X}$ and $\mathcal{Y}$ be classes of left $R$-modules such that $\mathcal{X}$ is closed under direct summands, finite direct sums and contains all projecive modules, and $\mathcal{Y}$ is closed under countable direct sums or countable direct products. Then the following assertions are equivalent:

1. (1)

$\operatorname{pd}_{\mathcal{X}}(Y)$* is finite for each $Y\in\mathcal{Y}$.* 2. (2)

$\operatorname{pd}_{\mathcal{X}}(\mathcal{Y})$* is finite.*

For cotorsion pairs and Tor-pairs we can compute relative dimensions using $\operatorname{Ext}$ and $\operatorname{Tor}$ functors respectively.

Lemma 1.8**.**

Let $n$ be a natural number and $M$ a right $R$-module.

1. (1)

If $(\mathcal{F},\mathcal{C})$ is a cotorsion pair in $\mathrm{Mod}\textrm{-}R$, then $\operatorname{pd}_{\mathcal{F}}(M)\leq n$ if and only if $\operatorname{Ext}^{n+1}_{R}(M,C)=0$ for each $C\in\mathcal{C}$. Moreover

[TABLE] 2. (2)

If $(\mathcal{T},\mathcal{S})$ is a Tor-pair, then $\operatorname{pd}_{\mathcal{T}}(M)\leq n$ if and only if $\operatorname{Tor}_{n+1}^{R}(M,S)=0$ for each $S\in\mathcal{S}$. Moreover,

[TABLE]

Proof.

Both proofs are similar. We shall prove (2). Let $K_{n-1}$ be a $(n-1)th$-syzygy of $M$. Then $\operatorname{pd}_{\mathcal{T}}(M)=n$ if and only if $K_{n-1}\in\mathcal{T}$ if and only if $\operatorname{Tor}_{1}^{R}(K_{n-1},S)=0$ for each $S\in\mathcal{S}$. But by [Rot09, Corollary] this is equivalent to $\operatorname{Tor}^{R}_{n+1}(M,S)=0$ for each $S\in\mathcal{S}$. ∎

Using this result it is easy to compute the dimension of the third module in a short exact sequence in the same way as it can be done for the classical projective dimension [Rot09, Exercise 8.5]. We shall use this result later.

Proposition 1.9**.**

Let $(\mathcal{T},\mathcal{S})$ be a Tor-pair and

[TABLE]

a short exact sequence of right modules. Then:

1. (1)

If $\operatorname{pd}_{\mathcal{T}}(B)<\operatorname{pd}_{\mathcal{T}}(A)$ then $\operatorname{pd}_{\mathcal{T}}(C)=\operatorname{pd}_{\mathcal{T}}(A)+1$. 2. (2)

If $\operatorname{pd}_{\mathcal{T}}(B)>\operatorname{pd}_{\mathcal{T}}(A)$ then $\operatorname{pd}_{\mathcal{T}}(C)=\operatorname{pd}_{\mathcal{T}}(B)$. 3. (3)

If $\operatorname{pd}_{\mathcal{T}}(B)=\operatorname{pd}_{\mathcal{T}}(A)$, then $\operatorname{pd}_{\mathcal{T}}(C)\leq\operatorname{pd}_{\mathcal{T}}(A)+1$.

Proof.

Given a nonzero natural number $n$ and $S\in\mathcal{S}$ we have, by [Rot09, Corollary 6.30], the exact sequence

[TABLE]

Set $n_{A}=\operatorname{pd}_{\mathcal{T}}(A)$ and $n_{B}=\operatorname{pd}_{\mathcal{T}}(B)$; take $S_{A},S_{B}\in\mathcal{S}$ with $\operatorname{Tor}_{n_{A}}^{R}(A,S_{A})\neq 0$ and $\operatorname{Tor}_{n_{B}}(B,S_{B})\neq 0$.

If $n_{B}<n_{A}$, then the sequence (i) for $n=n_{A}+1$ gives $\operatorname{Tor}_{n_{A}+2}(C,S)=0$ for each $S\in\mathcal{S}$. Morover, for $n=n_{A}$ and $S=S_{A}$, the sequence (i) gives $\operatorname{Tor}_{n_{A}+1}(C,S_{A})\neq 0$. Consequently $\operatorname{pd}_{\mathcal{T}}(C)=n_{A}+1$.

If $n_{A}<n_{B}$, then the sequence (i) for $n=n_{B}$ gives $\operatorname{Tor}_{n_{B}+1}(C,S)=0$ for all $S\in\mathcal{S}$. The same sequence for $n=n_{B}$ and $S=S_{B}$ gives that $\operatorname{Tor}_{n_{B}}^{R}(C,S_{B})\neq 0$, so that $\operatorname{pd}_{\mathcal{T}}(C)=n_{B}$.

Finally, if $n_{A}=n_{B}$, the sequence (i) for $n=n_{A}+1$ gives that $\operatorname{Tor}_{n_{A}+2}^{R}(C,S)=0$ for each $S\in\mathcal{S}$, so that $\operatorname{pd}_{\mathcal{T}}(C)\leq n_{A}+1$. ∎

Mittag-Leffler modules were introduced by Raynaud and Gruson in their seminal paper [RG71]. We shall work with the following relativization of the concept, introduced in [Rot94].

Definition 1.10**.**

Let $\mathcal{X}$ be a class of right $R$-modules and $M$ a left $R$-module. We say that $M$ is $\mathcal{X}$-Mittag-Leffler if for any family of modules in $\mathcal{X}$, $\{X_{i}:i\in I\}$, the canonical morphism from $\left(\prod_{i\in I}X_{i}\right)\otimes_{R}M$ to $\prod_{i\in I}(X_{i}\otimes_{R}M)$ is monic.

We shall denote by $\operatorname{ML}(\mathcal{X})$ the class consisting of all $\mathcal{X}$-Mittag-Leffler left $R$-modules. We are interested in Mittag-Leffler modules relative to a Tor-pair.

2. Tor pairs closed under products

As we have mentioned before, in [CI16] they are characterized rings for which direct products of flat modules have finite flat dimension. Let $(\mathcal{T},\mathcal{S})$ be a hereditary Tor-pair. In this section we study rings for which direct products of modules in $\mathcal{T}$ have finite $\mathcal{T}$-projective dimension. The main result relates this property with the $\operatorname{ML}(\mathcal{T})$-projective dimension of the class $\mathcal{S}$.

Theorem 2.1**.**

The following assertions are equivalent for a hereditary Tor-pair $(\mathcal{T},\mathcal{S})$ and a natural number $n$.

1. (1)

Each product of modules in $\mathcal{T}$ has $\mathcal{T}$-projective dimension less than or equal to $n$. 2. (2)

Each module in $\mathcal{S}$ has finite $\operatorname{ML}(\mathcal{T})$-projective dimension less than or equal to $n+1$.

Consequently:

[TABLE]

Proof.

Fix $\{T_{i}:i\in I\}$ a family of modules in $\mathcal{T}$ and $S$ an object of $\mathcal{S}$. Take a projective resolution of $S$,

[TABLE]

and consider the short exact sequence

[TABLE]

where $K_{n-1}=\ker d_{n-1}$ and $K_{-1}=S$ if $n=0$. Tensoring by $\prod_{i\in I}T_{i}$ we can construct the following commutative diagram with exact rows:

[TABLE]

(note that the last row is exact because $K_{n-1}\in\mathcal{S}$ by Proposition 1.2). Since $P_{n}$ is Mittag-Leffler, $h$ is monic and, consequently, $f$ is monic if and only if $g$ is monic. By [Rot09, Corollary 6.23 and Corollary 6.27] there exists a exact sequence

[TABLE]

so that $f$ is monic if and only if $\operatorname{Tor}_{n+1}^{R}\left(\prod_{i\in I}T_{i},S\right)=0$. The conclusion is that for a fixed family $\{T_{i}:i\in I\}$ in $\mathcal{T}$ and module $S\in\mathcal{S}$, $g$ is monic if and only if $\operatorname{Tor}_{n+1}^{R}\left(\prod_{i\in I}T_{i},S\right)=0$.

Now using that both $\{T_{i}:i\in I\}$ and $S$ are arbitrary we get, by Lemma 1.8, that all products of modules in $\mathcal{T}$ have $\mathcal{T}$-projective dimension less than or equal to $n$ if and only if each module in $\mathcal{S}$ has $\operatorname{ML}(\mathcal{T})$-projective dimension less than or equal to $n+1$. ∎

As an inmediate consequence we get the characterization of when the left hand class of a Tor-pair is closed under products.

Corollary 2.2**.**

The following assertions are equivalent for a hereditary Tor-pair $(\mathcal{T},\mathcal{S})$.

1. (1)

$\mathcal{T}$* is closed under products.* 2. (2)

Each module in $\mathcal{S}$ has $\operatorname{ML}(\mathcal{T})$-projective dimension less than or equal to $1$.

If we apply this result to the Tor-pair induced by the flat modules, we get the following well known results. Recall that the class of flat Mittag-Leffler modules is closed under extensions since, if

[TABLE]

is a short exact sequence in $R\textrm{-}\mathrm{Mod}$ with $K$ and $N$ flat and Mittag-Leffler, then $M$ is flat, the sequence is pure and, for each family of right $R$-modules $\{X_{i}:i\in I\}$ there exists a commutative diagram

[TABLE]

from which follows that $g$ is monic, as $f$ and $h$ are.

Corollary 2.3**.**

1. (1)

Pure submodules of flat Mittag-Leffler right modules are Mittag-Leffler. 2. (2)

$R$* is right coherent if and only if each submodule of a projective right module is Mittag-Leffler with respect to the flat modules.*

Proof.

(1) If we apply the previous result to the Tor-pair $(\mathrm{Flat}_{R},R\textrm{-}\mathrm{Mod})$ we get that each flat right module has $\operatorname{ML}(R\textrm{-}\mathrm{Mod})$-projective dimension less than or equal to $1$, as $R\textrm{-}\mathrm{Mod}$ is closed under products. Noting that $\mathrm{Flat}_{R}$ consists of all pure quotients of projective modules and that $\operatorname{ML}(R\textrm{-}\mathrm{Mod})$ is the class of all Mittag-Leffler modules, this is equivalent to all pure submodules of projective right modules being (flat) Mittag-Leffler modules.

Now let $M$ be a flat Mittag-Leffler right module and $K$ a pure submodule of $M$. Let $f:P\rightarrow\frac{M}{K}$ be an epimorphism with $P$ projective. Making pullback of $f$ along the projection $M\rightarrow\frac{M}{K}$ we get the following commutative diagram with exact rows and colummns:

[TABLE]

Since the first column is pure and $P$ is projective, $\operatorname{Ker}f$ is flat Mittag-Leffler by the previous proof. Then, as the class of flat Mittag-Leffler modules is closed under extensions, $Q$ is flat Mittag-Leffler as well. But the middle row is split, so that $K$ is isomorphic to a direct summand of $Q$. Thus, $K$ is flat Mittag-Leffler.

(2) If we consider the Tor-pair $(\mathrm{Mod}\textrm{-}R,{{}_{R}}\mathrm{Flat})$, we get that $R$ is right coherent if and only if ${{}_{R}}\mathrm{Flat}$ is closed under products if and only if (by the left version of Corollary 2.2) each right module has projective dimension relative to the Mittag-Leffler modules less than or equal to $1$. But this is equivalent to each submodule of a projective module being Mittag-Leffler with respect to the flat modules. ∎

Now, what about the class $\mathcal{T}_{m}$ where $m$ is a nonzero natural number? When is it closed under products? The following result, which extends [CI16, Proposition 4.1], gives the answer.

Proposition 2.4**.**

The following assertions are equivalent for a Tor pair $(\mathcal{T},\mathcal{S})$.

1. (1)

Each module in $\mathrm{Prod}(\mathcal{T})$ has finite $\mathcal{T}$-projective dimension. 2. (2)

$\operatorname{pd}_{\mathcal{T}}\left(\mathrm{Prod}(\mathcal{T})\right)$* is finite.* 3. (3)

There exists a natural number $m$ such that each module in $\mathrm{Prod}(\mathcal{T}_{m})$ has finite $\mathcal{T}$-projective dimension. 4. (4)

There exists a natural number $m$ such that $\operatorname{pd}_{\mathcal{T}}\left(\mathrm{Prod}(\mathcal{T}_{m})\right)$ is finite. 5. (5)

For any natural number $m$ each module in $\mathrm{Prod}(\mathcal{T}_{m})$ has finite $\mathcal{T}$-projective dimension. 6. (6)

For any natural number $m$, $\operatorname{pd}_{\mathcal{T}}\left(\mathrm{Prod}(\mathcal{T}_{m})\right)$ is finite.

Moreover, when all these conditions are satisfied then

[TABLE]

for each natural number $m$. If, in addition $\operatorname{pd}_{\mathcal{T}}(\mathrm{Prod}\mathcal{T})=0$ (that is, $\mathcal{T}$ is closed under products), then $\mathcal{T}_{m}$ is closed under products for each natural number $m$.

Proof.

(1) $\Leftrightarrow$ (2), (3) $\Leftrightarrow$ (4) and (5) $\Leftrightarrow$ (6) follow from Lemma 1.7.

(1) $\Leftrightarrow$ (4) and (5) $\Rightarrow$ (1) are trivial.

(1) $\Rightarrow$ (5) is proved by dimension shifting noting that, if the result is true for some natural number $m$ and $\{T_{i}:i\in I\}$ is a family of modules having $\mathcal{T}$-projective dimension less than or equal to $m+1$ then, for each $i\in I$ there exists a short exact sequence

[TABLE]

with $P_{i}$ projective and $K_{i}\in\mathcal{T}_{m}$. Since the direct product is an exact functor, these sequences give the exact sequence

[TABLE]

in which both the first and second term have finite $\mathcal{T}$-projective dimension by the induction hyphotesis. Then so has $\prod_{i\in I}T_{i}$ by Proposition 1.9.

In order to prove the last inequality we shall proceed by induction on $m$. Suppose that we have proved the result for some natural number $m$. The first inequality is trivial, since $\mathcal{T}_{m}\subseteq\mathcal{T}_{m+1}$. In order to prove the other one simply note that for any family of modules $\{T_{i}:i\in I\}$ in $\mathcal{T}_{m+1}$ we can construct, as above, a short exact sequence

[TABLE]

with $P\in\mathrm{Prod}(\mathcal{T})$ and $K\in\mathrm{Prod}(\mathcal{T}_{m})$. Using Proposition 1.9 and the induction hyphotesis we get the desired inequality.

Finally, if $\operatorname{pd}_{\mathcal{T}}(\mathrm{Prod}(\mathcal{T}))=0$ we induct on $m$. If $\mathcal{T}_{m}=\mathcal{T}_{m-1}$, the result follows from the induction hyphotesis. If $\mathcal{T}_{m}\neq\mathcal{T}_{m-1}$, then the preceeding inequality gives $\operatorname{pd}_{\mathcal{T}}(\mathrm{Prod}(\mathcal{T}_{m}))\leq m$. In addition $m\leq\operatorname{pd}_{\mathcal{T}}(\mathrm{Prod}(\mathcal{T}_{m}))$ as well, so that $\mathcal{T}_{m}$ is closed under products. ∎

As an application of this result we can characterize when the class $\mathcal{T}_{\omega}$ is closed under products:

Corollary 2.5**.**

The following assertions are equivalent for a hereditary Tor-pair $(\mathcal{T},\mathcal{F})$.

1. (1)

$\mathcal{T}_{\omega}$* is closed under direct products.* 2. (2)

$\operatorname{pd}_{\mathcal{T}}(\mathcal{T}_{\omega})$* and $\operatorname{pd}_{\operatorname{ML}(\mathcal{T})}(\mathcal{S})$ are finite. That is, the right finitistic $\mathcal{T}$-projective dimension is finite and each module in $\mathcal{S}$ has finite $\operatorname{ML}(\mathcal{T})$-projective dimension.*

Proof.

(1) $\Rightarrow$ (2). If $\mathcal{T}_{\omega}$ is closed under direct products, we can apply Lemma 1.7 to get that $\operatorname{pd}_{\mathcal{T}}(\mathcal{T}_{\omega})$ is finite. That is, $\mathcal{T}_{\omega}=\mathcal{T}_{n}$ for some natural number $n$. Now $\operatorname{pd}_{\operatorname{ML}(\mathcal{T})}(\mathcal{S})$ is finite as a consequence of Theorem 2.1 and Proposition 2.4.

(2) $\Rightarrow$ (1). Since $\operatorname{pd}_{\mathcal{T}}(\mathcal{T}_{\omega})$ is finite, there exists a natural number $n$ such that $\mathcal{T}_{\omega}=\mathcal{T}_{n}$. Now, as $\operatorname{pd}_{\operatorname{ML}(\mathcal{T})}(\mathcal{S})$, apply Corollary 2.2 to get that each product of modules in $\mathcal{T}$ has finite $\mathcal{T}$-projective dimension. By Proposition 2.4, $\mathcal{T}_{n}$ is closed under products as well. ∎

Recall that a class $\mathcal{X}$ of right $R$-modules is definable if it is closed under direct products, direct limits and pure submodules. As a consequence of the results of this section we can characterize when, fixed a Tor-pair $(\mathcal{T},\mathcal{S})$, the classes $\mathcal{T}_{m}$ are definable for each natural number $m$. The same proof of [CI16, Proposition 4.7] gives:

Proposition 2.6**.**

Let $(\mathcal{T},\mathcal{S})$ be a hereditary Tor-pair and $n$ a natural number. Then $\mathcal{T}_{n}$ is closed under direct limits and pure submodules.

Proof.

The closure under direct limits follows from Lemma 1.8 and the fact that the $\operatorname{Tor}_{n+1}^{R}$ functor commutes with direct colimits. In order to see that $\mathcal{T}_{n}$ is closed under pure submodules take $T\in\mathcal{T}_{n}$ and $T^{\prime}$ a pure submodule of $T$. Arguing as in [CI16, Proposition 4.7] we get, for each $S\in\mathcal{S}$, the exact sequence

[TABLE]

Now, if $T\in\mathcal{T}_{n}$ then so does $T^{\prime}$ by Lemma 1.8. ∎

Putting all things together, we charactize when $\mathcal{T}_{m}$ is a definable class for each $m\in\mathbb{N}$. This result extends [AHH08, Proposition 9.12]

Corollary 2.7**.**

The following assertions are equivalent for a hereditary Tor-pair $(\mathcal{T},\mathcal{S})$.

1. (1)

Each module in $\mathcal{S}$ has $\operatorname{ML}(\mathcal{T})$-projective dimension less than or equal to $1$. 2. (2)

$\mathcal{T}$* is closed under products.* 3. (3)

$\mathcal{T}_{m}$* is a definable category for each natural number $m$.*

Proof.

(1) $\Leftrightarrow$ (2) is Corollary 2.2. (2) $\Leftrightarrow$ (3) follows from propositions 2.4 and 2.6. ∎

3. Approximations by modules in $\mathcal{T}_{m}$

In this section we study the existence of approximations by modules in $\mathcal{T}_{m}$ for each natural number $m$. Let $\mathcal{X}$ be a class of right $R$-modules and $M$ a module. A $\mathcal{X}$-precover of $M$ is a morphism $f:X\rightarrow M$ with $X\in\mathcal{X}$ such that for each $X^{\prime}\in\mathcal{X}$, the induced morphism $\operatorname{Hom}_{R}(X^{\prime},X)\rightarrow\operatorname{Hom}_{R}(X^{\prime},M)$ is epic. The $\mathcal{X}$-precover $f$ is said to be a $\mathcal{X}$-cover if it is minimal in the sense that each endomorphism $g$ of $X$ satisfying $fg=f$ is an isomorphism. The class $\mathcal{X}$ is called precovering or covering if each right module has a $\mathcal{X}$-precover or a $\mathcal{X}$-cover respectively. Dually are defined $\mathcal{X}$-preenvelopes and $\mathcal{X}$-envelopes, and the corresponding preenveloping and enveloping classes.

Most of the known examples of classes providing for approximations are part of a “small” cotorsion pair $(\mathcal{F},\mathcal{C})$ (in the sense that it is generated by a set, i. e., there exists a set of modules $\mathcal{G}$ such that $\mathcal{C}=\mathcal{G}^{\perp}$). This is due to the fact that a cotorsion pair generated by a set always provide for precovers and preenvelopes, [GT06, Theorem 3.2.1] and [GT06, Lemma 2.2.6]. Moreover, by [GT06, Theorem 4.2.1], the left hand class of a cotorsion pair generated by a set is deconstructible (the definition will be precised later) and it has recently proved that deconstructible classes are precovering (see [SŠ11, Theorem 2.14] for a proof in exact categories and [Eno12, Theorem 5.5] for a proof in module categories), and that deconstructible classes closed under products are preenveloping [SŠ11, Theorem 4.19].

In this paper we are going to work with deconstructible classes. We are going to give easier proofs of the aforementioned results concerning deconstructible classes and approximations. Next we will use this results to prove that, if $(\mathcal{T},\mathcal{S})$ is a Tor-pair and $m$ a natural number, then $\mathcal{T}_{m}$ is always precovering, and is preenveloping povided it is closed under direct products, i. e., they are satisfied the conditions of Corollary 2.7.

Given a class of right $R$-modules $\mathcal{G}$, a $\mathcal{G}$-filtration of a module $M$ is a continuous chain of submodules of $M$, $(G_{\alpha}:\alpha<\kappa)$, where $\kappa$ is a cardinal, such that $M=\bigcup_{\alpha<\kappa}G_{\alpha}$, $G_{0}=0$ and $\frac{G_{\alpha+1}}{G_{\alpha}}\in\mathcal{G}$ for each $\alpha<\kappa$. We shall denote by $\operatorname{Filt}{-}\mathcal{G}$ the class of all $\mathcal{G}$-filtered modules. We shall say that a class of modules $\mathcal{X}$ is deconstructible if there exists a set of modules $\mathcal{G}$ such that $\mathcal{X}=\operatorname{Filt}{-}\mathcal{G}$.

We begin proving that a deconstructible class is precovering. Given a class of right modules $\mathcal{X}$ and a module $M$, the trace of $\mathcal{X}$ in $M$ is the submodule

[TABLE]

The module $M$ is said to be generated by $\mathcal{X}$ if there exists a family of modules in $\mathcal{X}$, $\{X_{i}:i\in I\}$, and an epimorphism $\varphi:\bigoplus_{i\in I}X_{i}\rightarrow M$. We shall denote by $\operatorname{Gen}(\mathcal{X})$ the class of all modules generated by $\mathcal{X}$. Recall that $M\in\operatorname{Gen}(\mathcal{X})$ if and only if $t_{\mathcal{X}}(M)=M$ [AF92, Proposition 8.12].

Lemma 3.1**.**

Let $\mathcal{X}$ be a class of right modules and $M$ a module. Suppose that $f:X\rightarrow M$ is a $\mathcal{X}$-precover of $M$. Then $\operatorname{Im}f=t_{\mathcal{X}}(M)$.

Proof.

Clearly $\operatorname{Im}f\leq t_{\mathcal{X}}(M)$. The other inclusion follows form the fact that for each morphism $g:X^{\prime}\rightarrow M$ with $X^{\prime}\in\mathcal{X}$, there exists $h:X^{\prime}\rightarrow X$ with $fh=g$ and, consequently, $\operatorname{Im}g\leq\operatorname{Im}f$. ∎

Lemma 3.2**.**

Let $\mathcal{X}$ be a class of right modules. Then $\mathcal{X}$ is precovering if and only if each module in $\operatorname{Gen}(\mathcal{X})$ has a $\mathcal{X}$-precover.

Proof.

Supppose that every module in $\operatorname{Gen}(\mathcal{X})$ has a $\mathcal{X}$-precover and let $M$ be any module. Since $t_{\mathcal{X}}(M)\in\operatorname{Gen}(\mathcal{X})$ by [AF92, Proposition 8.12], there exists a $\mathcal{X}$-precover $f:X\rightarrow t_{\mathcal{X}}(M)$. We claim that $if$ is a $\mathcal{X}$-precover of $M$, where $i:t_{\mathcal{X}}(M)\rightarrow M$ is the inclusion: for any $g:X^{\prime}\rightarrow M$ with $X^{\prime}\in\mathcal{X}$, since $\operatorname{Im}g\leq t_{\mathcal{X}}(M)$, $g$ factors through $t_{\mathcal{X}}(M)$. Then $g=i\overline{g}$ for some $\overline{g}:X^{\prime}\rightarrow t_{\mathcal{X}}(M)$. As $f$ is a $\mathcal{X}$-precover, there exists $h:X^{\prime}\rightarrow F$ with $fh=\overline{g}$. Then $fhi=\overline{g}i=g$. This proves the claim. ∎

As we mentioned before, if $(\mathcal{F},\mathcal{C})$ is a cotorsion pair generated by a set of modules $\mathcal{G}$, then $\mathcal{F}$ is precovering and $\mathcal{C}$ is preenveloping. More precisely, [GT06, Theorem 3.2.1] asserts that each module has a $\mathcal{C}$-preenvelope with cokernel in $\operatorname{Filt}{-}\mathcal{G}$. Using this result, Lemma 3.2 and the argument in Salce Lemma [GT06, Lemma 2.2.6], we can give an easy proof to the fact that any deconstructible class is precovering. This result was proved in [SŠ11] in exact categories and in [Eno12] using module theory techniques.

Theorem 3.3**.**

Any deconstructible class of right modules is precovering.

Proof.

Let $\mathcal{G}$ be any set of modules. In view of Lemma 3.2 we only have to see that each module in $\operatorname{Gen}(\operatorname{Filt}{-}\mathcal{G})$ has a $(\operatorname{Filt}{-}\mathcal{G})$-precover. Let $M\in\operatorname{Gen}(\operatorname{Filt}{-}\mathcal{G})$ and take $f:F\rightarrow M$ an epimorphism with $F\in\operatorname{Filt}{-}\mathcal{G}$. By [GT06, Theorem 3.2.1], there exists a short exact sequence

[TABLE]

with $P\in\mathcal{G}^{\perp}$ and $N\in\operatorname{Filt}{-}\mathcal{G}$. If we compute the pushout $i$ and the inclusion $j:\operatorname{Ker}f\rightarrow M$, we get the following commutative diagram with exact rows and colummns:

[TABLE]

Since $F$ and $N$ belong to $\operatorname{Filt}{-}\mathcal{G}$, then so does $Q$. But $P\in\mathcal{G}^{\perp}$, so that, by Ekolf Lemma [GT06, Lemma 3.1.2], $P\in\left(\operatorname{Filt}{-}\mathcal{G}\right)^{\perp}$ as well (note that if $X\in\operatorname{Filt}{-}\mathcal{G}$, then $X$ is ${{}^{\perp}}P$-filtered so that $X\in{{}^{\perp}}P$; i. e. $P\in X^{\perp}$). Consequently, $g$ is a $(\operatorname{Filt}{-}\mathcal{G})$-precover of $M$. ∎

Remark 3.4*.*

Given a class of modules $\mathcal{X}$ and a module $M$, a special $\mathcal{X}$-precover of $M$ is a morphism $f:X\rightarrow M$ with $\operatorname{Im}f=t_{\mathcal{X}}(M)$ and $\operatorname{Ker}f\in\mathcal{X}^{\perp}$ (note that we are not imposing that $f$ is epic [which cannot be if $\mathcal{X}$ is not generating!] as it is done in the classical definition of special preenvelopes [GT06, Definition 2.1.12]). With this definition, the preceeding theorem actually proves that any deconstructible class is special precovering.

Now we prove that a deconstructible class closed under products is preenveloping. In order to do this, we are going to use the following technical property which is employed in [EJ00, Corollary 6.2.2]. This property is related with the cardinality condition defined in [HJ08, Definition 1.1].

Definition 3.5**.**

Let $\mathcal{X}$ be a class of right modules and $\lambda$ an infinite cardinal. We say that $\mathcal{X}$ satisfies the property $\mathbb{P}_{\lambda}$ if there exists an infinite cardinal $\kappa_{\lambda}$ with the following property:

[TABLE]

We establish the relationship between this property and the existence preenvelopes, which was proved in [EJ00, Corollary 6.2.2] (see [HJ08, Proposition 1.2] too).

Proposition 3.6**.**

Let $\mathcal{X}$ be a class of right $R$-modules and $\lambda$ an infinite cardinal. If $\mathcal{X}$ is closed under products and satisfies the property $\mathbb{P}_{\lambda}$, then each module $M$ with cardinality less than or equal to $\lambda$ has a $\mathcal{X}$-preenvelope.

The idea of the proof of this theorem is to take, given a module $M$ with cardinality less than or equal to $\lambda$, a representing set $\mathcal{G}$ of the class of all modules in $\mathcal{X}$ with cardinality less than or equal to $\kappa_{\lambda}$. Then the canonical morphism from $M$ to $\prod_{G\in\mathcal{G}}G^{\operatorname{Hom}_{R}(M,G)}$ is trivially a $\mathcal{X}$-preenvelope as a consequence of the property $\mathbb{P}_{\lambda}$.

Remark 3.7*.*

We could consider the dual property of $\mathbb{P}_{\lambda}$: we say that $\mathcal{X}$ satisfies the property $\mathbb{Q}_{\lambda}$ if there exists an infinite cardinal $\kappa_{\lambda}$ satisfying

[TABLE]

The property $\mathbb{Q}_{\lambda}$ is related with the existence of precovers: if $\mathcal{X}$ is closed under direct sums and satisfies the property $\mathbb{Q}_{\lambda}$, then each module $M$ with cardinality less than or equal to $\lambda$ has a $\mathcal{X}$-precover. $\mathbb{Q}_{\lambda}$ is related with the co-cardinality condition defined in [HJ08, Definition 1.1].

One useful tool to deal with filtrations is Hill Lemma [GT06, Theorem 4.2.6]. Roughly speaking, it states that a filtration of a module can be enlarged to a class of submodules with certain properties. Recall that a cardinal $\kappa$ is regular if it is not the union of less than $\kappa$ sets with cardinality less than $\kappa$. Recall that given an infinite regular cardinal $\kappa$, a module $M$ is $\kappa$-presented if it there exists a presentation of $M$ with $\kappa$ generators and $\kappa$ relations.

Theorem 3.8**.**

Let $\kappa$ be an infinite regular cardinal and $\mathcal{G}$ a set of $<\kappa$-presented right modules. Let $M$ be a module with a $\mathcal{G}$-filtration, $M=\bigcup_{\alpha<\mu}M_{\alpha}$. Then there is a family $\mathcal{H}$ of submodules of $M$ such that:

1. (H1)

$M_{\alpha}\in\mathcal{H}$* for each $\alpha<\mu$.* 2. (H2)

$\mathcal{H}$* is closed under arbitrary sums and intersections.* 3. (H3)

Let $N,P\in\mathcal{H}$ such that $N\leq P$. Then $\frac{P}{N}$ is filtered by modules in $\left\{\frac{M_{\alpha+1}}{M_{\alpha}}:\alpha<\mu\right\}$. 4. (H4)

Let $N\in\mathcal{H}$ and $X$ a subset of $M$ of cardinality smaller than $\kappa$. Then there is a $P\in\mathcal{H}$ such that $N\cup X\leq P$ and $\frac{P}{N}$ is $<\kappa$-presented.

Using Hill lemma, we prove that a deconstructible class satisfies $\mathbb{Q}_{\lambda}$.

Proposition 3.9**.**

Let $\mathcal{X}$ be a deconstructible class of right modules. Then $\mathcal{X}$ satisfies $\mathbb{Q}_{\lambda}$ for each infinite cardinal $\lambda$.

Proof.

Since $\mathcal{X}$ is deconstructible, there exists a set $\mathcal{G}$ such that $\mathcal{X}=\mathcal{F}ilt-\mathcal{G}$. Let $\kappa$ be an infinite regular cardinal such that each module in $\mathcal{G}$ is $<\kappa$-presented.

Let $\lambda$ be an infinite cardinal and set $\kappa_{\lambda}:=\max\{\lambda,\kappa\}$. Let $X$ be a module in $\mathcal{X}$ and $S\leq X$ with $|X|\leq\lambda$. Now denote by $\mathcal{H}$ the family of submodules of $X$ given by the Hill Lemma.

If $\lambda<\kappa$ then $S$ is contained in a $<\kappa$-presented submodule of $M$ by (H4) which belongs to $\mathcal{X}$ by (H3). Then $\mathbb{P}_{\lambda}$ is satisfied since $\kappa_{\lambda}=\kappa$ in this case.

Suppose that $\lambda\geq\kappa$ so that $\kappa_{\lambda}=\lambda$. Write $S=\{s_{\alpha}:\alpha<\mu\}$ for some cardinal $\mu\leq\lambda$. Applying recursively (H4) and (H2) it is easy to construct a continuous chain of submodules of $M$, $\{S_{\alpha}:\alpha<\mu\}$, with $S_{\alpha}\in\mathcal{H}$, $\frac{S_{\alpha+1}}{S_{\alpha}}$ $<\kappa$-presented and $x_{\alpha}\in S_{\alpha+1}$ for each $\alpha<\mu$. Now, by (H3), $\frac{S_{\alpha+1}}{S_{\alpha}}\in\mathcal{X}$ so that $Y=\bigcup_{\alpha<\mu}S_{\alpha}$ is $\mathcal{X}$-filtered; by [SŠ11, Corollary 2.11], $Y$ belongs to $\mathcal{X}$. Moreover $Y$ has cardinality less than or equal to $\lambda=\kappa_{\lambda}$. Then $\mathcal{X}$ satisfies $\mathbb{P}_{\lambda}$. ∎

As a consequence of this result and Proposition 3.6 we inmediately get that a deconstructible class closed under product is preenveloping. This result was proved in [SŠ11, Theorem 4.19] for exact categories.

Theorem 3.10**.**

Each deconstructible class of right modules closed under products is preenveloping.

Now we apply these results to Tor-pairs

Corollary 3.11**.**

Let $(\mathcal{T},\mathcal{S})$ be a Tor-pair and $m$ a natural number. Then:

1. (1)

The class $\mathcal{T}_{m}$ is precovering. 2. (2)

If $R$ satisfies the equivalent condition of Corollary 2.7, $\mathcal{T}_{m}$ is preenveloping.

Proof.

By [ET00, Theorem 8], $\mathcal{T}$ is a deconstructible class, since $\mathcal{T}={{}^{\perp}}\mathcal{I}$, where $\mathcal{I}=\{S^{c}:S\in\mathcal{S}\}$ by [GT06, Lemma 2.2.3] ($S^{c}$ being the caracter module of $S$). By [CI16, Proposition 3.2], $\mathcal{T}_{m}$ is deconstructible as well. Then (1) follows from Theorem 3.3 and (2) from Theorem 3.10. ∎