# 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.

## Key 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.