Flat Mittag-Leffler modules over countable rings

TL;DR

This paper investigates the structure of flat Mittag-Leffler modules over countable rings, revealing their orthogonal classes and implications for module precovering properties, especially in relation to countable and right perfect rings.

Contribution

It characterizes the double Ext-orthogonal class of flat Mittag-Leffler modules over countable rings and shows the non-precovering nature of these modules in most cases.

Findings

Double Ext-orthogonal class contains all countable direct limits of flat Mittag-Leffler modules.

For countable rings, the orthogonal class equals all flat modules.

Flat Mittag-Leffler modules are not precovering unless the ring is right perfect.

Abstract

We show that over any ring, the double Ext-orthogonal class to all flat Mittag-Leffler modules contains all countable direct limits of flat Mittag-Leffler modules. If the ring is countable, then the double orthogonal class consists precisely of all flat modules and we deduce, using a recent result of \v{S}aroch and Trlifaj, that the class of flat Mittag-Leffler modules is not precovering in Mod-R unless R is right perfect.