Tree modules and limits of the approximation theory

TL;DR

This paper explores the construction of tree modules and their role in the limits of module approximation theory, including recent generalizations addressing classical problems in representation theory.

Contribution

It introduces a new construction of tree modules combined with tilting theory and Mittag-Leffler conditions, and applies recent generalizations to solve longstanding problems.

Findings

Construction of tree modules with approximation properties

Generalization addressing factorization and almost split sequences

Solution to Auslander’s old problem on almost split sequences

Abstract

In this expository paper, we present a construction of tree modules and combine it with (infinite dimensional) tilting theory and relative Mittag-Leffler conditions in order to explore limits of the approximation theory of modules. We also present a recent generalization of this construction due to Saroch which applies to factorization properties of maps, and yields a solution of an old problem by Auslander concerning existence of almost split sequences.