# Infinitely Generated Gorenstein Tilting Modules

## TL;DR

This paper advances the theory of infinitely generated Gorenstein tilting modules by introducing Gorenstein tilting approximations and exploring their implications for tilting classes, cotorsion pairs, and related homological conjectures.

## Contribution

It develops the concept of Gorenstein tilting approximations and applies them to analyze Gorenstein tilting classes and their cotorsion pairs, extending existing tilting theory.

## Key findings

- Established Gorenstein tilting approximations.
- Connected Gorenstein tilting modules to finitistic dimension conjectures.
- Provided criteria for the existence of complements to partial Gorenstein tilting modules.

## Abstract

The theory of finitely generated relative (co)tilting modules has been established in the 1980s by Auslander and Solberg, and infinitely generated relative tilting modules have recently been studied by many authors in the context of Gorenstein homological algebra. In this work, we build on the theory of infinitely generated Gorenstein tilting modules by developing "Gorenstein tilting approximations" and employing these approximations to study Gorenstein tilting classes and their associated relative cotorsion pairs. As applications of our results, we discuss the problem of existence of complements to partial Gorenstein tilting modules as well as some connections between Gorenstein tilting modules and finitistic dimension conjectures.