Gorenstein dimension of modules

TL;DR

This paper provides an accessible exposition of Gorenstein dimension and related concepts in modules over commutative Noetherian rings, simplifying the abstract theory originally developed in a more general non-commutative context.

Contribution

It adapts and simplifies the theory of Gorenstein dimension for commutative Noetherian rings, making it more accessible to algebraists familiar with Matsumura's commutative ring theory.

Findings

Simplified the theory of Gorenstein dimension for commutative Noetherian rings.

Connected Gorenstein dimension with classical module concepts like syzygies and torsionless modules.

Provided an accessible exposition aligned with standard commutative algebra textbooks.

Abstract

In these expository notes I discuss several concepts and results in the theory of modules over commutative rings, revolving around the Gorenstein dimension of modules. Some of the related notions are the Auslander dual, k-torsionless modules, and k-th syzygies. Essentially everything in these notes can be found, in one form or another, in the memoir "Stable module theory" by M. Auslander and M. Bridger (Mem. A.M.S., no. 94, 1969). The only difference is in presentation. In the Auslander-Bridger memoir many of the results are proved in the most general setting, e.g. over possibly non-commutative, non-Noetherian rings. The techniques used are quite abstract and unfamiliar to many commutative algebraists. Much space is devoted to the theory of satellites of functors which are exact only in the middle, etc. While such a degree of generality has many advantages, it does make the memoir difficult to read for the non-expert. My goal in writing these notes was to develop the theory in the context of commutative Noetherian rings, and to show that, in this important special case, the theory is fairly elementary and easy to build. As a practical matter, then, I wrote the notes using Matsumura's "Commutative ring theory" as the only pre-requisite; and indeed, my hope is that these notes can be read just like an extra chapter in Matsumura's book.