Claus Michael Ringel's main contributions to Gorenstein-projective modules

TL;DR

This paper reviews Claus Michael Ringel's significant contributions to the theory of Gorenstein-projective modules, highlighting his solutions to key problems, innovative techniques, and classifications within the field.

Contribution

It summarizes Ringel's novel methods, such as the $ ext{mho}$-quivers and algorithms for Nakayama algebras, and his comprehensive descriptions of module categories and reflexivity properties.

Findings

Solution to the independence problem of totally reflexivity conditions

Development of the $ ext{mho}$-quivers technique

Algorithm for Gorenstein-projective modules over Nakayama algebras

Abstract

In this article we try to recall Claus Michael Ringel's works on the Gorenstein-projective modules. This will involve but not limited to his fundamental contributions, such as in, the solution to the independence problem of totally reflexivity conditions; the technique of $\mho$-quivers; a fast algorithm to obtain the Gorenstein-projective modules over the Nakayama algebras; the one to one correspondence between the indecomposable non-projective perfect differential modules of a quiver and the indecomposable representations of this quiver; the description of the module category of the preprojective algebras of type $\mathbb A_n$ via submodule category; semi-Gorenstein-projective modules, reflexive modules, Koszul modules, as well as the $\Omega$-growth of modules, over short local algebras; and his negative answer to the question whether an algebra has to be self-injective in case all the simple modules are reflexive.