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

Nan Gao, Xue-Song Lu, Pu Zhang

TL;DR

This survey compiles Claus Michael Ringel's key contributions to Gorenstein-projective theory, including the independence of the defining conditions $(G1)$, $(G2)$, $(G3)$, the effective $\mho$-quiver technique, and explicit descriptions in Nakayama algebras and quiver-extended settings. It presents a unified framework linking semi-Gorenstein-projective, bi-semi-Gorenstein-projective, and Gorenstein-projective modules, with deep connections to submodule categories, differential module viewpoints, and preprojective algebras. Notable results include a fast algorithm for Nakayama algebras, a one-to-one correspondence between indecomposable non-projective GP $\Lambda$-modules and indecomposable $kQ$-modules, and a negative answer to whether all simple reflexive modules imply self-injectivity. The work also develops a robust Koszul and Omega-growth theory for short local algebras, providing numerical invariants, AR-theoretic implications, and criteria for left Koszulity, thereby enriching both representation theory and singularity theory with practical computational tools.

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 $Ω$-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.

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

TL;DR

This survey compiles Claus Michael Ringel's key contributions to Gorenstein-projective theory, including the independence of the defining conditions , , , the effective -quiver technique, and explicit descriptions in Nakayama algebras and quiver-extended settings. It presents a unified framework linking semi-Gorenstein-projective, bi-semi-Gorenstein-projective, and Gorenstein-projective modules, with deep connections to submodule categories, differential module viewpoints, and preprojective algebras. Notable results include a fast algorithm for Nakayama algebras, a one-to-one correspondence between indecomposable non-projective GP -modules and indecomposable -modules, and a negative answer to whether all simple reflexive modules imply self-injectivity. The work also develops a robust Koszul and Omega-growth theory for short local algebras, providing numerical invariants, AR-theoretic implications, and criteria for left Koszulity, thereby enriching both representation theory and singularity theory with practical computational tools.

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 -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 via submodule category; semi-Gorenstein-projective modules, reflexive modules, Koszul modules, as well as the -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.
Paper Structure (25 sections, 39 theorems, 39 equations)

This paper contains 25 sections, 39 theorems, 39 equations.

Table of Contents

  1. Semi-Gorenstein-projective modules and Gorenstein-projective modules
  2. Equivalent characterizations of the Gorenstein-projective modules
  3. Bi-semi-Gorenstein-projective modules which are not torsionless
  4. Independence problem
  5. 3-dimensional indecomposable $\Lambda(q)$-modules
  6. Left weakly Gorenstein algebras
  7. Reduced bi-semi-Gorenstein-projective modules
  8. The technique of $\mho$-quivers
  9. Gorenstein-projective modules over the Nakayama algebras
  10. Representations of quivers over the algebra of dual numbers
  11. Perfect differential $kQ$-modules
  12. Indecomposable perfect differential $kQ$-modules
  13. From submodule categories to preprojective algebras
  14. Preprojective algebras of type $\mathbb A_n$
  15. Objective triangle functors
  16. ...and 10 more sections

Key Result

Theorem 1.1

Let $A$ be an artin algebra, and $M\in A$-mod. Then the following are equivalent. $(1)$$M$ satisfies the conditions $(\text{\rm G}1), \ (\text{\rm G}2)$ and $(\text{\rm G}3):$${\rm (G1)}$$M$ is semi-Gorenstein-projective. ${\rm (G2)}$$M^* = \operatorname{Hom}(M, \ _AA)$ is semi-Gorenstein-projective

Theorems & Definitions (48)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Remark 1.6
  • Theorem 1.7
  • Proposition 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 38 more