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

2025/05/19 by Nan Gao, Gao, Nan, Xue-Song Lu +3 · 2 citations
Mathematics · #Algebraic structures and combinatorial models #Commutative Algebra and Its Applications #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2505.12637

openalex publication_date 2025/05/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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Cited by

• Weakly Gorensteinness of tensor algebras and Morita algebras
• A survey on Auslander-Gorenstein algebras

Related

• The Gorenstein projective modules for the Nakayama algebras. I
• Strongly Gorenstein projective, injective, and flat modules
• Gorenstein derived functors from special rings to arbitrary rings
• n-Strongly Gorenstein Projective, Injective and Flat modules
• Gorenstein-projective and semi-Gorenstein-projective modules. II