Auslander-Reiten's Cohen-Macaulay algebras and contracted preprojective algebras
Aaron Chan, Osamu Iyama, Rene Marczinzik

TL;DR
This paper establishes that contracted preprojective algebras of Dynkin type are Cohen-Macaulay, providing new examples and exploring their properties, including relations to singularities and syzygy categories.
Contribution
It introduces the first non-trivial class of Cohen-Macaulay algebras via contracted preprojective algebras of Dynkin type and examines their module categories.
Findings
Contracted preprojective algebras of Dynkin type are Cohen-Macaulay.
Stable endomorphism algebras of maximal Cohen-Macaulay modules over simple singularities are Cohen-Macaulay.
Counterexample to the equivalence of Cohen-Macaulay modules and syzygy categories in general.
Abstract
Auslander and Reiten called a finite dimensional algebra over a field Cohen-Macaulay if there is an -bimodule which gives an equivalence between the category of finitely generated -modules of finite projective dimension and the category of finitely generated -modules of finite injective dimension. For example, Iwanaga-Gorenstein algebras and algebras with finitistic dimension zero on both sides are Cohen-Macaulay, and tensor products of Cohen-Macaulay algebras are again Cohen-Macaulay. They seem to be all of the known examples of Cohen-Macaulay algebras. In this paper, we give the first non-trivial class of Cohen-Macaulay algebras by showing that all contracted preprojective algebras of Dynkin type are Cohen-Macaulay. As a consequence, for each simple singularity and a maximal Cohen-Macaulay -module , the stable endomorphism algebra is…
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Taxonomy
TopicsAlgebraic structures and combinatorial models · Advanced Topics in Algebra · Commutative Algebra and Its Applications
