2024/09/09 by Aaron Chan, Chan, Aaron, Osamu Iyama +3 · 1 citation
Mathematics · #13C60 #16E10 #16E35 #16G10 #18G80 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Representation Theory (math.RT) #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.2409.05603
openalex publication_date 2024/09/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Auslander and Reiten called a finite dimensional algebra A over a field Cohen-Macaulay if there is an A-bimodule W which gives an equivalence between the category of finitely generated A-modules of finite projective dimension and the category of finitely generated A-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 R and a maximal Cohen-Macaulay R-module M, the stable endomorphism algebra \underlineEndR(M) is Cohen-Macaulay. We also give a negative answer to a question of Auslander-Reiten asking whether the category CM A of Cohen-Macaulay A-modules coincides with the category of d-th syzygies, where d≥1 is the injective dimension of W. In fact, if A is a Cohen-Macaulay algebra that is additionally d-Gorenstein in the sense of Auslander, then CM A always coincides with the category of d-th syzygies.