Auslander-Reiten's Cohen-Macaulay algebras and contracted preprojective algebras
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arXiv
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| Natura: | Preprint |
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2024
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| _version_ | 1866916408198168576 |
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| author | Chan, Aaron Iyama, Osamu Marczinzik, Rene |
| author_facet | Chan, Aaron Iyama, Osamu Marczinzik, Rene |
| contents | 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 $\underline{End}_R(M)$ 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 $d\ge1$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_05603 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Auslander-Reiten's Cohen-Macaulay algebras and contracted preprojective algebras Chan, Aaron Iyama, Osamu Marczinzik, Rene Representation Theory Commutative Algebra Rings and Algebras 16G10, 16E10, 13C60, 18G80, 16E35 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 $\underline{End}_R(M)$ 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 $d\ge1$ 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. |
| title | Auslander-Reiten's Cohen-Macaulay algebras and contracted preprojective algebras |
| topic | Representation Theory Commutative Algebra Rings and Algebras 16G10, 16E10, 13C60, 18G80, 16E35 |
| url | https://arxiv.org/abs/2409.05603 |