Enhanced Auslander-Reiten duality and Morita theorem for singularity categories
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866917801437954048 |
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| author | Hanihara, Norihiro Iyama, Osamu |
| author_facet | Hanihara, Norihiro Iyama, Osamu |
| contents | We establish a Morita theorem to construct triangle equivalences between the singularity categories of (commutative and non-commutative) Gorenstein rings and the cluster categories of finite dimensional algebras over fields, and more strongly, quasi-equivalences between their canonical dg enhancements. More precisely, we prove that such an equivalence exists as soon as we find a quasi-equivalence between the graded dg singularity category of a Gorenstein ring and the derived category of a finite dimensional algebra which can be done by finding a single tilting object.
Our result is based on two key theorems on dg enhancements of cluster categories and of singularity categories, which are of independent interest. First we give a Morita-type theorem which realizes certain $\mathbb{Z}$-graded dg categories as dg orbit categories. Secondly, we show that the canonical dg enhancements of the singularity categories of symmetric orders have the bimodule Calabi-Yau property, which lifts the classical Auslander-Reiten duality on singularity categories.
We apply our results to such classes of rings as Gorenstein rings of dimension at most $1$, quotient singularities, and Geigle-Lenzing complete intersections, including finite or infinite Grassmannian cluster categories, to realize their singularity categories as cluster categories of finite dimensional algebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_14090 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Enhanced Auslander-Reiten duality and Morita theorem for singularity categories Hanihara, Norihiro Iyama, Osamu Representation Theory Commutative Algebra Algebraic Geometry Category Theory Rings and Algebras We establish a Morita theorem to construct triangle equivalences between the singularity categories of (commutative and non-commutative) Gorenstein rings and the cluster categories of finite dimensional algebras over fields, and more strongly, quasi-equivalences between their canonical dg enhancements. More precisely, we prove that such an equivalence exists as soon as we find a quasi-equivalence between the graded dg singularity category of a Gorenstein ring and the derived category of a finite dimensional algebra which can be done by finding a single tilting object. Our result is based on two key theorems on dg enhancements of cluster categories and of singularity categories, which are of independent interest. First we give a Morita-type theorem which realizes certain $\mathbb{Z}$-graded dg categories as dg orbit categories. Secondly, we show that the canonical dg enhancements of the singularity categories of symmetric orders have the bimodule Calabi-Yau property, which lifts the classical Auslander-Reiten duality on singularity categories. We apply our results to such classes of rings as Gorenstein rings of dimension at most $1$, quotient singularities, and Geigle-Lenzing complete intersections, including finite or infinite Grassmannian cluster categories, to realize their singularity categories as cluster categories of finite dimensional algebras. |
| title | Enhanced Auslander-Reiten duality and Morita theorem for singularity categories |
| topic | Representation Theory Commutative Algebra Algebraic Geometry Category Theory Rings and Algebras |
| url | https://arxiv.org/abs/2209.14090 |