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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2022
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2208.14712 |
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| _version_ | 1866911139132080128 |
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| author | Chen, Hongxing Xi, Changchang |
| author_facet | Chen, Hongxing Xi, Changchang |
| contents | Tachikawa's second conjecture predicts that a finitely generated, orthogonal module over a finite-dimensional self-injective algebra is projective. This conjecture is an important part of the Nakayama conjecture. Our principal motivation of this work is a systematic understanding of finitely generated, orthogonal generators over a self-injective Artin algebra from the view point of stable module categories. As a result, for an orthogonal generator M, we establish a recollement of the M-relative stable categories, describe compact objects of the right term of the recollement, and give equivalent characterizations of Tachikawa's second conjecture in terms of M-Gorenstein categories. Further, we introduce Gorenstein-Morita algebras and show that the Nakayama conjecture holds true for them. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_14712 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Homological theory of orthogonal modules Chen, Hongxing Xi, Changchang Representation Theory Rings and Algebras Primary 16E35, 18E30, 19D50, Secondary 16S10, 13B30, 20E06 Tachikawa's second conjecture predicts that a finitely generated, orthogonal module over a finite-dimensional self-injective algebra is projective. This conjecture is an important part of the Nakayama conjecture. Our principal motivation of this work is a systematic understanding of finitely generated, orthogonal generators over a self-injective Artin algebra from the view point of stable module categories. As a result, for an orthogonal generator M, we establish a recollement of the M-relative stable categories, describe compact objects of the right term of the recollement, and give equivalent characterizations of Tachikawa's second conjecture in terms of M-Gorenstein categories. Further, we introduce Gorenstein-Morita algebras and show that the Nakayama conjecture holds true for them. |
| title | Homological theory of orthogonal modules |
| topic | Representation Theory Rings and Algebras Primary 16E35, 18E30, 19D50, Secondary 16S10, 13B30, 20E06 |
| url | https://arxiv.org/abs/2208.14712 |