A recollement approach to Han's conjecture
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929482495950848 |
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| author | Wang, Ren Xu, Xiaoxiao Zhang, Jinbi Zhou, Guodong |
| author_facet | Wang, Ren Xu, Xiaoxiao Zhang, Jinbi Zhou, Guodong |
| contents | A conjecture due to Y. Han asks whether that Hochschild homology groups of a finite dimensional algebra vanish for sufficiently large degrees would imply that the algebra is of finite global dimension. We investigate this conjecture from the viewpoint of recollements of derived categories. It is shown that for a recollement of unbounded derived categories of rings which extends downwards (or upwards) one step, Han's conjecture holds for the ring in the middle if and only if it holds for the two rings on the two sides and hence
Han's conjecture is reduced to derived $2$-simple rings. Furthermore, this reduction result is applied to Han's conjecture for Morita contexts rings and exact contexts. Finally it is proved that Han's conjecture holds for skew-gentle algebras, category algebras of finite EI categories and Geiss-Leclerc-Schröer algebras associated to Cartan triples. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_00945 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A recollement approach to Han's conjecture Wang, Ren Xu, Xiaoxiao Zhang, Jinbi Zhou, Guodong Representation Theory K-Theory and Homology Rings and Algebras 16E40, 18G80, 19D55, 16E10, 16E35, 16E45, 16E45 A conjecture due to Y. Han asks whether that Hochschild homology groups of a finite dimensional algebra vanish for sufficiently large degrees would imply that the algebra is of finite global dimension. We investigate this conjecture from the viewpoint of recollements of derived categories. It is shown that for a recollement of unbounded derived categories of rings which extends downwards (or upwards) one step, Han's conjecture holds for the ring in the middle if and only if it holds for the two rings on the two sides and hence Han's conjecture is reduced to derived $2$-simple rings. Furthermore, this reduction result is applied to Han's conjecture for Morita contexts rings and exact contexts. Finally it is proved that Han's conjecture holds for skew-gentle algebras, category algebras of finite EI categories and Geiss-Leclerc-Schröer algebras associated to Cartan triples. |
| title | A recollement approach to Han's conjecture |
| topic | Representation Theory K-Theory and Homology Rings and Algebras 16E40, 18G80, 19D55, 16E10, 16E35, 16E45, 16E45 |
| url | https://arxiv.org/abs/2409.00945 |