Derived equivalence, recollements under H-Galois extensions
In this paper, assume that $ H $ is a Hopf algebra and $ A/B $ is an $ H $-Galois extension. Firstly, by introducing the concept of an $ H $-stable tilting complex $ T_{\bullet} $ over $ B $, we show that $ T_{\bullet}\otimes_BA $ is a tilting complex over $ A $ and a derived equivalence between two...
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| Format: | Article |
| Language: | English |
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AIMS Press
2023-01-01
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| Series: | AIMS Mathematics |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/math.2023165https://www.aimspress.com/article/doi/10.3934/math.2023165 |
| _version_ | 1797954193786404864 |
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| author | Jinlei Dong Fang Li Longgang Sun |
| author_facet | Jinlei Dong Fang Li Longgang Sun |
| author_sort | Jinlei Dong |
| collection | DOAJ |
| description | In this paper, assume that $ H $ is a Hopf algebra and $ A/B $ is an $ H $-Galois extension. Firstly, by introducing the concept of an $ H $-stable tilting complex $ T_{\bullet} $ over $ B $, we show that $ T_{\bullet}\otimes_BA $ is a tilting complex over $ A $ and a derived equivalence between two $ H $-module algebras can be extended to smash product algebras under some conditions. Then we observe that $ 0\rightarrow {\rm End}_{\mathcal{D}^b(B)}(T_{\bullet})\rightarrow {\rm End}_{\mathcal{D}^b(A)}(T_{\bullet}\otimes_BA) $ is an $ H $-Galois Frobenius extension if $ A/B $ is an $ H $-Galois Frobenius extension. Finally, for any perfect recollement of derived categories of $ H $-module algebras, we apply the above results to construct a perfect recollement of derived categories of their smash product algebras and generalize it to $ n $-recollements. |
| first_indexed | 2024-04-10T23:14:50Z |
| format | Article |
| id | doaj.art-ce9fb84c013548f394d8c4c20129ed8a |
| institution | Directory Open Access Journal |
| issn | 2473-6988 |
| language | English |
| last_indexed | 2024-04-10T23:14:50Z |
| publishDate | 2023-01-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | AIMS Mathematics |
| spelling | doaj.art-ce9fb84c013548f394d8c4c20129ed8a2023-01-13T02:04:31ZengAIMS PressAIMS Mathematics2473-69882023-01-01823210322510.3934/math.2023165Derived equivalence, recollements under H-Galois extensionsJinlei Dong0Fang Li1Longgang Sun2Department of Mathematics, Zhejiang University, Hangzhou, Zhejiang 310058, ChinaDepartment of Mathematics, Zhejiang University, Hangzhou, Zhejiang 310058, China Department of Mathematics, Zhejiang University, Hangzhou, Zhejiang 310058, ChinaIn this paper, assume that $ H $ is a Hopf algebra and $ A/B $ is an $ H $-Galois extension. Firstly, by introducing the concept of an $ H $-stable tilting complex $ T_{\bullet} $ over $ B $, we show that $ T_{\bullet}\otimes_BA $ is a tilting complex over $ A $ and a derived equivalence between two $ H $-module algebras can be extended to smash product algebras under some conditions. Then we observe that $ 0\rightarrow {\rm End}_{\mathcal{D}^b(B)}(T_{\bullet})\rightarrow {\rm End}_{\mathcal{D}^b(A)}(T_{\bullet}\otimes_BA) $ is an $ H $-Galois Frobenius extension if $ A/B $ is an $ H $-Galois Frobenius extension. Finally, for any perfect recollement of derived categories of $ H $-module algebras, we apply the above results to construct a perfect recollement of derived categories of their smash product algebras and generalize it to $ n $-recollements.https://www.aimspress.com/article/doi/10.3934/math.2023165https://www.aimspress.com/article/doi/10.3934/math.2023165tilting complexh-galois extensionh-frobenius extensionrecollement |
| spellingShingle | Jinlei Dong Fang Li Longgang Sun Derived equivalence, recollements under H-Galois extensions AIMS Mathematics tilting complex h-galois extension h-frobenius extension recollement |
| title | Derived equivalence, recollements under H-Galois extensions |
| title_full | Derived equivalence, recollements under H-Galois extensions |
| title_fullStr | Derived equivalence, recollements under H-Galois extensions |
| title_full_unstemmed | Derived equivalence, recollements under H-Galois extensions |
| title_short | Derived equivalence, recollements under H-Galois extensions |
| title_sort | derived equivalence recollements under h galois extensions |
| topic | tilting complex h-galois extension h-frobenius extension recollement |
| url | https://www.aimspress.com/article/doi/10.3934/math.2023165https://www.aimspress.com/article/doi/10.3934/math.2023165 |
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