Derived equivalence, recollements under $ H $-Galois extensions

• Published in: AIMS mathematics Vol. 8; no. 2; pp. 3210 - 3225
• Main Authors: Dong, Jinlei, Li, Fang, Sun, Longgang
• Format: Journal Article
• Language: English
• Published: AIMS Press 01.01.2023
• Subjects: h-frobenius extension; h-galois extension; recollement; tilting complex
• ISSN: 2473-6988; 2473-6988
• DOI: 10.3934/math.2023165

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.