Derived equivalence, recollements under $ H $-Galois extensions

Jinlei Dong; Fang Li; Longgang Sun; Jinlei Dong; Fang Li; Longgang Sun

doi:10.3934/math.2023165

AIMS Mathematics

2023, Volume 8, Issue 2: 3210-3225. doi: 10.3934/math.2023165

Previous Article Next Article

Research article

Derived equivalence, recollements under $ H $-Galois extensions

Department of Mathematics, Zhejiang University, Hangzhou, Zhejiang 310058, China

Received: 22 July 2022 Revised: 31 October 2022 Accepted: 31 October 2022 Published: 16 November 2022
MSC : 13B05, 13D09, 16E35, 16G10, 16S40

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.
- tilting complex,
- $ H $-Galois extension,
- $ H $-Frobenius extension,
- recollement
Citation: Jinlei Dong, Fang Li, Longgang Sun. Derived equivalence, recollements under $ H $-Galois extensions[J]. AIMS Mathematics, 2023, 8(2): 3210-3225. doi: 10.3934/math.2023165

Related Papers:

Abstract

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.

References

[1]	J. Rickard, Morita theory for derived categories, J. London Math. Soc., s2-39 (1989), 436–456. https://doi.org/10.1112/jlms/s2-39.3.436 doi: 10.1112/jlms/s2-39.3.436
[2]	I. Assem, N. Marmaridis, Tilting modules over split-by-nilpotent extensions, Commun. Algebra, 26 (1998), 1547–1555. https://doi.org/10.1080/00927879808826219 doi: 10.1080/00927879808826219
[3]	J. Miyachi, Tilting modules over trivial extensions of Artin algebras, Commun. Algebra, 13 (1985), 1319–1326. https://doi.org/10.1080/00927878508823223 doi: 10.1080/00927878508823223
[4]	J. Miyachi, Extensions of rings and tilting complexes, J. Pure Appl. Algebra, 105 (1995), 183–194. https://doi.org/10.1016/0022-4049(94)00145-6 doi: 10.1016/0022-4049(94)00145-6
[5]	J. Rickard, Lifting theorems for tilting complexes, J. Algebra, 142 (1991), 383–393. https://doi.org/10.1016/0021-8693(91)90313-W doi: 10.1016/0021-8693(91)90313-W
[6]	H. Tachikawa, T. Wakamatsu, Extensions of tilting functors and QF-$3$ algebras, J. Algebra, 103 (1986), 662–676. https://doi.org/10.1016/0021-8693(86)90159-6 doi: 10.1016/0021-8693(86)90159-6
[7]	J. Rickard, Derived equivalences as derived functors, J. London Math. Soc., s2-43 (1991), 37–48. https://doi.org/10.1112/jlms/s2-43.1.37 doi: 10.1112/jlms/s2-43.1.37
[8]	A. Marcus, Equivalences induced by graded bimodules, Commun. Algebra, 26 (1998), 713–731. https://doi.org/10.1080/00927879808826159 doi: 10.1080/00927879808826159
[9]	S. Caenepeel, S. Crivei, A. Marcus, M. Takeuchi, Morita equivalences induced by bimodules over Hopf-Galois extensions, J. Algebra, 314 (2007), 267–302. https://doi.org/10.1016/j.jalgebra.2007.02.033 doi: 10.1016/j.jalgebra.2007.02.033
[10]	P. Deligne, A. A. Beĭlinson, J. Bernstein, Faisceaux pervers, Astérisque, 1983.
[11]	E. Cline, B. Parshall, L. Scott, Finite-dimensional algebras and highest weight categories, J. Reine Angew. Math., 391 (1988), 85–99. https://doi.org/10.1515/crll.1988.391.85 doi: 10.1515/crll.1988.391.85
[12]	Y. Han, Recollements and Hochschild theory, J. Algebra, 397 (2014), 535–547. https://doi.org/10.1016/j.jalgebra.2013.09.018 doi: 10.1016/j.jalgebra.2013.09.018
[13]	S. Montgomery, Hopf algebras and their actions on rings, In: CBMS regional conference series in mathematics, American Mathematical Society, 1993. https://doi.org/10.1090/cbms/082
[14]	S. König, A. Zimmermann, Derived equivalences for group rings, In: Lecture notes in mathematics, Berlin: Springer-Verlag, 1998. https://doi.org/10.1007/BFb0096366
[15]	H. Abe, M. Hoshino, Derived equivalences for triangular matrix rings, Algebr. Represent. Theor., 13 (2010), 61–67. https://doi.org/10.1007/s10468-008-9098-1 doi: 10.1007/s10468-008-9098-1
[16]	M. Barot, H. Lenzing, One-point extensions and derived equivalence, J. Algebra, 264 (2003), 1–5. https://doi.org/10.1016/S0021-8693(03)00124-8 doi: 10.1016/S0021-8693(03)00124-8
[17]	H. Schneider, Representation theory of Hopf Galois extensions, Israel J. Math., 72 (1990), 196–231. https://doi.org/10.1007/BF02764620 doi: 10.1007/BF02764620
[18]	E. C. Dade, Group-graded rings and modules, Math. Z., 174 (1980), 241–262. https://doi.org/10.1007/BF01161413 doi: 10.1007/BF01161413
[19]	F. V. Oystaeyen, Y. Zhang, $H$-module endomorphism rings, J. Pure Appl. Algebra, 102 (1995), 207–219. https://doi.org/10.1016/0022-4049(94)00076-U doi: 10.1016/0022-4049(94)00076-U
[20]	D. Happel, Triangulated categories in the representation theory of finite-dimensional algebras, Cambridge University Press, 1988. https://doi.org/10.1017/CBO9780511629228
[21]	Y. Zhang, Hopf Frobenius extensions of algebras, Commun. Algebra, 20 (1992), 1907–1915. https://doi.org/10.1080/00927879208824439 doi: 10.1080/00927879208824439
[22]	Y. Qin, Y. Han, Reducing homological conjectures by $n$-recollements, Algebr. Represent. Theor., 19 (2016), 377–395. https://doi.org/10.1007/s10468-015-9578-z doi: 10.1007/s10468-015-9578-z
[23]	M. van den Bergh, A relation between Hochschild homology and cohomology for Gorenstein rings, Proc. Amer. Math. Soc., 126 (1998), 1345–1348. https://doi.org/10.1090/S0002-9939-98-04210-5 doi: 10.1090/S0002-9939-98-04210-5

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)