Abstract
Unlike Hochschild (co)homology and K-theory, global and dominant dimensions of algebras are far from being invariant under derived equivalences in general. We show that, however, global dimension and dominant dimension are derived invariant when restricting to a class of algebras with anti-automorphisms preserving simples. Such anti-automorphisms exist for all cellular algebras and in particular for many finite-dimensional algebras arising in algebraic Lie theory. Both dimensions then can be characterised intrinsically inside certain derived categories. On the way, a restriction theorem is proved, and used, which says that derived equivalences between algebras with positive ν-dominant dimension always restrict to derived equivalences between their associated self-injective algebras, which under this assumption do exist.
Funding statement: Ming Fang and Wei Hu are both partially supported by NSFC (11471315, 11321101, 11331006, 11471038, 11688101). Wei Hu is grateful to the Fundamental Research Funds for the Central Universities for partial support.
References
[1] F. W. Anderson and K. R. Fuller, Rings and categories of modules, 2nd ed., Grad. Texts in Math. 13, Springer, New York 1992. 10.1007/978-1-4612-4418-9Search in Google Scholar
[2] I. Assem, D. Simson and A. Skowroński, Elements of the representation theory of associative algebras. Vol. 1: Techniques of representation theory, London Math. Soc. Stud. Texts 65, Cambridge University Press, Cambridge 2006. 10.1017/CBO9780511614309Search in Google Scholar
[3] H. Chen and C. Xi, Dominant dimensions, derived equivalences and tilting modules, Israel J. Math. 215 (2016), no. 1, 349–395. 10.1007/s11856-016-1327-4Search in Google Scholar
[4] E. Cline, B. Parshall and L. Scott, Stratifying endomorphism algebras, Mem. Amer. Math. Soc. 124 (1996), no. 591, 1–119. 10.1090/memo/0591Search in Google Scholar
[5] A. S. Dugas and R. Martínez-Villa, A note on stable equivalences of Morita type, J. Pure Appl. Algebra 208 (2007), no. 2, 421–433. 10.1016/j.jpaa.2006.01.007Search in Google Scholar
[6] M. Fang, W. Hu and S. Koenig, Derived equivalences, restriction to self-injective subalgebras and invariance of homological dimensions, preprint (2016), https://arxiv.org/abs/1607.03513. Search in Google Scholar
[7] M. Fang, O. Kerner and K. Yamagata, Canonical bimodules and dominant dimension, Trans. Amer. Math. Soc. 370 (2018), no. 2, 847–872. 10.1090/tran/6976Search in Google Scholar
[8] M. Fang and S. Koenig, Endomorphism algebras of generators over symmetric algebras, J. Algebra 332 (2011), 428–433. 10.1016/j.jalgebra.2011.02.031Search in Google Scholar
[9] M. Fang and S. Koenig, Schur functors and dominant dimension, Trans. Amer. Math. Soc. 363 (2011), no. 3, 1555–1576. 10.1090/S0002-9947-2010-05177-3Search in Google Scholar
[10] M. Fang and S. Koenig, Gendo-symmetric algebras, canonical comultiplication, bar cocomplex and dominant dimension, Trans. Amer. Math. Soc. 368 (2016), no. 7, 5037–5055. 10.1090/tran/6504Search in Google Scholar
[11] M. Fang and H. Miyachi, Hochschild cohomology and dominant dimension, Trans. Amer. Math. Soc. 371 (2019), no. 8, 5267–5292. 10.1090/tran/7704Search in Google Scholar
[12] P. Gabriel and A. V. Roĭter, Representations of finite-dimensional algebras, Algebra. VIII, Encyclopaedia Math. Sci. 73, Springer, Berlin (1992), 1–177. 10.1007/978-3-642-58097-0Search in Google Scholar
[13] W. Hu, On iterated almost ν-stable derived equivalences, Comm. Algebra 40 (2012), no. 10, 3920–3932. 10.1080/00927872.2011.599085Search in Google Scholar
[14] W. Hu and C. Xi, Derived equivalences and stable equivalences of Morita type. I, Nagoya Math. J. 200 (2010), 107–152. 10.1215/00277630-2010-014Search in Google Scholar
[15] W. Hu and C. Xi, Derived equivalences for Φ-Auslander–Yoneda algebras, Trans. Amer. Math. Soc. 365 (2013), no. 11, 5681–5711. 10.1090/S0002-9947-2013-05688-7Search in Google Scholar
[16] W. Hu and C. Xi, Derived equivalences and stable equivalences of Morita type. II, Rev. Mat. Iberoam. 34 (2018), no. 1, 59–110. 10.4171/RMI/981Search in Google Scholar
[17] B. Keller, Deriving DG categories, Ann. Sci. Éc. Norm. Supér. (4) 27 (1994), no. 1, 63–102. 10.24033/asens.1689Search in Google Scholar
[18] O. Kerner and K. Yamagata, Morita algebras, J. Algebra 382 (2013), 185–202. 10.1016/j.jalgebra.2013.02.013Search in Google Scholar
[19] M. Linckelmann and B. Rognerud, On Morita and derived equivalences for cohomological Mackey algebras, Math. Z. 289 (2018), no. 1–2, 39–50. 10.1007/s00209-017-1942-8Search in Google Scholar
[20] Y. Liu and C. Xi, Constructions of stable equivalences of Morita type for finite dimensional algebras. II, Math. Z. 251 (2005), no. 1, 21–39. 10.1007/s00209-005-0786-9Search in Google Scholar
[21] R. Martínez-Villa, The stable equivalence for algebras of finite representation type, Comm. Algebra 13 (1985), no. 5, 991–1018. 10.1080/00927878508823203Search in Google Scholar
[22] R. Martínez-Villa, Properties that are left invariant under stable equivalence, Comm. Algebra 18 (1990), no. 12, 4141–4169. 10.1080/00927872.1990.12098256Search in Google Scholar
[23] V. Mazorchuk and S. Ovsienko, Finitistic dimension of properly stratified algebras, Adv. Math. 186 (2004), no. 1, 251–265. 10.1016/j.aim.2003.08.001Search in Google Scholar
[24] J.-I. Miyachi, Recollement and tilting complexes, J. Pure Appl. Algebra 183 (2003), no. 1–3, 245–273. 10.1016/S0022-4049(03)00072-0Search in Google Scholar
[25] K. Morita, Duality for modules and its applications to the theory of rings with minimum condition, Sci. Rep. Tokyo Kyoiku Daigaku Sect. A 6 (1958), 83–142. Search in Google Scholar
[26] B. J. Müller, The classification of algebras by dominant dimension, Canadian J. Math. 20 (1968), 398–409. 10.4153/CJM-1968-037-9Search in Google Scholar
[27] J. Rickard, Derived categories and stable equivalence, J. Pure Appl. Algebra 61 (1989), no. 3, 303–317. 10.1016/0022-4049(89)90081-9Search in Google Scholar
[28] J. Rickard, Morita theory for derived categories, J. London Math. Soc. (2) 39 (1989), no. 3, 436–456. 10.1112/jlms/s2-39.3.436Search in Google Scholar
[29] J. Rickard, Derived equivalences as derived functors, J. Lond. Math. Soc. (2) 43 (1991), no. 1, 37–48. 10.1112/jlms/s2-43.1.37Search in Google Scholar
[30] N. Spaltenstein, Resolutions of unbounded complexes, Compos. Math. 65 (1988), no. 2, 121–154. Search in Google Scholar
[31] C. C. Xi, Quasi-hereditary algebras with a duality, J. reine angew. Math. 449 (1994), 201–215. Search in Google Scholar
[32] K. Yamagata, Frobenius algebras, Handbook of algebra. Vol. 1, Elsevier/North-Holland, Amsterdam (1996), 841–887. 10.1016/S1570-7954(96)80028-3Search in Google Scholar
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