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Constructions of derived equivalences for algebras and rings
Changchang XI
Constructions of derived equivalences for algebras and rings
In this article, we shall survey some aspects of our recent (or related) constructions of derived equivalences for algebras and rings.
Derived equivalence / Frobenius-finite algebra / recollement / stable equivalence / tilting complex / Yoneda algebra
[1] |
Aihara T, Mizuno Y. Classifying tilting complexes over preprojective algebras of Dynkin type. Preprint, 2015, arXiv: 1509.07387
|
[2] |
Al-Nofayee S, Rickard J. Rigidity of tilting complexes and derived equivalences for selfinjective algebras. Preprint, 2013, arXiv: 1311.0504
|
[3] |
Asashiba H. The derived equivalence classification of representation-finite selfinjective algebras. J Algebra, 1999, 214: 182–221
CrossRef
Google scholar
|
[4] |
Asashiba H. On a lift of an individual stable equivalence to a standard derived equivalence for representation-finite self-injective algebras. Algebr Represent Theory, 2003, 6: 427–447
CrossRef
Google scholar
|
[5] |
Auslander M, Reiten I, Smalo S O. Representation Theory of Artin Algebras. Cambridge: Cambridge Univ Press, 1995
CrossRef
Google scholar
|
[6] |
Backelin J. On the rates of growth of the homologies of Veronese subrings. In: Roos J-E, ed. Algebra, Algebraic Topology and Their Interactions. Lecture Notes in Math, Vol 1183. Berlin: Springer, 1986, 79–100
CrossRef
Google scholar
|
[7] |
Baer D, Geigle W, Lenzing H. The preprojective algebra of a tame hereditary Artin algebra. Comm Algebra, 1987, 15(1-2): 425–457
CrossRef
Google scholar
|
[8] |
Barot M, Lenzing H. One-point extensions and derived equivalences. J Algebra, 2003, 264: 1–5
CrossRef
Google scholar
|
[9] |
Bazzoni S. Equivalences induced by infinitely generated tilting modules. Proc Amer Math Soc, 2010, 138: 533–544
CrossRef
Google scholar
|
[10] |
Beilinson A A. Coherent sheaves on Pn and problems of linear algebra. Funct Anal Appl, 1978, 12: 214–216
CrossRef
Google scholar
|
[11] |
Beilinson A A, Bernstein J, Deligne P. Faisceaux pervers. Asterisque, 1982, 100: 5–171
|
[12] |
Brenner S, Butler M C R. Generalizations of the Bernstein-Gelfand-Ponomarev reflection functors. In: Dlab V, Gabriel P, eds. Representation Theory II. Lecture Notes in Math, Vol 832. Berlin: Springer, 1980, 103–169
CrossRef
Google scholar
|
[13] |
Broué M. Isométríes de caractères et équivalences de Morita ou dérivées. Inst Hautes Études Sci Publ Math, 1990, 71: 45–63
CrossRef
Google scholar
|
[14] |
Chen H X, Xi C C. Good tilting modules and recollements of derived module categories. Proc Lond Math Soc, 2012, 104: 959–996
CrossRef
Google scholar
|
[15] |
Chen H X, Xi C C. Recollements of derived categories, I: Exact contexts. Preprint, 2012, arXiv: 1203.5168
|
[16] |
Chen H X, Xi C C. Good tilting modules and recollements of derived module categories II. Preprint, 2016, arXiv: 1206.0522
|
[17] |
Chen H X, Xi C C. Recollements of derived categories, II: Algebraic K-theory. Preprint, 2012, arXiv: 1212.1879
|
[18] |
Chen H X, Xi C C. Recollements of derived categories, III: Finitistic dimensions. J Lond Math Soc (to appear), arXiv: 1405.5090
|
[19] |
Chen H X, Xi C C. Dominant dimensions, derived equivalences and tilting modules. Israel J Math (to appear), DOI: 10.1007/s11856-016-1327-4, arXiv: 1503.02385
CrossRef
Google scholar
|
[20] |
Chen Y P. Derived equivalences in n-angulated categories. Algebr Represent Theory, 2013, 16: 1661–1684
CrossRef
Google scholar
|
[21] |
Chen Y P. Derived equivalences between subrings. Comm Algebra, 2014, 42: 4055–4065
CrossRef
Google scholar
|
[22] |
Chuang J, Rouquier R. Derived equivalences for symmetric groups and Sl2-categorification. Ann of Math, 2008, 167: 245–298
CrossRef
Google scholar
|
[23] |
Cline E, Parshall B, Scott L. Algebraic stratification in representation categories. J Algebra, 1988, 117: 504–521
CrossRef
Google scholar
|
[24] |
Dugas A. Tilting mutation of weakly symmetric algebras and stable equivalence. Algebr Represent Theory, 2014, 17: 863–884
CrossRef
Google scholar
|
[25] |
Dugas A. A construction of derived equivalent pairs of symmetric algebras. Proc Amer Math Soc, 2015, 143: 2281–2300
CrossRef
Google scholar
|
[26] |
Dugger D, Shipley B. K-theory and derived equivalences. Duke Math J, 2004, 124(3): 587–617
CrossRef
Google scholar
|
[27] |
Gelfand I M, Ponomarev V A. Model algebras and representations of graphs. Funktsional Anal i Prilozhen, 1979, 13(3): 1–12
CrossRef
Google scholar
|
[28] |
Han Y, Qin Y Y. Reducing homological conjectures by n-recollements. Preprint, 2014, arXiv: 1410.3223
|
[29] |
Happel D. Reduction techniques for homological conjectures. Tsukuba J Math, 1993, 17(1): 115–130
|
[30] |
Happel D. Triangulated categories in the representation theory of finite dimensional algebras. Cambridge: Cambridge Univ Press, 1988
CrossRef
Google scholar
|
[31] |
Happel D. The Coxeter polynomial for a one point extension algebra. J Algebra, 2009, 321: 2028–2041
CrossRef
Google scholar
|
[32] |
Hoshino M, Kato Y. Tilting complexes defined by idempotents. Comm Algebra, 2002, 30: 83–100
CrossRef
Google scholar
|
[33] |
Hoshino M, Kato Y. An elementary construction of tilting complexes. J Pure Appl Algebra, 2003, 177: 158–175
CrossRef
Google scholar
|
[34] |
Hu W, Koenig S, Xi C C. Derived equivalences from cohomological approximations, and mutations of Yoneda algebras. Proc Roy Soc Edinburgh Sect. A, 2013, 143(3): 589–629
|
[35] |
Hu W, Xi C C. D-split sequences and derived equivalences. Adv Math, 2011, 227: 292–318
CrossRef
Google scholar
|
[36] |
Hu W, Xi C C. Derived equivalences for Φ-Auslander-Yoneda algebras. Trans Amer Math Soc, 2013, 365: 589–629
CrossRef
Google scholar
|
[37] |
Hu W, Xi C C. Derived equivalences and stable equivalences of Morita type, I. Nagoya Math J, 2010, 200: 107–152
CrossRef
Google scholar
|
[38] |
Hu W, Xi C C. Derived equivalences and stable equivalences of Morita type, II. Rev Mat Iberoam (to appear), arXiv: 1412.7301
|
[39] |
Hu W, Xi C C. Derived equivalences constructed from pullback algebras. Preprint, 2015
|
[40] |
Kato Y. On derived equivalent coherent rings. Comm Algebra, 2002, 30(9): 4437–4454
CrossRef
Google scholar
|
[41] |
Keller B. Invariance and localization for cyclic homology of DG-algebras. J Pure Appl Algebra, 1998, 123: 223–273
CrossRef
Google scholar
|
[42] |
Koenig S. Tilting complexes, perpendicular categories and recollements of derived module categories of rings. J Pure Appl Algebra, 1991, 73: 211–232
CrossRef
Google scholar
|
[43] |
Ladkani S. On derived equivalences of lines, rectangles and triangles. J Lond Math Soc, 2013, 87: 157–176
CrossRef
Google scholar
|
[44] |
Lenzing H, Meltzer H. Sheaves on a weighted projective line of genus one, and representations of a tubular algebra. In: Representations of Algebras (Ottawa, ON, 1992). CMS Conf Proc, Vol 14. Providence: Amer Math Soc, 1993, 313–337
|
[45] |
Martinez-Villa R. The stable equivalence for algebras of finite representation type. Comm Algebra, 1985, 13(5): 991–1018
CrossRef
Google scholar
|
[46] |
Neeman A. Triangulated Categories. Ann of Math Stud, Vol 148. Princeton and Oxford: Princeton Univ Press, 2001
|
[47] |
Okuyama T. Some examples of derived equivalent blocks of finite groups. Preprint, 1997
|
[48] |
Pan S Y. Transfer of derived equivalences from subalgebras to endomorphism algebras. J Algebra Appl, 2016, 15(6): 1650100 (10 pp)
|
[49] |
Pan S Y, Peng Z. A note on derived equivalences for Φ-Green algebras. Algebr Represent Theory, 2014, 17: 1707–1720
CrossRef
Google scholar
|
[50] |
Pan S Y, Xi C C. Finiteness of finitistic dimension is invariant under derived equivalences. J Algebra, 2009, 322: 21–24
CrossRef
Google scholar
|
[51] |
Psaroudakis Ch. Homological theory of recollements of abelian categories. J Algebra, 2014, 398: 63–110
CrossRef
Google scholar
|
[52] |
Rickard J. Morita theory for derived categories. J Lond Math Soc, 1989, 39: 436–456
CrossRef
Google scholar
|
[53] |
Rickard J. Derived categories and stable equivalences. J Pure Appl Algebra, 1989, 64: 303–317
CrossRef
Google scholar
|
[54] |
Rickard J. Derived equivalences as derived functors. J Lond Math Soc, 1991, 43: 37–48
CrossRef
Google scholar
|
[55] |
Rickard J. The abelian defect group conjecture. In: Proceedings of the International Congress of Mathematicians, Vol II (Berlin, 1998) Doc Math, Extra Vol II. 1998, 121–128
|
[56] |
Rouquier R. Derived equivalences and finite dimensional algebras. In: International Congress of Mathematicians, Vol II. Zürich: Eur Math Soc, 2006, 191–221
|
[57] |
Verdier J L. Catégories dérivées, etat O. Lecture Notes in Math, Vol 569. Berlin: Springer-Verlag, 1977, 262–311
|
[58] |
Xi C C. Higher algebraic K-groups and D-split sequences. Math Z, 2013, 273: 1025–1052
CrossRef
Google scholar
|
[59] |
Zimmermann A. Representation Theory, A Homological Algebra Point of View. Algebr Appl, Vol 19. Cham: Springer International Publishing Switzerland, 2014
|
[60] |
Zvonareva A O. Two-term tilting complexes over Brauer tree algebras. J Math Sci, 2015, 209(4): 568–587
CrossRef
Google scholar
|
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