Constructions of derived equivalences for algebras and rings
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Abstract
In this article, we shall survey some aspects of our recent (or related) constructions of derived equivalences for algebras and rings.
Keywords
Derived equivalence Frobenius-finite algebra recollement stable equivalence tilting complex Yoneda algebraMSC
18E30 16G10 16S50 18G15Preview
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References
- 1.Aihara T, Mizuno Y. Classifying tilting complexes over preprojective algebras of Dynkin type. Preprint, 2015, arXiv: 1509.07387Google Scholar
- 2.Al-Nofayee S, Rickard J. Rigidity of tilting complexes and derived equivalences for selfinjective algebras. Preprint, 2013, arXiv: 1311.0504Google Scholar
- 3.Asashiba H. The derived equivalence classification of representation-finite selfinjective algebras. J Algebra, 1999, 214: 182–221MathSciNetCrossRefMATHGoogle 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: 427–447MathSciNetCrossRefMATHGoogle Scholar
- 5.Auslander M, Reiten I, Smalø S O. Representation Theory of Artin Algebras. Cambridge: Cambridge Univ Press, 1995CrossRefMATHGoogle 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, 79–100CrossRefGoogle Scholar
- 7.Baer D, Geigle W, Lenzing H. The preprojective algebra of a tame hereditary Artin algebra. Comm Algebra, 1987, 15(1–2): 425–457MathSciNetCrossRefMATHGoogle Scholar
- 8.Barot M, Lenzing H. One-point extensions and derived equivalences. J Algebra, 2003, 264: 1–5MathSciNetCrossRefMATHGoogle Scholar
- 9.Bazzoni S. Equivalences induced by infinitely generated tilting modules. Proc Amer Math Soc, 2010, 138: 533–544MathSciNetCrossRefMATHGoogle Scholar
- 10.Beilinson A A. Coherent sheaves on Pn and problems of linear algebra. Funct Anal Appl, 1978, 12: 214–216MathSciNetCrossRefMATHGoogle Scholar
- 11.Beilinson A A, Bernstein J, Deligne P. Faisceaux pervers. Asterisque, 1982, 100: 5–171MathSciNetGoogle Scholar
- 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, 103–169CrossRefGoogle Scholar
- 13.Broué M. Isométríes de caractères et équivalences de Morita ou dérivées. Inst Hautes Études Sci Publ Math, 1990, 71: 45–63CrossRefMATHGoogle Scholar
- 14.Chen H X, Xi C C. Good tilting modules and recollements of derived module categories. Proc Lond Math Soc, 2012, 104: 959–996MathSciNetCrossRefMATHGoogle Scholar
- 15.Chen H X, Xi C C. Recollements of derived categories, I: Exact contexts. Preprint, 2012, arXiv: 1203.5168Google Scholar
- 16.Chen H X, Xi C C. Good tilting modules and recollements of derived module categories II. Preprint, 2016, arXiv: 1206.0522MATHGoogle Scholar
- 17.Chen H X, Xi C C. Recollements of derived categories, II: Algebraic K-theory. Preprint, 2012, arXiv: 1212.1879Google Scholar
- 18.Chen H X, Xi C C. Recollements of derived categories, III: Finitistic dimensions. J Lond Math Soc (to appear), arXiv: 1405.5090Google Scholar
- 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: 1503.02385Google Scholar
- 20.Chen Y P. Derived equivalences in n-angulated categories. Algebr Represent Theory, 2013, 16: 1661–1684MathSciNetCrossRefMATHGoogle Scholar
- 21.Chen Y P. Derived equivalences between subrings. Comm Algebra, 2014, 42: 4055–4065MathSciNetCrossRefMATHGoogle Scholar
- 22.Chuang J, Rouquier R. Derived equivalences for symmetric groups and Sl2-categorification. Ann of Math, 2008, 167: 245–298MathSciNetCrossRefMATHGoogle Scholar
- 23.Cline E, Parshall B, Scott L. Algebraic stratification in representation categories. J Algebra, 1988, 117: 504–521MathSciNetCrossRefMATHGoogle Scholar
- 24.Dugas A. Tilting mutation of weakly symmetric algebras and stable equivalence. Algebr Represent Theory, 2014, 17: 863–884MathSciNetCrossRefMATHGoogle Scholar
- 25.Dugas A. A construction of derived equivalent pairs of symmetric algebras. Proc Amer Math Soc, 2015, 143: 2281–2300MathSciNetCrossRefMATHGoogle Scholar
- 26.Dugger D, Shipley B. K-theory and derived equivalences. Duke Math J, 2004, 124(3): 587–617MathSciNetCrossRefMATHGoogle Scholar
- 27.Gelfand I M, Ponomarev V A. Model algebras and representations of graphs. Funktsional Anal i Prilozhen, 1979, 13(3): 1–12MathSciNetGoogle Scholar
- 28.Han Y, Qin Y Y. Reducing homological conjectures by n-recollements. Preprint, 2014, arXiv: 1410.3223MATHGoogle Scholar
- 29.Happel D. Reduction techniques for homological conjectures. Tsukuba J Math, 1993, 17(1): 115–130MathSciNetMATHGoogle Scholar
- 30.Happel D. Triangulated categories in the representation theory of finite dimensional algebras. Cambridge: Cambridge Univ Press, 1988CrossRefMATHGoogle Scholar
- 31.Happel D. The Coxeter polynomial for a one point extension algebra. J Algebra, 2009, 321: 2028–2041MathSciNetCrossRefMATHGoogle Scholar
- 32.Hoshino M, Kato Y. Tilting complexes defined by idempotents. Comm Algebra, 2002, 30: 83–100MathSciNetCrossRefMATHGoogle Scholar
- 33.Hoshino M, Kato Y. An elementary construction of tilting complexes. J Pure Appl Algebra, 2003, 177: 158–175MathSciNetCrossRefMATHGoogle 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): 589–629MathSciNetCrossRefMATHGoogle Scholar
- 35.Hu W, Xi C C. D-split sequences and derived equivalences. Adv Math, 2011, 227: 292–318MathSciNetCrossRefMATHGoogle Scholar
- 36.Hu W, Xi C C. Derived equivalences for Φ-Auslander-Yoneda algebras. Trans Amer Math Soc, 2013, 365: 589–629MathSciNetMATHGoogle Scholar
- 37.Hu W, Xi C C. Derived equivalences and stable equivalences of Morita type, I. Nagoya Math J, 2010, 200: 107–152MathSciNetCrossRefMATHGoogle Scholar
- 38.Hu W, Xi C C. Derived equivalences and stable equivalences of Morita type, II. Rev Mat Iberoam (to appear), arXiv: 1412.7301Google Scholar
- 39.Hu W, Xi C C. Derived equivalences constructed from pullback algebras. Preprint, 2015Google Scholar
- 40.Kato Y. On derived equivalent coherent rings. Comm Algebra, 2002, 30(9): 4437–4454MathSciNetCrossRefMATHGoogle Scholar
- 41.Keller B. Invariance and localization for cyclic homology of DG-algebras. J Pure Appl Algebra, 1998, 123: 223–273MathSciNetCrossRefMATHGoogle Scholar
- 42.Koenig S. Tilting complexes, perpendicular categories and recollements of derived module categories of rings. J Pure Appl Algebra, 1991, 73: 211–232MathSciNetCrossRefMATHGoogle Scholar
- 43.Ladkani S. On derived equivalences of lines, rectangles and triangles. J Lond Math Soc, 2013, 87: 157–176MathSciNetCrossRefMATHGoogle 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, 313–337Google Scholar
- 45.Martinez-Villa R. The stable equivalence for algebras of finite representation type. Comm Algebra, 1985, 13(5): 991–1018MathSciNetCrossRefMATHGoogle Scholar
- 46.Neeman A. Triangulated Categories. Ann of Math Stud, Vol 148. Princeton and Oxford: Princeton Univ Press, 2001Google Scholar
- 47.Okuyama T. Some examples of derived equivalent blocks of finite groups. Preprint, 1997Google Scholar
- 48.Pan S Y. Transfer of derived equivalences from subalgebras to endomorphism algebras. J Algebra Appl, 2016, 15(6): 1650100 (10 pp)MathSciNetCrossRefMATHGoogle Scholar
- 49.Pan S Y, Peng Z. A note on derived equivalences for Φ-Green algebras. Algebr Represent Theory, 2014, 17: 1707–1720MathSciNetCrossRefMATHGoogle Scholar
- 50.Pan S Y, Xi C C. Finiteness of finitistic dimension is invariant under derived equivalences. J Algebra, 2009, 322: 21–24MathSciNetCrossRefMATHGoogle Scholar
- 51.Psaroudakis Ch. Homological theory of recollements of abelian categories. J Algebra, 2014, 398: 63–110MathSciNetCrossRefMATHGoogle Scholar
- 52.Rickard J. Morita theory for derived categories. J Lond Math Soc, 1989, 39: 436–456MathSciNetCrossRefMATHGoogle Scholar
- 53.Rickard J. Derived categories and stable equivalences. J Pure Appl Algebra, 1989, 64: 303–317MathSciNetCrossRefMATHGoogle Scholar
- 54.Rickard J. Derived equivalences as derived functors. J Lond Math Soc, 1991, 43: 37–48MathSciNetCrossRefMATHGoogle 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, 121–128Google Scholar
- 56.Rouquier R. Derived equivalences and finite dimensional algebras. In: International Congress of Mathematicians, Vol II. Zürich: Eur Math Soc, 2006, 191–221Google Scholar
- 57.Verdier J L. Catégories dérivées, etat O. Lecture Notes in Math, Vol 569. Berlin: Springer-Verlag, 1977, 262–311Google Scholar
- 58.Xi C C. Higher algebraic K-groups and D-split sequences. Math Z, 2013, 273: 1025–1052MathSciNetCrossRefMATHGoogle Scholar
- 59.Zimmermann A. Representation Theory, A Homological Algebra Point of View. Algebr Appl, Vol 19. Cham: Springer International Publishing Switzerland, 2014Google Scholar
- 60.Zvonareva A O. Two-term tilting complexes over Brauer tree algebras. J Math Sci, 2015, 209(4): 568–587MathSciNetCrossRefMATHGoogle Scholar
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