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Approximations, ghosts and derived equivalences

Part of: Abelian groups Representation theory of rings and algebras Abelian categories Homological algebra

Published online by Cambridge University Press:  26 January 2019

Yiping Chen  and
Wei Hu
Yiping Chen
Affiliation:
School of Mathematics and Statistics, Wuhan University, Wuhan, 430072China ([email protected])
Wei Hu
Affiliation:
School of Mathematical Sciences, Laboratory of Mathematics and Complex Systems, Beijing Normal University, Beijing, 100875China ([email protected])
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Abstract

Approximation sequences and derived equivalences occur frequently in the research of mutation of tilting objects in representation theory, algebraic geometry and noncommutative geometry. In this paper, we introduce symmetric approximation sequences in additive categories and weakly n-angulated categories which include (higher) Auslander-Reiten sequences (triangles) and mutation sequences in algebra and geometry, and show that such sequences always give rise to derived equivalences between the quotient rings of endomorphism rings of objects in the sequences modulo some ghost and coghost ideals.

Keywords

derived equivalencemutationghost idealtilting complex

MSC classification

Primary: 18E30: Derived categories, triangulated categories 20K40: Homological and categorical methods
Secondary: 18G35: Chain complexes 16G10: Representations of Artinian rings
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 2 , April 2020 , pp. 813 - 840
Copyright
Copyright © Royal Society of Edinburgh 2019

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References

1Aihara, T. and Iyama, O.. Silting mutation in triangulated categories. J. Lond. Math. Soc. (2) 85 (2012), 633–668.CrossRefGoogle Scholar
2Assem, I., Simson, D. and Skowroński, A.. Elements of the representation theory of associative algebras. London Mathematical Society Student Texts 65, vol. 1 (Cambridge: Cambridge University Press, 2006).CrossRefGoogle Scholar
3Auslander, M. and Smalø, S. O.. Preprojective modules over artin algebras. J. Algebra 66 (1980), 61–122.CrossRefGoogle Scholar
4Auslander, M., Reiten, I. and Smalø, S. O.. Representation theory of Artin algebras. Cambridge Studies in Advanced Mathematics, vol. 36 (Cambridge: Cambridge University Press, 1995).CrossRefGoogle Scholar
5Buan, A. B., Marsh, R., Reineke, M., Reiten, I. and Todorov, G.. Tilting theory and cluster combinatorics. Adv. Math. 204 (2006), 572–618.CrossRefGoogle Scholar
6Chen, Y.. Derived equivalences in n-angulated categories. Algebr. Represent. Theory 16 (2013), 1661–1684.CrossRefGoogle Scholar
7Geiss, C., Keller, B. and Oppermann, S.. n-angulated categories. J. Reine Angew. Math. 675 (2013), 101–120.Google Scholar
8Happel, D.. Triangulated categories in the representation theory of finite-dimensional algebras. London Mathematical Society Lecture Note Series, vol. 119 (Cambridge: Cambridge University Press, 1988).CrossRefGoogle Scholar
9Happel, D. and Unger, L.. Almost complete tilting modules. Proc. Amer. Math. Soc. 107 (1989), 603–610.CrossRefGoogle Scholar
10Happel, D. and Unger, L.. Complements and the generalized Nakayama conjecture. In Algebras and modules, II (Geiranger, 1996), CMS Conf. Proc., vol. 24, pp. 293–310 (Providence, RI: Amer. Math. Soc., 1998).Google Scholar
11Hoshino, M. and Kato, Y.. An elementary construction of tilting complexes. J. Pure Appl. Algebra 177 (2003), 159–175.CrossRefGoogle Scholar
12Hu, W. and Xi, C. C.. 𝒟-split sequences and derived equivalences. Adv. Math. 227 (2011), 292–318.CrossRefGoogle Scholar
13Hu, W. and Xi, C. C.. Derived equivalences for Φ-Auslander-Yoneda algebras. Trans. Amer. Math. Soc. 365 (2013), 5681–5711.CrossRefGoogle Scholar
14Hu, W., Koenig, S. and Xi, C. C.. Derived equivalences from cohomological approximations and mutations of Φ-Yoneda algebras. Proc. Roy. Soc. Edinburgh Sect. A 143 (2013), 589–629.CrossRefGoogle Scholar
15Iyama, O. and Wemyss, M.. Maximal modifications and Auslander-Reiten duality for non-isolated singularities. Invent. Math. 197 (2014), 521–586.CrossRefGoogle Scholar
16Keller, B.. Deriving DG categories. Ann. Sci. École Norm. Sup. 27 (1994), 63–102.CrossRefGoogle Scholar
17Keller, B.. On triangulated orbit categories. Doc. Math. 10 (2005), 551–581.Google Scholar
18Krause, H.. Krull-Schmidt categories and projective covers. Expo. Math. 33 (2015), 535–549.CrossRefGoogle Scholar
19Ladkani, S.. Perverse equivalences, BB-tilting, mutations and applications, Preprint (2010).Google Scholar
20Rickard, J.. Morita theory for derived categories. J. London Math. Soc. 39 (1989), 436–456.CrossRefGoogle Scholar
21Rudakov, A. N.. Exceptional collections, mutations and helices. In Helices and vector bundles, London Math. Soc. Lecture Note Ser., vol. 148, pp. 1–6 (Cambridge: Cambridge Univ. Press, 1990).CrossRefGoogle Scholar
22Van den Bergh, M.. Non-commutative crepant resolutions. In The legacy of Niels Henrik Abel, pp. 749–770 (Berlin: Springer-Verlag, 2004).CrossRefGoogle Scholar