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DIFFERENTIAL GRADED ENDOMORPHISM ALGEBRAS, COHOMOLOGY RINGS AND DERIVED EQUIVALENCES

Part of: Rings and algebras arising under various constructions Representation theory of rings and algebras Abelian categories Homological algebra

Published online by Cambridge University Press:  12 September 2018

SHENGYONG PAN ,
ZHEN PENG  and
JIE ZHANG
SHENGYONG PAN*
Affiliation:
Department of Mathematics, Beijing Jiaotong University, Beijing 100044, ChinaBeijing Center for Mathematics and Information Interdisciplinary Sciences, Beijing 100048, China e-mail: [email protected]
ZHEN PENG*
Affiliation:
School of Mathematics and Statistics, Anyang Normal University, Anyang 455000, China e-mail: [email protected]
JIE ZHANG*
Affiliation:
Department of Mathematics, Beijing Institute of Technology, Beijing 100081, China e-mail: [email protected]
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Abstract

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In this paper, we will consider derived equivalences for differential graded endomorphism algebras by Keller's approaches. First, we construct derived equivalences of differential graded algebras which are endomorphism algebras of the objects from a triangle in the homotopy category of differential graded algebras. We also obtain derived equivalences of differential graded endomorphism algebras from a standard derived equivalence of finite dimensional algebras. Moreover, under some conditions, the cohomology rings of these differential graded endomorphism algebras are also derived equivalent. Then we give an affirmative answer to a problem of Dugas (A construction of derived equivalent pairs of symmetric algebras, Proc. Amer. Math. Soc. 143 (2015), 2281–2300) in some special case.

MSC classification

Primary: 18E30: Derived categories, triangulated categories
Secondary: 16G10: Representations of Artinian rings 16S10: Rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.) 18G15: Ext and Tor, generalizations, Künneth formula
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 61 , Issue 3 , September 2019 , pp. 557 - 573
Copyright
Copyright © Glasgow Mathematical Journal Trust 2018 

References

REFERENCES

Bridgeland, T., Equivalences of derived categories and Fourier–Mukai transforms, Bull. Lond. Math. Soc. 31 (1) (1999), 25–34.CrossRefGoogle Scholar
Chen, X. and Yang, D., Homotopy categories, Leavitt path algebras and Gorenstein projective modules, Intrer. Math. Res. Notice 2015 (10) (2015), 2597–2633.CrossRefGoogle Scholar
Dugas, A., A construction of derived equivalent pairs of symmetric algebras, Proc. Amer. Math. Soc. 143 (6) (2015), 2281–2300.CrossRefGoogle Scholar
Efimov, A. I., Lunts, V. A. and Orlov, D. O., Deformation theory of objects in homotopy and derived categories I: General theory, Adv. Math. 222 (1) (2009), 359–401.CrossRefGoogle Scholar
Happel, D., Triangulated categories in the representation theory of finite dimensional algebras, London Mathematical Society Lecture Note Series, 119. (Cambridge University Press, Cambridge, 1988).CrossRefGoogle Scholar
Hoshino, M. and Kato, Y., Tilting complexes associated with a sequence of idempotents, J. Pure Appl. Algebra 183 (1) (2003), 105–124.CrossRefGoogle Scholar
Hu, W. and Pan, S. Y., Stable functors of derived equivalences and Gorenstein projective modules, Math. Nach. 290 (10) (2017), 1512–1530.CrossRefGoogle Scholar
Hu, W. and Xi, C. C., Derived equivalences and stable equivalences of Morita type, I, Nagoya Math. J. 200 (2010), 107–152.CrossRefGoogle Scholar
Hu, W. and Xi, C. C., Derived equivalences for Φ-Auslander–Yoneda algebras, Trans. Amer. Math. Soc. 365 (11) (2013), 5681–5711.CrossRefGoogle Scholar
Iyama, O. and Takahashi, R., Tilting and cluster tilting for quotient singularities, Math. Ann. 356 (3) (2013), 1065–1105.CrossRefGoogle Scholar
Keller, B., Deriving dg categories, Ann. Sci. École Norm. Sup. 27 (1) (1994), 63–102.CrossRefGoogle Scholar
Keller, B., On differential graded categories, in International congress of mathematicians, vol. II (Eur. Math. Soc., Zürich, 2006), 151–190.Google Scholar
Neeman, A., Triangulated categories, Annals of Mathematics Studies, vol. 148 (Princeton University Press, Princeton, NJ, 2001).CrossRefGoogle Scholar
Nicolas, P. and Saorin, M., Generalized tilting theory. Appl. Categ. Structures, 26 (2) (2018), 309–368.CrossRefGoogle Scholar
Nicolas, P. and Saorin, M., Classical derived functors as fully faithful embeddings, preprint, arXiv:1403.4726 (2014).Google Scholar
Pan, S. Y., Derived equivalences for Φ-Cohen–Macaulay Auslander–Yoneda algebras, Algebr. Represent. Theory 17 (3) (2014), 885–903.CrossRefGoogle Scholar
Porta, M., Shaul, L. and Yekutieli, A., Completion by derived double centralizer, Algebr. Represent. Theor. 17 (2) (2014), 481–494.CrossRefGoogle Scholar
Rickard, J., Morita theory for derived categories, J. London Math. Soc. 39 (3) (1989), 436–456.CrossRefGoogle Scholar
Rickard, J., Derived equivalences as derived functors, J. London Math. Soc. 43 (1) (1991), 37–48.CrossRefGoogle Scholar
Zheng, R., On the Auslander–Yoneda algebras of modules over k[x]/(xn), Algebra Collq. 22 (1) (2015), 147–162.CrossRefGoogle Scholar