Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Connected cochain DG algebras of Calabi-Yau dimension 0


Authors: J.-W. He and X.-F. Mao
Journal: Proc. Amer. Math. Soc. 145 (2017), 937-953
MSC (2010): Primary 16E10, 16E30, 16E45, 16E65
DOI: https://doi.org/10.1090/proc/13081
Published electronically: November 29, 2016
MathSciNet review: 3589295
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ A$ be a connected cochain differential graded (DG, for short) algebra. This note shows that $ A$ is a 0-Calabi-Yau DG algebra if and only if $ A$ is a Koszul DG algebra and $ \mathrm {Tor}_A^0(\Bbbk _A,{}_A\Bbbk )$ is a symmetric coalgebra. Let $ V$ be a finite dimensional vector space and $ w$ a potential in $ T(V)$. Then the minimal subcoalgebra of $ T(V)$ containing $ w$ is a symmetric coalgebra, which implies that a locally finite connected cochain DG algebra is 0-CY if and only if it is defined by a potential $ w$.


References [Enhancements On Off] (What's this?)

  • [AFH] L. L. Avramov, H.-B. Foxby, and S. Halperin, Differential graded homologcial algebra, (in Preparation).
  • [BGS] Alexander Beilinson, Victor Ginzburg, and Wolfgang Soergel, Koszul duality patterns in representation theory, J. Amer. Math. Soc. 9 (1996), no. 2, 473-527. MR 1322847 (96k:17010), https://doi.org/10.1090/S0894-0347-96-00192-0
  • [BSW] Raf Bocklandt, Travis Schedler, and Michael Wemyss, Superpotentials and higher order derivations, J. Pure Appl. Algebra 214 (2010), no. 9, 1501-1522. MR 2593679 (2011e:16022), https://doi.org/10.1016/j.jpaa.2009.07.013
  • [BV] A. Bondal and M. van den Bergh, Generators and representability of functors in commutative and noncommutative geometry, Mosc. Math. J. 3 (2003), no. 1, 1-36, 258 (English, with English and Russian summaries). MR 1996800 (2004h:18009)
  • [CDN] F. Castaño Iglesias, S. Dăscălescu, and C. Năstăsescu, Symmetric coalgebras, J. Algebra 279 (2004), no. 1, 326-344. MR 2078404 (2005d:16059), https://doi.org/10.1016/j.jalgebra.2004.05.007
  • [FHT] Yves Félix, Stephen Halperin, and Jean-Claude Thomas, Rational homotopy theory, Graduate Texts in Mathematics, vol. 205, Springer-Verlag, New York, 2001. MR 1802847 (2002d:55014)
  • [Gin] V. Ginzburg, Calabi-Yau algebras, arXiv:math/0612139.
  • [Mac] S. Maclane, Homology, Berlin, FRG: Springer-Verlag, (1963).
  • [Kel] Bernhard Keller, Deriving DG categories, Ann. Sci. École Norm. Sup. (4) 27 (1994), no. 1, 63-102. MR 1258406 (95e:18010)
  • [HVZ] Ji-Wei He, Fred Van Oystaeyen, and Yinhuo Zhang, Hopf algebra actions on differential graded algebras and applications, Bull. Belg. Math. Soc. Simon Stevin 18 (2011), no. 1, 99-111. MR 2809906 (2012d:16031)
  • [HW] J.-W. He and Q.-S. Wu, Koszul differential graded algebras and BGG correspondence, J. Algebra 320 (2008), no. 7, 2934-2962. MR 2442004 (2009h:16011), https://doi.org/10.1016/j.jalgebra.2008.06.021
  • [HMS] Dale Husemoller, John C. Moore, and James Stasheff, Differential homological algebra and homogeneous spaces, J. Pure Appl. Algebra 5 (1974), 113-185. MR 0365571 (51 #1823)
  • [FIJ] Anders Frankild, Srikanth Iyengar, and Peter Jørgensen, Dualizing differential graded modules and Gorenstein differential graded algebras, J. London Math. Soc. (2) 68 (2003), no. 2, 288-306. MR 1994683 (2004f:16013), https://doi.org/10.1112/S0024610703004496
  • [Jor] Peter Jørgensen, Amplitude inequalities for differential graded modules, Forum Math. 22 (2010), no. 5, 941-948. MR 2719763 (2012d:16032), https://doi.org/10.1515/FORUM.2010.049
  • [L] K. Lefèvre-Hasegawa, Sur les $ A_\infty $-Catégories, Université Paris 7, Thése de Doctorat, 2003.
  • [MW1] X.-F. Mao and Q.-S. Wu, Homological invariants for connected DG algebras, Comm. Algebra 36 (2008), no. 8, 3050-3072. MR 2440300 (2009i:16022), https://doi.org/10.1080/00927870802110870
  • [MW2] XueFeng Mao and QuanShui Wu, Compact DG modules and Gorenstein DG algebras, Sci. China Ser. A 52 (2009), no. 4, 648-676. MR 2504967 (2010d:16009), https://doi.org/10.1007/s11425-008-0175-z
  • [Pr] Stewart B. Priddy, Koszul resolutions, Trans. Amer. Math. Soc. 152 (1970), 39-60. MR 0265437 (42 #346)
  • [VdB] M. Van den Bergh, Calabi-Yau algebras and superpotentials, Sel. Math. New Ser., DOI 10.1007/s00029-014-0166-6.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 16E10, 16E30, 16E45, 16E65

Retrieve articles in all journals with MSC (2010): 16E10, 16E30, 16E45, 16E65


Additional Information

J.-W. He
Affiliation: Department of Mathematics, Hangzhou Normal University, 16 Xuelin Road, Hangzhou Zhejiang 310036, People’s Republic of China
Email: jwhe@hznu.edu.cn

X.-F. Mao
Affiliation: Department of Mathematics, Shanghai University, Shanghai 200444, People’s Republic of China
Email: xuefengmao@shu.edu.cn

DOI: https://doi.org/10.1090/proc/13081
Keywords: Differential graded algebra, Koszul, Calabi-Yau, potential
Received by editor(s): August 27, 2014
Received by editor(s) in revised form: March 4, 2015, June 1, 2015, and January 4, 2016
Published electronically: November 29, 2016
Communicated by: Lev Borisov
Article copyright: © Copyright 2016 American Mathematical Society

American Mathematical Society