Connected cochain DG algebras of Calabi-Yau dimension 0
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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
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Additional Information
- J.-W. He
- Affiliation: Department of Mathematics, Hangzhou Normal University, 16 Xuelin Road, Hangzhou Zhejiang 310036, People’s Republic of China
- MR Author ID: 710882
- Email: jwhe@hznu.edu.cn
- X.-F. Mao
- Affiliation: Department of Mathematics, Shanghai University, Shanghai 200444, People’s Republic of China
- MR Author ID: 846632
- Email: xuefengmao@shu.edu.cn
- 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
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 145 (2017), 937-953
- MSC (2010): Primary 16E10, 16E30, 16E45, 16E65
- DOI: https://doi.org/10.1090/proc/13081
- MathSciNet review: 3589295