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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
Published electronically: November 29, 2016
MathSciNet review: 3589295
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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$.

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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

X.-F. Mao
Affiliation: Department of Mathematics, Shanghai University, Shanghai 200444, People’s Republic of China

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

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