Connected cochain DG algebras of Calabi-Yau dimension 0

He, J.-W.; Mao, X.-F.

doi:10.1090/proc/13081

Connected cochain DG algebras of Calabi-Yau dimension 0
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by J.-W. He and X.-F. Mao PDF

Proc. Amer. Math. Soc. 145 (2017), 937-953 Request permission

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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