Connected cochain DG algebras of Calabi-Yau dimension 0

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.