Connected cochain DG algebras of Calabi-Yau dimension 0

Abstract

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