We prove a version of Koszul duality and the induced derived equivalence for Adams connected A_∞-algebras that generalizes the classical Beilinson-Ginzburg-Soergel Koszul duality. As an immediate consequence, we give a version of the Bernstein-Gel'fand-Gel'fand correspondence for Adams connected A_∞-algebras.

We give various applications. For example, a connected graded algebra A is Artin-Schelter regular if and only if its Ext-algebra Ext^∗_A(k,k) is Frobenius. This generalizes a result of Smith in the Koszul case. If A is Koszul and if both A and its Koszul dual A^! are noetherian satisfying a polynomial identity, then A is Gorenstein if and only if A^! is. The last statement implies that a certain Calabi-Yau property is preserved under Koszul duality.

The first author is supported by the NSFC (project 10571152) of China and the NSF of Zhejiang Province (J20080154). The third author is supported by the NSFC (key project 10331030) of China and Doctorate Foundation (No. 20060246003), Ministry of Education of China. The fourth author is supported by the NSF of USA and the Royalty Research Fund of the University of Washington.