ar X iv : 0 71 0 . 54 92 v 1 [ m at h . R A ] 2 9 O ct 2 00 7 Koszul Equivalences in A ∞ - Algebras

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 Bernšte˘ ın-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.