Koszul Equivalences in $A_{\infty }$-Algebras

Abstract

We prove a version of Koszul duality and the induced derived equivalence for Adams connected $A_{\infty }$-algebras that generalizes the classical Beilinson-Ginzburg-Soergel Koszul duality. As an immediate consequence, we give a version of the Bernšte{\uı}n-Gel'fand-Gel'fand correspondence for Adams connected $A_{\infty }$-algebras. We give various applications. For example, a connected graded algebra $A$ is Artin-Schelter regular if and only if its Ext-algebra $\Ext^\ast_A(k,k)$ 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.