Koszul Equivalences in $A_\infty$-Algebras

TL;DR

This paper extends Koszul duality to Adams connected $A_ infty$-algebras, establishing derived equivalences and generalizing classical dualities, with applications to Artin-Schelter regularity, Gorenstein properties, and Calabi-Yau conditions.

Contribution

It generalizes classical Koszul duality to $A_ infty$-algebras and derives new equivalences and applications in algebraic properties.

Findings

Artin-Schelter regularity characterized by Frobenius Ext-algebra

Gorenstein property equivalence under Koszul duality for noetherian P.I. algebras

Calabi-Yau property preserved under Koszul duality

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 Bern\v{s}te{\u\i}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.