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arXiv · 0710.5492

Koszul Equivalences in $A_\infty$-Algebras

D. -M. Lu·J. H. Palmieri·Q. -S. Wu·J. J. Zhang
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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.

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BibTeXRIS

D. -M. Lu, J. H. Palmieri, Q. -S. Wu, J. J. Zhang. 2007-10-29. Koszul Equivalences in $A_\infty$-Algebras. https://arxiv.org/abs/0710.5492

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