 

Di Ming Lu, John H. Palmieri, Quan Shui Wu, and
James J. Zhang
Koszul equivalences in A_{∞}algebras


Published: 
August 23, 2008 
Keywords: 
A_{∞}algebra, graded algebra, ArtinSchelter regular algebra, Koszul duality, derived equivalence, Gorenstein property 
Subject: 
16A03,16A62,16E65 


Abstract
We prove a version of Koszul duality and the induced
derived equivalence for Adams connected A_{∞}algebras that
generalizes the classical BeilinsonGinzburgSoergel Koszul duality.
As an immediate consequence, we give a version of the
BernsteinGel'fandGel'fand correspondence for Adams
connected A_{∞}algebras.
We give various applications. For example, a connected graded
algebra A is ArtinSchelter regular if and only if its Extalgebra
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 CalabiYau property is preserved under Koszul
duality.


Acknowledgements
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.


Author information
Di Ming Lu:
Department of Mathematics, Zhejiang University, Hangzhou 310027, China
dmlu@zju.edu.cn
John H. Palmieri:
Department of Mathematics, Box 354350, University of Washington, Seattle, WA 98195, USA
palmieri@math.washington.edu
Quan Shui Wu:
Institute of Mathematics, Fudan University, Shanghai, 200433, China
qswu@fudan.edu.cn
James J. Zhang:
Department of Mathematics, Box 354350, University of Washington, Seattle, WA 98195, USA
zhang@math.washington.edu

