Dualities in Koszul graded AS Gorenstein algebras

Abstract

The paper is dedicated to the study of certain non commutative graded AS Gorenstein algebras $\Lambda $. The main result of the paper is that for Koszul algebras $\Lambda $ with Yoneda algebra $\Gamma $, such that both $\Lambda $ and $\Gamma $ are graded AS Gorenstein noetherian of finite local cohomology dimension on both sides, there are dualities of triangulated categories: \underline{$gr$}$_{\Lambda}[\Omega ^{-1}]$ $\cong D^{b}(Qgr_{\Gamma})$ and \underline{$gr$}$_{\Gamma}[\Omega ^{-1}]$ $\cong D^{b}(Qgr_{\Lambda})$ where, and $Qgr_{\Gamma}$ is the category of tails, this is: the category of finitely generated graded modules $gr_{\Gamma}$ divided by the modules of finite length, and $D^{b}(Qgr_{\Gamma})$ the corresponding derived category and \underline{$gr$}$_{\Lambda}[\Omega^{-1}]$ the stabilization of the category of finetely generated graded $\Lambda $-modules, module the finetely generated projective modules.