Dualities in Koszul graded AS Gorenstein algebras

TL;DR

This paper establishes dualities between certain stabilized categories of modules over Koszul graded AS Gorenstein algebras and their Yoneda algebras, revealing deep categorical symmetries in noncommutative algebra.

Contribution

It proves dualities of triangulated categories for Koszul graded AS Gorenstein algebras with their Yoneda algebras, under specific noetherian and cohomological conditions.

Findings

Dualities between stabilized module categories and derived categories of tails.

Conditions under which these dualities hold for Koszul AS Gorenstein algebras.

Extension of known dualities to noncommutative graded Gorenstein settings.

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.