Koszul equivalences and dualities.
The subject of Koszul equivalences and dualities is connected mainly with the context of derived categories, especially when we deal with Koszul algebras. It was shown by \textit{A. Beilinson, V. Ginzburg} and \textit{W. Soergel} [J. Am. Math. Soc. 9, No. 2, 473-526 (1996; Zbl 0864.17006)] that there is an equivalence between certain full triangulated subcategories of the derived categories \({\mathcal D}(_\Lambda\text{Gr})\) and \({\mathcal D}(_\Gamma\text{Gr})\) and also by composing with the canonical duality defined by \(\Hom_{\Lambda_0}(-,\Lambda_0)\) a duality between suitable subcategories of \({\mathcal D}(_\Lambda\text{Gr})\) and \({\mathcal D}(_\Gamma\text{Gr})\) is obtained. The authors extend these results to some Abelian categories arising from categories associated to positively graded algebras. It is shown that for such an algebra the categories of linear complexes of projectives and almost injectives are both naturally equivalent to the category of graded modules over the quadratic dual algebra \(A^!\). Also in the case where \(A=\Lambda\) is a graded factor of a path algebra with Yoneda algebra \(\Gamma\), it is shown that the category of linear complexes of finitely generated right projectives over \(\Gamma\) is dual to the category of locally finite graded left modules over the quadratic algebra \(\widetilde\Lambda\) associated to \(\Lambda\).
scientific article; zbMATH DE number 2101158