Koszul equivalences and dualities.
type: Item
instance of: scholarly article
title: Koszul equivalences and dualities.
zbMATH Open document ID: 1057.16026
author: Manuel Saorín
author: Roberto Martinez-Villa
published in: Pacific Journal of Mathematics
publication date: 2004-09-17T00:00:00Z
review text: 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 \(\Lambda\).
reviewed by: Constantin Năstăsescu
Mathematics Subject Classification ID: 16S37
Mathematics Subject Classification ID: 18E10
Mathematics Subject Classification ID: 16D90
Mathematics Subject Classification ID: 18E30
Mathematics Subject Classification ID: 16W50
Mathematics Subject Classification ID: 16E30
zbMATH DE Number: 2101158
zbMATH Keywords: Koszul equivalences
zbMATH Keywords: Abelian categories
zbMATH Keywords: graded algebras
zbMATH Keywords: quadratic algebras
zbMATH Keywords: Koszul dualities
zbMATH Keywords: derived categories
MaRDI profile type: Publication
OpenAlex ID: W2120006515
DOI: 10.2140/PJM.2004.214.359
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description: scientific article; zbMATH DE number 2101158