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Dedualizing Complexes of Bicomodules and MGM Duality Over Coalgebras

Authors
  • Positselski, Leonid1, 2, 3, 4
  • 1 University of Haifa, Department of Mathematics, Faculty of Natural Sciences, Mount Carmel, Haifa, 31905, Israel , Haifa (Israel)
  • 2 National Research University Higher School of Economics, Laboratory of Algebraic Geometry, Moscow, 119048, Russia , Moscow (Russia)
  • 3 Institute for Information Transmission Problems, Sector of Algebra and Number Theory, Moscow, 127051, Russia , Moscow (Russia)
  • 4 Charles University, Faculty of Mathematics and Physics, Department of Algebra, Sokolovská 83, 186 75 Prague 8, Prague, Czech Republic , Prague (Czechia)
Type
Published Article
Journal
Algebras and Representation Theory
Publisher
Springer Netherlands
Publication Date
Oct 12, 2017
Volume
21
Issue
4
Pages
737–767
Identifiers
DOI: 10.1007/s10468-017-9736-6
Source
Springer Nature
Keywords
License
Yellow

Abstract

We present the definition of a dedualizing complex of bicomodules over a pair of cocoherent coassociative coalgebras C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {C}$\end{document} and D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {D}$\end{document}. Given such a complex ℬ∙\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {B}^{\bullet }$\end{document}, we construct an equivalence between the (bounded or unbounded) conventional, as well as absolute, derived categories of the abelian categories of left comodules over C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {C}$\end{document} and left contramodules over D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {D}$\end{document}. Furthermore, we spell out the definition of a dedualizing complex of bisemimodules over a pair of semialgebras, and construct the related equivalence between the conventional or absolute derived categories of the abelian categories of semimodules and semicontramodules. Artinian, co-Noetherian, and cocoherent coalgebras are discussed as a preliminary material.

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