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Algebras and Representation Theory

, Volume 21, Issue 4, pp 737–767 | Cite as

Dedualizing Complexes of Bicomodules and MGM Duality Over Coalgebras

  • Leonid Positselski
Article
  • 37 Downloads

Abstract

We present the definition of a dedualizing complex of bicomodules over a pair of cocoherent coassociative coalgebras \(\mathcal {C}\) and \(\mathcal {D}\). Given such a complex \(\mathcal {B}^{\bullet }\), we construct an equivalence between the (bounded or unbounded) conventional, as well as absolute, derived categories of the abelian categories of left comodules over \(\mathcal {C}\) and left contramodules over \(\mathcal {D}\). 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.

Keywords

Artinian coalgebras Co-Noetherian coalgebras Cocoherent coalgebras Comodules Contramodules Semialgebras Semimodules Semicontramodules Derived categories Derived comodule-contramodule correspondence MGM duality/equivalence Dedualizing complexes 

Mathematics Subject Classification (2010)

16T15 16D90 16E35 16P70 

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Copyright information

© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Natural SciencesUniversity of HaifaHaifaIsrael
  2. 2.Laboratory of Algebraic GeometryNational Research University Higher School of EconomicsMoscowRussia
  3. 3.Sector of Algebra and Number TheoryInstitute for Information Transmission ProblemsMoscowRussia
  4. 4.Faculty of Mathematics and Physics, Department of AlgebraCharles UniversityPragueCzech Republic

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