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Dedualizing Complexes of Bicomodules and MGM Duality Over Coalgebras

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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.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Natural Sciences, University of Haifa, Mount Carmel, Haifa, 31905, Israel

    Leonid Positselski

  2. Laboratory of Algebraic Geometry, National Research University Higher School of Economics, Moscow, 119048, Russia

    Leonid Positselski

  3. Sector of Algebra and Number Theory, Institute for Information Transmission Problems, Moscow, 127051, Russia

    Leonid Positselski

  4. Faculty of Mathematics and Physics, Department of Algebra, Charles University, Sokolovská 83, 186 75 Prague 8, Prague, Czech Republic

    Leonid Positselski

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  1. Leonid Positselski
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Correspondence to Leonid Positselski.

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Presented by Michel Van den Bergh.

While preparing this paper, the author’s research was supported by a fellowship from the Lady Davis Foundation at the Technion, the Israel Science Foundation grant # 446/15 at the University of Haifa, and the Grant Agency of the Czech Republic under the grant P201/12/G028 at Charles University in Prague.

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Positselski, L. Dedualizing Complexes of Bicomodules and MGM Duality Over Coalgebras. Algebr Represent Theor 21, 737–767 (2018). https://doi.org/10.1007/s10468-017-9736-6

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  • DOI: https://doi.org/10.1007/s10468-017-9736-6

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