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