Dedualizing complexes of bicomodules and MGM duality over coalgebras

Abstract

This paper is based on the material of Section 4 and Appendix C in arXiv:1503.05523v6, which was excluded from the subsequent versions of arXiv:1503.05523. 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.

LaTeX 2e with pb-diagram and xy-pic; 34 pages, 2 commutative diagrams; v.2: Section 3 added, Example 2.4 expanded; v.3: paragraph inserted in Section 0.2, several references added, several misprints corrected; v.4: more misprints corrected, a reference updated, the numbering of sections shifted to agree with the journal version; v.5: several misprints corrected in Lemmas 2.5, 2.10, and 2.11

Cited by in corpus (4)

• Derived Categories
• Derived, coderived, and contraderived categories of locally presentable abelian categories
• Pseudo-dualizing complexes of bicomodules and pairs of t-structures
• Homological full-and-faithfulness of comodule inclusion and contramodule forgetful functors