Dedualizing complexes of bicomodules and MGM duality over coalgebras

TL;DR

This paper introduces dedualizing complexes of bicomodules over coalgebras and constructs equivalences between derived categories of comodules and contramodules, extending to semimodules and semicontramodules.

Contribution

It defines dedualizing complexes over coalgebras and semialgebras and establishes derived category equivalences, advancing duality theory in coalgebra contexts.

Findings

Defined dedualizing complexes of bicomodules and bisemimodules.

Constructed equivalences between derived categories of comodules and contramodules.

Extended duality concepts to semimodules and semicontramodules.

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.