Dedualizing Complexes of Bicomodules and MGM Duality Over Coalgebras

Abstract

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

The Tilting–Cotilting Correspondence

“…We conclude the section with a rather concrete example of a tilting equivalence in commutative algebra, which is closely related to more classical equivalences of Matlis and of Greenlees and May (see [67,66]) and which we later extend in Example 8.4 in the case of commutative Noetherian rings of Krull dimension one.…”

“…(1) A is a commutative ring, I ⊂ A is a weakly proregular finitely generated ideal and the powers of I form a base of neighborhoods of zero (see [67,Corollary 1.4]), or (2) A is a commutative ring, S ⊂ A is a multiplicative subset satisfying certain conditions and {As | s ∈ S} is a base of neighborhoods of zero (see [66, Theorem 6.6 (a)]). In both these situations, there are examples of tilting equivalences in our sense.…”

Dedualizing complexes and MGM duality

“…Still, from the point of view of many a reader the necessity to delve into an additional, largely unrelated background and recall the basics of coalgebras on the way to the desired definition from commutative algebra may feel like an unnecessary burden. Therefore, we decided to move our treatment of the MGM duality for coalgebras to a separate paper [28], while restricting ourselves here to a brief mention of this theory and its main result in this section of the introduction.…”

“…The definition of a dedualizing complex as a complex of bicomodules over two coalgebras satisfying a list of conditions is presented in [28]. Given a dedualizing complex B • for a pair of cocoherent coalgebras C and D, an equivalence between the bounded or unbounded, conventional or absolute derived categories of left comodules over C and left contramodules over D is obtained,…”