The Galois Theory of Matrix C-rings

Abstract

ABSTRACT. A theory of monoids in the category of bicomodules of a coalgebra C or Crings is developed. This can be viewed as a dual version of the coring theory. The notion of a matrix ring context consisting of two bicomodules and two maps is introduced and the corresponding example of a C-ring (termed a matrix C-ring) is constructed. It is shown that a matrix ring context can be associated to any bicomodule which is a one-sided quasifinite injector. Based on this, the notion of a Galois module is introduced and the structure theorem, generalising Schneider's Theorem II [H.-J. Schneider, Israel J. Math., 72 (1990), 167-195], is proven. This is then applied to the C-ring associated to a weak entwining structure and a structure theorem for a weak A-Galois coextension is derived. The theory of matrix ring contexts for a firm coalgebra (or infinite matrix ring contexts) is outlined. A Galois connection associated to a matrix C-ring is constructed.

Galois theory in bicategories

“…Moreover, the functor − • B d : Hom(Ω, B) → Rcom(Ω, D) preserves equalizers, since it is the right adjoint to the forgetful functor U Ω,D . Therefore the equalizer (12) exists in Rcom(Ω, D) and is preserved by the forgetful functor. Applying this argument a second time we obtain the same result for the equalizer (13).…”

“…We know by assumption that the equalizer (12) exists in Hom(Ω, B). Moreover, the functor − • B d : Hom(Ω, B) → Rcom(Ω, D) preserves equalizers, since it is the right adjoint to the forgetful functor U Ω,D .…”

“…If we consider the comonad, then we can apply our general theory (or in fact, the special case of the coendomorphism corings), if we consider the monad, then we have to apply the dual version of the theory. This dual Galois theory is recently developed in [12], the monad M ⊗ D N is termed a matrix C-ring.…”

“…Our work intends to provide not only a transparent view on the interactions between the different approaches in the recent development on Galois comodules, but we also hope to shed new light on the relationship between the coring theory and the theory of comonads [30]. Moreover several other versions and generalizations of Hopf-Galois theory, which have been formulated during the last years (such as equivalences between categories of comodules [9], [52], Galois theory for C-rings [12] and Galois theory for group-corings [18]) fit perfectly within our general framework.…”