Comatrix Corings and Galois Comodules over firm rings

Galois corings with a group-like element [4] provide a neat framework to understand the analogies between several theories like the Faithfully Flat Descent for (noncommutative) ring extensions [26], Hopf-Galois algebra extensions [27], or noncommutative Galois algebra extensions [23, 15]. A Galois coring is isomorphic in a canonical way to the Sweedler’s canonical coring A ⊗B A associated to a ring extension B ⊆ A, where B is a subring of “coinvariants”. Canonical means here that the group-like of the Galois coring corresponds to the group-like 1⊗B1 of A⊗BA, where 1 denotes the unit element of the ring A. Canonical corings were used in [29] to formulate a predual to the Jacobson-Bourbaki theorem for division rings. Comatrix corings were introduced in [16] to make out the structure of cosemisimple corings over an arbitrary ground ring A [16, Theorem 4.4]. The construction has been used in different “Galois-type” contexts, where no group-like element is available, as the characterization of corings having a projective generator [16, Theorem 3.2]; the tightly related non-commutative descent for modules [16, Theorem 3.2,Theorem 3.10], [9, Theorem 3.7, Theorem 3.8, Theorem 3.9], [8, 18.27]; the formulation of a predual to the JacobsonBourbaki correspondence for simple artinian rings [13]; or the construction of a Brauer Group using corings [11]. The original definition of a comatrix coring was built on a bimodule Σ over unital rings B and A such that ΣA is finitely generated and projective. This finiteness condition seems to be, at a first glance, essential to define comatrix corings and to introduce the notion of a Galois coring (see [6]). Nevertheless, it was soon discovered [17] that the concepts and results from [16, §2, §3], including comatrix and Galois corings (or Galois comodules, as in [8, 5]) and the Descent Theorem, can be developed for a functor from a small category to a category of finitely generated and projective modules. These “infinite” comatrix corings have been recently constructed in a more general functorial setting in [14].