Galois corings and groupoids acting partially on algebras

Bagio and Paques [Partial groupoid actions: globalization, Morita theory and Galois theory, Comm. Algebra 40 (2012) 3658–3678] developed a Galois theory for unital partial actions by finite groupoids. The aim of this note is to show that this is actually a special case of the Galois theory for corings, as introduced by Brzeziński [The structure of corings, Induction functors, Maschke-type theorem, and Frobenius and Galois properties, Algebr. Represent. Theory 5 (2002) 389–410]. To this end, we associate a coring to a unital partial action of a finite groupoid on an algebra [Formula: see text], and show that this coring is Galois if and only if [Formula: see text] is an [Formula: see text]-partial Galois extension of its coinvariants.

On three special types of partial Galois extensions

In this paper, we study the following three special types of partial Galois extensions: DeMeyer–Kanzaki partial Galois extension, partial Galois Azumaya extension and commutator partial Galois extension.

Partial Galois cohomology and related homomorphisms

For a partial Galois extension of commutative rings, we give a seven terms sequence, which is an analog of the Chase–Harrison–Rosenberg sequence.

The structure of a partial Galois extension II

Let [Formula: see text] be a partial Galois extension where [Formula: see text] is a partial action of a finite group on a ring [Formula: see text] such that the associated ideals are generated by central idempotents. We determine the set of all Galois extensions in [Formula: see text], and give an orthogonality criterion for nonzero elements in the Boolean semigroup generated by those central idempotents. These results lead to a structure theorem for [Formula: see text].