We interpret the collection of invertible bimodules as a groupoid and call it the Picard groupoid. We use this groupoid to generalize the classical construction of crossed products to what we call groupoid crossed products, and show that these coincide with the class of strongly groupoid graded rings. We then use groupoid crossed products to obtain a generalization from the group graded situation to the groupoid graded case of the bijection from a second cohomology group, defined by the grading and the functor from the groupoid in question to the Picard groupoid, to the collection of equivalence classes of rings strongly graded by the groupoid.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm106-1-1,
author = {Patrik Lundstr\"om},
title = {Strongly groupoid graded rings and cohomology},
journal = {Colloquium Mathematicae},
volume = {106},
year = {2006},
pages = {1-13},
zbl = {1126.16031},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm106-1-1}
}
Patrik Lundström. Strongly groupoid graded rings and cohomology. Colloquium Mathematicae, Tome 106 (2006) pp. 1-13. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm106-1-1/