Group-graded rings and duality

Abstract

We give an alternative construction of the duality between finite group actions and group gradings on rings which was shown by Cohen and Montgomery in [1]. This duality is then used to extend known results on skew group rings to corresponding results for large classes of group-graded rings. Finally we modify the construction slightly to handle infinite groups.

A generalization of Gabrielʼs Galois covering functors and derived equivalences

“…The latter gives a generalization of a categorical version of [6, Theorem 2.6] of Beattie. We note that the definition of smash products given in [9] is easy to handle and very useful, and that we can regard it as a categorical version of the definition of smash products by Quinn [22] (when the group is finite), and it enables us to formulate the covering construction by Green [15], and recovers the usual smash product of a k-algebra and the k-dual of a group algebra. In this paper we formulated the categorical version of Cohen-Montgomery duality as much as possible in the scope of categories.…”

Chain conditions and semigroup graded rings

“…Again by Proposition 1.1, Q e = Qe is a left Noetherian ring. Hence, by Quinn's results[16], Q is left Noetherian; and thus left Goldie. EXAMPLE.…”

“…These turn out to be very difiBcult questions as they are even unsolved for group rings. However some results on Noetherianess are known if the ring is graded by a (polycyclic-by-) finite group (see [4], [16]). Therefore we are able to solve both questions (Section 1) for rings weakly or strongly graded by a cancellative monoid which is contained in a polycyclic-by-finite (unique product) group.…”

Coinduction functor and simple comodules

“…We thus identify A with its image A. Proof (i) This was proved in [15] and [12], using the fact that End( A C) is isomorphic to the ring of finite rows |G| × |G|-matrices over A. Here, we can restrict ourselves to the colinear endomorphism ring End( C C).…”

“…This pairing is constructed using the smash product [6,15] and the canonical coring arising from a group-graded base ring. Using this functor, we will apply the results of Sect.…”