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Homological finiteness conditions for modules over strongly group-graded rings

Published online by Cambridge University Press:  24 October 2008

Jonathan Cornick
Affiliation:
Centre de Recerca Matematica, Institut d'estudis Catalans, Apartat 50, E 08193 Bellaterra, Spain
Peter H. Kropholler
Affiliation:
School of Mathematical Sciences, Queen Mary and Westfield College, Mile End Road, London E1 4NS

Extract

Throughout this paper, k denotes a commutative ring. We will develop a theory of homological finiteness conditions for modules over certain graded k-algebras which generalizes known theory for group algebras. The simplest of our results, Theorem A below, generalizes certain results of Aljadeff and Yi on crossed products of polycyclic-by-finite groups (cf. [1, 11]), but also applies to many other crossed products in cases where little was previously known. Before stating the results, we recall definitions of graded and strongly graded rings. Let G be a monoid. Naively, a G-graded k-algebra is a k-algebra R which admits a k-module decomposition,

in such a way that Rg Rh ⊆ for all g, hG. If R is a G-graded k-algebra and X is any subset of G, then we write Rx for the k-submodule of R supported on X; that is

Note that if H is a submonoid of G then RH is a subalgebra of R.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1996

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References

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