Global dimensions of crossed products
type: Item
instance of: scholarly article
title: Global dimensions of crossed products
zbMATH Open document ID: 0584.16014
DOI: 10.1016/0022-4049(86)90033-2
author: Eli Aljadeff
author: Shmuel Rosset
published in: Journal of Pure and Applied Algebra
publication date: 1986-01-01T00:00:00Z
review text: Let \(K_ t^{\alpha}\Gamma\) be the crossed product of the group \(\Gamma\) over the field K associated with the action \(t: \Gamma\) \(\to Aut K\) of \(\Gamma\) on K and the cohomology class \(\alpha \in H^ 2(\Gamma,K^*)\). For each normal subgroup \(\Gamma_ 0\) of \(\Gamma\), the present article gives an upper bound for the global dimension of \(K_ t^{\alpha}\Gamma\) in terms of the global dimensions of related crossed products for \(\Gamma_ 0\) and \(G=\Gamma /\Gamma_ 0\). In particular, one always has \[ gl.\dim K_ t^{\alpha}\Gamma \leq gl.\dim K_ t\Gamma \leq gl.\dim K\Gamma (=cd_ K\Gamma), \] where \(K\Gamma\) is the ordinary group algebra and \(K_ t\Gamma\) is the skew group ring associated with t. Moreover, if \(cd_ K\Gamma <\infty\) then gl.dim \(K_ t\Gamma =cd_ K\Gamma\), but otherwise the above inequalities are strict, in general. The principal technical tool in the proof is a spectral sequence which generalizes the usual Hochschild-Serre spectral sequence for the cohomology of the group extension \(1\to \Gamma_ 0\to \Gamma \to G\to 1\) to the setting of crossed products.
Mathematics Subject Classification ID: 16E10
Mathematics Subject Classification ID: 16Exx
Mathematics Subject Classification ID: 16S34
zbMATH DE Number: 3935253
zbMATH Keywords: cohomological dimension
zbMATH Keywords: crossed product
zbMATH Keywords: action
zbMATH Keywords: cohomology class
zbMATH Keywords: global dimension
zbMATH Keywords: group algebra
zbMATH Keywords: skew group ring
zbMATH Keywords: Hochschild-Serre spectral sequence
reviewed by: Martin Lorenz
MaRDI profile type: Publication
full work available at URL: 0022-4049(86)90033-2
OpenAlex ID: W2077592907
cites work: Q4143495
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description: scientific article; zbMATH DE number 3935253