Global dimensions of crossed products
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.
scientific article; zbMATH DE number 3935253