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Homological dimension of skew group rings and crossed products. (English) Zbl 0803.16027
Let \(R\) be a commutative Noetherian ring, let \(G\) be a finite group acting on \(R\), and let \(S = R * G\) be the skew group ring. For each ideal \(I\) of \(R\), set \(G(I) = \{g \in G : r^ g - r \in I \text{ for all } r\in R\}\), and let gl.dim denote global homological dimension. A special case of the key results proved here states that \(\text{gl.dim}(S) < \infty\) if and only if \(\text{gl.dim}(R) < \infty\) and, for each maximal ideal \(M\) of \(R\) with \(\text{char}(R/M) = p > 0\), \(G(M)\) contains no element of order \(p\). The more general results apply to fully bounded Noetherian coefficient rings \(R\), although the exact analogue of the result quoted above is no longer valid. For, as an interesting example cited in the paper shows, complications may arise when the characteristic of a simple Artinian factor of \(R\) divides the uniform rank of the factor. Extensions of the above to skew group rings and crossed products of polycyclic-by-finite groups are also given; these make use of results of E. Aljadeff and S. Rosset [J. Pure Appl. Algebra 40, 103- 113 (1986; Zbl 0584.16014)] and E. Aljadeff [J. Lond. Math. Soc., II. Ser. 44, 47-54 (1991; Zbl 0688.16010)].

16S35 Twisted and skew group rings, crossed products
16E10 Homological dimension in associative algebras
16P40 Noetherian rings and modules (associative rings and algebras)
16D25 Ideals in associative algebras
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