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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apply to fully bounded Noetherian coefficient rings $$R$$, although the exact analogue of the result quoted above is no longer valid.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)].

##### MSC:
 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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