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Partial actions and Galois theory. (English) Zbl 1142.13005
A partial action \(\alpha\) of a finite group \(G\) on a unital algebra \(S\), defined by M. Dokuchaev and R. Exel [Trans. Am. Math. Soc. 357, No. 5, 1931–1952 (2005; Zbl 1072.16025)], is a collection of ideals \(S_{\sigma}\) for \(\sigma\) in \(G\), together with isomorphisms \(\alpha_{\sigma}: S_{\sigma^{-1}} \longrightarrow S_{\sigma}\) that satisfy conditions of compatibility with the group. If the ideals \(S_{\sigma}\) are generated by central idempotents \(1_{\sigma}\), then the partial action possesses an enveloping action, which implies that there is a (global) action of \(G\) on the algebra \(S' = \sum_{\sigma \in G} \sigma(S)\) so that \(\alpha_{\sigma} = \sigma\) on \(S_{\sigma^{-1}}\). In this setting the authors define \(S\) to be a partial Galois extension of \[ R = S^{\alpha} = \{ x \in S | \alpha_{\sigma}(x1_{\sigma^{-1}}) = 1_{\sigma}x \text{ for all } \sigma \text{ in } G\} \] by generalizing the Galois coordinates definition from S. U. Chase, D. K. Harrison and A. Rosenberg [Galois theory and Galois cohomology of commutative rings, Mem. Am. Math. Soc. 52, 15–33 (1965; Zbl 0143.05902), Theorem 1.3, (b)]. They show that \(S\) is a partial Galois extension of \(R\) iff \(S'\) is a Galois extension of \(R' = {S'}^G\) and extend the fundamental theorem of Galois theory of [loc.cit.], Theorem 2.3 to partial Galois extensions.

MSC:
13B05 Galois theory and commutative ring extensions
13A50 Actions of groups on commutative rings; invariant theory
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