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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
Keywords:
Galois theory; partial action
Full Text:
References:
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