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A CHARACTERISATION FOR A GROUPOID GALOIS EXTENSION USING PARTIAL ISOMORPHISMS

Part of: Special categories Ring extensions and related topics

Published online by Cambridge University Press:  06 March 2017

WAGNER CORTES  and
THAÍSA TAMUSIUNAS
WAGNER CORTES
Affiliation:
Instituto de Matemática, Universidade Federal do Rio Grande do Sul, Porto Alegre-RS, Av. Bento Gonçalves, 9500, 91509-900, Brazil email wocortes@gmail.com
THAÍSA TAMUSIUNAS*
Affiliation:
Departamento de Ciências Exatas e Sociais Aplicadas, Universidade Federal de Ciências da Saúde de Porto Alegre, Porto Alegre-RS, Rua Sarmento Leite, 245, 90050-170, Brazil email trtamusiunas@yahoo.com.br
*
✉trtamusiunas@yahoo.com.br
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Abstract

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Let $S|_{R}$ be a groupoid Galois extension with Galois groupoid $G$ such that $E_{g}^{G_{r(g)}}\subseteq C1_{g}$, for all $g\in G$, where $C$ is the centre of $S$, $G_{r(g)}$ is the principal group associated to $r(g)$ and $\{E_{g}\}_{g\in G}$ are the ideals of $S$. We give a complete characterisation in terms of a partial isomorphism groupoid for such extensions, showing that $G=\dot{\bigcup }_{g\in G}\text{Isom}_{R}(E_{g^{-1}},E_{g})$ if and only if $E_{g}$ is a connected commutative algebra or $E_{g}=E_{g}^{G_{r(g)}}\oplus E_{g}^{G_{r(g)}}$, where $E_{g}^{G_{r(g)}}$ is connected, for all $g\in G$.

Keywords

groupoidgroupoid Galois extensionpartial isomorphism

MSC classification

Primary: 13B05: Galois theory
Secondary: 18B40: Groupoids, semigroupoids, semigroups, groups (viewed as categories)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 96 , Issue 1 , August 2017 , pp. 59 - 68
Copyright
© 2017 Australian Mathematical Publishing Association Inc. 

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