1 results
A CHARACTERISATION FOR A GROUPOID GALOIS EXTENSION USING PARTIAL ISOMORPHISMS
- Part of
-
- Journal:
- Bulletin of the Australian Mathematical Society / Volume 96 / Issue 1 / August 2017
- Published online by Cambridge University Press:
- 06 March 2017, pp. 59-68
- Print publication:
- August 2017
-
- Article
-
- You have access
- Export citation
-
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 
$g\in G$.
