A CHARACTERISATION FOR A GROUPOID GALOIS EXTENSION USING PARTIAL ISOMORPHISMS

Let S vertical bar(R) be a groupoid Galois extension with Galois groupoid G such that E-g(Gr(g)) subset of C1(g), for all g is an element of G, where C is the centre of S, G(r(g)) is the principal group associated to r(g) and {E-g}(g is an element of G) are the ideals of S. We give a complete characterisation in terms of a partial isomorphism groupoid for such extensions, showing that G = (boolean OR) over dot(g is an element of G)Isom(R)(Eg-1, E-g) if and only if E-g is a connected commutative algebra or E-g = E-g(Gr(g)) circle plus E-g(Gr(g)) where E-g(Gr(g)) is connected, for all g is an element of G.