ABSTRACT
In [MV93] the basic facts on regulating subgroups and regulators are “dual ized” . Note that X < C, C completely decomposable, and e(C /X ) = 0 is equivalent to eC C X , eC completely decomposable, and e(X /eC ) = 0. Instead of studying completely decomposable subgroups of finite index in a given almost completely decomposable group X, they consider completely decomposable groups contain ing X as a subgroup of finite index. The theory is complicated by the fact that there is no natural universe containing the completely decomposable overgroups of X , although one might fix a divisible hull QX and consider only the completely decomposable groups C with X < C < Q X and C /X finite. Co-Butler decom positions play the role of Butler decompositions and embeddings X C C must be compared with the split exact sequence (see (7.2.2))
A typical result is the following.
