On unit groups of algebraic number fields (Q1118646)

In a paper to be published in Nagoya Math. J. the author proved that if K/k is a Galois extension of an algebraic number field, U is the group of unit ideles of K, \(E_ K, E_ k\) are the groups of units of K resp. k, N is the norm map of K-ideles to k-ideles and Un is the group of ideles of K of unit norm then the groups \((E_ k\cap N(K^*))/N(E_ K)\) and \((Un\cap UK^*)/(Un\cap U)(Un\cap K^*)\) are isomorphic. Now he shows that this result can be obtained as a special case of a cohomological lemma, which he proves by diagram chasing.