Notes on the dual of the ideal class groups of CM-fields

In this paper, for a CM abelian extension K/k of number fields, we propose conjecture which describes completely the Fitting ideal minus part Pontryagin dual T-ray class group K set T primes as Gal(K/k)-module. Here, emphasize that consider full group, and do not throw away ramifying (the object study is quotient by subgroup generated classes primes). We prove our consequence equivariant Tamagawa conjecture, also Iwasawa theoretic version conjecture.