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In this paper, for a CM abelian extension $K/k$ of number fields, we proposea conjecture which describes completely the Fitting ideal of the minus part ofthe Pontryagin dual of the $T$-ray class group of $K$ for a set $T$ of primesas a ${\rm Gal}(K/k)$-module. Here, we emphasize that we consider the fullclass group, and do not throw away the ramifying primes (namely, the object westudy is not the quotient of the class group by the subgroup generated by theclasses of ramifying primes). We prove that our conjecture is a consequence ofthe equivariant Tamagawa number conjecture, and also prove that the Iwasawatheoretic version of our conjecture holds true under the assumption $\mu=0$without assuming eTNC.

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