  We completely calculate the Fitting ideal of the classical $p$-ramified
Iwasawa module for any abelian extension $K/k$ of totally real fields, using
the shifted Fitting ideals recently developed by the second author. This
generalizes former results by the first and third authors where we had to
assume that only $p$-adic places may ramify in $K/k$. One of the important
ingredients is the computation of some complexes in appropriate derived
categories.

Number Theory