Let K/k be a finite abelian CM-extension and T a suitable finite set of finite primes of k. In this paper, we determine the Fitting ideal of the minus component of the T -ray class group of K, except for the 2-component, assuming the validity of the equivariant Tamagawa number conjecture. As an application, we give a necessary and sufficient condition for the Stickelberger element to lie in that Fitting ideal.
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 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.
ContentsS 1.3. Shifted Fitting ideals 1.4. Decomposition of group rings 2. Proof of main result (I) 2.1. Some facts on arithmetic complexes 2.2. The algebraic part of the proof 2.3. Principality of Fitt R (X Sp ) 3. Proof of main result (II) 3.1. The determinant homomorphism 3.2. Description of C S by p-adic L-functions 3.3. Proof of Theorem 0.1 3.4. Integrality of θ mod S 4. A strategy for computing Fitt [1] R (Z 0 S ′ ) 4.1. The algebraic problem 4.2. How to attack Problem 4.1: an idea 4.3. The most general situation 4.4. A first special setting 4.5. Another special setting 5. Description of the Fitting ideal in the case that Gal(K/k) is cyclic References [21] J. Tate. The cohomology group of tori in finite Galois extensions of number fields.
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