Fitting ideals of $p$-ramified Iwasawa modules over totally real fields

Abstract

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.

Fitting ideals in two-variable equivariant Iwasawa theory and an application

“…Lemma 2.9(2) does not play a practical role in this paper, but it clarifies the analogy with the one-variable setting in Remark 1.2. In the one-variable setting, the Fitting ideal of X Sp(k) (k ∞ ) is described in [4] by Fitt [1] of Z 0 at non p-adic primes. In our two-variable setting, Theorem 1.3 and Lemma 2.9 (2) show that the analogy holds for X {p} (L ∞ ).…”

“…In this section, we review facts on Galois cohomology complexes (this section does not have novelty). We follow the notations in [4], and refer to Nekovář [11] for detail.…”

“…In this subsection, we define and study a complex C Σ,S which will play an important role in the proof of Theorem 1.3. The idea behind the definition is the same as [4] in the one-variable case.…”

“…For each finite place v of K outside p, let T v ⊂ G be the inertia subgroup and σ v ∈ G/T v the Frobenius automorphism. Proposition 5.5 ([4,Proposition 3.13]). For every finite place v of K outside p, there exists a unique element f v ∈ Frac(R) × satisfying the following.…”

“…It is a more recent work [4] that describes the Fitting ideal of X Sp(k) (k ∞ ) in general, using Fitt [1] of a bit more complicated modules. Our second main result in this paper below is an analogue of this result.…”