Appendix : about the Λ-freeness of C ∞.
Jean-Robert Belliard
Université de Franche-Comté, Laboratoire de mathématiques UMR 6623,
16 route de Gray, 25030 Besançon cedex, FRANCE.
belliard@math.univ-fcomte.fr

Let F be an abelian totally real number field (without any further restriction). In the paper[Bel98] we gave a sufficient condition to Λ-freeness of the Λ-module C ∞ associated to the cyclotomic
Zp -extension F∞ /F . There amongst other hypotheses we assumed that p was unramified in thebase field denoted K there and F here. Now due to the applications given in the main part ofthis paper, it seems worthwhile to write down precisely the relevant condition without assuming
anything about ramification at p.
Lemma 0.1 Let L ⊂ F∞ be the maximal subfield of F∞ such that p is (at most) tamely ramified
in L (under hypothesis (R), L is the field F tam of ??). Put ΛL := Zp [[Gal(L∞ /L)]], where L∞ /L
is the cyclotomic Zp -extension of L. Then we have :1. As a group, C ∞ only depends on F∞ not on F .
2. L∞ = F∞ .
3. C ∞ is Λ-free if and only if C ∞ is ΛL -free.
Proof. 1 is clear. 2 is lemma 1.2 of [Bel05]. By theorem 2.2 of [Bel02], the Λ-freeness is equivalent
to an asymptotic condition (namely the ”asymptotic Galois Descent” property for the C n ’s). But
there exists some N ∈ N and some b ∈ Z such that for all n ≥ N we have Fn = Ln+b : theequivalence in 3 follows.
¤In the light of the lemma 0.1, as far as the Λ-freeness of C ∞ is concerned, we can (but shallnot) in the sequel assume without loss of generality that p is (at most) tamely ramified in F . We
could then just refer to [Bel98] for all proofs. But we thought it would be better to give themhere including most of the details, taking this opportunity to simplify the arguments of [Bel98] bytranslating them at infinite level.
We need a few more notations. They are close to those used in [Bel98] but formally not exactly
same. Let P be the set of rational primes l 6= p which are ramified in F/Q. The case P = ∅ isobvious but allowed. For all supernatural
Q numbers t we put Q(t) = Q(ζt ). For all J ⊂ P andall n ∈ N ∪ {∞}, we put Fn (J) := Q( l∈J l∞ p∞ ) ∩ Fn . We shall abbreviate by F (J) := F0 (J),G(J) = Gal(F∞ (J)/B∞ ), where B∞ is the cyclotomic Zp -extension of Q, and G = G(P). For any
abelian number field K we write cond(K) for its conductor, that is the minimal n ∈ N such thatK ⊂ Q(n). For all n ∈ N and all J ⊂ P we will denote by εn (J) the following cyclotomic number(unit if J 6= ∅) :εn (J) := NQ(cond(Fn (J)))/Fn (J) (1 − ζcond(Fn (J)) ).
It follows from the classical distribution relations that for n ≥ 1 the εn (J)’s form a norm coherent
sequence. Therefore, fixing a γ generating Gal(B∞ /Q) we may define ε∞ (J) ∈ C ∞ by the following
formula :½(εn (J))n≥1 if J 6= ∅ε∞ (J) :=(εn (∅)(γ−1) )n∈N else.

2We shall examine the ΛL [G]-module structure of C ∞ (recall that G = G(P) = Gal(F∞ /B∞ )),and for that it is more convenient to write additively the multiplication in C ∞ and to keep multiplication for the action of Λ[G]. Consistently for B∞ ⊂ N ⊂ K ⊂ F∞ we will denote by TK/N theGal(F∞ /K)

trace (actually the algebraic norm map) acting on elements x ∈ C ∞. The elements ε∞ (J)are subject to the following distribution relations :µ¶A
be the (global) Artin symbol for any abelian extension N/K and (fracLemma 0.2 LetN/Ktional) ideal A of K. For all J ⊂ P and all I ⊂ J, put PI,J for the set of rational primes l dividingcond(F∞ (J)) but not dividing cond(F∞ (I)). We have :
Ãµ
¶−1 !
Y(l)
 ε∞ (I).
1−TF∞ (J)/F∞ (I) (ε∞ (J)) = 
F∞ (I)/Ql∈PI,JProof. The analogous relations at finite levels are well known. We then just take inverse limits.
¤Lemma 0.3 The system {ε∞ (J), J ⊂ P} generates C ∞ over ΛL [Gal(F∞ /B∞ )].
Proof. This follows from distribution relations (e.g. lemma 0.2) and lemma 2.3 of [Gre92].
¤We now state the ad hoc hypothesis that will ensure the freeness of C ∞ . This ”generalized
hypothesis (B)” is an immediate generalization of (HB) in [Bel98].
Definition 0.4 For all J ⊂ P we call norm ideal of J and denote by N (J) the ideal of Zp [G(J)]generated by traces­®
N (J) := TF∞ (J)/F∞ (J−{l}) ; l ∈ J .
We say that the pair (F, p) satisfies ”generalized Hypothesis B” ((gHB) for short) if and only if for
all J ⊂ P the quotient Zp [G(J)]/N (J) is torsion free.
This hypothesis is quite technical, but very natural regarding the proof of theorem 0.6. The
following proposition shows that (gHB) holds true in many usual cases. Indeed all known cases of
freedom of C ∞ are consequence of (gHB) via theorem 0.6.
Proposition 0.5 For all l ∈ P let us denote by Il ⊂ G the inertia subgroup for l in F∞ /B∞ .
Recall that for any finite abelian group H, we write H for its p-Sylow subgroup (considered also
as the maximal p-quotient of H). The pair (F, p) satisfies (gHB) as soon as one of the following
properties holds :L
1. The I l , l ∈ P are mutually direct summands. In other words the natural map l∈P I l −→ Gis injective.
2. dimFp (G/pG) ≤ 1, or equivalently G is cyclic.
3. #P ≤ 2
Proof. These sufficient conditions are the exact analogues of those in section IV.1 of [Bel98]. Notethat they concern only the maximal p-quotient of the Galois groups involved. Since the (unique)prime above p of B∞ is tamely ramified in F∞ /B∞ , everything goes as if p were unramified in thebase field F . Taking this into account, we can repeat verbatim the proofs in [Bel98]. ¤Theorem 0.6 Let us denote by r(I) the Zp -rank of Zp [G(I)]/N (I). Assume that (F, p) satisfies(gHB). Fix any total order ≤ on the set of all subsets of P extending the inclusion. Then :1. ∀J ⊂ P we have isomorphisms of ΛL [G]-modules :Zp [G(J)]hε∞ (J)ihε∞ (J)i∼TT⊗Zp ΛL ∼;==N (J)hε∞ (I); I ⊂ J, I 6= Ji hε∞ (J)ihε∞ (I); I ≤ J, I 6= Ji hε∞ (J)ihence all three ΛL [G]-modules are free over ΛL (of ranks r(J)).

REFERENCES

3Lr(I)
2. C ∞ ∼= I⊂P ΛL as ΛL -modules; in particular C ∞ is ΛL -free of rank [L : Q] and Λ-free ofrank [F : Q].
Proof. C ∞ has the same Λ-rank (resp. ΛL -rank) as U ∞ , namely [F : Q] (resp. [L : Q] = [F∞ :Z [G(J)]B∞ ]) : see e.g. [Kuz72], [Bel02] ... By (gHB) the modules pN (J) ⊗Zp ΛL are ΛL -free. We firstprove that 1 implies 2. From the tautological (split over ΛL under (gHB) by 1) exact sequences :ES(J)hε∞ (I); I ≤ J, I 6= Ji / hε∞ (I); I ≤ Ji//hε∞ (J)i \hε∞ (I); I ≤ J, I 6= Ji hε∞ (J)iwe see that 1 implies 2 by induction. Let us prove 1.
By lemma 0.2 and by the inclusion hε∞ (I); I ⊂ J, I 6= Ji ⊂ hε∞ (I); I ≤ J, I 6= Ji we havesurjective morphisms :(Sur)Zp [G(J)]hε∞ (J)ihε∞ (J)iTT⊗Zp ΛL ³³.
N (J)hε∞ (I); I ⊂ J, I 6= Ji hε∞ (J)ihε∞ (I); I ≤ J, I 6= Ji hε∞ (J)i

Therefore the required isomorphisms will follow from equalities of ΛL -ranks. Now :Lemma 0.7 Recall that for all I, r(I) is the Zp -rank of Zp [G(I)]/N (I). Put d(I) = [F∞ (I) : B∞ ].
1. For all J ⊂ P we haver(J) =X(−1)#J−#I d(I)I⊂J

2.
[L : Q] = [F∞ : B∞ ] = d(P) =Xr(J)J⊂P

Proof. 1 is proven using character theory and following the same way as proposition 2.11 of [Bel98] :just replace the function fn (I) there by d(I) here. 2 follows from 1 and the combinatorial lemma
2.14 of [Bel98]. Details are left to the reader.
¤We resume the proof of theorem 0.6. For any ΛL -module M let us denote its ΛL -rank byrankΛL (M ). Putµ¶hε∞ (J)i
Tt(J) := rankΛL.
hε∞ (I); I ≤ J, I 6= Ji hε∞ (J)i
From the surjections (Sur) we see thatP r(J) ≥ t(J). On the other hand,P summing ranks in all thesequences (ES(J)), we recover that J t(J) = rankΛL (C ∞ ) = d(P) = J r(J) by 2 of lemma 0.7.
Therefore all inequalities r(J) ≥ t(J) are actually equalities. This proves theorem 0.6.
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