Minimal resolutions of Iwasawa modules

In this paper, we study the module-theoretic structure of classical Iwasawa modules. More precisely, for a finite abelian $p$-extension $K/k$ of totally real fields and the cyclotomic $\mathbb{Z}_p$-extension $K_{\infty}/K$, we consider $X_{K_{\infty},S}={\rm Gal}(M_{K_{\infty},S}/K_{\infty})$ where $S$ is a finite set of places of $k$ containing all ramifying places in $K_{\infty}$ and archimedean places, and $M_{K_{\infty},S}$ is the maximal abelian pro-$p$-extension of $K_{\infty}$ unramified outside $S$. We give lower and upper bounds of the minimal numbers of generators and of relations of $X_{K_{\infty},S}$ as a $\mathbb{Z}_p[[{\rm Gal}(K_{\infty}/k)]]$-module, using the $p$-rank of ${\rm Gal}(K/k)$. This result explains the complexity of $X_{K_{\infty},S}$ as a $\mathbb{Z}_p[[{\rm Gal}(K_{\infty}/k)]]$-module when the $p$-rank of ${\rm Gal}(K/k)$ is large. Moreover, we prove an analogous theorem in the setting that $K/k$ is non-abelian. We also study the Iwasawa adjoint of $X_{K_{\infty},S}$, and the minus part of the unramified Iwasawa module for a CM-extension. In order to prove these theorems, we systematically study the minimal resolutions of $X_{K_{\infty},S}$.


Introduction
Throughout this paper we fix a prime number p.We write F ∞ for the cyclotomic Z pextension of F for any number field F .
Let K/k be a finite abelian p-extension of totally real fields (see Theorem 3.3 for the nonabelian case).We consider the abelian extension K ∞ /k, whose Galois group we denote by G = Gal(K ∞ /k).Suppose that S is a finite set of places of k, containing all archimedean places and all places that ramify in K ∞ .In particular, S contains all p-adic places.Let M K∞,S denote the maximal abelian pro-p-extension of K ∞ unramified outside S. Our main purpose in this paper is to study the classical Iwasawa module X K∞,S = Gal(M K∞,S /K ∞ ) over the Iwasawa algebra R = Z p [[G]].
Define I G to be the augmentation ideal of R = Z p [[G]], namely I G = Ker(Z p [[G]] → Z p ).We write Q(R) for the total quotient ring of R. We consider an R-submodule R ∼ of Q(R), which consists of elements x ∈ Q(R), satisfying xI G ⊂ R.This is the module of pseudo-measures of G in the sense of Serre.The p-adic L-function of Deligne and Ribet is an element g K∞/k,S in R ∼ , satisfying the following property.Suppose that κ : G → Z × p is the cyclotomic character.For a character ψ of G of finite order with values in an algebraic closure Q p of Q p and for a positive integer n, one can extend a character κ n ψ : G → Q × p to a ring homomorphism R → Q p , and also to R ∼ → Q p .Then g K∞/k,S satisfies κ n ψ(g F∞/k,S ) = L S (1 − n, ψω −n ) for any character ψ of G of finite order and for any positive integer n ∈ Z >0 , where L S (s, ψω −n ) is the S-truncated L-function, and ω is the Teichmüller character.
In [5,Theorem 3.3] and [6,Theorem 4.1], as a refinement of the usual main conjecture, Greither and the second author computed the Fitting ideal of X K∞,S as an R-module to obtain Fitt R (X K∞,S ) = a G I G g K∞/k,S , where a G is a certain ideal of R which is determined only by the group structure of G.The explicit description of a G is obtained in [7, §1.2] by Greither, Tokio and the second author.We do not explain this ideal a G in this paper, but only mention two facts.If s is the p-rank of G (i.e., s = dim Fp (F p ⊗ Z G)) and m R is the maximal ideal of R, then we have a G ⊂ m s(s−1)/2 R .Also, if G is isomorphic to (Z/p m ) ⊕s , then a G = (p m R + I G ) s(s−1)/2 .We also note here that the classical main conjecture in Iwasawa theory studies the character component X ψ K∞,S for K which corresponds to the kernel of ψ.In this case, G is cyclic, and only the case s = 1 is studied.
The above computation of Fitt R (X K∞,S ) suggests that X K∞,S is complicated as an Rmodule when the p-rank s of G is large.To understand such complicatedness, we study in this paper the minimal numbers of generators and relations of X K∞,S .Let gen R (X K∞,S ) (resp.r R (X K∞,S )) be the minimal number of generators (resp. of relations) of X K∞,S as an R-module.
In order to state the main result of this paper, we need the maximal abelian pro-p-extension M k,S of k unramified outside S. By our choice of S, we have Now we state the main result of this paper.For any abelian group A, we define its p-rank by rank p A = dim Fp (F p ⊗ Z A), which is finite in all cases we consider in this paper.
Theorem 1.1.Let us write s = rank p Gal(K ∞ /k ∞ ) = rank p G, t = rank p Gal(M k,S /K ∞ ) for the p-ranks of the Galois groups.Then we have Remark 1.2.We note that Leopoldt's conjecture for k is equivalent to that the extension M k,S /k ∞ is finite though we do not assume it in Theorem 1.1.We get the equalities of the minimal numbers of generators in the following special cases.
(1) If Leopoldt's conjecture holds and p ≥ 3, then we may take K so that K ∞ = M k,S .
In this case, we have t = 0, so the theorem says where s = rank p Gal(M k,S /k ∞ ).
(2) In case K ∞ = k ∞ , we have s = 0, so the theorem says Indeed, this follows directly from Lemma 6.1.Except for these cases, we have no theoretical method to determine the exact value of gen R (X K∞,S ) so far.
Remark 1.3.In §7 we give several numerical examples for p = 2, 3. We take k = Q and K/Q which is a real abelian extension such that Gal(K/Q) ≃ (Z/pZ) ⊕s .Here, we pick up some typical examples from §7.
In this paper, we prove not only the above Theorem 1.1 but also its non-abelian generalization in Theorem 3.3.We also study and determine the minimal numbers of generators and relations of the dual (Iwasawa adjoint) of X K∞,S (see Theorem 3.4).This is relatively easier than Theorem 3.3.Also, we give in §3.3 some applications to the minus part of certain Iwasawa modules of CM-fields (see Corollary 3.5), using Kummer duality.
A key to the proof of our theorems is the existence of certain exact sequences, called Tate sequences.We remark here that Greither also used a different kind of Tate sequence in [3] to get information on the minimal numbers of generators of class groups of number fields.Our method of using the Tate sequences is totally different from Greither's.
This paper is organized as follows.After algebraic preliminaries in §2, we will state the main results in §3.The proof is given in § §4-6.Finally in §7, we will observe numerical examples.

Acknowledgments
The authors would like to thank Yuta Nakamura, who computed gen R (X K∞,S ) for several examples in his master's thesis in a slightly different situation from ours.They also thank Cornelius Greither heartily for his interest in the subject of this paper and for giving them some valuable comments.The first and the second authors are supported by JSPS KAKENHI Grant Numbers 22K13898 and 22H01119, respectively.

Algebraic preliminaries
2.1.Minimal resolutions.Let R be a Noetherian local ring, which we do not assume to be commutative.Let m be the Jacobson radical of R, that is, m is the maximal left (right) ideal of R. For simplicity, let us assume that k := R/m is a commutative field.We will often consider the case R = Z p [[G]] for a pro-p group G, in which case R is indeed local and we have k = F p (see [13,Proposition 5.2.16 (iii)]).Definition 2.1.For a finitely generated (left) R-module M, we write gen R (M) for the minimal number of generators of M as an R-module.Also, we write r R (M) for the minimal number of relations of M as an R-module (see Definition 2.4 below).
Remark 2.2.The following observations will be often used.
(1) By Nakayama's lemma (e.g., [8,Corollary 13.12]), we have for any two-sided ideal I ⊂ m of R. In particular, we have Therefore, for a finitely generated Z p -module M, we have (2) If we have an exact sequence The proof is standard.(3) In item (2) above, if we assume that R is a discrete valuation ring (DVR), the formula is refined as This follows from the structure theorem for finitely generated modules over principal ideal domains.
Example 2.3.Let us observe an example for which the formula in item (3) above does not hold when R is not a DVR ] and its submodule with n ≥ 1.Then we have gen R (M ′ ) = n + 1 and gen R (M) = 1, so gen R (M ′ ) ≤ gen R (M) does not hold.
Next we introduce the minimal resolutions of modules.
Definition 2.4.Let M be a finitely generated R-module.We can construct an exact sequence of R-modules . Such a sequence is called a minimal resolution of M. In this case, since R r n+1 → R rn induces the zero map on (R/m) r n+1 → (R/m) rn , by the definition of the Tor functor, the integer r n coincides with In particular, the integer r n is independent of the choice of minimal resolutions.By definition we have n (F p , Z p ) and the formula in Definition 2.4.

Group homology.
In this subsection, we summarize facts about group homology.
Let G be a finite group.The following lemma is well-known.
Lemma 2.6.We have H 1 (G, Z) ≃ G ab , the abelianization of G, and As for the second homology groups, if G is abelian, it is known that H 2 (G, Z) is isomorphic to 2 G (see [1, Chap.V, Theorem 6.4 (iii)]).If G is not abelian, H 2 (G, Z) is much harder to study, which is also known as the Schur multiplier of G (cf. [11]).
For now, we observe a relation between H n (G, Z) and H n (G, Z/MZ) for a p-power M.
Lemma 2.7.Let n ≥ 2. For any m ≥ 1, we have In particular, as the right hand side is independent from m, we have Proof.This follows from the universal coefficient theorem (see [1, Chap.I, Proposition 0.8], for example), which says in our case that In case G is abelian, it is not hard to compute the p-rank of the n-th homology group: Lemma 2.8.Suppose G is abelian and put s = rank p G. Then we have n! for n ≥ 0 (when n = 0, the right hand side is understood to be 1).
Proof.By replacing G by its p-Sylow subgroup, we may assume that G is a p-group.As in [5, §1.2] or [12, §4.3], we can construct an explicit minimal free resolution of Z p as an R-module Thus, the lemma follows from Lemma 2.5.
We also need the following duality theorem between the cohomology groups and the homology groups (see [1, Chap VI Proposition 7.1], for example).
Lemma 2.9.Let G be a finite group and M a (discrete) G-module.We define its Pontryagin dual M ∨ by M ∨ = Hom(M, Q/Z).Then for any n ∈ Z ≥0 , we have an isomorphism between H n (G, M) and Hom(H n (G, M ∨ ), Q/Z).

3.1.
Setting.As in §1, let p be any prime number, k a totally real field, and k ∞ its cyclotomic Z p -extension.For a finite set S of places of k such that S contains all the archimedean places and all p-adic places, we write M k,S for the maximal abelian pro-p-extension of k unramified outside S.
Let K ∞ /k be a pro-p Galois extension of totally real fields such that K ∞ contains k ∞ and the extension K ∞ /k ∞ is finite.We do not assume that K ∞ /k is abelian, but we have to assume the following.

Assumption 3.1. There exists an intermediate finite Galois extension
In other words, the map induced by the restriction maps modules and the target is free, f has a section.We take a section and define K to be the fixed field of the image of the section.A point is that K/k is then automatically Galois as K ∞ /k is abelian.
We take an S such that K ∞ /k is unramified outside S. Let M K∞,S /K ∞ be the Galois group of the maximal abelian pro-p-extension of K ∞ that is unramified outside places lying above S, and X K∞,S = Gal(M K∞,S /K ∞ ) as in the Introduction.Then it is known that X K∞,S is a finitely generated torsion module over the 3.2.The statements.We use the notation in §3.1.To state the result, let us put For instance, we have s 0 = 1 and s 1 = rank p G ab by Lemma 2.6.Recall that Lemma 2.8 tells us an explicit formula of s n in case K ∞ /k ∞ is abelian; in particular, we have s 2 = s(s + 1)/2 and s 3 = s(s + 1)(s + 2)/6 with s = rank p G(= s 1 ).
The following is the main result, which contains a non-abelian generalization of Theorem 1.1.
Theorem 3.3.When Assumption 3.1 is satisfied, the following inequalities and equalities hold. ( (2) We have It is easy to see that Theorem 3.3 implies Theorem 1.1, thanks to Lemma 3.2.
We also prove corresponding theorems concerning the dual (Iwasawa adjoint) of X K∞,S .For a finitely generated torsion R-module M, we define the dual (Iwasawa adjoint) of M by We are interested in the R-module X * K∞,S .It is known that the structure of X * K∞,S is often simpler than X K∞,S itself (e.g., when we are concerned with their Fitting ideals).The following theorem implies that we encounter such a phenomenon when we are concerned with the minimal resolutions.Theorem 3.4.When Assumption 3.1 is satisfied, the following equalities hold. ( and K∞,S ) = 0 for n ≥ 2 and r 1 (X * K∞,S ) − r 0 (X * K∞,S ) = 0.In §5, we will prove s 2 ≤ gen R (X K ∞,S ) in Theorem 3.3(1), Theorem 3.3(2), and Theorem 3.4(2).These parts follow only from the existence of the Tate sequence introduced in §4.The rest of the statements (t ≤ gen R (X K∞,S ) ≤ s 2 + t in Theorem 3.3(1) and Theorem 3.4(1)) will be proved in §6.

3.3.
Applications for the minus parts of Iwasawa modules for CM-extensions.In this subsection we apply the main theorems in the previous subsection to CM-extensions.We keep the notation in §3.1, so K ∞ /k is an extension of totally real fields satisfying Assumption 3.1.Only in this subsection we assume that p is odd, which is mainly for making the functor of taking the character component exact for characters of Gal(K ∞ (µ p )/K ∞ ).
We consider the field K ∞ = K ∞ (µ p ) obtained by adjoining all p-th roots of unity to K ∞ .So K ∞ /k is a CM-extension.We also use an intermediate field K n of the Z p -extension K ∞ /K(µ p ) such that [K n : K(µ p )] = p n for each n ≥ 0. Let L n be the maximal abelian pro-p-extension of K n unramified everywhere.So Gal(L n /K n ) is isomorphic to the p-component A Kn of the ideal class group of K n by class field theory.We denote by Defining L ∞ to be the maximal abelian pro-p-extension of K ∞ unramified everywhere, we know that X K∞ = Gal(L ∞ /K ∞ ).Let n 0 be the smallest integer such that all p-adic places are totally ramified in Put ∆ = Gal(K ∞ /K ∞ ), which is of order prime to p by our assumption p = 2 in this subsection.Therefore, since Gal( where χ runs over all characters of ∆ with values in Z × p , and M χ is the χ-component of M defined by Note that each M χ is an R-module.Let ω : ∆ → Z × p be the Teichmüller character, giving the action on µ p .Using our main results in §3.2, we study A ω K∞ and Y ω K∞ .Note that ω is an odd character, so the complex conjugation acts on these modules as −1. Let S p be the set of all p-adic places and all archimedean places.Recall that we write (−) ∨ for the Pontryagin dual.By Kummer pairing (see [13,Theorem 11.4.3] or [15,Proposition 13.32]), we have an isomorphism ) is Tate twist.Also, by [13,Theorem 11.1.8]we have For any finitely generated torsion R-module M which has no nontrivial finite submodule, we know (M * ) * ≃ M (see, for example, [13, Proposition 5.5.8 (iv)]).Since X K∞,Sp has no nontrivial finite submodule, so does Y ω K∞ .Therefore, it follows from the previous isomorphism that X * K∞,Sp ≃ Y ω K∞ (−1) is an isomorphism.
Thus, from Theorems 3.3 and 3.4 we get Corollary 3.5.In Theorem 3.3, we further assume that p > 2 and S = S p (so K ∞ /k ∞ is unramified outside p).
(1) Then we have

The Tate sequence
A key ingredient to prove Theorems 3.3 and 3.4 is an exact sequence that X K∞,S satisfies, which is often called the Tate sequence.Indeed, as noted in the final paragraph of §3.2, parts of main theorems can be deduced from the existence of the Tate sequence only.On the other hand, the other parts require additional arithmetic study that we will do in §6.The Tate sequence also played a key role in computing the Fitting ideal of X K∞,S in the work [5], [6], and [7] that we mentioned in §1.
In order to prove the main theorems, we need the Tate sequence of the following type.
Theorem 4.1.There exists an exact sequence of R-modules where P and Q are finitely generated torsion R-modules whose projective dimensions are ≤ 1.
Moreover, this sequence is functorial when K ∞ varies.More precisely, for a finite normal subgroup H of Gal(K ∞ /k), we have an exact sequence where P H , Q H denote the H-coinvariant modules, and φ H the homomorphism induced by φ.Proposition 1.20].Note that this complex works well even for p = 2 (here, we use our assumption that S contains all archimedean places).Taking the project limit, we get a perfect complex concentrated on degrees 0, 1, 2 with C i finitely generated projective over R and whose cohomology groups are (see [2, page 86, line 6]) and H i (C • ) = 0 for i = 1, 2, where we used the weak Leopoldt conjecture which is proven in this case (see [13,Theorem 10.3.25]).By the definition of cohomology groups, we have an exact sequence where we regard C 0 as a submodule of C 1 via d 0 and Φ is induced by d 1 .Take a non-zerodivisor f in the center of R that annihilates Z p .Then the image of Φ contains f C 2 , so by the projectivity of C 2 we can construct a commutative diagram of R-modules Then defining P and Q as the cokernel of these vertical maps respectively, we obtain the Tate sequence as claimed.The functoriality follows from that of RΓ c (O K∞,S , Z p (1)).

Abstract Tate sequences
Let G be a (not necessarily abelian) finite p-group.Motivated by Theorem 4.1, we study a Λ[G]-module X that satisfies an abstract Tate sequence, that is: Setting 5.1.There exists an exact sequence of Λ[G]-modules where both P and Q are finitely generated torsion Λ[G]-modules whose projective dimensions are ≤ 1.
In this section, we show that the existence of a Tate sequence gives a severe constraint on the integers gen Λ[G] (X) = r 0 (X), r Λ[G] (X) = r 1 (X), and r n (X) (n ≥ 2).

The statements. To state the result, let us define
for n ≥ 0.
The following are the main theorems in this section.As noted in the final paragraph of §3.2, those are enough to show parts of Theorems 3.3 and 3.4.Theorem 5.2.Let X be a Λ[G]-module that satisfies a Tate sequence as in Setting 5.1.
Claims (1) and (2) will be proved respectively in §5.3 and §5.4.For a Λ[G]-module M, we define its dual (Iwasawa adjoint) by ).The corresponding theorem for the dual is: Theorem 5.3.Let X be a Λ[G]-module that satisfies a Tate sequence as in Setting 5.1.
If G is trivial, then we have r n (X * ) = 0 for n ≥ 2 and r 1 (X * ) − r 0 (X * ) = 0.This theorem will be proved in §5.6.The idea is basically the same as that of Theorem 5.2.However, we need an additional algebraic proposition shown in §5.5.

Specialization. We consider modules over Λ = Z p [[T ]].
As explained in Example 2.3, gen Λ (−) does not behave very well for short exact sequences.A key idea to prove the main theorems is to apply specialization method to reduce to modules over DVRs.
We define Here, a monic distinguished polynomial is by definition a polynomial of the form where a 1 , . . ., a e ∈ pZ p .By the Weierstrass preparation theorem, any prime element of Λ can be written as the product of a unit element and an element of F in a unique way.
For each f ∈ F , put which is a domain.We define a subset F 0 ⊂ F by The following lemma tells us a concrete description of F 0 .Although the lemma is unnecessary for the proof of the main results, we include it in this paper to clarify the situation.Lemma 5.4.We have F 0 = {p} ∪ F 1 ∪ F 2 , where we put Proof.It is clear that {p} ⊂ F 0 and F 1 ⊂ F 0 .Also, F 2 ⊂ F 0 holds by the Eisenstein irreducibility criterion.Therefore, it remains to only show Moreover, since the residue field of O f is the same as that of Λ, namely F p , we see that the extension K f /Q p is totally ramified.In case the extension K f /Q p is trivial, we have deg(f ) = 1, so we obtain f ∈ F 1 .In case K f /Q p is non-trivial, the image of T in O f must be a uniformizer of O f , so its minimal polynomial f is in F 2 (see [14,Chap. I,Proposition 18]).This completes the proof.

Proof of Theorem 5.2(1).
Let us now study a Λ[G]-module X satisfying a Tate sequence as in Setting 5.1.We define a Λ-module X (G) by where P G and Q G denote the G-coinvariant modules and φ G denotes the induced homomorphism.Note that X (G) does not coincide with the coinvariant module X G in general; in fact, the difference is what we shall investigate from now on.
The following proposition is a key to prove the main theorem.
Proposition 5.5.Let f ∈ F be an element that is prime to both char Λ (P ) and char Λ (Q), where char Λ (−) denotes the characteristic polynomial.We set m = ord p (f (0)) ≥ 1.Then we have an exact sequence of finitely generated torsion O f -modules Proof.Firstly note that f (0) = 0 since char Λ (Q) is divisible by char Λ (Z p ) = (T ).By taking modulo f of the sequence (5.1), we obtain an exact sequence of finitely generated torsion Let L denote the image of the map φ : P/f → Q/f .Since both P/f and Q/f are Gcohomologically trivial, taking the G-homology, we obtain exact sequences and also an isomorphism We can combine these observations with the exact sequence obtained by taking modulo f of sequence (5.2) to construct a diagram This is a commutative diagram of finitely generated torsion O f -modules.By applying the snake lemma, we obtain the proposition.
Proof of Theorem 5.2(1).In Proposition 5.5, we take f so that f ∈ F 0 , i.e., O f is a DVR.Then the injective homomorphism from H 2 (G, Z p /p m Z p ) to (X/f ) G in Proposition 5.5 implies where the first equality follows from Nakayama's lemma.Since where the second equality follows from Lemma 2.7.Combining these formulas, we obtain gen Λ[G] (X) ≥ s 2 , as claimed.
Remark 5.6.This argument also shows with π as in Proposition 5.5.Then the full statement of Theorem 3.3(1) follows if we can take f so that gen O f (Ker(π)) coincides with the t in Theorem 3.3(1).This is indeed possible, but in §6 we will give a more direct proof for the rest of Theorem 3.3(1) instead.
Proof.By the assumption on P , there exists a presentation of P of the form 0 → Λ[G] a → Λ[G] a → P → 0. The lemma follows immediately from this.Lemma 5.8.Let 0 → M ′ → P → M → 0 be a short exact sequence of finitely generated torsion Λ[G]-modules such that the projective dimension of P is ≤ 1.Then we have r n (M ′ ) = r n+1 (M) 5.5.An algebraic proposition.This subsection provides preliminaries to the proof of Theorem 5.3.Let C be the category of finitely generated torsion Λ[G]-modules whose projective dimension over Λ is ≤ 1, that is, those that do not have nontrivial finite submodules.
We also write P for the subcategory of C that consists of modules whose projective dimension over Λ[G] is ≤ 1.
For a module M ∈ C, it is known that the dual ) is also in C and (M * ) * ≃ M ([13, Propositions 5.5.3 (ii) and 5.5.8 (iv)]).Moreover, if P ∈ P, we have P * ∈ P.These facts are also explained in [12, §3.1].
In this subsection, we prove the following proposition.
Proposition 5.10.Let d ≥ 0 be an integer.Let us consider exact sequences Remark 5.11.It is easy to deduce claim (1) from Lemma 5.8.Indeed, we have for n ≥ 2 and On the other hand, claim (2) cannot be deduced from Lemma 5.8.Roughly speaking, claims (1) and ( 2) are respectively what we need to prove Theorems 5.2 and 5.3.
To prove Proposition 5.10, it is convenient to use the concept of axiomatic Fitting invariants introduced by the first author [12].More concretely, inspired by [4, §3.2], we use the notion of P-trivial Fitting invariant defined as follows.
Definition 5.12.A P-trivial Fitting invariant is a map F : C → Ω, where Ω is a commutative monoid, satisfying the following properties: (i) If P ∈ P, we have F (P ) is the identity element of Ω.
(ii) For a short exact sequence 0 → M ′ → M → P → 0 in C with P ∈ P, we have (iii) For a short exact sequence 0 → P → M → M ′ → 0 in C with P ∈ P, we have It is an important fact [12, Proposition 3.17] that conditions (ii) and (iii) are equivalent to each other (assuming (i)).Note that in this setting we do not have to assume Ω is a commutative monoid, and instead a pointed set structure suffices.
A fundamental example of a Fitting invariant is of course given by the Fitting ideal; more precisely, the Fitting ideal modulo principal ideals satisfies the axioms of P-trivial Fitting invariants.
The following proposition introduces another kind of Fitting invariants.
Proof.We check the conditions (i) and (ii).Firstly, (i) is a restatement of Lemma 5.7.
Secondly, (ii) follows from the associated long exact sequence, taking Lemma 5.7 into account again.Indeed, the long exact sequence collapses into isomorphisms for degree ≥ 2 and a 6term exact sequence for degrees 0, 1.
Note that (iii) cannot be shown in a similar manner.This is because the lower degree part of the associated long exact sequence becomes an 8-term exact sequence.It is important that (iii) follows from (i) and (ii).
Proof of Proposition 5.10.By Proposition 5.13, it is enough to show where we used Z * p ≃ Z p .As in §5.3, we use the specialization method.Let us take any element f ∈ F 0 that is coprime to char Λ (P ) and char Λ (Q).Put m = ord p (f (0)) ≥ 1.Then (5.3) yields an exact sequence 0 → Z p /p m Z p → Q * /f → P * /f → X * /f → 0. Observe that both P * /f and Q * /f are G-cohomologically trivial.So we have where Ĥ−1 denotes the Tate cohomology group.By definition, Ĥ−1 (G, X * /f ) is a submodule of H 0 (G, X * /f ), so the above isomorphism shows as O f is a DVR.By Nakayama's lemma, the left hand side is equal to gen Λ[G] (X * ).Also, as in the proof of Theorem 5.2(1) in §5.3, where the second equality follows from Lemma 2.9 and the third from Lemma 2.7.Thus we obtain (1).
(2) In case G is trivial, since the projective dimension of X as a Λ-module is ≤ 1, we may apply Lemma 5.7 to obtain the assertion.
From now on, we assume G is non-trivial.Since We truncate it to an exact sequence Since its dual (−) * is also exact and we have Z * p ≃ Z p and Z p [G] * ≃ Z p [G], we obtain an exact sequence (5.5) By comparing (5.3) and (5.5), Proposition 5.10(2) implies Let us compute r n (Ker(d 1 ) * ) for any n ≥ 0. We combine (5.5) with (5.4) to an exact sequence By construction, this is a minimal resolution of Ker(d 1 ) * as a Z p [G]-module.For this we need the hypothesis that G is non-trivial; the map ε * • ε can be identified with the map By applying Lemma 5.9, we obtain This completes the proof.
6. Proof of the rest of Theorems 3.3(1) and 3.4(1) Now we return to the arithmetic situation described in §3.1.
To prove Theorem 3.4(1), we also need the following general lemma.Proof.As in Lemma 5.7, the minimal resolution of M is of the form 0 which is again a minimal resolution.Thus we obtain the lemma.
Proof of Theorem 3.4 (1).Applying the dual of the Tate sequence introduced in (5.3) to our setting, we obtain an exact sequence By the compatibility of the Tate sequences, we obtain an isomorphism (X * K∞,S ) G ≃ X * k∞,S .Therefore, we have gen R (X * K∞,S ) = gen Λ (X * k∞,S ).By Lemma 6.2 (with G trivial), this is then equal to gen Λ (X k∞,S ).By Lemma 6.1, this completes the proof.6.2.Proof of Theorem 3.3 (1).Note that applying Theorem 5.2(1) to the Tate sequence introduced in Theorem 4.1, we already obtained the inequality s 2 ≤ gen R (X K∞,S ).Therefore, it remains only to prove t ≤ gen R (X K∞,S ) ≤ s 2 + t.Lemma 6.3.We have an exact sequence Proof.By the Hochschild-Serre spectral sequence we have an exact sequence where we used the weak Leopoldt conjecture H 2 (O K∞,S , Q p /Z p ) = 0.For i = 1, 2, the Pontryagin dual of H i (G, Q p /Z p ) is the projective limit of H i (G, Z/p m ) by Lemma 2.9.But since G is a finite p-group, it is finite and isomorphic to H i (G, Z).Therefore, taking the Pontryagin dual of the above exact sequence and using Lemma 2.6, we get the conclusion.
On the other hand, by taking the Γ = Gal(k ∞ /k)-coinvariant of the second sequence, we obtain an exact sequence G ab → Y Γ → (X k∞,S ) Γ → G ab → 0.
By Lemma 6.1, this is reformulated as This sequence, together with M k∞,S ∩ K ∞ = M k,S ∩ K ∞ and the definition of t, implies that (6.2) t ≤ gen Λ (Y ) ≤ gen Zp (G ab ) + t.

Numerical Examples
In this section, we numerically check the inequality concerning gen R (X K∞,S ) in Theorem 1.1 by using the computer package PARI/GP.We consider k = Q and its finite abelian pextension K that is totally real.Let S be a finite set of places of Q containing p and the archimedean place ∞, such that K/Q is unramified outside S.
We observe that M K,S is the union of the maximal p-extensions of K in the ray class fields of modulus p m v∈S\{p} v for all m ≥ 0. Note here that, since K is abelian over Q, the Leopoldt conjecture is shown to be true for K by work of Brumer (see [13,Theorem 10.3.16]).Therefore, Gal(M K,S /K ∞ ) is finite, and we can compute it by computing the ray class groups for finitely many m.In this way we can determine the quantity gen Zp (Gal(M K,S /K ∞ ) Gal(K/Q) ).
7.1.The case p = 3.Let us take p = 3, though the discussion is basically valid for any odd prime p.We write S \ {p, ∞} = {ℓ 1 , . . ., ℓ s }.By the theorem of Kronecker-Weber, the Galois group of the maximal abelian extension of Q unramified outside S is isomorphic to Z × p × s i=1 Z × ℓ i , so we have As is well-known, we may assume ℓ i ≡ 1(mod p) for 1 ≤ i ≤ s without loss of generality.
for any P-trivial Fitting invariant F .For this, we apply the theory of shifts F d (−), F −d (−) of Fitting invariants of the first author [12, Theorem 3.19].By the exact sequence involving M, N, and P i , the definition of the shifts implies F d (M) = F (N), F −d (N) = F (M), and similarly for M ′ , N ′ .Then what we have to show is just a reformulation of the welldefinedness of the shifts, which is already established by the first author in [12, Theorem 3.19].5.6.Proof of Theorem 5.3.Proof of Theorem 5.3.(1) By taking the dual of the Tate sequence, we obtain an exact sequence(5.3)