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

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.


Introduction
One of the most important themes in Iwasawa theory is to study the relationship between p-adic analytic objects and p-adic algebraic objects, usually formulated as "main conjectures," in which the algebraic objects are described by characteristic ideals of suitable arithmetic modules. However, we are recently understanding that there exists closer relationship between analytic and algebraic objects beyond characteristic ideals. For example, such relationship can be described by using Fitting ideals.
Namely, in certain cases, using the p-adic L-functions corresponding to the arithmetic objects, we can describe the Fitting ideals of certain arithmetic modules, which give more information than the characteristic ideals. But in such cases, modified Iwasawa modules had been always used; see for example, [10], [3], etc.
In this paper we study a much more difficult and subtle object, the Fitting ideals of nonmodified classical Iwasawa modules. We prove that they can be described by the analytic objects and certain ideals constructed from simple objects. We think it is remarkable that the Fitting ideals of classical Iwasawa modules can be also described by some variants of p-adic L-functions.
In order to explain slightly more details, we introduce here notation in this paper. Throughout this paper, we fix an odd prime number p. We consider a finite abelian extension K/k of totally real number fields and the cyclotomic Z p -extension K ∞ of K. Let S p be the set of p-adic places of k. For any algebraic extension F/k, let S ram (F/k) be the set of finite places of k which are ramified in F . For any finite set S of primes of k, let X K∞,S be the S-ramified Iwasawa module, which is by definition the Galois group of the maximal pro-p-abelian extension of K ∞ unramified outside S. Recall that X K∞,S is a module over the Iwasawa algebra R = Z p [[G]], where G = Gal(K ∞ /k) is the profinite Galois group in this setting. We simply write X S for X K∞,S when no confusion arises.
The main theme in this paper is to compute Fitt R (X Sp ), the Fitting ideal of the Iwasawa module X Sp . The module X Sp has been important in Iwasawa theory. For example, we have the Kummer duality (see Remark 5.3) between X Sp and A ω K(µ p ∞ ) , the Teichmüller character component of the inductive limit A K(µ p ∞ ) of the p-parts of the ideal class groups (full class groups) of K(µ p n ). The Fitting ideals of the minus part (A − K(µ p ∞ ) ) ∨ are known outside the Teichmüller character component (for example the method of the first author [6] can be applied to the Iwasawa theoretic situation without assuming the ETNC, see also [15] by the third author for the T -modified class group version and the Iwasawa theoretic version), and we know that the Teichmüller character component A ω K(µ p ∞ ) , which is dual to our X Sp , is a subtle and interesting object. Note that this duality plays practically no role in the present paper.
In the papers [7], [8] by the first and the third author, and in the paper [9] with Tokio, we determined Fitt R (X S ) when S contains S ram (K ∞ /k) = S ram (K/k) ∪ S p . Therefore, Fitt R (X Sp ) was determined in [7], [8], [9] under the assumption that S ram (K/k) ⊂ S p , that is, K/k is unramified outside p. But the assumption S ram (K/k) ⊂ S p , is a pretty severe constraint. In the present paper we completely remove the assumption S ram (K/k) ⊂ S p , and determine Fitt R (X Sp ) for any finite abelian extension K/k of totally real fields. Thus we are mainly concerned with the case S ram (K ∞ /k) S p .
The main result of this paper is the following.
Theorem 0.1. Let S be a finite set of finite places of k such that S ⊃ S p ∪ S ram (K/k) and S = S p . Put S ′ = S \ S p = ∅. Suppose the µ-invariant of X Sp vanishes. Then we have The definitions of the R-module Z 0 S ′ , of the element θ mod S , and of Fitt [1] R will be given in Section 1. We introduce in this paper an integral element θ mod S ∈ R, which is a kind of (modified) equivariant Iwasawa power series. The shifted Fitting ideal Fitt [1] R was introduced by the second author in [12]. It is defined by using a certain type of resolutions and the syzygies produced by them. The main point of the theorem is that all quantities on the right hand side are computable in principle.
The crucial point in this study is the case when Gal(K/k) is a p-group. In fact, the ring R = Z p [[G]] is semi-local, and decomposed into direct product of local rings. Let G = G (p ′ ) × G (p) be the decomposition of G such that G (p ′ ) is a finite group of order prime to p and G (p) is a pro-p group. Then each local component of R corresponds to an equivalence class of characters of G (p ′ ) (see Subsection 1.4), and accordingly the statement of Theorem 0.1 can be decomposed. On the one hand, the trivial character component is the most difficult, and the statement is equivalent to that for the pro-p extension (K ∞ ) G (p ′ ) /k with Galois group G (p) . On the other hand, the non-trivial character components are easier to handle; for example, those components of Fitt [1] R (Z 0 S ′ ) can be computed easily (see Corollary 1.10). In that sense the case that Gal(K/k) is a p-group is essential. However, the proof of Theorem 0.1 does not involve an explicit reduction to that case.
The proof of our main result occupies Sections 2 and 3; indeed the proof splits naturally into an algebraic part and an arithmetic part. The former constructs a certain complex C S via an exact triangle, whose other two terms come from complexes that arise in global and local Galois cohomology respectively. This produces a short exact sequence 0 → X Sp → H 1 (C S ) → Z 0 S ′ → 0, as in Proposition 2.9. Since the middle term turns out to be cohomologically trivial, this already gives a formula for Fitt R (X Sp ) in terms of Fitt [1] R (Z 0 S ′ ): these two quantities differ by a principal factor governed by the complex C S . In the second part of the proof, this factor is then identified with the (equivariant, modified) p-adic L-function θ mod S . In Subsection 2.3, we also discuss the natural question under what circumstances Fitt R (X Sp ) is principal. The rough answer is: very rarely (see Proposition 2.12).
In Section 4 we present several attempts to make our determination of Fitt R (X Sp ) really explicit. The module Z 0 S ′ that occurs in the main result appears to be fairly explicit, but a closer look quickly shows that (unless the extension K/k is very small in a way) an honestly explicit description of its first shifted Fitting ideal is not obvious at all, and in fact turns out to be pretty hard in general. We present a general method to attack the problem, and show that it produces in some nice cases a truly explicit result, that is, a concrete list of generators for Fitt [1] (Z 0 S ′ ). In the final Section 5 we compute Fitt [1] R (Z 0 S ′ ) explicitly to determine the Fitting ideal of X Sp in the case that K/k is cyclic and satisfies some mild conditions (see Theorems 5.1 and 5.4). Especially, these results give generalizations of the main result in [14] by the third author where only the case that K/k is of degree p was treated. We think that this new look at the third author's previous result is a good way to use our main result and to test the techniques of Section 4.
Remark 0.2. Large parts of this paper, as they are written now, make an essential use of homological algebra. More precisely speaking, we need the theory of complexes including the cone construction, and some theory of derived categories. We would like to mention here that in the earliest stages of this manuscript we used different and more elementary methods. Actually, as far as the proof of the main result is concerned, one might call those other methods old-fashioned, since they mimicked and partially repeated ingenious arguments of Tate [21], which are over fifty years old. It is interesting to note that already in those old arguments one can perceive some central ideas of homological algebra like the use of Ext groups, but the theory of complexes was not used in the way we know it today. Anyway, it may be reassuring to know that alternative arguments exist, but we think that using the framework of Galois cohomology and complexes leads to shorter arguments and to a better logical structure, so this is what the reader will actually see in the body of this paper.

Ingredients for the main result
Our main result in this paper is Theorem 0.1 in Introduction. In this section we define the R-module Z 0 S ′ , the element θ mod S ∈ R, and Fitt [1] R which appeared in the statement of Theorem 0.1, and also give detailed explanation of several statements mentioned in Introduction. We recall some important notation from the introduction. Let p be an odd prime number, K/k a finite abelian extension of totally real fields, and K ∞ the cyclotomic Z p -extension of K. Put G = Gal(K ∞ /k) and R = Z p [[G]]. We denote by S p the set of p-adic primes of k, and by S ram (K/k) the set of primes of k which are ramified in K/k. Let X Sp = X Sp (K ∞ ) be the S p -ramified Iwasawa module for K ∞ .
1.1. Definition of Z 0 S ′ . As in Theorem 0.1, let S be a finite set of finite places of k such that S ⊃ S p ∪ S ram (K/k) and S = S p . Put S ′ = S \ S p = ∅.
For each finite place v of k outside p, let G v be the decomposition subgroup of G at v.
which is regarded as an R-module; note that it is a finitely generated free Z p -module. Moreover, put where the map Z S ′ → Z p is defined to be the augmentation map on each summand Z v . Note that this map is onto for the precise reason that we assume S ′ to be nonempty.

S
. Again let S be a finite set of finite places of k such that S ⊃ S p ∪ S ram (K/k), but we do not assume S = S p in this subsection. Definition 1.1. Let v be a finite place of k outside p. We denote by N(v) the cardinality of the residue field of k at v. Let T v ⊂ G v be the inertia group, which is finite. Let σ v ∈ G/T v be the N(v)-th power Frobenius automorphism.
where v runs over the finite places of k that are not in S. This infinite product converges on the half plane ℜ(s) > 1 and L S (ψ, s) has a meromorphic continuation to the whole plane C.
We fix embeddings Q ֒→ Q p and Q ֒→ C. Then each finite character ψ : G → C × can be regarded to have values in Q p × . Thus ψ induces a continuous Z p -algebra homomorphism denote the cyclotomic character, and ω : Gal(k(µ p )/k) ֒→ Z × p denote the Teichmüller character. The Z p -algebra homomorphisms induced by them are also written by the same letters. The S-truncated p-adic L-functions θ S are defined via interpolation properties, as follows. Definition 1.3. Let θ S = θ S,K∞/k ∈ Frac(R) × be the element characterized by (κ n cyc ψ)(θ S ) = L S (1 − n, ψω −n ) for each finite character ψ of G and positive integers n with n ≡ 0 mod [K(µ p ) : K]. The existence of θ S follows from Deligne-Ribet [5]. Moreover, it is known that θ S is a pseudomeasure in the sense of Serre [20], that is, we have Ann R (Z p )θ S ⊂ R.
We define a modification θ mod S of θ S as follows.
for ψ and n as in Definition 1.3, where v runs over the elements in S ′ such that ψ is unramified at v.
It is easy to see that θ mod S is uniquely determined by the interpolation properties. It is also easy to deduce the existence of θ mod S from that of θ S as an element of Frac(R) × . Furthermore, we show later in Subsection 3.4 the following.
We note that a variant of this element θ mod S was called "Greither's Stickelberger element" in [14, Theorem 0.1], and its integrality was proved in [14, Lemma 2.1] for a special type of extension K/k studied there.
We also note that we do not use Theorem 1.5 in the proof of Theorem 0.1. Let us first discuss some basic properties of θ mod S . Lemma 1.6. (1) Let S 1 be a finite set which contains S. Then we have (2) Recall that, for any intermediate field (1) We note that v ∈ S 1 \ S is unramified in K ∞ /k because S contains all ramifying primes. By the definition of θ S , we have Then by the definition of θ mod S , we obtain The claim in (2) follows from the interpolation properties of θ mod S,K∞/k and θ mod S,M∞/k . Theorem 1.7. Let n be a non-negative integer and X a finitely generated torsion R-module. Take an n-step resolution 0 → Y → P 1 → · · · → P n → X → 0 of X, in which all modules are finitely generated torsion over R and such that pd R (P i ) ≤ 1 for i = 1, . . . , n. If we put then the fractional ideal Fitt R (X) of R is independent of the choice of the n-step resolution. In this sense, Fitt R (X) is well defined.
1.4. Decomposition of group rings. In general, suppose that ∆ is a finite abelian group of order prime to p. Then we have a decomposition where χ runs over equivalence classes of p-adic characters of ∆ (two characters χ 1 , χ 2 are equivalent if and only if σχ 1 = χ 2 for some σ ∈ Gal(Q p /Q p )), and O χ = Z p [Im(χ)] is a Z p [∆]-module on which ∆ acts via χ. According to this decomposition, each Z p [∆]-module M can be decomposed as We also have where the first component of the right hand side corresponds to the trivial character χ = 1.
Here we give a description of Fitt [1] R (Z v ). Let v be a finite place of k outside p. Recall (Definition 1.1) that T v is the inertia group in K ∞ /k, and σ v ∈ G/T v is the Frobenius automorphism. Let be the map induced by the multiplication by the norm element N Tv = σ∈Tv σ.
Proof. The statement of the lemma can be decomposed according to characters χ of G (p ′ ) . We consider the decomposition , we have Z χ v = 0 and χ(ν v ) = 0, so the equation holds. Therefore, we only have to deal with χ which is trivial on T v . Note that, by local class field theory, T v is a quotient of the unit group O × kv , so in particular T is a cyclic group. Hence we can take a generator δ v of T v . Take a lift σ v ∈ G of σ v ∈ G/T v . Then G v is topologically generated by σ v and δ v , so we have . Thus we have an exact sequence Observe that the cokernel of N Tv here has a presentation (N Tv , σ v −1) as an R-module. Hence we obtain Now we give an explicit description of Fitt [1] R (Z 0 S ′ ) for non-trivial character components. Proposition 1.9. For any non-trivial character χ of G (p ′ ) , we have Then the assertion follows from Proposition 1.8 immediately.
Note that N Tv ∈ R goes to 0 in R χ unless χ is trivial on T v , that is, χ(v) = 1. Therefore, only places v ∈ S ′ with χ(v) = 1 contribute in the product.
Using Theorem 0.1 and the Proposition 1.9, we get a complete description of the Fitting ideal of the non-trivial character component of X Sp . Corollary 1.10. For any non-trivial character χ of G (p ′ ) , we have

Proof of main result (I)
2.1. Some facts on arithmetic complexes. We collect some facts on local and global arithmetic complexes. A comprehensive reference is Nekovář [16]. Let k S /k be the maximal S-ramified algebraic extension. For each finite place v of k, let k v be the completion at v. Fix an algebraic closure k v of k v and an inclusion k S ֒→ k v over k. Then any representation of Gal(k S /k) will yield a representation of Gal(k v /k v ).
We denote by (−) ∨ the Pontryagin dual of a module. This symbol will also be used for the corresponding construction in derived categories. As usual, we denote by µ p m the group of p m -th roots of unity. Let Z p (1) = lim ← −m µ p m be the Tate module. Let χ G : Gal(k S /k) ։ Gal(K ∞ /k) = G ֒→ R × be the tautological representation. We consider , which is an R-module of rank one with a certain action of Gal(k S /k).
We shall study the complexes which are defined using the continuous cochain complexes for the profinite groups Gal(k S /k) and Gal(k v /k v ) (see [16, (3.4.1)]). We denote by D perf (R) the derived category of perfect complexes of R-modules, and by D perf tor (R) the subcategory of objects whose cohomology groups are torsion as R-modules. We will see that most of the complexes we treat in this paper are objects of D perf tor (R). First we recall the following fact. The following two propositions are interpretations of the local Tate duality and the global Poitou-Tate exact sequence, respectively. It would be certain that they have been known since Grothendieck's works, and are explicitly mentioned in Nekovář [16]. We use [ where the first morphism is obtained by the localization, and the second morphism by the localization and the duality in Proposition 2.2.
Remark 2.4. It should also be possible to deduce this exact triangle from the paper [1]. The notation there is closer in spirit to ours than Nekovář's, but there is the disadvantage that everything is formulated at finite level, and we have not checked whether the transition to the projective limit offers problems. A little more precisely: The definition of the cone in [1], formula (3) on p.1345, gives an exact triangle where C denotes the cone. Then with the method of loc.cit. p.1357, see equation (36) in particular, it should be possible to identify C with RΓ(k S /k, Z p (1) ∨ (1)) ∨ [−2]. Again, we gloss over some technical problems and we do not try to discuss the passage from Z p (1) (finite level) to T (infinite level).
Next we compute the cohomology groups of the global and local complexes.
where µ p ∞ (K n ⊗ k k v ) denotes the p-primary subgroup of (K n ⊗ k k v ) × and the inverse limit is taken with respect to the norm maps. Then J v is naturally an R-module and its structure is as in Remark 2.6. Put Let X S = X S (K ∞ ) be the S-ramified Iwasawa module. From global class field theory, we have an exact sequence where the injectivity of J S ′ → X S follows from the weak Leopoldt conjecture.
Remark 2.6. Take a place w of K ∞ above v, and put Here K n,w denotes the completion of K n at the place below w.
If µ p ∞ (K ∞,w ) = 0, then we have J w = 0 and thus J v = 0. Otherwise, we have µ p ∞ ⊂ (K ∞,w ) × and J w ≃ Z p . In the latter case, the action of G v on J w is given by the cyclotomic Proposition 2.7. We have and Proof. We feel that it should also be possible to assemble a proof from suitable references to Nekovář's book [16], but we will write out a direct proof for the reader's convenience.
We have Then where the third isomorphism follows from Shapiro's lemma. The weak Leopoldt conjecture, which says that H 2 (k S /K ∞ , Q p /Z p ) vanishes, is known to be true. This implies the first assertion of Proposition 2.7.
For the second assertion, we use Proposition 2.2 to see that A computation similar to the global case that we just have done shows (G v,n is an ad hoc abbreviation for Gal(K n,w /k v )): This implies the assertion for i = 1. For i = 1, the above computation implies For each positive integer m, the exact sequence . By taking the inverse limit with respect to m and n, we obtain lim ← −n H 1 (K n,w , Z p (1)) ≃ J w . This completes the proof.
Proof. Propositions 2.1 and 2.3 imply that these complexes are objects of D perf (R). By Proposition 2.7, the cohomology groups are torsion.

2.2.
The algebraic part of the proof. We define a complex , namely define it such that it fits into a distinguished triangle where the first morphism is induced by the restriction, using Proposition 2.2. By Corollary 2.8, C S is actually an object of D perf tor (R). Proposition 2.9. We have H i (C S ) = 0 unless i = 1, and an exact sequence Proof. Taking the long exact sequence associated to (2.2) and using Proposition 2.7, we obtain an exact sequence Then the assertion follows from the exact sequences (1.1) and (2.1).
Corollary 2.10. The projective dimension of H 1 (C S ) is at most one, and we have Proof. Since C S is perfect, the first statement of Proposition 2.9 tells us that pd R (H 1 (C S )) < ∞. By the exact sequence (2.3), H 1 (C S ) does not contain any non-trivial finite submodule. Hence we have pd R (H 1 (C S )) ≤ 1. The formula (2.4) is therefore a consequence of (2.3) and the definition of Fitt . At the end of this section we put the preceding result into perspective by discussing the exact conditions under which the ideal Fitt R (X Sp ) is principal. Keep the setup of preceding sections.
Consider the commutative diagram with exact rows and columns Using these homomorphisms, define a homomorphism s : By construction, s is a section of the natural projection Z S ′ → Z S ′ \{v * } , and moreover the image of s is contained in Z 0 S ′ . Therefore s gives a splitting of the left vertical sequence, which completes the proof.
. Then the following are equivalent.
Proof. The equivalence (i) ⇔ (ii) follows from the argument of [4,Proposition 4]. The equivalence (ii) ⇔ (iii) follows from the short exact sequence (2.3) and the first line of Corollary 2.10. The equivalence (iv) ⇔ (v) is clear, and the implication (iv) ⇒ (iii) is trivial. Now we show the implication (iii) ⇒ (iv). Put H = Gal(K ∞ /k ∞ ), which is a non-trivial p-group by assumption. We note that, for any R-module M which is free of finite rank over First suppose that all quotients G/G v with v ∈ S ′ are non-trivial. Then the Z p -rank of It is easy to check that, for each For completeness, we note the following.
Lemma 2.13. Suppose that K/k is a p-extension and that S ram Proof. As already used in the proof of Proposition 2.12, the ideal Fitt R (X Sp ) is principal if and only if pd R (X Sp ) ≤ 1. By Propositions 2.1 and 2.7, we see that pd R (X Sp ) ≤ 1 is equivalent to pd R (Z p ) ≤ 1, which is true exactly when K ∞ = k ∞ .

Proof of main result (II)
In this section, we complete the proof of Theorem 0.1 by determining Fitt R (H 1 (C S )). This is in a certain way the arithmetic part of the proof. We need a few preliminaries concerning determinants and Fitting ideals.
3.1. The determinant homomorphism. This subsection is devoted to the homological algebra related to the determinant functor. Let I(R) be the commutative group of invertible fractional ideals of R. We shall introduce a group homomorphism, called the determinant, Here K 0 denotes the Grothendieck group of a triangulated category. We refer to Knudsen-Mumford [13] for more on the theory of determinants.
Let Ch perf (R) be the abelian category of perfect complexes of R-modules. More precisely, Ch perf (R) consists of bounded complexes F of R-modules such that F i is finitely generated and projective for all i. Let Ch perf tor (R) be the subcategory of complexes with torsion cohomology groups.
where L is an invertible Rmodule and α : Spec(R) → Z is a locally constant map. Two graded invertible R-modules (L, α) and (L ′ , α ′ ) are said to be isomorphic if α = α ′ and L and L ′ are isomorphic as R-modules. For two graded invertible R-modules (L, α), (L ′ , α ′ ), we define Then (Hom R (L, R), −α) is the inverse of (L, α). Definition 3.2. For a finitely generated projective R-module F , let rank(F ) denote the (locally constant) rank of F , and define the determinant of F by (2) Let F and F ′ be finitely generated projective R-modules. Then we have a canonical isomorphism , which is locally given by Here r and r ′ denote the local rank of F and F ′ , respectively.
The appearance of the sign is the reason of introducing the information of the rank in the definition of the determinant.
Thanks to Lemma 3.3(2), this is independent from the ordering of Z. We denote by Det −1 R (F ) its inverse.
Lemma 3.5. The following hold true. ( Proof. (1) and (2) follow from Lemma 3.3(1). (1) and (2) imply Here we disregard the degree since it is zero. From now on, we identify Det R (F ) with its image in Frac(R). This defines a mapping Det R from the set of isomorphism classes of objects of D perf tor (R) to the set of fractional ideals of R. Lemma 3.7. The map Det R that was just defined induces a group homomorphism Det R : Proof. This follows from Lemma 3.5 (1) and (3).
As a preparation for the main arguments, we now formulate two lemmas, relating determinants to Fitting ideals.
tor (R) be a complex and n be an integer. Suppose that we have H i (F ) = 0 for any i = n and pd R (H n (F )) ≤ 1. Let Q be the foll ring of quotients of R and let λ = λ F Q be the canonical trivialization Det Q (F Q ) → Q. Then we have Proof. By translation, we may and will assume that n = 0. By using truncations, we see that the complex F is quasi-isomorphic to the complex H 0 (F )[0]. Taking a projective resolution of H 0 (F ) of length 2, we can construct a perfect complex F ′ ∈ Ch perf tor (R) which is quasiisomorphic to F such that (F ′ ) i = 0 for i = −1, 0. We can assume that both (F ′ ) −1 and (F ′ ) 0 are free R-modules of the same rank a.
We take bases e 1 , . . . , e a of (F ′ ) −1 and f 1 , . . . , f a of (F ′ ) 0 . Then we can identify the homomorphism d : On the other hand, one may verify Det R (F ′ ) −1 = (det(A)). This is a standard fact, but we give a sketch of the proof for completeness. Let f * 1 , . . . , f * a be the dual basis of f 1 , . . . , f a . Put Q = Frac(R) for notational simplicity. Then we have a natural isomorphism has (e 1 ∧ · · · ∧ e a ) ⊗ (f * 1 ∧ · · · ∧ f * a ) as a basis over R and it goes to by the trivialization. This proves Det R (F ′ ) −1 = (det(A)).
Lemma 3.9. Let F ∈ D perf tor (R) be a complex and n be an integer. Suppose that we have H i (F ) = 0 for i = n, n + 1 and H i (F ) does not contain any nonzero finite submodule for i = n, n + 1. Then we have where the superscript (−) * denotes the Iwasawa adjoint.
Proof. By translation, we may assume that n = 0. By using truncations, we see that the complex F is quasi-isomorphic to a complex F ′ such that (F ′ ) i = 0 for i = 0, 1. Moreover, the construction of the truncations allows us to assume (F ′ ) 1 is a projective R-module. Then we have an exact sequence We can construct a projective R-module F and a homomorphism F → (F ′ ) 0 such that the composition F → (F ′ ) 0 → (F ′ ) 1 is an injective homomorphism with torsion cokernel. Then, by defining P 1 and P 2 as the cokernel of F → (F ′ ) 0 and F → (F ′ ) 1 respectively, we have an exact sequence 0 → H 0 (F ) → P 0 → P 1 → H 1 (F ) → 0. By the assumption, none of these modules contain any nonzero finite submodule. Then by the construction, we deduce that pd R (P i ) ≤ 1 for i = 0, 1. By a purely algebraic result (see [2,Lemma 5] or [12,Remark 4.8]), we have On the other hand, by construction, the complex F is quasi-isomorphic to the complex [P 0 → P 1 ] located at degrees 0 and 1. Hence the distinguished triangle The final equation follows from Lemma 3.8. This completes the proof.
3.2. Description of C S by p-adic L-functions. Let us now go back to the arithmetic situation. Recall that it is our goal to determine the Fitting ideal of H 1 (C S ). We first express it as a combination of determinants of one global and some local complexes.

Lemma 3.10. We have
Proof. By Lemma 3.8, we have But the definition (2.2) of C S and Lemma 3.7 imply This completes the proof, since the shift by −2 does not change the determinant. Now we deal with the global term in the preceding lemma. The following is a formulation of an abelian equivariant main conjecture.
Proof. We use results of Ritter and Weiss; see [18, §4]. They constructed a certain exact sequence 0 → X S (K ∞ ) → Cok(Ψ) → Cok(ψ) → Z p → 0 of finitely generated torsion R-modules with pd R (Cok(Ψ)) ≤ 1, pd R (Cok(ψ)) ≤ 1 (we do not give the definitions of Ψ and ψ here). Moreover, the equality known as "equivariant main conjecture" is an established theorem (assuming the vanishing of the µ-invariant), thanks to Ritter-Weiss. Note that even in the non-commutative case, Ritter and Weiss [19] and Kakde [11] proved analogous results, again assuming the vanishing of the µ-invariant. In preparation for the final part of the proof, we state another lemma, which by now seems to be well known.
. For a prime ideal q of Λ, let R q be the localization of R with respect to the multiplicative set Λ \ q. (1) We have Det R (RΓ(k v , T)) = (f v ).
(2) For any continuous character ψ : G → Q p × such that ψ| Gv is non-trivial, we have Proof. Observe that property (2) ensures the uniqueness of f v .
This reduces the problem to a completely local statement. We will in fact find f v in the ring is of order prime to p. Then as in Subsection 1.
v ]] where χ runs over equivalence classes of p-adic characters of G where χ runs over the equivalent classes of characters of G using the identification (3.1). We shall show that this element satisfies the desired properties (1) and (2); let us begin with the latter.
Property (2): Let ψ : G v → Q p × be a non-trivial continuous character.
First suppose that ψ is unramified at v. Then ψ is trivial on Hence Now suppose that ψ is ramified at v. If ψ is non-trivial on T ′ v , then ψ(f v ) = 1 by the definition of f v . Otherwise, ψ is non-trivial on T Therefore Property (1): By Proposition 2.7 and Remark 2.6, we can apply Lemma 3.9 to obtain Note that we have (J w ) * ≃ J w by the simple description in Remark 2.6. By Lemma 3.12, this formula characterizes Det Rv (RΓ(k v , T v )). Hence it is enough to show that Recall the identification (3.1). Since the actions of T ′ v on Z p and J w are trivial, the equation (3.2) for the χ-part with χ |T ′ v = 1 holds trivially. Thus we have only to worry about , whose augmentation ideal is generated by as ideals. Here we used Remark 2.6 and κ v (σ v ) = N(v) in the latter case.
We shall show that , respectively. Thus, it is enough to show that the difference of these elements, Since N(v) − 1 is an element of p nv Z p by the definition of n v , this implies (N(v) − 1)(δ v − 1) ∈ (δ v − 1) 2 . This completes the proof.

3.3.
Proof of Theorem 0.1. By comparing the values given by the respective interpolation formulas, we have where f v is introduced in Proposition 3.13. Therefore, Lemma 3.10, Theorem 3.11, and Proposition 3.13 imply (3.4) Fitt R (H 1 (C S )) = (θ mod S ). Then Theorem 0.1 follows immediately from Corollary 2.10.
Remark 3.14. Let us give a direct argument showing that the right hand side of Theorem 0.1 is actually independent of the choice of S. Since this independence is a logical consequence of our main result, this verification is not strictly necessary, but we think that doing it anyway gives a nice consistence check for our result.
Let S 1 ⊃ S be another finite set and put S ′ 1 = S 1 \ S p . By the exact sequence we have an exact sequence

R is again a Fitting invariant by [12, Theorem 2.6]). For
Hence, using Lemma 1.6(1), we obtain This completes the proof of independence from the choice of S.

3.4.
Integrality of θ mod S . In this subsection we prove Theorem 1.5. We can immediately deduce Theorem 1.5 from the above equation (3.4). However, the proof of (3.4) relies on the deep result on the equivariant main conjecture (Theorem 3.11), which is only established under µ = 0. For this reason, it is meaningful to prove Theorem 1.5 directly.
Before the proof, we would like to insert a general remark.
Remark 3.15. Recall that K/k is a finite abelian extension of totally real fields and K ∞ is the cyclotomic Z p -extension of K. Then it is easy to show that there is a finite abelian extension K ′ of k such that K ′ ∩ k ∞ = k and K ∞ is the cyclotomic Z p -extension of K ′ . In fact, since the p-component of Gal(K ∞ /k) is a finitely generated Z p -module of rank 1, it is decomposed into Z p × (a finite abelian p-group). Therefore, Gal(K ∞ /k) can be decomposed into Z p × (a finite abelian group), which proves the existence of K ′ . We can see that many objects in this paper depend only on the extension K ∞ /k and not on the specific field K. In particular, the statements of Theorems 0.1 and 1.5 do not depend on K. Therefore, we may assume to begin with that K ∩ k ∞ = k, whenever we are concerned with statements of this type.
If χ is a non-trivial character of G (p ′ ) , the integrality of the χ-component of θ mod S , namely the fact that the image of θ mod v ]], can be easily chacked by a slight modification of the argument we give below. So we only consider the trivial character component in the following, and thus we may assume K/k is a p-extension.
We begin with the following lemma, which gives an alternative description of the element f v defined in Proposition 3.13. Note that we are now assuming G (p ′ ) = 1, so G In particular, the right hand side does not depend on the choice of σ v .
We note that the numerator of the right hand side of the equation in Lemma 3. 16 Proof. We denote by α v the right hand side. Let ψ be any character of G and we compare the interpolation property of α v with Proposition 3.13 (2). If ψ is non-trivial on T v , then We prepare an algebraic lemma.  Assume to the contrary that r(T ) = 0. Then there is a character ψ of H such that ψ(r(T )) = 0. Since the degree of ψ(r(T )) is less than m, we see that ψ(r(T )) is not divisible by ψ(F * (T )). This contradicts ψ(ϑ) ∈ O ψ [[T ]]. Hence we have r(T ) = 0.
Proof of Theorem 1.5. Recall that we are assuming G is a pro-p-group. As we explained in Remark 3.15, we may and do assume K ∩ k ∞ = k. Put H = Gal(K ∞ /k ∞ ), which is a finite abelian p-group by our assumption. By fixing a topological generator of Gal(K ∞ /K), we may and do identify R with By the formula (3.3) and Lemma 3.16, we have θ mod and Then we know that the µ-invariant of F (T ) is zero in the sense of [2, §2]. Moreover, since S ′ is non-empty and θ S is a pseudo-measure, we have G( by Lemma 1.6(1). Since S ′ = ∅ and θ Sp is a pseudo-measure, we have θ mod S ∈ R. Second suppose that S ram (K/k) ⊂ S p and ψ is faithful as a character of H, so H is cyclic in this case. By Lemma 1.6(1), we may assume that S = S ram (K/k) ∪ S p . Since ψ is then ramified at all v ∈ S ′ = S ram (K/k) \ S p , we have ψ(θ mod S ) = ψ(θ S ) by Definitions 1.3 and 1.4. Since ψ is non-trivial and θ S is a pseudo-measure, the right hand Finally suppose that ψ is not faithful as a character of H.

A strategy for computing Fitt
In this section, we look at methods of computing Fitt [1] R (Z 0 S ′ ). The motivation for this is fairly obvious: without any concrete information on this Fitting ideal, our main result would remain rather abstract and impractical. As one application among others, we will reprove in Section 5 a previous result of the third author [14].
Throughout this section, we assume that K/k is a p-extension.

4.1.
The algebraic problem. We propose an algebraic problem whose full understanding (if it can be achieved) will help a lot in computing Fitt [1] R (Z 0 S ′ ). Let p be a prime number and G a finite abelian p-group. We denote the group ring by R = Z p [G]. Take subgroups G 1 , . . . , G r of G with r ≥ 1. We consider the R-module and the R-submodule Z 0 of Z, defined by the exact sequence where the surjective map is the augmentation map. Now the algebraic problem is the following. In the subsequent sections, we will try to solve this problem. Before that, we explain how to utilize a solution of Problem 4.1 for a computation of Fitt [1] R (Z 0 S ′ ). In the arithmetic situation as in Theorem 0.1, let K n be the n-th layer of K ∞ /K. Take n sufficiently large such that no places in S ′ split in K ∞ /K n . With this choice, put G = Gal(K n /k) and let G v be the decomposition group of v in G. Then we can identify Let us moreover put R = Z p [G], and note that pd R (R) ≤ 1, since R = R/(γ p n K − 1)R with γ K ∈ Γ K = Gal(K ∞ /K) a topological generator.
For a matrix B over a commutative ring and a non-negative integer e, let Min e (B) denote the ideal generated by the e-minors of B.

Proposition 4.2. In the above situation, let
0 be an exact sequence over R. We identify A with a matrix and take a lift A of A over R.

Then we have
Proof. The short exact sequence (Cok(A)). Furthermore there is an exact sequence By the definition of Fitting ideals, we obtain

4.2.
How to attack Problem 4.1: an idea. We explain a very general idea which will be essential in the subsequent sections. We shall construct a homological complex D of R-modules such that: (a) the components of D are finitely generated free R-modules; (b) D is located in degrees ≥ 0, that is, all components in degree ≤ −1 are zero (remember that the numbering is homological, so the degrees increase when we go to the left); (c) D is exact except in degree 1, and We have to warn our readers right away that we have to write the degrees as superscripts, not as subscripts (which would be much more customary), in our homological complexes. We will need the subscript position later, to distinguish different complexes of similar type.
Such a complex D gives a way to compute Fitt [1] R (Z 0 S ′ ) from a complex D as follows.

Proposition 4.3. Consider the arithmetic situation as in Proposition 4.2. Let
be a complex over R satisfying the above conditions (a)(b)(c). Put t n = rank R (D n ) for n ≥ 0.
We regard A as a matrix over R by choosing bases of D 3 and D 2 , and take a lift A over R.
Then we have Proof. By the properties (b) and (c), we have exact sequences By the first sequence, the module Ker(d 1 ) is free of rank t 1 − t 0 . Now Proposition 4.2 implies the assertion.
In order to construct such a complex D, we will first construct complexes C and C i (i = 1, . . . , r) which have similar properties. More precisely, (a) and (b) will hold without change; and (c) is modified to (c ′ ): C and C i are exact except in degree 0, satisfying H 0 (C) ≃ Z p , and H 0 (C i ) = Z p [G/G i ] for i = 1, . . . , r.
Moreover, we will construct a morphism of complexes which induces the augmentation homomorphism in degree 0 homology. Then, roughly speaking, D can be constructed by either taking the mapping cone of f or the cokernel of f ; the choice between these two options will depend on the precise setting.
Moreover, define ε : B 0 (G) → Z p by sending the empty tuple (which is by definition the only basis element of B 0 (G)) to 1.
The following is well known.
Proposition 4.5. The sequence Therefore, the complex satisfies the conditions described above. Similarly, for each 1 ≤ i ≤ r, the complex satisfies the required conditions, because then For every i ∈ {1, . . . , r} there is a natural morphism C i → C induced by where the vertical arrow in degree n sends the basis element (g 1 , . . . , g n ) ∈ B n (G i ) to the "same" basis element (g 1 , . . . , g n ) ∈ B n (G). Thus we have a morphism f as claimed in (4.1).
Let D = Cone(f ) be the mapping cone of f , so we have an exact sequence Then the conditions (a)(b)(c) for D hold by construction.

4.4.
A first special setting. As we said above, the above construction tends to lead to a free resolution of Z 0 with extremely unwieldy terms. Motivated by this, we consider in this subsection the (fairly rare) case where G = G 1 × · · · × G r and moreover G i is cyclic for each i. In this case, we shall obtain an alternative construction of C, C i , and D involving much smaller modules.

4.4.1.
Definition of C. The construction of C will closely follow the approach of the first and the third author in [7]. For all i, we choose a generator σ i of G i , and we denote by Then E i is exact except for degree 0, and H 0 (E i ) = Z p . We define which satisfies the conditions (a)(b)(c), since H 0 (C) = Z p ⊗ Zp · · · ⊗ Zp Z p = Z p .
The structure of C is fully described in [7]. In particular, for each n ≥ 0, the n-th component of C is the free R-module on the set of monomials {x l 1 . . . x ln | 1 ≤ l 1 ≤ · · · ≤ l n ≤ r}.

Definition of
Note that the structure of C i is quite easy to understand. In a way, C i arises from E i via base change from the smaller group ring Z p [G i ] to the big group ring Z p [G]. For each n ≥ 0, the n-th component of C i is a free R-module of rank one, and the differentials are "the same" as in the complex E i . The definition of the complexes C i is arranged in this particular way in order to make it possible to construct a map f of complexes below. 4.4.3. Definition of f . For each j = i, we have a unique morphism E ′ j → E j which is identity in degree 0. Together with the identity morphism E i → E i , we get a morphism C i → C. Thus we obtain a morphism f as claimed in (4.1).
It is not hard to see that in degree n, the morphism f sends the canonical basis element of C i to the basis element x n i of C n . This is a very special basis element, labeled by a power of one single variable x i ; recall that the general basis element of C n is labeled by a general monomial of degree n in x 1 , . . . , x r .

Definition of D.
We can take the mapping cone of f to construct D as in Section 4.3. However, in our special case, it is much more efficient to consider the "cokernel" of f . The quotation marks are supposed to draw attention to the minor problem that f is not injective in degree 0. In fact, in degree 0, the morphism f looks like Σ : On the other hand, in all strictly positive degrees, the morphism f : r i=1 C i → C is fortunately injective and the cokernel is free over R = Z p [G]. These facts can be read off from the description of f just given, in terms of the bases.
To avoid the minor problem in degree 0, we modify C to → 0] is concentrated in degrees 1 and 0. Then we can extend f to an injective morphism f ′ : satisfies the conditions (a)(b)(c). More precisely, we stipulate that in degree 0, the new component of f ′ , that is, the additional morphism of complexes r i=1 C i → Y , is simply the identity morphism on r i=1 Z p [G]. This completes the construction of D. Moreover the construction gives us a nice description of D at no expense at all. Indeed, the component D n of D in any degree n ≥ 2 is the free R-module on the set {x l 1 . . . x ln | 1 ≤ l 1 ≤ · · · ≤ l n ≤ r} \ {x n 1 , . . . , x n r }. (Here is a catch phrase describing this set: Take all monomials of degree n and throw out the pure powers.) The structure morphisms of D are canonically induced by those of C.

Arithmetic situation. Let us get back to the arithmetic situation.
We assume K ∩ k ∞ = k and also that every v ∈ S ′ is inert in K ∞ /K. By this assumption, we can put G = Gal(K/k) (see the text before Proposition 4.2). We label S ′ = {v 1 , . . . , v r }, put the decomposition groups G i = G v i for i = 1, . . . , r, and we suppose that G = G 1 ×· · ·×G r and each G i is cyclic. Then, using the above information, it is possible to obtain much more precise information on Fitt R (Z 0 S ′ ). Example 4.6. Here is a modest numerical example. We take p = 3, k = Q and K the compositum of the cubic extensions of Q with conductor 7 and 223 respectively. The primes 7 and 223 stay inert in k ∞ because they are congruent to 1 modulo 3 but not modulo 9.
We take S ′ = {7, 223}. The group G = Gal(K/Q) is the product of the two decomposition groups at 7 and 223, since these two primes are cubic residues modulo each other. This shows that we are indeed in the setting considered in this subsection with r = 2. We have not made any effort to determine the modified Stickelberger element (even approximately), but this should be possible, in principle, by going over to the minus side and using classical cyclotomic Stickelberger elements.
Returning to the general case, we construct the complex D as above. Then the ranks t n = rank R (D n ) satisfy t 0 = 1, t 1 = r, t 2 = r(r − 1)/2. Let A be the matrix that describes the differential d 3 : D 3 → D 2 in the complex D constructed above, in the canonical bases. Then the rows (columns) of A are indexed by the monomials in x 1 , . . . , x r of degree 3 (degree 2 respectively), with the extra restriction that the pure powers x 3 i (x 2 i resp.) are omitted. In [7] and [9], the differential C 3 → C 2 was studied. It was described by a matrixM r , whose rows and columns were indexed in exactly the same fashion, with the only difference that the pure cubes and squares were still present as labels. Since D is obtained as a homomorphic image of C as discussed above, it is very simple to describe the new matrix A. It is obtained fromM r in [7], [9] by eliminating all rows with labels x 3 i and all columns with labels x 2 j (with 1 ≤ i, j ≤ r).
The entries of A are all of the form ν i or τ i (neglecting signs), where we put ν i = N G i , the norm element, and τ i = σ i − 1 for compatibility with [7]. Note that τ i ν i = 0. Since we are assuming K ∩ k ∞ = k, the natural map H = Gal(K ∞ /k ∞ ) → G = Gal(K/k) is an isomorphism, so the matrix A can be lifted to a matrix A uniquely as a matrix over Z p [H]. We denote this matrix A simply by A. Similarly by abuse of notation, we use the same symbols σ i , ν i , τ i in H, which are the canonical lifts from G. Then the entries of the lifted matrix A = A have the same description in terms of ν i and τ i . Now by Proposition 4.3, we have T −e Min e (A), where we put T = γ K − 1.
Let us first give examples where r = 2 or r = 3. When we write presentation matrices in this section, we use the row vector convention.
Example 4.8. Suppose r = 3. We then get: Let us compute the 3-minors of A. To begin with, we easily see that neither ν 2 1 τ 2 nor ν 2 1 τ 3 appears as a minor. But ν 1 ν 2 τ 3 does appear. By a not overly tedious complete verification, one obtains the following result. Let be the ideal of R generated by the given list of elements. Then we find that Thus we obtain in the case r = 3: Let us now go back to general r. In the situation of the paper [9], which deals with the somewhat larger matrixM r instead of A, the minors ofM r are completely determined for general r. Since A is a submatrix ofM r , this certainly gives an upper bound for Fitt [1] R (Z 0 ), which is already something to start with. We will be a little more ambitious and state a conjectural equality in a moment.
Our conjectural decription involves the notion of "admissible" ν-monomials, which was already crucial in [9]. We quickly review the definition here and refer to that paper ([9, Section 1.2]) for more information.
A ν-monomial is, by definition, any expression ν f 1 1 . . . ν fr r with exponents f 1 , . . . , f r ≥ 0. Similarly, one defines τ -monomials. A (τ, ν)-monomial is, by definition, a product z = xy with x a τ -monomial and y a ν-monomial. We also demand that the events τ i appears in x, and ν i appears in y do not happen simultaneously for any i (because if that happens, we get z = 0). The ν-part of such a (τ, ν)-monomial z is simply y; we write ν(z) = y. Definition 4.9. We say that a ν-monomial y = ν f 1 1 . . . ν fr r is admissible if there is a permutation σ of {1, 2, . . . , r} satisfying for any 1 ≤ i ≤ r.
Putting it very roughly: to be an admissible monomial, the exponents must not be distributed too unevenly. Suppose r = 4 and let us look at ν-monomials of degree 6. Then for instance no exponent in an admissible monomial can exceed 3; but this does not suffice, since for example ν 3 1 ν 3 2 is not admissible either. On the other hand, ν 3 1 ν 2 2 ν 3 is admissible, and many other ν-monomials as well.
The following is proved in [9]. Theorem 4.11 ([9]). For all e ≥ 0, the e-th minor ideal Min d (M r ) is generated by all (τ, ν)-monomials of degree e whose ν-part is admissible.
To state our conjecture, a little extra notation will be useful. For any ν-monomial y, let n(y) denote the number of indices i such that ν i occurs in y. Define M(d, ℓ) to be the set of (τ, ν)-monomials z of degree d such that ν(z) is admissible and n(ν(z)) ≥ ℓ (more simply put: which contain at least ℓ different ν i ). Finally, recall that t 2 = r(r − 1)/2. Note that for all e ≥ t 2 − r − 1, the second argument of M(e, −) occurring here is nonpositive, so the restricting condition concerning the number of ν i that must show up in the monomials is vacuously satisfied.
By the equation (4.3), this conjecture implies the following statement. We could call it the weak form of the above conjecture (which then would be called the strong form): Conjecture 4.13. We keep the same notation and assumptions. Then the union of the following sets generates the ideal Fitt [1] R (Z 0 S ′ ): Here the last exponent t 2 +1−r can be rewritten (r−1)(r−2)/2, and the last set M(0, r−1−t 2 ) only consists of the trivial monomial 1.
It is not difficult to see that these conjectures agree with the result of our calculations in the cases r = 2 (Example 4.7) or r = 3 (Example 4.8). We think that we have verified it completely for r = 4 as well. However, a general proof for all r would probably require to revisit most of the technical arguments in [9], and we haven't tried to do this.
We point out that while the latter conjecture seems logically weaker than the former, it is hard to image a proof (even partial) of the weaker conjecture which does not pass through a proof of the strong one.

4.5.
Another special setting. In this subsection, apart from the setting in Subsection 4.4, we suppose that r = 1, so we are given only one subgroup G 1 ⊂ G. In this case, we can decompose G as an abelian group into where all factors G (j) are non-trivial cyclic. Hence s is the p-rank of G. Moreover, we can arrange the decomposition so that . Note that this is where we use r = 1; in the general case, we cannot expect a decomposition of G into cyclic factors that is compatible with all subgroups G i . Remark 4.14. An important remark is in order. In our arithmetical setting for abelian p-extension K/k, G 1 is always the decomposition group G v of a tamely ramified prime v in some abelian extension K n /k with group G. Since the ramification group of such a prime is always cyclic, G v is always generated by at most two elements as an abelian p-group. Thus in the last formula we can arrange things so as to have 1 . But in practice, this reduction is possibly not too helpful, see a further comment below, where we do some calculations for the case s = 2.
4.5.1. Definition of C, C 1 , f, D. Exactly as in Section 4.4.1, we fix a generator σ (j) of G (j) , for 1 ≤ j ≤ s. We define a complex E (j) for every j by (4.2), with G i replaced by G (j) , and a complex C = E (1) ⊗ · · · ⊗ E (s) . Moreover, we may define E 1 → E (j) explicitly. We stipulate that f is identity in every even degree, and f is multiplication by µ (j) = 1 + σ (j) + · · · + (σ (j) ) m j −1 in every odd degree. We omit some easy verifications that certain squares commute, making f into a morphism of complexes.
Hence we obtain a morphism f : C 1 → C. By taking the mapping cone of f , it is again possible to construct a complex D satisfying the conditions (a)(b)(c). However, the matrices representing the maps C n 1 → C n in each degree n will contain entries involving factors µ (j) . Since these elements are in general neither zero nor units in R, it cannot be expected (and indeed does not happen in general) that the degree-wise cokernels of f are again free over R. So it does not seem possible to replace the cone of f by the cokernel, and this makes the calculations more difficult. 4.5.2. Arithmetic situation. As in Subsubsection 4.4.5, we assume K ∩ k ∞ = k and that every v ∈ S ′ is inert in K ∞ /K. Put G = Gal(K/k), which admits an isomorphism from H = Gal(K ∞ /k ∞ ). Suppose that S ′ consists of a single place v 1 and put G 1 = G v 1 . Let s be the p-rank of G. Example 4.15. First suppose s = 1; we omit all super-and subscripts j, since j = 1 is the only value. For example, σ is a generator of G, τ = σ − 1, and m = (G : G 1 ). Let ν be the norm element of G 1 .
One can check that the ranks t n of D n are given by t n = 2 for n ≥ 1 and t 0 = 1. The differential from D 3 to D 2 is given by the square matrix ν 1 0 τ . Let T = γ K − 1 be defined as in Subsubsection 4.4.5. By Proposition 4.3 again, we obtain: . For this case it is quite possible (and actually quicker) to calculate the left hand side directly, by finding a simple resolution of Z 0 by hand; we did it, and the results agree. Example 4.16. Next suppose s = 2. This case should be prototypical in a certain way, since (as pointed out in Remark 4.14) we can always assume that G 1 has only two cyclic summands G (1) 1 and G (2) 1 ; of course s can be larger than 2. Even letting s = 2, we found the calculation rather cumbersome. Again, we need to take the cone, not the cokernel.
The R-ranks t n of D n are given by t n = 2n + 1 for n ≥ 0. We determined the matrix A of the differential D 3 → D 2 . To write it down we have to review notation: For j = 1, 2, σ (j) is a generator of G (j) and m j = (G (j) : G where n j = ord(σ (j) ) = ♯G (j) . The outcome is the following 7 times 5 matrix: up to sign. Using ideals generated by the minors of this matrix A, it is again possible to write down an expression that gives Fitt [1] R (Z 0 ). But we have strong doubts whether such an expression would be very enlightening. All 5-minors are 0 (unless we made an error), but the minors of degree 4 and less seem to be so numerous and hard-to-describe that it is not really helpful to write them all down; in fact we gave up somewhere along the way.

Description of the Fitting ideal in the case that Gal(K/k) is cyclic
In this section, we concentrate on cyclic p-extensions K/k and we describe Fitt R (X Sp ) more explicitly from Theorem 0.1. We deduce two results from Theorem 0.1 (see Theorems 5.1 and 5.4). Both are a generalization of the main result of the third author in [14] where the case [K : k] = p was studied. We keep the notations of Theorem 0.1. As in Remark 3.15, we will be assuming throughout K ∩ k ∞ = k. Moreover, we assume that the µ-invariant of X Sp vanishes.
For each finite place v of k outside p, as in Subsection 1.4, ) is the map induced by the multiplication by N Tv = σ∈Tv σ. Recall that σ v ∈ G/T v is the Frobenius automorphism (Definition 1.1).
Put Γ k = Gal(k ∞ /k) and H = Gal(K ∞ /k ∞ ). Then N H = h∈H h also induces a map , which we denote by ν H . Choose a topological generator γ of Γ k .
Theorem 5.1. Suppose that K/k is a cyclic p-extension and K ∩ k ∞ = k. Assume that we have a place v * ∈ S ′ such that v * is totally ramified in K/k and that G v * ⊃ G v holds for all v ∈ S ′ . Then we have Note that, since v * is totally ramified in K/k, we have ν H = ν v * .
Before we prove this theorem, let us deduce a corollary which is exactly Theorem 0.1 (1) in [14].
In general, for the cyclotomic Z p -extension k ∞ /k, let k n denote its n-th layer. For each finite place v of k outside p, let n v be the non-negative integer such that the decomposition field of v in k ∞ /k is k nv . This use of n v cancels an earlier meaning of n v in the proof of Proposition 3.13. Note that, if v is totally ramified in K ∞ /k ∞ , then σ v is an element of Γ k = Gal(k ∞ /k) which generates the same subgroup as γ p nv .
Corollary 5.2 ([14, Theorem 0.1(1)]). Suppose that K/k is a cyclic extension of degree p and that K ∩ k ∞ = k. Take S = S p ∪ S ram (K/k) and suppose that S ′ = S \ S p = ∅ 1 . Then we have Proof. In this situation, any v ∈ S ′ is totally ramified in K/k since its degree is p. Therefore, we have ν v = ν H . Moreover, if we take v * ∈ S ′ such that n v * is the minimum of the n v , v ∈ S ′ , then v * satisfies the condition in Theorem 5.1. Then it is not hard to see that Thus Theorem 5.1 implies the corollary. • The statement of [14] concerns (A ω K(µ p ∞ ) ) ∨ , where A K(µ p ∞ ) is the inductive limit of the p-parts of the ideal class groups of K(µ p n ) and ω denotes the Teichmüller character. Thanks to the Kummer duality between A ω K(µ p ∞ ) and X Sp = X Sp (K ∞ ), we can translate the statement of [14] into a statement about X Sp . • In the present paper, the hypothesis that the µ-invariant of X Sp vanishes is slightly weaker than that of [14]. • The modified Stickelberger element ϑ K(µ p ∞ ) is defined in [14, (2.3.4)], using a modifying factor ξ. This factor ξ corresponds to our modifying factor v∈S ′ f −1 v .
Proof of Theorem 5.
By Lemma 2.11, we have Fitt [1] R (Z 0 S ′ ) = Fitt since shifted Fitting ideals are multiplicative on direct sums. The second term in the right hand side is computed in Proposition 1.8. For the first term, it is enough to show Fitt [1] R (Z p [Gal(k n /k)] 0 ) = 1, ν H γ − 1 γ p n − 1 for any non-negative integer n.
In the rest of this calculation we will neglect some signs; this will play no role in the calculation of Fitting ideals via minors of certain matrices, and it spares us the effort of being precise about the signs of morphisms in the tensor product of complexes. These sign questions are certainly important in many settings, but for us they are inessential and would mess up some arguments.
We apply the arguments of Sections 4.4 and 4.5 to G = Gal(K n /k) = Gal(K n /k n ) × Gal(K n /K).
Take a generator δ of H = Gal(K ∞ /k ∞ ), which is identified with Gal(K n /k n ) ⊂ G. Let γ K ∈ Gal(K ∞ /K) be the lift of γ ∈ Γ k . As in Section 4.4, define complexes and C = [. . .

K
We have a natural injective homomorphism C 1 → C whose cokernel D looks like . . are obtained by removing both the final row and the final column of d 2 , d 3 , . . . . From this complex D, we obtain an exact sequence Explicitly we can write down and it is easy to see that the ideal generated by its 2 × 2 minors is (γ p n K − 1, N H (γ K − 1)). Hence Fitt [1] R (Z p [Gal(k n /k)] 0 ) = (γ p n K − 1) −1 (γ p n K − 1, N H (γ K − 1)) = 1, ν H γ − 1 γ p n − 1 .
This completes the proof.
Finally there is another variant, where there is no privileged place v * and still the degree of the cyclic p-extension K/k can be arbitrary. The proof again uses the techniques of Section 4; but this time the reduction lemma 2.11 cannot be used. So one has to work with the full direct sum of "local" complexes C i , not just one of them, and this makes it inevitable to work with the cone, not the cokernel, of the map i C i → C of complexes. This makes the proof more complicated, but it will be given in very explicit terms.
Theorem 5.4. Suppose that K/k is a cyclic p-extension with K ∩ k ∞ = k as before, and let δ denote a generator of H = Gal(K ∞ /k ∞ ). Assume that the inertial degrees of all v ∈ S ′ in 4.3, we have short exact sequences 0 → Ker(d 1 ) → D 1 → D 0 → 0 and 0 → Cok(d 3 ) → Ker(d 1 ) → Z 0 S ′ → 0. Since we have pd R (D n ) ≤ 1 for any degree n, the first exact sequence implies pd R (Ker(d 1 )) ≤ 1 and Then the second exact sequence implies (5.1) Fitt (Cok(d 3 )).
By the description of d 3 above, we easily see that the module Cok(d 3 ) has a presentation as an R-module (all blank entries being zero). To obtain Fitt R (Cok(d 3 )), we have to compute the maximal minors of B. As a consequence, we shall show that Fitt R (Cok(d 3 )) is generated by the following elements: where J runs through all proper subsets of {1, 2, . . . , r}.
Let V run through all subsets of {1, 2, . . . , 2r + 2} of cardinality r + 1, and let d V be the determinant of the submatrix of B picking up the v-th rows for v ∈ V (we ignore the sign throughout). For each 1 ≤ i ≤ r, the i-th column in B is zero except for the i-th and (r + 1 + i)-th rows. Hence d V = 0 only if, for each 1 ≤ i ≤ r, we have either i ∈ V or r + 1 + i ∈ V . We divide the argument into two cases. Case 1. There is (a unique) 1 ≤ l ≤ r such that both l ∈ V and r + 1 + l ∈ V hold. In this case, putting J = (V ∩ {1, 2, . . . , r}) \ {l}, we can see that where i ∈ J means i runs through {1, 2, . . . , r} \ J. In this way, we obtain the third family of elements in (5.2). Case 2. For each 1 ≤ i ≤ r, exactly one of i ∈ V or r + 1 + i ∈ V holds. Put J = {1, 2, . . . , r} ∩ V . Then exactly one of r + 1 ∈ V or 2r + 2 ∈ V holds, and accordingly we obtain These elements are in the ideal generated by (5.2). Moreover, letting V = {1, 2, . . . , r, r + 1} and V = {1, 2, . . . , r, 2r + 2} respectively, we can produce the first two elements in (5.2) among d V .