On the non-abelian Brumer–Stark conjecture and the equivariant Iwasawa main conjecture

We show that for an odd prime p, the p-primary parts of refinements of the (imprimitive) non-abelian Brumer and Brumer–Stark conjectures are implied by the equivariant Iwasawa main conjecture (EIMC) for totally real fields. Crucially, this result does not depend on the vanishing of the relevant Iwasawa μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document}-invariant. In combination with the authors’ previous work on the EIMC, this leads to unconditional proofs of the non-abelian Brumer and Brumer–Stark conjectures in many new cases.

. The material presented up until this point then allows us to state the main theorem of this article (discussed above) in Sect. 5. The next three sections are then devoted to the proof of this result. First we present further auxiliary results on Iwasawa algebras and the EIMC in Sect. 6. We then complete the proof in Sects. 7 and 8 by working with complexes at the finite and infinite levels, respectively. In Sect. 9 we recall the notion of hybrid p-adic group rings introduced in [24]. This notion was further developed in [25] where it played a key role in obtaining the first unconditional proofs of the EIMC in cases where the vanishing of the relevant μ-invariant is not known. Finally, in Sect. 10 we combine these results on the EIMC with the main result of this article to give unconditional proofs of the non-abelian Brumer and Brumer-Stark conjectures in many new cases.

Notation and conventions
All rings are assumed to have an identity element and all modules are assumed to be left modules unless otherwise stated. We fix the following notation: R × the group of units of a ring R ζ(R) the centre of a ring R Ann R (M) the annihilator of the R-module M M m×n (R) the set of all m × n matrices with entries in a ring R ζ n a primitive nth root of unity K ∞ the cyclotomic Z p -extension of the number field K μ K the roots of unity of a field K cl K the class group of a number field K K c an algebraic closure of a field K K + the maximal totally real subfield of a field K embeddable into C Irr F (G) the set of F-irreducible characters of the (pro)-finite group G (with open kernel) where F is a field of characteristic 0 χ the character contragredient to χ 2 Algebraic preliminaries

Algebraic K-theory
Let R be a noetherian integral domain with field of fractions E. Let A be a finite-dimensional semisimple E-algebra and let A be an R-order in A. Let PMod(A) denote the category of finitely generated projective (left) A-modules. We write K 0 (A) for the Grothendieck group of PMod(A) (see [10,Sect. 38]) and K 1 (A) for the Whitehead group (see [10,Sect. 40]). Let K 0 (A, A) denote the relative algebraic K -group associated to the ring homomorphism A → A. We recall that K 0 (A, A) is an abelian group with generators [X , g, Y ] where X and Y are finitely generated projective A-modules and g : E ⊗ R X → E ⊗ R Y is an isomorphism of A-modules; for a full description in terms of generators and relations, we refer the reader to [45, p. 215]. Moreover, there is a long exact sequence of relative K -theory (see [45,Chapter 15]) The reduced norm map nr = nr A : A → ζ(A) is defined componentwise on the Wedderburn decomposition of A and extends to matrix rings over A (see [9, Sect. 7D]); thus it induces a map K 1 (A) → ζ(A) × , which we also denote by nr. Let C b (PMod(A)) be the category of bounded complexes of finitely generated projective A-modules. Then K 0 (A, A) identifies with the Grothendieck group whose generators are [C • ], where C • is an object of the category C b tor (PMod(A)) of bounded complexes of finitely generated projective A-modules whose cohomology modules are R-torsion, and the relations are as follows: in C b tor (PMod(A)) (see [49,Chapter 2] or [44,Sect. 2], for example). Let D(A) be the derived category of A-modules. A complex of A-modules is said to be perfect if it is isomorphic in D(A) to an element of C b (PMod(A)). We denote the full triangulated subcategory of D(A) comprising perfect complexes by D perf (A), and the full triangulated subcategory comprising perfect complexes whose cohomology modules are R-torsion by D perf tor (A). Then any object of D perf tor (A) defines an element in K 0 (A, A). In particular, a finitely generated R-torsion A-module M of finite projective dimension considered as a complex concentrated in degree 0 defines an element [M] ∈ K 0 (A, A).

Denominator ideals
Let R be a noetherian integrally closed domain with field of fractions E. Let A be a finitedimensional separable E-algebra and let A be an R-order in A. We choose a maximal R-order M such that A ⊆ M ⊆ A. Following [23,Sect. 3.6], for every matrix H ∈ M b×b (A) there is a generalised adjoint matrix H * ∈ M b×b (M) such that H * H = H H * = nr(H ) · 1 b×b (note that the conventions in [23,Sect. 3.6] slightly differ from those in [29]). IfH ∈ M b×b (A) is a second matrix, then (HH ) * =H * H * . We define The first claim of the following result is a special case of [23,Proposition 4.4]. The second claim then follows easily from (2.3). The importance of the ζ(A)-module H(A) comes from its relation to non-commutative Fitting invariants, which we introduce now.

Non-commutative Fitting invariants
For further details on the following material we refer the reader to [29] and [23]. Let A be a finite-dimensional separable algebra over a field E and A be an R-order in A, where R is an integrally closed complete commutative noetherian local domain with field of fractions E. For example, if p is a prime we can take A to be a p-adic group ring Z p [G] where G is a finite group or to be a completed group algebra Z p G where G is a one-dimensional p-adic Lie group.
Let X and Y be two ζ(A)-submodules of an R-torsionfree ζ(A)-module. Then X and Y are said to be nr(A)-equivalent if there exists a positive integer n and a matrix U ∈ GL n (A) such that X = nr(U ) · Y . We denote the corresponding equivalence class by [X ] nr(A) . We say that X is nr(A)-contained in Y (and write [X ] nr(A) ⊆ [Y ] nr(A) ) if for all X ∈ [X ] nr(A) there exists Y ∈ [Y ] nr(A) such that X ⊆ Y . Note that it suffices to check this property for one X 0 ∈ [X ] nr(A) . We will say that x is contained in We identify the homomorphism h with the corresponding matrix in M a×b (A) and define In the case a = b we call (2.4) a quadratic presentation. The Fitting invariant of h over A is defined to be We call Fitt A (h) a Fitting invariant of M over A. One defines Fitt max A (M) to be the unique Fitting invariant of M over A which is maximal among all Fitting invariants of M with respect to the partial order "⊆". If M admits a quadratic presentation h, we set (2.5) which can be shown to be independent of the chosen quadratic presentation. Now let C • ∈ D perf tor (A) and recall from Sect. 2.1 that C • defines an element [C • ] in the relative algebraic K -group K 0 (A, A). Recall the long exact sequence of K -theory (2.1). If Note that this is well-defined by the exactness of (2.1). ) (this is the case in the situation of (2.2), for example) it is straightforward to show that To put this in context, we note that if C • is isomorphic in D(A) to a complex P −1 → P 0 concentrated in degrees −1 and 0 such that P −1 and P 0 are both finitely generated R-torsion A-modules of projective dimension at most 1, then We list some properties of non-commutative Fitting invariants which we will use later.
Proof The first equality follows from (2.8). We consider A and B as complexes concentrated in degree 0. Then we have a short exact sequence of complexes

Ray class groups
Let L/K be a finite Galois extension of number fields with Galois group G. For each place v of K we fix a place w of L above v and write G w and I w for the decomposition group and inertia subgroup of L/K at w, respectively. When w is a finite place, we choose a lift φ w ∈ G w of the Frobenius automorphism at w; moreover, we write P w for the associated prime ideal in L and ord w for the associated valuation. For any set S of places of K , we write S(L) for the set of places of L which lie above those in S. Now let S be a finite set of places of K containing the set S ∞ = S ∞ (K ) of archimedean places and let T be a second finite set of places of K such that S ∩ T = ∅.

Equivariant Artin L-values
Let S be a finite set of places of K containing S ∞ . Let Irr C (G) denote the set of complex irreducible characters of G. For χ ∈ Irr C (G), let V χ be a left C[G]-module with character χ. If T is a second finite set of places of K such that S ∩ T = ∅, we define We set denotes the anti-involution induced by g → g −1 for g ∈ G.
The functions S,T (s) are the so-called (S, T )-modified G-equivariant L-functions and we define Stickelberger elements Note that a priori we only have θ T S ∈ ζ(C[G]), but by a result of Siegel [43] we know that θ T S in fact belongs to ζ(Q [G]). If T is empty, we abbreviate θ T S to θ S . Let p be a prime and let ι : C p → C be a field isomorphism. Then the image of θ T S under the canonical maps and this is independent of the choice of ι. We shall henceforth consider θ T S as an element of ζ( 3) when convenient. Moreover, we shall often drop ι and ι −1 from the notation.

Reduction to CM-extensions
Let χ ∈ Irr C (G). The order of vanishing formula for L S (s, χ) at s = 0 (see [46,Chapter I,Proposition 3.4]) shows that if either χ is non-trivial and S contains an (infinite) place v such that V G w χ = 0 or χ is trivial and |S| > 1 then the χ-part of θ T S vanishes. Hence if θ T S is non-trivial, precisely one of the following possibilities occurs: (i) K is totally real and L is totally complex, (ii) K is an imaginary quadratic field, L/K is unramified and S = S ∞ or (iii) L = K = Q and S = S ∞ . In case (iii), the Brumer-Stark conjecture is trivial. For case (ii), see [17,Remark 6.3] for G abelian; the situation in which G is non-abelian has been considered recently by Nomura [35]. Finally, case (i) can often be reduced to the case that L is a CM-field (see [17,Proposition 6.4] for G abelian; the same argument works for general G under the assumptions of [34, Proposition 4.1] with r = 0). Therefore, we shall henceforth assume that L/K is a CM-extension, that is, L is a CM-field, K is totally real and complex conjugation induces a unique automorphism j of L lying in the centre of G.

The non-abelian Brumer and Brumer-Stark conjectures
Assume that L/K is a CM-extension and let S ram = S ram (L/K ) be the set of all places of K that ramify in L/K .
Hypothesis Let S and T be finite sets of places of K . We say that Remark 3.1 Condition (iii) means that there are no roots of unity of L congruent to 1 modulo all primes in T (L). In particular, this will be satisfied if T contains primes of two different residue characteristics or at least one prime of sufficiently large norm.
We choose a maximal order . For a fixed choice of S we define A S to be the ζ(Z[G])-submodule of ζ(M(G)) generated by the elements δ T (0), where T runs through the finite sets of places of K such that Hyp(S ∪ S ram ∪ S ∞ , T ) is satisfied. The following conjecture was formulated in [30] and is a non-abelian generalisation of Brumer's conjecture.   Burns [6] has also formulated a conjecture which generalises many refined Stark conjectures to the non-abelian situation. In particular, it implies Conjecture 3.2 (see [6, Proposition 3.5.1]).

Remark 3.5
For α ∈ L × we define We call α an anti-unit if α 1+ j = 1. Let ω L := nr(|μ L |). The following is a non-abelian generalisation of the Brumer-Stark conjecture ([30, Conjecture 2.7]). Then ω L · θ S ∈ I(G) and for each x ∈ H(G) and each fractional ideal a of L, there is an anti-unit α = α(x, a, S) ∈ L × such that a x·ω L ·θ S = (α) and for each finite set T of primes of K such that Hyp(S ∪ S α , T ) is satisfied there is an for each z ∈ H(G).

A criterion involving Pontryagin duals and Fitting invariants
For an abstract abelian group A we write A ∨ for Hom(A, Q/Z). This induces an equivalence between the categories of abelian profinite groups and discrete abelian torsion groups (see [37,Theorem 1.1.11] and the discussion thereafter). For a finitely generated , and this is endowed with the contragredient G-action For a G-module M we write M + and M − for the submodules of M upon which j acts as 1 and −1, respectively. In particular, we shall be interested in (cl T L,S ( p)) − for odd primes p; we will abbreviate this module to A T L,S when p is clear from context. Note that A T L,S is a finite module over the ring Z p [G] − := Z p [G]/(1 + j). We shall need the following variant of [30, Proposition 3.9]. Proposition 3.9 Let S be a finite set of places of K containing S ram ∪ S ∞ and let p be an odd prime. Suppose that for every finite set T of places of K such that Then both B S(L/K , S, p) and B(L/K , S, p) are true.

Remark 3.10
The containment (3.5) may be considered as a 'dual version' of the so-called strong Brumer-Stark property, which is

Certain one-dimensional p-adic Lie extensions
Let p be an odd prime and let K be a number field. Let L/K be a Galois extension such that L contains the cyclotomic Z p -extension K ∞ of K and [L : K ∞ ] is finite. Then the Galois group G := Gal(L/K ) is a one-dimensional p-adic Lie group. Let H = Gal(L/K ∞ ) and let K = Gal(K ∞ /K ). Let γ K be a topological generator of K Z p . The argument given in [39,Sect. 1] shows there exists a lift γ ∈ G of γ K that induces a splitting of the short exact sequence Thus we obtain a semidirect product G = H where Z p is the pro-cyclic subgroup of G topologically generated by γ . Since any homomorphism → Aut(H ) must have open kernel, we may choose a natural number n such that γ p n is central in G. We fix such an n and set 0 := p n ; hence 0 Z p is contained in the centre of G.

The Iwasawa algebra as an order
The Iwasawa algebra of G is where the inverse limit is taken over all open normal subgroups There is a ring isomorphism R : Hence O (G) is finitely generated as an R-module and is an R-order in the separable E :=

An exact sequence of algebraic K-groups
Specialising (2.1) to the situation A = Q(G) and A = (G) and applying [50, Corollary 3.8] (see [25,Sect. 4.1] for further explanation) we obtain the exact sequence

Remark 4.1
If M is a finitely generated R-torsion (G)-module of projective dimension at most 1, then combining the triviality of ρ in (4.2) with Remark 2.2 shows that M admits a quadratic presentation.

Characters and central primitive idempotents
Fix a character χ ∈ Irr Q c p (G) (i.e. an irreducible Q c p -valued character of G with open kernel) and let η be an irreducible constituent of res G H χ. Then G acts on η as η g (h) = η(g −1 hg) for g ∈ G, h ∈ H , and following [39, Sect. 2] we set .
In fact, every primitive central idempotent of Q c (G) is of this form and e χ = e χ if and only if χ = χ ⊗ ρ for some character ρ of G of type W (i.e. res G H ρ = 1). Let w χ = [G : St(η)] and note that this is a power of p since H is a subgroup of St(η).
Let F/Q p be a finite extension over which both characters χ and η have realisations. Let V χ denote a realisation of χ over F. By [39, Propositions 5 and 6], there exists a unique element γ χ ∈ ζ(Q F (G)e χ ) such that γ χ acts trivially on V χ and γ χ = g χ c χ with g χ ∈ G mapping to γ w χ K mod H and with c χ ∈ (F[H ]e χ ) × . Moreover, γ χ generates a pro-cyclic p-subgroup χ of Q F (G)e χ and induces an isomorphism

Determinants and reduced norms
Following [39, Proposition 6], we define a map where the last arrow is induced by mapping γ χ to γ w χ K . It follows from ibid. that j χ is independent of the choice of γ K and that for every matrix ∈ M n×n (Q(G)) we have which is easily seen to be independent of the choice of γ . Hence the map for all Galois automorphisms σ ∈ Gal(Q c p /Q p ). By [39, Proof of Theorem 8] we have an isomorphism By [39,Theorem 8] such that we obtain a commutative triangle (4.4)

The p-adic cyclotomic character and its projections
Let χ cyc be the p-adic cyclotomic character defined by σ (ζ ) = ζ χ cyc (σ ) for any σ ∈ Gal(L(ζ p )/K ) and any p-power root of unity ζ .
Let ω and κ denote the composition of χ cyc with the projections onto the first and second factors of the canonical decomposition Z × p = μ p−1 × (1 + pZ p ), respectively; thus ω is the Teichmüller character. We note that κ factors through K (and thus also through G) and by abuse of notation we also use κ to denote the associated maps with these domains. We put u := κ(γ K ). For r ∈ N 0 divisible by p − 1 (or more generally divisible by the degree [L(ζ p ) : L]), up to the natural inclusion map of codomains, we have χ r cyc = κ r .

Admissible one-dimensional p-adic Lie extensions
We henceforth assume that L/K is an admissible one-dimensional p-adic Lie extension. In other words, in addition to the existing assumptions that p is an odd prime, K is a number field, L/K is a Galois extension of K such that L contains the cyclotomic Z p -extension K ∞ of K and [L : K ∞ ] is finite, we now further assume that L is totally real. Clearly, this forces K to be totally real, which had not been assumed previously.

Power series and p-adic Artin L-functions
Fix a character χ ∈ Irr Q c p (G). Each topological generator γ K of K permits the definition of a power series G χ,S (T ) ∈ Q c p ⊗ Q p Quot(Z p T ) by starting out from the Deligne-Ribet power series for one-dimensional characters of open subgroups of G (see [12]; also see [1,8]) and then extending to the general case by using Brauer induction (see [18]). One then has an equality where L p,S (s, χ) denotes the 'S-truncated p-adic Artin L-function' attached to χ constructed by Greenberg [18], and where, for irreducible χ, one has Now [39,Proposition 11] implies that is independent of the topological generator γ K and lies in Hom for every χ ∈ Irr Q c p (G).

The = 0 hypothesis
Let S ∞ be the set of archimedean places of K and let S p be the set of places of K above p. Let S ram = S ram (L/K ) be the (finite) set of places of K that ramify in L/K ; note that S p ⊆ S ram . Let S be a finite set of places of K containing S ram ∪ S ∞ . Let M ab S ( p) be the maximal abelian prop-extension of L unramified outside S and let X S = Gal(M ab S ( p)/L). As usual G acts on X S by g · x =gxg −1 , where g ∈ G, andg is any lift of g to Gal(M ab S ( p)/K ). This action extends to a left action of (G) on X S . Since L is totally real, a result of Iwasawa [22] shows that X S is finitely generated and torsion as a ( 0 )-module.

Definition 4.2
We say that L/K satisfies the μ = 0 hypothesis if X S is finitely generated as a Z p -module.
The μ = 0 hypothesis is conjecturally always true and is known to hold when L/Q is abelian as follows from work of Ferrero and Washington [14]. For the relation to the classical Iwasawa μ = 0 conjecture see [25,Remark 4.3], for instance. In the sequel, we shall not assume the μ = 0 hypothesis for L/K except where explicitly stated.

A canonical complex
Let O L,S denote the ring of integers O L in L localised at all primes above those in S. There is a canonical complex It follows from [13, Proposition 1.
In particular, . Note that C • S (L/K ) and the complex used by Ritter and Weiss (as constructed in [39]) become isomorphic in D( (G)) by [33,Theorem 2.4] (see also [47] for more on this topic). Hence it makes no essential difference which of these complexes we use.

A reformulation of the equivariant Iwasawa main conjecture (EIMC)
We can now state a slight reformulation of the EIMC given in [25]. The relation of this version to the framework [7] (as used in [26]) will be discussed in Sect. 6.2. Recall that p is an odd prime and L/K is an admissible one-dimensional p-adic Lie extension.

Conjecture 4.3 (EIMC) There exists
It is also conjectured that ζ S is unique, but we shall not be concerned with this issue here. Moreover, it can be shown that the truth of Conjecture 4.3 is independent of the choice of S, provided that S is finite and contains S ram ∪ S ∞ . Crucially, this version of the EIMC does not require the μ = 0 hypothesis for its formulation. The following theorem has been shown independently by Ritter and Weiss [40] and Kakde [26]. By considering the cases in which the μ = 0 hypothesis is known, we obtain the following corollary (see [25,Corollary 4.6] for further details).

Corollary 4.5
Let P be a Sylow p-subgroup of G. If L P /Q is abelian then the EIMC holds for L/K .
In [25], the present authors prove the EIMC in a number of cases where the μ = 0 hypothesis is not known. In Sect. 10, these will be combined with main results of the present article (see Sect. 5.3) to give unconditional proofs of the non-abelian Brumer-Stark conjecture in many new cases. For now, we only note the following result which relies heavily on a result of Ritter and Weiss [39,Theorem 16].

p-adic Artin L-functions and the interpolation property
Let K be a totally real number field and let G K = Gal(K c /K ) be its absolute Galois group. Let p be an odd prime and let S p denote the set of places of K above p. Let S ∞ denote the set of archimedean places of K and let S be a finite set of places of K such that S p ∪ S ∞ ⊆ S.
Let χ ∈ Irr Q c p (G K ) and let K χ be the extension of K attached to χ; thus χ may be considered as a character attached to a faithful representation of Gal(K χ /K ) and K χ /K is of finite degree since χ has open kernel. Assume that K χ is totally real. Let ι : C p → C be a choice of field isomorphism and let L S (s, ι , which is in fact independent of the choice of ι (compare with the discussion of Sect. 3.2). If χ is one-dimensional then for r ≥ 1 we have where ω : Gal(K (ζ p )/K )) −→ μ p−1 ⊆ Z × p is the Teichmüller character. Using Brauer induction, (5.1) can be extended to the case where χ is of arbitrary degree provided that r ≥ 2 (see [18,Sect. 4]). However, if χ(1) > 1 and r = 1, this argument fails due to the potential presence of trivial zeros. Nevertheless, it seems plausible that the identity holds in general. As both sides are well-behaved with respect to direct sum, inflation and induction of characters, one can show that (5.2) does hold when χ is a monomial character, i.e., a character induced from a one-dimensional character of a subgroup (also see the discussion in [20,Sect. 2]). From recent work of Burns [5, Theorem 5.2 (i)] it follows that the left hand side of (5.2) vanishes whenever the right hand side does.

Equivariant p-adic Artin L-values
Let L/K be a finite Galois CM-extension of number fields with Galois group G. Let j ∈ G denote complex conjugation. Let L + = L j be the maximal totally real subfield of L and let G + = Gal(L + /K ) G/ j . Let p be an odd prime. Recall that χ ∈ Irr C (G) or Irr C p (G) is said to be even when χ( j) = χ(1), and odd when χ( j) = −χ(1). Let S be a finite set of places of K such that S p ∪ S ∞ ⊆ S and let T be a second set of finite places of K such that S ∩ T = ∅. In this situation, we define p-adic Stickelberger elements by A finite group is said to be monomial if each of its (complex) irreducible characters is monomial. Note that every finite metabelian or supersoluble group is monomial by [48,Sect. 4.4, Theorem 4.8 (1)]).

Lemma 5.1 If G + is monomial then θ T p,S = θ T S .
Proof Dropping ι from the notation (as we may), we recall from Sect. 3.2 that Thus for χ ∈ Irr C p (G) it suffices to show that L S (0,χ) = 0 when χ is even and L S (0,χ) = L p,S (0,χω) when χ is odd. Since χ is even if and only ifχ is even, we may replaceχ by χ in the previous sentence. Suppose that χ is odd and set ψ = χω. Then it suffices to show that L p,S (0, ψ) = L S (0, ψω −1 ). Moreover, ψ is even and so may be considered as a character of the monomial group G + . Hence the desired equality now follows from the discussion in Sect. 5.1 and the appropriate substitution of symbols.

Statement of the main theorem and corollary
We are now in a position to state the main results of this article. In Sect. 10, these will be combined with the authors' previous work on the EIMC [25] to give unconditional proofs of the non-abelian Brumer-Stark conjecture in many new cases. The main reason why some version of the μ = 0 hypothesis is assumed in these results is of course to ensure that the EIMC holds (see also [25,Remark 4.3]). However, both results use a version of the EIMC that requires a μ = 0 hypothesis even for its formulation. For this reason our proof is very different from (though partly inspired by) those in [17,33]. Note that in [33,Sect. 4] the identity (5.2) is implicitly assumed to hold.

Certain maps between Iwasawa algebras
We assume the setup and notation of Sects. 4.1, 4.2, 4.3, 4.4, 4.5 and 4.6. Fix a character χ ∈ Irr Q c p (G) and let η be an irreducible constituent of res G H χ. Let F/Q p be a finite extension over which both characters χ and η have realisations. Recalling the notation of Sects. 4.4 and 4.6 in particular, for r ∈ Z we define maps where the last arrow is induced by mapping γ χ to (u r γ K ) w χ . Note that j 0 χ = j χ (see Sect. 4.5). Now assume that ζ p ∈ L. For s ∈ Z let x → t s cyc (x) and x → t s ω (x) be the automorphisms on Q F (G) induced by g → χ s cyc (g)g and g → ω s (g)g for g ∈ G, respectively. Lemma 6.1 Let r , s ∈ Z. Then for every x ∈ ζ(Q F (G)) we have j r χ (t s cyc (x)) = j r +s χω s (x). Proof It follows easily from the definitions that w χ = w χω . We claim that t 1 ω (γ χω ) = γ χ . (6.1) Write γ χω = g χω c χω with g χω ∈ G and c χω ∈ (F[H ]e χω ) × where g χω and c χω satisfy the defining properties of γ χω as given in Sect. 4.4. Put g χ := g χω and c χ := ω(g χω )t 1 ω (c χω ). It is then easily checked that g χ c χ has the defining properties of γ χ , and thus t 1 ω (γ χω ) = g χ c χ = γ χ . This establishes (6.1). Recalling that u = κ(γ K ) we compute t 1 cyc (γ χω ) = χ cyc (g χω )g χω t 1 ω (c χω ) = u w χω t 1 ω (γ χω ) = u w χ γ χ , where we have used (6.1) for the last equality. Finally, we have that j r χ (t s cyc (γ χω s )) = j r χ (u w χ ·s γ χ ) = u w χ ·s (u r γ K ) w χ = (u r +s γ K ) w χ = j r +s χω s (γ χω s ) for every r , s ∈ Z as desired.

Relation to the framework of [7]
We now discuss induced by the continuous group homomorphism where σ denotes the image of σ in G/H = K . By [7, Lemma 3.3] the homomorphism (6.2) extends to a ring homomorphism and this in turn induces a homomorphism Let aug : O ( K ) O be the augmentation map and put p = ker(aug). Writing O ( K ) p for the localisation of O ( K ) at p, it is clear that aug naturally extends to a homomorphism aug : O ( K ) p → F. One defines an evaluation map It is straightforward to show that for r ∈ Z we have If ζ is an element of K 1 (Q(G)), we define ζ(π) to be φ( π (ζ )). Conjecture 4.3 now implies that there is an element ζ S ∈ K 1 (Q(G)) such that ∂(ζ S ) = −[C • S (L/K )] and for each r ≥ 1 divisible by p − 1 and every irreducible Artin representation π χ of G with character χ we have where the first equality follows from [32, Lemma 2.3] (for the last equality see Sect. 5.1).

Non-commutative Fitting invariants over Iwasawa algebras
Let p be an odd prime and let G = H be a one-dimensional p-adic Lie group. Let For the next result we note that the maps j χ and φ of Sects. 6.1 and 6.2 are purely algebraic in nature and thus do not depend on the underlying Galois extension. denote the étale sheaf defined by the group of units of a scheme X . The closed immersion ι : Z T → U S induces a canonical morphism G m,U S → ι * G m,Z T , which can be shown to be surjective by considering stalks. Let G T m,U S denote the kernel of this morphism; then we have an exact sequence of étale sheaves If w is a finite place of L, we let O L,w be the localisation of O L at w. We denote the field of fractions of the Henselisation O h L,w of O L,w by L w . If w is archimedean, we let L w be the completion of L at w. In both cases we let Br(L w ) be the Brauer group of L w .
The main purpose of this subsection is to generalise the following result.

Proposition 7.1 Let S be a finite set of places of K containing S ∞ . Then
Proof This is [28, Chapter II, Proposition 2.1]. Let be a subset of S containing S ∞ . We shall consider the natural map

Lemma 7.2 Let S and T be as above. Then R
Here, the set T may be empty, in which case we put G ∅ m,U S := G m,U S and similarly with G m,U . Sequence (7.1) and Lemma 7.2 for S and induce a commutative diagram where the rows are exact triangles. Let D T ,S = D T ,S (L) be the cone of ψ T ,S . The diagram shows that D T ,S does not in fact depend on T and thus we denote it by D ,S .

Proposition 7.3 Let S, T and be as above. We have
Proof As D T ,S = D ,S does not depend on T , we can and do assume that T is empty. Let Z ,S := U − U S ; then Z ,S is a closed subscheme of U and by [27, Chapter III, Proposition 1.25] we have an isomorphism where the righthand side denotes cohomology with support on Z ,S . We now apply [27, Chapter III, Corollary 1.28] and [28, Chapter II, Proposition 1.5] to obtain the desired result. We now change notation as follows: let = S for the choice of S as in the statement of the proposition; and enlarge S in such a way that S is finite and disjoint from T and that cl T L,S vanishes. The same reasoning as above shows that H 1 et (U S , G T m,U S ) also vanishes. Therefore the long exact cohomology sequence induced by (7.2) yields an exact sequence 2) yields an exact sequence whose first terms coincide with (3.1). Similarly, taking global sections in (7.1) gives a long exact sequence in cohomology whose first terms coincide with (3.2).

Flat cohomology
Let X be an affine scheme and consider the multiplicative group scheme G m/X over X . We let μ n/X be the kernel of multiplication by n ∈ N and for a prime p we put μ p ∞ /X := lim − → j μ p j /X which is an ind-X -group scheme.
By the flat site on X we shall mean the site of all quasi-finite flat schemes of finite presentation over X with the f pp f -topology; so coverings are surjective families of flat morphisms that are locally of finite presentation. Note that our definition agrees with the f pq f -site in [42], but differs from the flat site in [27]; however, this will play no decisive role in the following by [27, Chapter III, Proposition 3.1]. The group schemes G m/X and μ n/X represent abelian sheaves for the flat site on X and we denote the corresponding cohomology groups by H i fl (X , G m ) and H i fl (X , μ n ), respectively. As G m is smooth over X , we have an isomorphism by [27,Chapter III,Theorem 3.9]. For a prime p we write μ p (L) for the group of p-power roots of unity in L. We now specialise to the case X = U L := U S ∞ and p odd. Then μ p ∞ /U L also represents a sheaf for the flat site on U L . As the Kummer sequences are exact for the flat site on U L for all n ∈ N, we have an exact triangle ). By Proposition 7.1 and (7.5) we thus obtain isomorphisms and an exact sequence

A reduction step
We now begin the proof of the main theorem of this article.

Proof of Theorem 5.2
This proof will occupy the rest of Sect. 7 and all of Sect. 8. We first prove a reduction step that will allow us to make certain simplifying assumptions. Put L := L(ζ p ) and C := Gal(L /L). Then C is a cyclic group whose order divides p−1.
The idempotent e C := |C| −1 c∈C c belongs to the group ring Z p [Gal(L /K )] and so by Lemma 2.6 (iii). As Stickelberger elements also behave well under base change, i.e., e C θ T p,S (L /K ) = θ T p,S (L/K ), we may assume without loss of generality that ζ p ∈ L. Note that as we are considering the p-parts we only need that E T L,S ( p) is torsionfree, as opposed to the stronger requirement that E T L,S is torsionfree. Since S p ⊆ S and Hyp(S, T ) holds, this hypothesis is unaffected by replacing L with L(ζ p ).
For clarity, we now list the assumptions that we shall use for the rest of this proof (including the lemmas and propositions proved along the way). We can make these assumptions either for the reasons just explained or because they are direct consequences of our hypotheses. Note that S ram (L/K ) ∪ S p = S ram (L ∞ /K ).
Assumptions We henceforth assume that S, T are finite sets of places of K and that (i)

Complexes at the finite level
Now taking p-minus parts of sequence (3.2) for S = S ∞ yields an exact sequence of The middle arrow Note that taking p-minus parts is an exact functor as p is odd.

Proposition 7.6
There are isomorphisms In particular, the isomorphism class of C T • (L/K ) does not depend on T . Proof Proposition 7.1 describes the cohomology of R ét (U L , G m,U L )( p) − as follows. First, we have isomorphisms As L is totally complex, we have H í et (U L , G m,U L ) = 0 for every i ≥ 4. Finally, we have an exact sequence However, the Brauer groups Br(L w ) = Br(C) vanish for all archimedean places w of L and (Q/Z) − = 0. We therefore have H í et (U L , G m,U L )( p) − = 0 for every i ≥ 2. Using Proposition 7.4 and the assumption that E T L,S ( p) is torsionfree, we likewise find that Now sequence (7.1) and Lemma 7.2 together give an exact triangle and thus we obtain the first required isomorphism in D( ). Moreover, the exact triangle (7.6) and the above considerations show that the natural map Therefore we obtain the second required isomorphism in D(Z p [G] − ).

Setup and notation
We now work at the 'infinite level' and use the techniques of Iwasawa theory. Let G := Gal(L ∞ /K ), which we may write as G = H where Z p and H := Gal(L ∞ /K ∞ ) naturally identifies with a normal subgroup of G. Let 0 be an open subgroup of that is central in G and recall from (4.1) that (G) := Z p G is a free R := Z p 0 -order in Q(G). Let j ∈ G denote complex conjugation (this an abuse of notation because its image in the quotient group G := Gal(L/K ) is also denoted by j) and let G + := G/ j = Gal(L + ∞ /K ). Then j ∈ H and so again (G + ) is a free R-order in Q(G + ). Moreover, (G) − := (G)/(1 + j) is also a free R-module of finite rank. For any (G)-module M we write M + and M − for the submodules of M upon which j acts as 1 and −1, respectively, and consider these as modules over (G + ) and (G) − , respectively. We note that M is R-torsion if and only if both M + and M − are R-torsion. Furthermore, any R-module that is finitely generated as a Z p -module is necessarily R-torsion.
Let χ cyc : G → Z × p denote the p-adic cyclotomic character (recall the assumption that ζ p ∈ L). Let μ p n = μ p n (L ∞ ) denote the group of p n th roots of unity in L × ∞ and let μ p ∞ be the nested union (or direct limit) of these groups. Let hence M(r ) is simply M with the modified G-action g · m = χ cyc (g) r g(m) for g ∈ G and m ∈ M. In particular, we have Q p /Z p (1) μ p ∞ and (G + )(−1) (G) − . Recall that for a Z-module M we previously defined M( p) := Z p ⊗ Z M; we shall use both notations M( p) and M(r ) in the sequel, believing the meaning to be clear from context. We note that the property of being R-torsion is preserved under taking Tate twists.
For every place v of K we denote the decomposition subgroup of G at a chosen prime w ∞ above v by G w ∞ (everything will only depend on v and not on w ∞ in the following). We note that the index [G : G w ∞ ] is finite when v is a finite place of K .

Complexes at the infinite level
Let L n be the nth layer in the cyclotomic Z p -extension of L. Then lim − →n Lemma 8. 1 We have isomorphisms Proof Recall that C T • (L n /K ) is defined by the middle arrow of the sequence (7.7) for the layer L n . Taking the direct limit over all n gives an exact sequence of (G) − -modules is the complex in degrees 0 and 1 given by the middle arrow of (8.1), giving the desired result. For For a finite set S of places of K we let U ∞,S := Spec(O L ∞ ,S∪S ∞ ) and put U L ∞ := U ∞,S ∞ . The following proposition can be viewed as a 'derived version' of results that are well-known at the level of cohomology. It seems possible that this result is known to experts, but the authors were unable to locate a proof in the literature.

Proposition 8.2
There are isomorphisms In particular, the isomorphism class of C T • (L ∞ /K ) does not depend on T .

Proof
The assumption that ζ p ∈ L is crucial in this proof. Even though it is not strictly necessary, we first check that the two complexes C T • (L ∞ /K ) and R ét (U ∞,S p , μ p ∞ ) − compute the same cohomology. Let M S p be the maximal profinite extension of L ∞ that is unramified outside S p and let M ab S p ( p) be the maximal abelian prop-extension of L ∞ inside M S p . We put H S p := Gal(M S p /L ∞ ) and X S p := Gal(M ab There is a canonical isomorphism with Galois cohomology where the last isomorphism is Kummer duality [37,Theorem 11.4.3]. If we compare this with Lemma 8.1, we see that R ét (U ∞,S p , μ p ∞ ) − and C T • (L ∞ /K ) compute the same cohomology.
We now establish the derived version of this result and also consider flat cohomology. By Proposition 7.6 we have isomorphisms for each layer L n in the cyclotomic Z p -extension. Taking direct limits over all n yields the first required isomorphism by [27,Chapter III,Lemma 1.16] which holds for the flat topology as well (see [27,Chapter III,Remark 1.17 (d)] and [19, p. 172]). Finally, the natural map is an isomorphism by [42, Part II, Lemma 3] (note that this result is formulated only in terms of cohomology, but the proof actually shows that the cone of this map is acyclic).
For a finite set S of places of K containing S p ∪ S ∞ recall the definition of the complex which occurs in the EIMC. For an integer m we let C • Proof Using (8.1) and Corollary 8.3 we see that for every finite set T of places of K such that Hyp(S p ∪ S ram ∪ S ∞ , T ) is satisfied, the extension class of the complex C • S p (L + ∞ /K )(−1) in Ext 2 (G) − (Z p (−1), X + S p (−1)) may be represented by the exact sequence Let Z S be the kernel of the natural surjection X + S (−1) X + S p (−1) and let W T be the kernel of the right-most surjection in (8.5). Then the short exact sequences , induce long exact sequences in cohomology, which by Lemma 8.4 (ii) give the following diagram where we have omitted the subscript (G) − from all Ext-groups, and all rows and columns are exact. Note that the top two squares are commutative (see [21, (7 Then the fact that C • S p (L + ∞ /K )(−1) is represented by (8.5) shows that α p maps ε to . Now a diagram chase shows that there exists a preimage of ε under γ that is mapped to The proof of the following lemma explains why we write Y T S (−1) rather than Y T S . Note that every (G) − -module M may be written as an n-fold Tate twist for every n ∈ Z; simply write M = M(−n)(n).
We put Note that this slightly differs from the corresponding element S,T in [33].
The exact sequence (8.3) shows that each (ind G G w∞ Z p (−1)) − has a quadratic presentation and that its Fitting invariant is generated by ξ v . Hence Lemma 2.6 (ii) gives and we therefore obtain the desired result. We now suppose that the EIMC holds for L + ∞ /K . The surjection (8.4), together with Lemma 2.6 (i) and Proposition 8.7 then imply that As the transition maps in the direct limit A T L ∞ = lim − →n A T L n are injective by [17,Lemma 2.9], the transition maps in the projective limit Hom(A T L ∞ , Q p /Z p ) = lim ← −n (A T L n ) ∨ are surjective. As L := Gal(L ∞ /L) clearly acts trivially on (A T L ) ∨ , we have a surjection Fix an odd character χ ∈ Irr Q c p (G) and view χ as an irreducible character of G with open kernel. We have φ( j χ (t 1 cyc ( S ))) = φ( j 1 χω ( S )) = L p,S (0, χω), where the first and second equalities follow from Lemma 6.1 and (6.4), respectively. Moreover, φ( j χ ( v∈T ξ v )) = δ T (0,χ). As Fitting invariants behave well under base change by Proposition 6.2, we have where we have again used Lemma 2.6 (i). This completes the proof of Theorem 5.2.

Hybrid p-adic group rings
We recall material on hybrid p-adic group rings from [24,Sect. 2] and [25,Sect. 2]. We shall sometimes abuse notation by using the symbol ⊕ to denote the direct product of rings or orders.
Let p be a prime and let G be a finite group. For a normal subgroup N G, let e N = |N | −1 σ ∈N σ be the associated central trace idempotent in the group algebra Q p [G]. Then there is a ring isomorphism Z p [G]e N Z p [G/N ]. We now specialise [24,Definition 2.5] to the case of p-adic group rings (we shall not need the more general case of N -hybrid orders).
We say that the p-adic group ring For n ∈ N, let C n denote the cyclic group of order n, let A n denote the alternating group on n letters and let S n denote the symmetric group on n letters. Let V 4 denote the subgroup of A 4 generated by double transpositions. We now recall two examples from [24,Sect. 2.3] (also see [25,Sect. 2.2]).

Example 9.5
Let p < q be distinct primes and assume that p | (q − 1). Then there is an embedding C p → Aut(C q ) and so there is a fixed-point-free action of C p on C q . Hence the corresponding semidirect product G = C q C p is a Frobenius group (see [24,Theorem 2.12] or [41,Sect. 4.6], for example), and so Z p [G] is N -hybrid with N = C q . Example 9.6 Let q be a prime power and let F q be the finite field with q elements. The group Aff(q) of affine transformations on F q is the group of transformations of the form x → ax +b with a ∈ F × q and b ∈ F q . Let G = Aff(q) and let Proof Suppose H is monomial. Let χ ∈ Irr C (G). If N ≤ ker χ then χ is inflated from some ϕ ∈ Irr C (G/N ). Otherwise N ker χ and so χ is induced from some ψ ∈ Irr C (N ) by Theorem 9.3 (iv). The Frobenius complement H G/N is monomial by assumption. Moreover, the Frobenius kernel N is nilpotent by Theorem 9.3 (ii) and thus is monomial by [9,Theorem 11.3]. However, induction is transitive and inflation commutes with induction (as in [48,Theorem 4.2 (3)], for example) so that in both cases χ is induced from a linear character. Therefore G is monomial. The converse follows from the fact that any quotient of a monomial group is monomial (this can easily be proved using that inflation commutes with induction; also see [3, Chapter 2, Sect. 4]). The last claim follows since H is monomial in these cases by [48,Sect. 4.4, Theorem 4.8 (1)]).

New p-adic hybrid group rings from old
We recall two results from [   is also coprime to p. Hence by Theorem 4.6 the EIMC holds for L(ζ p ) + ∞ /K , and so the desired result follows from Corollary 5.4.

Further unconditional cases of the EIMC
We now apply the results of [25] to give criteria for the EIMC to hold unconditionally in cases of interest to us. Theorem 10.2 Let L/K be a finite Galois extension of totally real number fields with Galois group G. Let p be an odd prime and let L ∞ be the cyclotomic Z p -extension of L. Let N be a normal subgroup of G and let P be a Sylow p-subgroup of G := Gal(L N /K ) G/N . Suppose that Z p [G] is N -hybrid and that (L N ) P /Q is abelian. Let K /K be a field extension such that K is totally real, K /Q is abelian and p [K : K ] < ∞. Let L ∞ = L ∞ K . Then the EIMC holds for both L ∞ /K and L ∞ /K .

Remark 10.3
It is straightforward to see that for every prime p and every finite group G, the p-adic group ring Z p [G] is {1}-hybrid. Hence, in particular, Theorem 10.2 can be applied in the case that N is trivial.

Remark 10.4
The hypothesis that (L N ) P /Q is abelian forces K /Q to be abelian, and thus one can take K to be the compositum of K with another finite abelian extension K /Q such that p [K : Q]. In particular, Theorem 10.2 can be applied with K = K (ζ p ) + and L ∞ = L ∞ (ζ p ) + = L(ζ p ) + ∞ .

Proof of Theorem 10.2
The EIMC holds for L ∞ /K by [25,Theorem 4.17 (v)]. Let F = L N and put F = F K and L = L K . Let G = Gal(L /K ) and N = Gal(L /F ). Then Z p [G ] is N -hybrid by Remark 9.11. Let P be a Sylow p-subgroup of G := G /N . Then ((L ) N ) P = (F ) P = F P K = (L N ) P K , which is an abelian extension of Q as it is the compositum of two such extensions. Therefore the EIMC holds for L ∞ /K by a further application of [25,Theorem 4.17 (v)]. Proof First note that G is monomial by Lemma 9.7 since V is abelian. Suppose that p |U |. Let N = U and F = (L + ) N . Then Z p [G] is N -hybrid by Proposition 9.4. Hence the desired result follows from Theorem 10.5 in this case since F/Q is abelian, which forces F P /Q to be abelian. Suppose that U is a p-group. Taking N = {1} and F = L + (see Remark 10.6) we apply Theorem 10.5 with G = G and P = U to obtain the desired result.

Example 10.8
In particular, U is an -group in Corollary 10.7 in the following cases: • G Aff(q), where q is a prime power (see Example 9.6), • G C C q , where q < are distinct primes such that q | ( − 1) and C q acts on C via an embedding C q → Aut(C ) (see Example 9.5), • G is isomorphic to any of the Frobenius groups constructed in [25,Example 2.11]. Corollary 10.9 Let L/K be a finite Galois CM-extension of number fields. Let p be an odd prime and let S be a finite set of places of K such that S p ∪ S ram (L/K ) ∪ S ∞ ⊆ S. Then both B S(L/K , S, p) and B(L/K , S, p) are true when L + /K is any of the extensions considered in [25,Examples 4.21,4.22 or 4.23].
Proof The group in [25,Examples 4.21] is monomial by an application of Lemma 9.7 and the group in [25,Example 4.22] is monomial by an application of [3, Chapter 2, Theorem 3.10]. For [25,Examples 4.23] it is straightforward to check that S 4 is a monomial group. That the remaining hypotheses of Theorem 10.5 are satisfied in each case are verified in the cited examples themselves.
In certain situations, we can also remove the condition that S p ⊆ S. To illustrate this, we conclude with the following result.