A Conductor Formula

for Completed Group Algebras

Andreas Nickel^{1}

Received: November 25, 2013

Communicated by Otmar Venjakob

Abstract. Let 𝔬 be the ring of integers in a finite extension of ℚ_{p}. If G is a finite group and Γ is a maximal 𝔬-order containing the group ring 𝔬[G], Jacobinski’s conductor formula gives a complete description of the central conductor of Γ into 𝔬[G] in terms of characters of G. We prove a similar result for completed group algebras 𝔬[[G]], where G is a p-adic Lie group of dimension 1. We will also discuss several consequences of this result.

2010 Mathematics Subject Classification: 16H10, 16H20, 11R23 Keywords and Phrases: central conductor, completed group algebras, extensions of lattices, Fitting invariants

Introduction

Let 𝔬 be the ring of integers in a number field K (or in a finite extension K of ℚ_{p}) and consider the group ring 𝔬[G] of a finite group G over 𝔬. The central conductor F(𝔬[G]) consists of all elements x in the center of 𝔬[G] such that xΓ ⊆ 𝔬[G], where Γ ⊆ K[G] is a chosen maximal 𝔬-order containing 𝔬[G], i.e.

F(𝔬[G]) = {x ∈ ζ(𝔬[G]) | xΓ ⊆ 𝔬[G]} .

Here, we write ζ(Λ) for the center of a ring Λ. A result of Jacobinski [Ja66] (see also [CR81, Theorem 27.13]) gives a complete description of the central conductor in terms of the irreducible characters of G. More precisely, we have

F(𝔬[G]) = ⨁_{χ} \frac{|G|}{χ(1)}𝔇^{−1}(𝔬[χ]/𝔬),  (1)

1The author acknowledges financial support provided by the DFG

Documenta Mathematica 19 (2014) 601–627

where 𝔇^{−1}(𝔬[χ]/𝔬) denotes the inverse different of 𝔬[χ], the ring of integers in K(χ) := K(χ(g) | g ∈ G), with respect to 𝔬, and the sum runs through all absolutely irreducible characters of G modulo the following Galois action: If χ is an absolutely irreducible character of G and σ belongs to Gal(K(χ)/K), then σ acts on χ as^{σ}χ(g) = σ(χ(g)) for every g ∈ G. Jacobinski’s main interest was in determining annihilators of Ext; in fact, he showed that

F(𝔬[G]) · Ext_{𝔬[G]}^{1}(M, N) = 0

for all 𝔬[G]-lattices M and 𝔬[G]-modules N. For instance, it can be deduced from this result that |G|/χ(1) annihilates Ext_{𝔬[G]}^{1}(M_{χ}, N) if M_{χ} is an 𝔬[G]- lattice such that K ⊗_{𝔬} M_{χ} is absolutely simple with character χ. Later, Roggenkamp [Ro71] showed that the annihilators obtained in this way are in fact the best possible in a certain precise sense.

In this article we consider completed group algebras 𝔬[[G]], where 𝔬 denotes the ring of integers in a finite extension K of ℚ_{p} and G is a p-adic Lie group of dimension 1. Hence G may be written as a semi-direct product H ⋊ Γ with finite H and a cyclic pro-p-group Γ, isomorphic to ℤ_{p}. We will exclude the special case p = 2, as we will make heavily use of results of Ritter and Weiss [RW04] (where the underlying prime is assumed to be odd) on the total ring of fractions Q^{K}(G) of 𝔬[[G]]. However, it turns out that the results provided by Ritter and Weiss are not sufficient for our purposes and so we shall have to determine the structure of Q^{K}(G) in more detail, thereby generalizing and extending results of Lau [La12] (where K = ℚ_{p} and G is pro-p). We will do this in the first section. In section 2 we provide the necessary preparations for our main theorem including results on reduced traces and conductors. The main theorem will then be stated and proved in section 3. More precisely, if we define the central conductor in complete analogy to the group ring case, then we have an inclusion

⨁_{χ/∼} \frac{|H|w_{χ}}{χ(1)} · 𝔇^{−1}(𝔬_{χ}/𝔬)𝔬_{χ}[[Γ′_{χ}]] ⊆ F(𝔬[[G]]),  (2)

A Conductor Formula for Completed Group Algebras 603

where the sum runs through all absolutely irreducible characters of G with open kernel up to a certain explicit equivalence relation. Moreover, w_{χ} is the index of a certain subgroup (depending on χ) in G and 𝔬_{χ} denotes the ring of integers in K_{χ} := K(χ(h) | h ∈ H). Finally, Γ′_{χ} is a cyclic pro-p-group which has an explicitly determined topological generator. We will give precise definitions later in the text. The inclusion (2) is not far from being an equality. In fact, we show that its ‘χ-part’ becomes an equality after localization at any height 1 prime ideal of 𝔬_{χ}[[Γ′_{χ}]] that does not contain p. It is an equality if G = H × Γ is a direct product or if no skewfields occur in the Wedderburn decomposition of Q^{K}(G) (in fact it suffices to suppose that the Schur indices are not divisible by p). Moreover, we will also explicitly determine the central conductor whenever G is a pro-p-group; we use this to give an example where (2) is a proper inclusion.

The proof of Jakobinski’s central conductor formula does not carry over unchanged to the present situation for two reasons. First, the completed group algebra is an order over the power series ring 𝔬[[T]], but there is no canonical choice of embedding of 𝔬[[T]] into ζ(𝔬[[G]]). Secondly and more seriously, the ring 𝔬[[T]] is a regular local ring, but it is not a Dedekind domain. Furthermore, even if we localize at a height one prime ideal, the residue field will not be finite. Hence we do not have the well elaborated theory of maximal orders over discrete valuation rings with finite residue field at our disposal. In the aforementioned two cases, when G = H × Γ is a direct product or when no skewfields occur in the Wedderburn decomposition, we will overcome this problem by replacing our chosen maximal 𝔬[[T]]-order by a suitable maximal 𝔬-order. When G is a pro-p-group, it is the explicit description of the occurring skewfields due to Lau [La12] which allows us to determine the central conductor.

Finally, we derive some consequences in section 4. In particular, we obtain results for the corresponding Ext-groups in analogy to the group ring case. We also apply our main result to the theory of non-commutative Fitting invariants introduced by the author [Ni10] and further developed in [JN13]. This theory may be applied to 𝔬[[G]]-modules even if G is non-abelian, but in contrast to the commutative case, the Fitting invariant of a finitely presented 𝔬[[G]]-module M might not be contained in the annihilator of M. To obtain annihilators one has to multiply by a certain ideal H(𝔬[[G]]) of ζ(𝔬[[G]]) which is hard to determine in general. However, it is easily seen that H(𝔬[[G]]) always contains the central conductor so that our main theorem provides a method to compute explicit annihilators of a finitely presented 𝔬[[G]]-module, at least, if we are able to compute its Fitting invariant.

Acknowledgement. The author is indebted to Henri Johnston for his many suggestions and helpful remarks.

1 The total ring of fractions of a completed group algebra

Let p be an odd prime and let G be a profinite group containing a finite normal subgroup H such that G/H ≃ Γ for a pro-p-group Γ, isomorphic to ℤ_{p}; thus G can be written as a semi-direct product H ⋊ Γ and is a p-adic Lie group of dimension 1. We denote the completed group algebra ℤ_{p}[[G]] by Λ(G) that is

Λ(G) := ℤ_{p}[[G]] = \underleftarrow{lim} ℤ_{p}[G/N],

where the inverse limit is taken over all open normal subgroups N of G. If K is a finite field extension of ℚ_{p} with ring of integers 𝔬, we put Λ^{𝔬}(G) := 𝔬⊗_{ℤ_{p}} Λ(G) = 𝔬[[G]]. We fix a topological generator γ of Γ and choose a natural number n such that γ^{p^{n}} is central in G. Since we also have that Γ^{p^{n}} ≃ ℤ_{p}, there is an isomorphism 𝔬[[Γ^{p^{n}}]] ≃ 𝔬[[T]] induced by γ^{p^{n}} ↦ 1 + T. Here, R := 𝔬[[T]] denotes the power series ring in one variable over 𝔬. If we view Λ^{𝔬}(G) as an R-module (or more generally as a left R[H]-module), there is a decomposition

Λ^{𝔬}(G) = ⨁_{i=0}^{p^{n}−1} R[H]γ^{i}.

Hence Λ^{𝔬}(G) is finitely generated and free as an R-module and is an R-order in the separable L := Quot(R)-algebra Q^{K}(G) := L ⊗_{R} Λ^{𝔬}(G). Note that Q^{K}(G) is obtained from Λ^{𝔬}(G) by inverting all regular elements. By [RW04, Lemma 1] we have Q^{K}(G) = K ⊗_{ℚ_{p}} Q(G), where Q(G) := Q^{ℚ_{p}}(G) denotes the total ring of fractions of Λ(G). Let ℚ_{p}^{c} be an algebraic closure of ℚ_{p} and denote by Irr (G) the set of absolutely irreducible ℚ_{p}^{c}-valued characters of G with open kernel. Fix χ ∈ Irr (G). Let η be an irreducible constituent of res _{H}^{G}χ. Then G acts on η as η^{g}(h) = η(g^{−1}hg) for g ∈ G, h ∈ H, and we set

St(η) := \{g ∈ G : η^{g} = η\}, e(η) := \frac{η(1)}{|H|} ∑_{h∈H} η(h^{−1})h, e_{χ} := ∑_{η|res _{H}^{G}χ} e(η).

Choose a finite Galois extension E of K such that the character χ has a realization V_{χ} over E. By [RW04, Corollary to Proposition 6] e_{χ} is a primitive central idempotent of Q^{E}(G). In fact, it is shown that every primitive central idempotent of Q^{c}(G) := ℚ_{p}^{c} ⊗_{ℚ_{p}} Q(G) is an e_{χ}, and e_{χ} = e_{χ′} if and only if χ = χ′ ⊗ ρ for some character ρ of G of type W (i.e. res _{H}^{G}ρ = 1). By Clifford theory [CR81, Proposition 11.4] the irreducible constituents of res _{H}^{G}χ are precisely the conjugates of η under the action of G, each occurring with the same multiplicity z_{χ}. By [RW04, Lemma 4] we have z_{χ} = 1 and thus equalities

res _{H}^{G}χ = ∑_{i=0}^{w_{χ}−1} η^{γ^{i}}, e_{χ} = ∑_{i=0}^{w_{χ}−1} e(η^{γ^{i}}) = \frac{χ(1)}{|H|w_{χ}} ∑_{h∈H} χ(h^{−1})h,  (3)

where w_{χ} := [G : St(η)]. Note that w_{χ} is a power of p as H is a subgroup of St(η). We now put

K_{χ} := K(χ(h) | h ∈ H) ⊆ K(η) := K(η(h) | h ∈ H)

and note that K_{χ} = K_{χ⊗ρ} whenever ρ is of type W. As g^{−1}Hg = H, we have K(η^{g}) = K(η) for every g ∈ G and thus K(η) does not depend on the particular choice of irreducible constituent of res _{H}^{G}χ. We let σ ∈ Gal(ℚ_{p}^{c}/K) act on χ as ^{σ}χ(g) = σ(χ(g)) for all g ∈ G and similarly on characters of H. Note that the actions on res _{H}^{G}χ and η factor through Gal(K_{χ}/K) and Gal(K(η)/K), respectively.

Lemma 1.1. The degree [K(η) : K_{χ}] divides w_{χ}; in particular, it is a power of p.

Proof. Fix σ in Gal(K(η)/K_{χ}). As σ acts trivially on res _{H}^{G}χ, the character ση also is an irreducible constituent of res _{H}^{G}χ. Hence there are irreducible constituents η_{1}, …, η_{s} of res _{H}^{G}χ such that res _{H}^{G}χ may be written as

res _{H}^{G}χ = ∑_{i=1}^{s} ∑_{σ∈Gal(K(η_{i})/K_{χ})}^{σ}η_{i}.

Then w_{χ}η(1) = χ(1) = ∑_{i=1}^{s}[K(η_{i}) : K_{χ}]η_{i}(1) = s[K(η) : K_{χ}]η(1) as desired.

By [RW04, Proposition 5] there is a unique element γ_{χ} ∈ ζ(Q^{c}(G)e_{χ}) such that γ_{χ} acts trivially on V_{χ} and γ_{χ} = γ^{w_{χ}} · c = c · γ^{w_{χ}}, where c ∈ (ℚ_{p}^{c}[H]e_{χ})^{×}. An analysis of the proof in fact shows that c ∈ (E[H]e_{χ})^{×} if both χ and η have realizations over E. We can and do assume this in the following. Again by [RW04, Proposition 5] the element γ_{χ} generates a procyclic p-subgroup Γ_{χ} of (Q^{E}(G)e_{χ})^{×}. Moreover, γ_{χ} induces an isomorphism Q^{E}(Γ_{χ}) ⟶^{≃} ζ(Q^{E}(G)e_{χ}) by [RW04, Proposition 6]. We let σ ∈ Gal(E/K) act on Q^{E}(G) = E ⊗_{K} Q^{K}(G) as σ(x ⊗ y) = σ(x) ⊗ y for all x ∈ E and y ∈ Q^{K}(G). We then have

σ(e_{χ}) = e_{^{σ}χ}, σ(γ_{χ}) = γ_{^{σ}χ}  (4)

for every σ ∈ Gal(E/K); here, the latter equality follows from the uniqueness of γ_{^{σ}χ}. By (3) we have σ(e_{χ}) = e_{χ} whenever σ ∈ Gal(E/K_{χ}); in particular, we have an action of Gal(E/K_{χ}) on ζ(Q^{E}(G)e_{χ}). For a finite extension F of ℚ_{p} let U_{F}^{1} denote the group of principal units in F.

Lemma 1.2. Fix χ ∈ Irr (G). Then there is a principal unit x = x_{χ} ∈ U_{E}^{1} with the following properties.

(i) The element γ′_{χ} := xγ_{χ} is invariant under the action of Gal(E/K_{χ}) on ζ(Q^{E}(G)e_{χ}).

(ii) Let m ≥ 0 be an integer. If γ^{p^{m}} acts trivially on V_{χ}, then x^{p^{m}} belongs to U_{K_{χ}}^{1}.

Moreover, if Γ′_{χ} denotes the procyclic p-subgroup generated by γ′_{χ}, then

Q^{E}(Γ_{χ})^{Gal(E/K_{χ})} = Q^{K_{χ}}(Γ′_{χ}).

Proof. We first observe that all claims do not depend on the choice of E. To see this let F be a second finite Galois extension of K such that the character χ can be realized over F. Replacing F by EF we may assume E ⊆ F. If there is an x ∈ U_{E}^{1} with (i) and (ii), then we may use the same x for F as U_{E}^{1} ⊆ U_{F}^{1} . Conversely, suppose there is an x ∈ U_{F}^{1} which fulfills (i) and (ii) with E replaced by F. Let σ ∈ Gal(F/E) be arbitrary. Then σ acts trivially on γ_{χ} as γ_{χ} ∈ ζ(Q^{E}(G)e_{χ}), and it acts trivially on γ′_{χ} by (i). Hence σ(x) = σ(γ′_{χ})σ(γ_{χ})^{−1} = γ′_{χ}γ_{χ}^{−1} = x and thus x ∈ U_{F}^{1} ∩ E = U_{E}^{1} as desired. Finally, as Gal(F/E) acts trivially on γ_{χ}, we have Q^{F} (Γ_{χ})^{Gal(F/E)} = Q^{E}(Γ_{χ}), and thus the last statement of the Lemma does not depend on E either. We now may assume that E is obtained from K_{χ} by adjoining roots of unity. More precisely, the group G/ ker(χ) is finite as χ has open kernel, and by [CR81, Theorem 15.16] we may take E = K_{χ}(ζ), where ζ is a root of unity of order |G/ ker(χ)|. Let us denote by μ_{p}(E) the group of all p-power roots of unity in E. Let σ ∈ Gal(E/K_{χ}) be arbitrary. Then res _{H}^{G}χ = res _{H}^{G}^{σ}χ and thus σχ = χ ⊗ ρ_{σ} for some character ρ_{σ} of type W. Hence σ(γ_{χ}) = γ_{^{σ}χ} = ζ_{σ}γ_{χ} with ζ_{σ} := ρ_{σ}(γ)^{−w_{χ}} ∈ μ_{p}(E) by [RW04, Proposition 5]. The assignment

Gal(E/K_{χ}) → μ_{p}(E), σ ↦ ζ_{σ} = γ_{χ}^{σ−1}

is a 1-cocycle. Let E_{0} be the maximal unramified extension of K_{χ} in E. Then E = E_{0}(μ_{p}(E)) and thus H^{1}(Gal(E/E_{0}), μ_{p}(E)) = 1 by [NSW08, Proposition 9.1.6]. By the inflation-restriction sequence this yields an isomorphism

H^{1}(Gal(E_{0}/K_{χ}), μ_{p}(E_{0})) ≃ H^{1}(Gal(E/K_{χ}), μ_{p}(E)).  (5)

The natural map H^{1}(Gal(E_{0}/K_{χ}), μ_{p}(E_{0})) → H^{1}(Gal(E_{0}/K_{χ}), U_{E_{0}}^{1}) is trivial, as the latter group vanishes by [NSW08, Proposition 7.1.2]. This and (5) imply that there is an a ∈ μ_{p}(E) · U_{E_{0}}^{1} ⊆ U_{E}^{1} such that ζ_{σ} = a^{σ−1} for all σ ∈ Gal(E/K_{χ}). We put x := a^{−1}; then γ′_{χ} := xγ_{χ} is easily seen to be invariant under the action of Gal(E/K_{χ}). Now let m ≥ 0 be an integer such that γ^{p^{m}} acts trivially on V_{χ}. Then also c^{p^{m}} = (γ_{χ}γ^{−w_{χ}})^{p^{m}} acts trivially on V_{χ} and belongs to (E[H]e_{χ})^{×}. So we must have c^{p^{m}} = e_{χ}. We deduce that γ_{χ}^{p^{m}} = γ^{p^{m}w_{χ}}e_{χ} and thus σ(γ_{χ}^{p^{m}}) = γ_{χ}^{p^{m}} for every σ ∈ Gal(E/K_{χ}). We obtain

σ(x^{p^{m}}) = σ((γ′_{χ})^{p^{m}})σ(γ_{χ}^{p^{m}})^{−1}
= (γ′_{χ})^{p^{m}}(γ_{χ}^{p^{m}})^{−1}
= x^{p^{m}};

hence x^{p^{m}} belongs to U_{E}^{1} ∩ K_{χ} = U_{K_{χ}}^{1}. For the last assertion, we observe that U_{E}^{1} is a ℤ_{p}-module and therefore

Λ^{𝔬_{E}}(Γ_{χ}) = Λ^{𝔬_{E}}(Γ′_{χ})  (6)

as (γ′_{χ})^{z} = x^{z}γ_{χ}^{z} for every z ∈ ℤ_{p} (see also [Hi93, Lemma 1, p.199]). Hence also Q^{E}(Γ_{χ}) = Q^{E}(Γ′_{χ}) and thus

Q^{E}(Γ_{χ})^{Gal(E/K_{χ})} = Q^{E}(Γ′_{χ})^{Gal(E/K_{χ})} = Q^{K_{χ}}(Γ′_{χ})

as desired.

Remark 1.3. The principal unit x and the element γ′_{χ} are unique up to a principal unit in K_{χ}. To see this let y ∈ U_{E}^{1} be a second principal unit which fulfills (i) and (ii) of Lemma 1.2 with γ′_{χ} replaced by γ′′_{χ} := yγ_{χ}. Then yx^{−1} = γ′′_{χ}(γ′_{χ})^{−1} is a principal unit which is invariant under Gal(E/K_{χ}) and thus belongs to U_{K_{χ}}^{1}.

Example. Assume that G = H × Γ is a direct product. Then w_{χ} = 1 and res _{H}^{G}χ = η is irreducible. If χ(γ) = χ(1), then γ_{χ} = γe_{χ} and we may choose γ′_{χ} = γ_{χ}. If χ(γ) ≠ χ(1), we may write χ = χ′ ⊗ ρ with χ′(γ) = χ′(1) = χ(1) and some character ρ of type W. By [RW04, Proposition 5] we have γ_{χ} = γ_{χ′}ρ(γ)^{−1} = ρ(γ)^{−1}γe_{χ}. Then x = ρ(γ) fulfills (i) and (ii) of Lemma 1.2 and we find that γ′_{χ} = γ_{χ′} = γe_{χ} = γ′_{χ′}.

Definition 1.4. Let χ, ψ ∈ Irr (G). We say that χ and ψ are equivalent over K (and write χ ∼_{K} ψ) if there is σ ∈ Gal(K_{χ}/K) such that ^{σ}(res _{H}^{G}χ) = res _{H}^{G}ψ.

Note that if χ ∼_{K} ψ, then we have K_{χ} = K_{ψ} and via (4) an isomorphism Q^{K_{χ}}(Γ′_{χ}) ≃ Q^{K_{ψ}}(Γ′_{ψ}). Moreover, we have χ ∼_{K} χ⊗ρ whenever ρ is a character of type W.

Proposition 1.5. Let G be a p-adic Lie group of dimension 1 and let K be a finite extension of ℚ_{p}. Then there is an isomorphism

ζ(Q^{K}(G)) ≃ ⨁_{χ∈Irr (G)/∼_{K}} Q^{K_{χ}}(Γ′_{χ}),

where the sum runs through all χ ∈ Irr (G) up to the above equivalence relation.

Proof. Since there are only finitely many central primitive idempotents e_{χ} of Q^{c}(G), we may choose a finite Galois extension E of K such that E[H] contains each e_{χ}. We will use the fact that the center of Q^{K}(G) coincides with the Gal(E/K)-invariants of ζ(Q^{E}(G)). For this let σ be an element of Gal(E/K). Fix χ, ψ ∈ Irr (G). Then σ(e_{χ}) = e_{^{σ}χ}, and σ(e_{χ}) = e_{ψ} if and only if res _{H}^{G}ψ = ^{σ}(res _{H}^{G}χ). In particular, the action of Gal(E/K) on e_{χ} factors through Gal(K_{χ}/K), and we may write

ζ(Q^{K}(G)) = ⨁_{χ∈Irr (G)/∼_{K}} ζ(Q^{K}(G)ε_{χ}), ε_{χ} := ∑_{σ∈Gal(K_{χ}/K)}σ(e_{χ}) ∈ ζ(Q^{K}(G)).

(7)

Now let β ∈ ζ(Q^{K}(G)ε_{χ}). We view β as an element in ζ(Q^{E}(G)ε_{χ}) which is invariant under Gal(E/K). We may therefore write

β = (β_{σ})_{σ} ∈ ⨁_{σ∈Gal(K_{χ}/K)} ζ(Q^{E}(G)σ(e_{χ})) ≃ ⨁_{σ∈Gal(K_{χ}/K)} Q^{E}(Γ_{^{σ}χ})

where, by abuse of notation, σ denotes also a chosen lift of σ in Gal(E/K). As we have already mentioned before, the uniqueness of γ_{χ} implies that γ_{^{σ}χ} = σ(γ_{χ}) for every σ ∈ Gal(E/K); thus β is determined by β_{1}, and β_{1} lies in Q^{E}(Γ_{χ})^{Gal(E/K_{χ})} = Q^{K_{χ}}(Γ′_{χ}), where the equality is Lemma 1.2. Now β ↦ β_{1} induces an isomorphism ζ(Q^{K}(G)ε_{χ}) ≃ Q^{K_{χ}}(Γ′_{χ}).

Remark 1.6. In the special case, where K = ℚ_{p} and G is a pro-p-group, a similar result has been established by Lau [La12] using a different method. The same is true for Corollary 1.9 below.

Corollary 1.7. Let K be a finite extension of ℚ_{p} with ring of integers 𝔬. Choose a maximal R-order Λ̃^{𝔬}(G) containing Λ^{𝔬}(G). Then

ζ(Λ̃^{𝔬}(G)) ≃ ⨁_{χ∈Irr (G)/∼_{K}} Λ^{𝔬_{χ}}(Γ′_{χ}),

where 𝔬_{χ} denotes the ring of integers in K_{χ}.

Proof. This follows from Proposition 1.5 as Λ^{𝔬_{χ}}(Γ′_{χ}) is the integral closure of R in Q^{K_{χ}}(Γ′_{χ}).

Remark 1.8. Here, Λ̃^{𝔬}(G) is an order over R = 𝔬[[T]], where we have identified 1+T with γ^{p^{n}} for a chosen large n. Let us fix a χ ∈ Irr (G) in each equivalence class over K. If m = m(χ) is sufficiently large, then γ^{p^{m}} acts trivially on V_{χ} and hence γ_{χ}^{p^{m}} = (γ^{p^{m}})^{w_{χ}}e_{χ}. Enlarging n and m if necessary (for the finitely many χ), we may assume that p^{n} = p^{m} · w_{χ}. Hence we may also assume that the inclusion

R = 𝔬[[T]] ↣ Λ^{𝔬_{χ}}(Γ′_{χ})

is induced by 1 + T ↦ γ_{χ}^{p^{n}/w_{χ}} = x^{−p^{m}}(γ′_{χ})^{p^{m}} ∈ Λ^{𝔬_{χ}}(Γ′_{χ}), where x^{−p^{m}} belongs to U_{K_{χ}}^{1} by Lemma 1.2. We will fix such an n for the rest of the paper.

Corollary 1.9. The algebra Q^{K}(G) has Wedderburn decomposition

Q^{K}(G) ≃ ⨁_{χ∈Irr (G)/∼_{K}}(D_{χ})_{n_{χ}×n_{χ}},

where n_{χ} ∈ ℕ and D_{χ} is a skewfield with center Q^{K_{χ}}(Γ′_{χ}). If s_{χ} denotes the Schur index of D_{χ}, then we have an equality χ(1) = n_{χ}s_{χ}.

Proof. All assertions are immediate from Proposition 1.5 apart from the last equality. Let us denote the simple component (D_{χ})_{n_{χ}×n_{χ}} by A_{χ}. With E sufficiently large as in Proposition 1.5 we compute

(n_{χ}s_{χ})^{2} = dim_{Q^{Kχ} (Γ′_{χ})}(A_{χ})
= [K_{χ} : K]^{−1} · dim_{Q^{K}(Γ′_{χ})}(A_{χ})
{(i) \atop =} [K_{χ} : K]^{−1} · dim_{Q^{E}(Γ_{χ})}(E ⊗_{K} A_{χ})
{(ii) \atop =} [K_{χ} : K]^{−1} · ∑_{σ∈Gal(K_{χ}/K)} dim_{Q^{E}(Γ_{χ})}(Q^{E}(G)e_{^{σ}χ})
{(iii) \atop =} [K_{χ} : K]^{−1} · ∑_{σ∈Gal(K_{χ}/K)}^{σ}χ(1)^{2}
= χ(1)^{2}

Here, (i) follows from the equality Q^{E}(Γ′_{χ}) = Q^{E}(Γ_{χ}) which was established in the proof of Lemma 1.2 (confer equation (6)). The isomorphism E ⊗_{K} A_{χ} ≃ ⨁_{σ∈Gal(K_{χ}/K)} Q^{E}(G)e_{^{σ}χ} implies (ii), and (iii) is shown in the proof of [RW04, Proposition 6].

Remark 1.10. In the case K = ℚ_{p} and G a pro-p-group one can determine the occurring skewfields explicitly (see [La12, Theorem 1] and also (15) below).

Theorem 1.11. Fix χ ∈ Irr (G) and let A_{χ} = (D_{χ})_{n_{χ}×n_{χ}} be the corresponding simple component of Q^{K}(G). Then E ⊗_{K} A_{χ} splits whenever one (and hence every) irreducible constituent η of res _{H}^{G}χ can be realized over E. More precisely, we then have an isomorphism

E ⊗_{K} A_{χ} ≃ ⨁_{σ∈Gal(K_{χ}/K)} (Q^{E}(Γ′_{χ}))_{χ(1)×χ(1)} .

Proof. Recall that L = Q^{K}(Γ^{p^{n}}) for our fixed sufficiently large n. Let E be a finite extension of K and let L′ := E ⊗_{K} L = Q^{E}(Γ^{p^{n}}). As L′-vector space and more generally as left L′[H]-module, we have a decomposition

Q^{E}(G) = ⨁_{i=0}^{p^{n}−1} L′[H]γ^{i}.

Now fix χ ∈ Irr (G) and let η be an irreducible constituent of res _{H}^{G}χ. Suppose that η has a realization over E. Recall the definition (7) of ε_{χ}. As K_{χ} ⊆ K(η) ⊆ E, we have a decomposition

E ⊗_{K} A_{χ} = Q^{E}(G)ε_{χ} = ⨁_{σ∈Gal(K_{χ}/K)} Q^{E}(G)σ(e_{χ}).

As the centers of Q^{E}(G)σ(e_{χ}), σ ∈ Gal(K_{χ}/K) are isomorphic fields via (4) (compare also the proof of Proposition 1.5), it suffices to show that Q^{E}(G)e_{χ} splits.

Since E is a subfield of L′ we have L′[H]e(η) ≃ L′_{η(1)×η(1)} and similarly for every other irreducible constituent of res _{H}^{G}χ. We obtain

Q^{E}(G)e_{χ} = ⨁_{i=0}^{p^{n}−1} L′[H]e_{χ}γ^{i}

= ⨁_{i=0}^{p^{n}−1} ⨁_{j=0}^{w_{χ}−1} L′[H]e(η^{γ^{j}})γ^{i}

≃ ⨁_{i=0}^{p^{n}−1} ⨁_{j=0}^{w_{χ}−1} L′_{η(1)×η(1)}γ^{i},

where we have used equation (3) for the second equality. We now choose an indecomposable idempotent f_{η} = f_{η}e(η) of L′[H]e(η) ≃ L′_{η(1)×η(1)}. Observe that for a second indecomposable idempotent f′_{η} of L′[H]e(η) we have an isomorphism f_{η}L′[H]f′_{η} ≃ L′. As Q^{E}(G)e_{χ} is a simple algebra over its center Q^{E}(Γ′_{χ}) by Corollary 1.9, and f_{η} is also an idempotent in Q^{E}(G)e_{χ}, it suffices to show that f_{η}Q^{E}(G)e_{χ}f_{η} is a field, namely Q^{E}(Γ′_{χ}). For this, we first observe that f_{η,i} := γ^{i}f_{η}γ^{−i} is also an indecomposable idempotent for every 0 ≤ i < p^{n}, and belongs to L′[H]e(η^{γ^{i}}) as γ^{i}e(η)γ^{−i} = e(η^{γ^{i}}) and H is normal in G. However, e(η) = e(η^{γ^{i}}) if and only if w_{χ} divides i, and thus

f_{η}L′[H]f_{η,i} ≃ \{ \begin{matrix}L′ & if w_{χ} | i \\ 0 & otherwise. \end{matrix}

Since e(η^{γ^{j}})γ^{i}f_{η} = f_{η,i}e(η^{γ^{j}})γ^{i} we therefore have

f_{η}Q^{E}(G)e_{χ}f_{η} ≃ ⨁_{{i=0 \atop wχ|i}}^{p^{n}−1} L′γ^{i} = ⨁_{i=0}^{w_{χ}^{−1}p^{n}−1} L′γ^{w_{χ}i}.

We conclude that dim_{L′}(f_{η}Q^{E}(G)e_{χ}f_{η}) = w_{χ}^{−1}p^{n}. However, recall that L′ = Q^{E}(Γ^{p^{n}}) and γ^{p^{n}} identifies with γ_{χ}^{p^{n}/w_{χ}} = x^{−p^{n}/w_{χ}}(γ′_{χ})^{p^{n}/w_{χ}} by Remark 1.8, where x^{−p^{n}/w_{χ}} ∈ U_{K_{χ}}^{1} ⊆ (L′)^{×}. As L′-vector space we therefore have a decomposition

Q^{E}(Γ′_{χ}) = ⨁_{i=0}^{w_{χ}^{−1}p^{n}−1} L′(γ′_{χ})^{i}.

Thus we also have dim_{L′}(Q^{E}(Γ′_{χ})) = w_{χ}^{−1}p^{n}. Since Q^{E}(Γ′_{χ}) = ζ(Q^{E}(G)e_{χ}) is contained in f_{η}Q^{E}(G)e_{χ}f_{η}, we are done.

Corollary 1.12. There is a finite Galois extension E of K such that

Q^{E}(G) ≃ ⨁_{χ∈Irr (G)/W}(Q^{E}(Γ_{χ}))_{χ(1)×χ(1)},

where the sum runs through all χ ∈ Irr (G) modulo twists with characters of type W. In particular, no skewfields occur in the Wedderburn decomposition of the semi-simple algebra Q^{E}(G).

Proof. This is an immediate consequence of Theorem 1.11 and Corollary 1.9 once we observe that Q^{E}(Γ_{χ}) = Q^{E}(Γ′_{χ}) by (6) if E is sufficiently large.

Corollary 1.13. Write η(1) = s_{η}n_{η}, where s_{η} denotes the Schur index of η. Then s_{χ} divides s_{η}[K(η) : K_{χ}] and n_{η} divides n_{χ}.

Proof. By definition of the Schur index there is a field E of minimal degree s_{η} over K(η) such that η can be realized over E. However,

E ⊗_{K} A_{χ} ≃ E ⊗_{K} Q^{K_{χ}}(Γ′_{χ}) ⊗_{Q^{Kχ} (Γ′_{χ})} A_{χ}
≃ ⨁_{σ∈Gal(K_{χ}/K)} E ⊗_{K_{χ}} Q^{K_{χ}}(Γ′_{χ}) ⊗_{Q^{Kχ} (Γ′_{χ})} A_{χ}
≃ ⨁_{σ∈Gal(K_{χ}/K)} Q^{E}(Γ′_{χ}) ⊗_{Q^{Kχ} (Γ′_{χ})} A_{χ}

splits by Theorem 1.11. Now [Re03, Theorem 28.5] implies that s_{χ} divides

[Q^{E}(Γ′_{χ}) : Q^{K_{χ}}(Γ′_{χ})] = [E : K_{χ}] = s_{η}[K(η) : K_{χ}].

Moreover, we have

w_{χ}s_{η}n_{η} = w_{χ}η(1) = χ(1) = n_{χ}s_{χ} | n_{χ}s_{η}[K(η) : K_{χ}] | n_{χ}s_{η}w_{χ},

where the last divisibility is due to Lemma 1.1. Thus n_{η} divides n_{χ} as claimed.

Example. Assume that H is a p-group. Then s_{η} = 1 for all irreducible characters η of H by a result of Roquette [Ro58]. However, Lau [La12] gives examples, where s_{χ} = [K(η) : K_{χ}] is non-trivial.

Let us denote the global dimension of a ring Λ by gl.dim (Λ). The following lemma is only needed to justify Remark 3.6 below and can be skipped on a first reading.

Lemma 1.14. Let A be a separable L-algebra of finite dimension over L and let Λ be a maximal R-order in A. If A is split, then there is a maximal 𝔬-order Δ in Λ such that R ⊗_{𝔬} Δ = Λ (and thus L ⊗_{𝔬} Δ = A).

Proof. There are natural numbers k > 0 and n_{i}, 1 ≤ i ≤ k such that A = ⨁_{i=1}^{k} A_{i} and A_{i} ≃ L_{n_{i}×n_{i}}. We put Λ̃_{i} := R_{n_{i}×n_{i}} and Λ̃ := ⨁_{i=1}^{k} Λ̃_{i}. Observe that this is a maximal R-order in A by [Re03, Theorem 8.7]. Then Δ̃ := ⨁_{i=1}^{k} 𝔬_{n_{i}×n_{i}} is a maximal 𝔬-order in ⨁_{i=1}^{k} K_{n_{i}×n_{i}} and has R ⊗_{𝔬} Δ̃ = Λ̃ and L ⊗_{𝔬} Δ̃ = A. Since global dimension is invariant under Morita equivalence (cf. [Ra69, Corollary, p. 476]), we have

gl.dim (Λ̃_{i}) = gl.dim (R) = 2.

Moreover, Λ̃_{i}/rad(Λ̃_{i}) is a matrix ring over the residue field of R and thus a simple artinian ring. As likewise Λ decomposes into Λ = ⨁_{i=1}^{k} Λ_{i}, where each Λ_{i} is a maximal R-order in A_{i}, we may apply [Ra69, Theorem 5.4] in each simple component: there is a unit a_{i} ∈ A_{i} such that Λ_{i} = a_{i}^{−1}Λ̃_{i}a_{i}. We put a := ∑_{i=1}^{k} a_{i} so that Λ = a^{−1}Λ̃a. We further put Δ := a^{−1}Δ̃a. Then Δ is a maximal 𝔬-order in Λ, and we have

R ⊗_{𝔬} Δ = a^{−1}(R ⊗_{𝔬} Δ̃)a = a^{−1}Λ̃a = Λ

as desired.

2 Traces and conductors

Let R be a noetherian integrally closed domain with quotient field L. If A is a simple L-algebra, we denote by tr_{A/L} the reduced trace from A to L. If A is separable, then by [Re03, Theorem 9.26] the reduced trace gives rise to a symmetric associative nondegenerate bilinear form A × A → L which sends (a, b) to tr_{A/L}(ab). Note that if A = L′ is a field, then tr_{L′/L} = Tr_{L′/L} is the ordinary trace of fields. Now let Λ be an R-order in A. By a Λ-lattice we mean a finitely generated Λ-module which is torsionfree as R-module. We consider duality with respect to the trace form: For each full left Λ-lattice M in A (that is A ⊗_{Λ} M = A) we associate the dual right Λ-lattice

D(M/R) := {x ∈ A | tr_{A/L}(xM) ⊆ R} .

In particular, applying this construction to the two-sided Λ-lattice Λ we obtain a two-sided Λ-lattice D(Λ/R) which (by abuse of language) we will call the inverse different of Λ with respect to R. If D(Λ/R) happens to be invertible, we call 𝔇(Λ/R) := D(Λ/R)^{−1} the different of Λ with respect to R.

Lemma 2.1. Let R be a noetherian integrally closed domain with quotient field L and let A be a simple separable L-algebra. Let L ⊆ L′ ⊆ ζ(A) be fields and denote the integral closure of R in L′ by R′. Then for every maximal R-order Λ in A, we have

D(Λ/R) ⊇ D(Λ/R′)D(R′/R)

with equality if D(R′/R) is invertible.

Proof. As Λ is also a maximal R′-order by [Re03, Theorem 10.5], we have Λ = R′Λ and D(Λ/R′) is defined. Let a ∈ D(Λ/R′) and b ∈ D(R′/R) be arbitrary. Then

tr_{A/L}(abΛ) = tr_{L′/L}(tr_{A/L′}(abΛ)) = tr_{L′/L}(btr_{A/L′}(aΛ)) ⊆ tr_{L′/L}(bR′) ⊆ R

as desired. Now assume that D(R′/R) = 𝔇(R′/R)^{−1} is invertible. We conclude

x ∈ D(Λ/R) ⟺ tr_{L′/L}(tr_{A/L′}(xΛ)) ⊆ R
⟺ tr_{L′/L}(tr_{A/L′}(xR′Λ)) ⊆ R
⟺ tr_{L′/L}(R′tr_{A/L′}(xΛ)) ⊆ R
⟺ tr_{A/L′}(xΛ) ⊆ D(R′/R)
⟺ tr_{A/L′}(x𝔇(R′/R)Λ) ⊆ R′
⟺ x𝔇(R′/R) ⊆ D(Λ/R′)
⟺ x ∈ D(Λ/R′)D(R′/R).

Now let A be a semisimple L-algebra and let Tr be the ordinary trace map from A to L. For every x ∈ A we associate the homomorphism Tr_{x} ∈ Hom_{L}(A, L) defined by Tr_{x}(a) := Tr (xa) for all a ∈ A. Let Λ be an R-order in A. For a full left Λ-lattice M in A we may also consider duality with respect to the ordinary trace form:

D_{ord} (M/R) := {x ∈ A | Tr (xM) ⊆ R} = {x ∈ A | Tr_{x}(M) ⊆ R}

is a right Λ-lattice in A. We have a canonical homomorphism of right Λ-modules

δ_{M} : D_{ord} (M/R) → M^{+} := Hom_{R}(M, R), x ↦ Tr_{x}|_{M},  (8)

where λ ∈ Λ acts on f ∈ M^{+} as f^{λ}(m) := f(λm) for all m ∈ M. Similar observations hold for full right Λ-lattices in A. Recall that an R-module M is called reflexive if the canonical map M → M^{++}, m ↦ [f ↦ f(m)] is an isomorphism.

Proposition 2.2. Let R be a noetherian integrally closed domain with quotient field L. Let A be a semisimple L-algebra and let Λ be an R-order in A. If the ordinary trace Tr gives rise to a nondegenerate bilinear form A × A → L, (a, b) ↦ Tr (ab), then the homomorphism δ_{M} in (8) is an isomorphism for every full Λ-lattice M in A. In particular, D_{ord} (D_{ord} (M/R)/R) = M if and only if M is reflexive.

Proof. As the bilinear form A × A → L, (a, b) ↦ Tr (ab) is nondegenerate, we have Tr_{x} = Tr_{y} if and only if x = y. It follows that the map δ_{A} : A → Hom_{L}(A, L), x ↦ Tr_{x} is injective and thus an isomorphism of L-vector spaces as both sides have the same dimension over L. It is also a homomorphism of right A-modules. Now let M be a full left Λ-lattice in A. Then δ_{M} is just the restriction of δ_{A} to D_{ord} (M/R) and thus injective. However, by [NSW08, Remark p.268] restriction to M yields an isomorphism

M^{+} ≃ {ϕ ∈ Hom_{L}(A, L) | ϕ(M) ⊆ R}
= {Tr_{x} | x ∈ A, Tr_{x}(M) ⊆ R}
= {Tr_{x} | x ∈ D_{ord} (M/R)} .

Hence the image of δ_{M} is M^{+} as claimed.

We now return to the case L = Quot(R), where R = 𝔬[[T]] is the power series ring in one variable over the ring of integers 𝔬 in K. Recall that as an L-vector space we have Q^{K}(G) = ⨁_{i=0}^{p^{n}−1} L[H]γ^{i} = ⨁_{i=0}^{p^{n}−1} Lγ^{i}[H]. We denote by Tr the ordinary trace map from Q^{K}(G) to L.

Lemma 2.3. The elements γ^{i}h, 0 ≤ i < p^{n}, h ∈ H form an L-basis of Q^{K}(G) and

Tr (γ^{i}h) = \{ \begin{matrix}p^{n}|H| & if γ^{i} = h = 1 \\ 0 & otherwise. \end{matrix}

Its dual basis with respect to Tr is given by (p^{n}|H|)^{−1}h^{−1}γ^{−i}, 0 ≤ i < p^{n}, h ∈ H.

Proof. It is clear that these elements form an L-basis of Q^{K}(G). Now let 0 ≤ i < p^{n} and h ∈ H. We use the same basis to compute Tr (γ^{i}h). For every 0 ≤ j < p^{n} and h′ ∈ H, we have to write γ^{i}hγ^{j}h′ as an L-linear combination of these basis elements. However, γ^{i}hγ^{j}h′ = γ^{i+j}h_{j}h′ with h_{j} := γ^{−j}hγ^{j} ∈ H. This actually belongs to this basis if i + j < p^{n}; if i + j ≥ p^{n}, then γ^{i+j}h_{j}h′ = γ^{p^{n}}γ^{i+j−p^{n}}h_{j}h′ with γ^{p^{n}} ∈ L = Q^{K}(Γ^{p^{n}}). Since only the diagonal entries of the corresponding p^{n}|H|×p^{n}|H| matrix contribute to Tr (γ^{i}h), we now suppose that γ^{j}h′ = γ^{i+j}h_{j}h′ if i + j < p^{n}, and that γ^{j}h′ = γ^{i+j−p^{n}}h_{j}h′ if i + j ≥ p^{n}. In both cases we must have h_{j} = 1 and thus h = 1. In the first case, we also have j = i + j which forces i = 0, whereas in the second case j = i + j − p^{n} which gives i = p^{n}. This contradicts i < p^{n} and so the latter case does not occur. We see that this matrix has all diagonal entries equal to zero if γ^{i}h ≠ 1; hence Tr (γ^{i}h) = 0 in this case. That Tr (1) = p^{n}|H| is clear. It is now easily checked that (p^{n}|H|)^{−1}h^{−1}γ^{−i}, 0 ≤ i < p^{n}, h ∈ H is the dual basis.

Lemma 2.4. Let K′ be a finite field extension of K with ring of integers 𝔬′ and let a be a principal unit in some finite extension of K′. Let n ∈ ℕ_{0} and suppose that a^{p^{n}} belongs to U_{K′}^{1}. Consider Λ^{𝔬′}(Γ) as R-order via the embedding

ι_{a,n} : R ↣ Λ^{𝔬′}(Γ), 1 + T ↦ (aγ)^{p^{n}}.

Then the inverse different

D(Λ^{𝔬′}(Γ)/R) = {x ∈ Q^{K′}(Γ) | Tr_{Q^{K′} (Γ)/L}(xΛ^{𝔬′}(Γ)) ⊆ R}

is given by

D(Λ^{𝔬′}(Γ)/R) = p^{−n}𝔇^{−1}(𝔬′/𝔬)Λ^{𝔬′}(Γ),

where 𝔇^{−1}(𝔬′/𝔬) denotes the usual inverse different of 𝔬′ with respect to 𝔬. In particular, D(Λ^{𝔬′}(Γ)/R) is invertible.

Proof. First, we consider the case a = 1 and n = 0. If x_{1}, …, x_{k} form an 𝔬basis of 𝔬′, then x_{1}, …, x_{k} are also an 𝔬[[T]]-basis of 𝔬′[[T]] which is isomorphic to Λ^{𝔬′}(Γ) via 1 + T ↦ γ. Hence its dual basis with respect to the ordinary trace Tr_{K′/K} of fields, is also a dual basis with respect to Tr_{Q^{K′} (Γ)/L}. Hence D(Λ^{𝔬′}(Γ)/R) equals 𝔇^{−1}(𝔬′/𝔬)Λ^{𝔬′}(Γ) and is invertible in this case. For the general case let ι′_{a,n} : Λ^{𝔬′}(Γ) ↣ Λ^{𝔬′}(Γ) be induced from γ ↦ (aγ)^{p^{n}}. Then ι_{a,n} = ι′_{a,n} ∘ ι_{1,0}. We may therefore assume 𝔬 = 𝔬′ by the first part of the proof and an application of Lemma 2.1. The inverse different only depends upon the image of R in Λ^{𝔬}(Γ). However, ι_{a,n}(R) = ι_{1,n}(R) as a^{p^{n}} ∈ U_{K′}^{1}. Thus we may also assume a = 1. But in this case the result follows from Lemma 2.3 with G = Γ.

By Corollary 1.9 we may write Q^{K}(G) = ⨁_{χ∈Irr (G)/∼_{K}} A_{χ}, where A_{χ} = (D_{χ})_{n_{χ}×n_{χ}}, n_{χ} ∈ ℕ and D_{χ} is a skewfield with Schur index s_{χ} and center Q^{K_{χ}}(Γ′_{χ}). Again by Corollary 1.9, we have χ(1) = s_{χ}n_{χ}. Hence the ordinary trace may be written as

Tr = ∑_{χ∈Irr (G)/∼_{K}} χ(1)tr_{χ},  (9)

where tr_{χ} denotes the reduced trace from A_{χ} to L. Moreover, we have

tr_{χ} = Tr_{Q^{Kχ} (Γ′_{χ})/L} ∘ tr_{A_{χ}/Q^{Kχ} (Γ′_{χ})},  (10)

where Tr_{Q^{Kχ} (Γ′_{χ})/L} denotes the ordinary trace of fields and tr_{A_{χ}/Q^{Kχ} (Γ′_{χ})} denotes the reduced trace from A_{χ} into its center. Recall from Remark 1.8 that we have fixed a sufficiently large integer n ≥ 0 such that R = 𝔬[[T]] embeds into Λ^{𝔬_{χ}}(Γ′_{χ}) via 1 + T ↦ (x^{−1}γ′_{χ})^{p^{n}/w_{χ}}. Now Lemma 2.1 and Lemma 2.4 (with the embedding ι_{a,m}, where a = x^{−1} and p^{m} = p^{n}/w_{χ}) imply that the following definition does not depend on the choice of the embedding.

Definition 2.5. Choose a maximal R-order Λ̃^{𝔬}(G) containing Λ^{𝔬}(G). We have a decomposition Λ̃^{𝔬}(G) = ⨁_{χ∈Irr (G)/∼_{K}} Λ̃_{χ}^{𝔬}(G), where each Λ̃_{χ}^{𝔬}(G) is a maximal R-order in A_{χ}. For sufficiently large n we call the two-sided Λ̃_{χ}^{𝔬}(G)- lattice

D_{χ}(Λ̃^{𝔬}(G)) = D_{norm}(Λ̃_{χ}^{𝔬}(G)/R) := p^{n}D(Λ̃_{χ}^{𝔬}(G)/R) =

= p^{n} · {x ∈ A_{χ} | tr_{χ}(xΛ̃_{χ}^{𝔬}(G)) ⊆ R}

the normalized inverse different.

Definition 2.6. Let Λ ⊆ Λ̃ be a pair of rings. Then

(Λ̃ : Λ)_{l} := {x ∈ Λ̃ | xΛ̃ ⊆ Λ}

is called the left conductor of Λ̃ into Λ. Similarly,

(Λ̃ : Λ)_{r} := {x ∈ Λ̃ | Λ̃x ⊆ Λ}

is called the right conductor of Λ̃ into Λ.

Proposition 2.7. Let Λ̃^{𝔬}(G) be a maximal R-order containing Λ^{𝔬}(G). Then Λ^{𝔬}(G), the maximal order Λ̃^{𝔬}(G) as well as the left conductor (Λ̃^{𝔬}(G) : Λ^{𝔬}(G))_{l} and the right conductor (Λ̃^{𝔬}(G) : Λ^{𝔬}(G))_{r} are reflexive (and thus free) Rmodules.

Proof. We first observe that R is a 2-dimensional regular local ring and thus every reflexive R-module is free (and vice versa) by [NSW08, Proposition 5.1.9]. As Λ^{𝔬}(G) is free over R, it is also reflexive. Moreover, every maximal R-order is reflexive by [Re03, Theorem 11.4]. Finally, let M := (Λ̃^{𝔬}(G) : Λ^{𝔬}(G))_{r} (the argument for (Λ̃^{𝔬}(G) : Λ^{𝔬}(G))_{l} is similar). We denote by P (R) the set of prime ideals of R of height 1. As M is R-torsionfree, the proof of [NSW08, Proposition 5.1.8] shows that we have an injection M ↣ M^{++} = ⋂_{𝔭∈P (R)} M_{𝔭}, where the equality is [NSW08, Lemma 5.1.2]. Now let x ∈ ⋂_{𝔭∈P (R)} M_{𝔭} be arbitrary; we have to show that x ∈ M. For every 𝔭 ∈ P (R) we have x ∈ Λ̃^{𝔬}(G)_{𝔭} and Λ̃^{𝔬}(G)_{𝔭}x ⊆ Λ^{𝔬}(G)_{𝔭}. Thus x ∈ ⋂_{𝔭∈P (R)} Λ̃^{𝔬}(G)_{𝔭} = Λ̃^{𝔬}(G) and

Λ̃^{𝔬}(G)x = ⋂_{𝔭∈P (R)} Λ̃^{𝔬}(G)_{𝔭}x ⊆ ⋂_{𝔭∈P (R)} Λ^{𝔬}(G)_{𝔭} = Λ^{𝔬}(G),

that is x ∈ M.

We now establish the following analogue of Jacobinski’s conductor formula [CR81, Theorem 27.8].

Theorem 2.8. Let Λ̃^{𝔬}(G) be a maximal R-order containing Λ^{𝔬}(G). Then

(Λ̃^{𝔬}(G) : Λ^{𝔬}(G))_{l} = (Λ̃^{𝔬}(G) : Λ^{𝔬}(G))_{r} = ⨁_{χ∈Irr (G)/∼_{K}} \frac{|H|}{χ(1)}D_{χ}(Λ̃^{𝔬}(G)).

Proof. Put M := (Λ̃^{𝔬}(G) : Λ^{𝔬}(G))_{r}; then M is the largest left Λ̃^{𝔬}(G)-lattice in Λ^{𝔬}(G). So its ordinary dual D_{ord} (M/R) is the smallest right Λ̃^{𝔬}(G)-lattice containing D_{ord} (Λ^{𝔬}(G)/R). Since Λ^{𝔬}(G) is free over R with basis γ^{i}h, 0 ≤ i < p^{n}, h ∈ H, we therefore have

D_{ord} (M/R) = D_{ord} (Λ^{𝔬}(G)/R)Λ̃^{𝔬}(G) = (p^{n}|H|)^{−1}Λ̃^{𝔬}(G),

where we have used Lemma 2.3 for the second equality. However, M is reflexive by Proposition 2.7 and so Proposition 2.2 yields

M = D_{ord} (D_{ord} (M/R)/R) = = D_{ord} ((p^{n}|H|)^{−1}Λ̃^{𝔬}(G)/R) = p^{n}|H|D_{ord} (Λ̃^{𝔬}(G)/R).

The theorem now follows from the equalities

D_{ord} (Λ̃^{𝔬}(G)/R) = ⨁_{χ∈Irr (G)/∼_{K}} D_{ord} (Λ̃_{χ}^{𝔬}(G)/R)
D_{ord} (Λ̃_{χ}^{𝔬}(G)/R) = χ(1)^{−1}D(Λ̃_{χ}^{𝔬}(G)/R)
= (χ(1)p^{n})^{−1}D_{χ}(Λ̃^{𝔬}(G)),

as a similar argument works for (Λ̃^{𝔬}(G) : Λ^{𝔬}(G))_{l}.

3 A formula for the central conductor

Definition 3.1. Let Λ̃^{𝔬}(G) be a maximal R-order containing Λ^{𝔬}(G). Then the central conductor of Λ̃^{𝔬}(G) into Λ^{𝔬}(G) is defined to be

F(Λ^{𝔬}(G)) = F(Λ̃^{𝔬}(G)/Λ^{𝔬}(G)) := ζ(Λ^{𝔬}(G)) ∩ (Λ̃^{𝔬}(G) : Λ^{𝔬}(G))_{l}
= ζ(Λ^{𝔬}(G)) ∩ (Λ̃^{𝔬}(G) : Λ^{𝔬}(G))_{r}
= {x ∈ ζ(Λ^{𝔬}(G)) | xΛ̃^{𝔬}(G) ⊆ Λ^{𝔬}(G)} .

Remark 3.2. We will see below that the central conductor only depends on Λ^{𝔬}(G) and not on the choice of maximal order containing Λ^{𝔬}(G). Hence the notation F(Λ^{𝔬}(G)) is justified.

By Corollary 1.9 we may write Q^{K}(G) = ⨁_{χ∈Irr (G)/∼_{K}} A_{χ}, where each A_{χ} ≃ (D_{χ})_{n_{χ}×n_{χ}} is simple. Similarly, Λ̃^{𝔬}(G) decomposes into ⨁_{χ∈Irr (G)/∼_{K}} Λ̃_{χ}^{𝔬}(G), where each Λ̃_{χ}^{𝔬}(G) is a maximal R-order in A_{χ} with center Λ^{𝔬_{χ}}(Γ′_{χ}). The central conductor is an ideal in ζ(Λ̃^{𝔬}(G)) = ⨁_{χ∈Irr (G)/∼_{K}} Λ^{𝔬_{χ}}(Γ′_{χ}) and so we may write

F(Λ̃^{𝔬}(G)/Λ^{𝔬}(G)) = ⨁_{χ∈Irr (G)/∼_{K}} F_{χ}(Λ̃^{𝔬}(G)/Λ^{𝔬}(G)),

where each F_{χ}(Λ̃^{𝔬}(G)/Λ^{𝔬}(G)) is an ideal in Λ^{𝔬_{χ}}(Γ′_{χ}). We define

d_{χ} := Q^{K_{χ}}(Γ′_{χ}) ∩ D(Λ̃_{χ}^{𝔬}(G)/Λ^{𝔬_{χ}}(Γ′_{χ})).

We denote by P_{χ} the set of prime ideals of Λ^{𝔬_{χ}}(Γ′_{χ}) of height 1.

Lemma 3.3. The fractional Λ^{𝔬_{χ}}(Γ′_{χ})-ideal d_{χ} is a reflexive Λ^{𝔬_{χ}}(Γ′_{χ})-module which does not depend on the choice of maximal order Λ̃^{𝔬}(G).

Proof. We first show that d_{χ} is reflexive. As it is torsionfree, we have an injection d_{χ} ↣ d_{χ}^{++} = ⋂_{𝔭∈P_{χ}}(d_{χ})_{𝔭}. Now let x ∈ ⋂_{𝔭∈P_{χ}}(d_{χ})_{𝔭}. Then clearly x ∈ Q^{K_{χ}}(Γ′_{χ}) and

tr_{A_{χ}/Q^{Kχ} (Γ′_{χ})}(xΛ̃_{χ}^{𝔬}(G)_{𝔭}) ⊆ Λ^{𝔬_{χ}}(Γ′_{χ})_{𝔭}

for all 𝔭 ∈ P_{χ}. This implies

tr_{A_{χ}/Q^{Kχ} (Γ′_{χ})}(xΛ̃_{χ}^{𝔬}(G)) ⊆ ⋂_{𝔭∈P_{χ}} tr_{A_{χ}/Q^{Kχ} (Γ′_{χ})}(xΛ̃_{χ}^{𝔬}(G)_{𝔭})
⊆ ⋂_{𝔭∈P_{χ}} Λ^{𝔬_{χ}}(Γ′_{χ})_{𝔭}
= Λ^{𝔬_{χ}}(Γ′_{χ}).

Thus x belongs to Q^{K_{χ}}(Γ′_{χ}) ∩ D(Λ̃_{χ}^{𝔬}(G)/Λ^{𝔬_{χ}}(Γ′_{χ})) = d_{χ} as claimed. Now let Λ̆^{𝔬}(G) be a second maximal R-order containing Λ^{𝔬}(G). Then likewise Λ̆^{𝔬}(G) decomposes into ⨁_{χ∈Irr (G)/∼_{K}} Λ̆_{χ}^{𝔬}(G), where each Λ̆_{χ}^{𝔬}(G) is a maximal R-order in A_{χ}. We put

d̆_{χ} := Q^{K_{χ}}(Γ′_{χ}) ∩ D(Λ̆_{χ}^{𝔬}(G)/Λ^{𝔬_{χ}}(Γ′_{χ})).

Fix a height 1 prime ideal 𝔭 ∈ P_{χ}. By [AG60, Proposition 3.5] there is a unit a_{χ} ∈ A_{χ} such that Λ̆_{χ}^{𝔬}(G)_{𝔭} = a_{χ}Λ̃_{χ}^{𝔬}(G)_{𝔭}a_{χ}^{−1}. Now let x ∈ Q^{K_{χ}}(Γ′_{χ}). Then

tr_{A_{χ}/Q^{Kχ} (Γ′_{χ})}(xΛ̆_{χ}^{𝔬}(G)_{𝔭}) = tr_{A_{χ}/Q^{Kχ} (Γ′_{χ})}(xa_{χ}Λ̃_{χ}^{𝔬}(G)_{𝔭}a_{χ}^{−1})
= tr_{A_{χ}/Q^{Kχ} (Γ′_{χ})}(a_{χ}xΛ̃_{χ}^{𝔬}(G)_{𝔭}a_{χ}^{−1})
= tr_{A_{χ}/Q^{Kχ} (Γ′_{χ})}(xΛ̃_{χ}^{𝔬}(G)_{𝔭}),

where the second equality holds, since x is central in A_{χ}. Hence

tr_{A_{χ}/Q^{Kχ} (Γ′_{χ})}(xΛ̆_{χ}^{𝔬}(G)_{𝔭}) ⊆ Λ^{𝔬_{χ}}(Γ′_{χ})_{𝔭}⟺tr_{A_{χ}/Q^{Kχ} (Γ′_{χ})}(xΛ̃_{χ}^{𝔬}(G)_{𝔭}) ⊆ Λ^{𝔬_{χ}}(Γ′_{χ})_{𝔭},

that is (d_{χ})_{𝔭} = (d̆_{χ})_{𝔭} for all 𝔭 ∈ P_{χ}. As both d_{χ} and d̆_{χ} are reflexive, this implies d_{χ} = d̆_{χ} as desired.

Let π_{χ} be a prime element in 𝔬_{χ} and put 𝔭_{χ} := π_{χ}Λ^{𝔬_{χ}}(Γ′_{χ}) ∈ P_{χ}.

Lemma 3.4. There is an integer r_{χ} ≤ 0 such that d_{χ} = 𝔭_{χ}^{r_{χ}}. If the Schur index s_{χ} is not divisible by p, then r_{χ} = 0 and thus d_{χ} = Λ^{𝔬_{χ}}(Γ′_{χ}).

Proof. Let Δ_{χ} be a maximal order in D_{χ}. Then (Δ_{χ})_{n_{χ}×n_{χ}} is a maximal order in A_{χ} by [Re03, Theorem 8.7]. Let α ∈ (Δ_{χ})_{n_{χ}×n_{χ}} be the matrix with a 1 in the upper left corner and zeros everywhere else. Let x ∈ d_{χ}. Then in particular

tr_{A_{χ}/Q^{Kχ} (Γ′_{χ})}(xα) = x · tr_{A_{χ}/Q^{Kχ} (Γ′_{χ})}(α) = x · s_{χ} ∈ Λ^{𝔬_{χ}}(Γ′_{χ}).  (11)

Here, the first equality follows from linearity of the reduced trace as x belongs to Q^{K_{χ}}(Γ′_{χ}). For the second equality we compute

tr_{A_{χ}/Q^{Kχ} (Γ′_{χ})}(α) = tr_{D_{χ}/Q^{Kχ} (Γ′_{χ})}(1) = s_{χ}.

As s_{χ} is an integer, it follows from (11) that x ∈ Λ^{𝔬_{χ}}(Γ′_{χ})_{𝔭} for every 𝔭 ∈ P_{χ}, 𝔭 ≠ 𝔭_{χ}. However, d_{χ} is reflexive by Lemma 3.3 and thus we must have d_{χ} = 𝔭_{χ}^{r_{χ}} for some integer r_{χ}. Since obviously Λ^{𝔬_{χ}}(Γ′_{χ}) ⊆ d_{χ}, we have r_{χ} ≤ 0. Finally, if p does not divide s_{χ}, then by (11) we have x ∈ Λ^{𝔬_{χ}}(Γ′_{χ})_{𝔭} for every 𝔭 ∈ P_{χ} including 𝔭_{χ} and thus r_{χ} = 0 in this case.

Theorem 3.5. Let Λ̃^{𝔬}(G) be a maximal order containing Λ^{𝔬}(G). Then F(Λ̃^{𝔬}(G)/Λ^{𝔬}(G)) does not depend on the choice of maximal order Λ̃^{𝔬}(G) and we have an equality

F(Λ^{𝔬}(G)) = F(Λ̃^{𝔬}(G)/Λ^{𝔬}(G)) = ⨁_{χ∈Irr (G)/∼_{K}} \frac{|H|w_{χ}}{χ(1)} · 𝔇^{−1}(𝔬_{χ}/𝔬)d_{χ}.

Here, d_{χ} = 𝔭_{χ}^{r_{χ}} for some integer r_{χ} ≤ 0. In particular, we have an inclusion

\frac{|H|w_{χ}}{χ(1)} · 𝔇^{−1}(𝔬_{χ}/𝔬)Λ^{𝔬_{χ}}(Γ′_{χ}) ⊆ F_{χ}(Λ̃^{𝔬}(G)/Λ^{𝔬}(G)),  (12)

which becomes an equality after localization at every 𝔭 ∈ P_{χ} different from 𝔭_{χ}. Moreover, the inclusion (12) is an equality in each of the following cases:

(i) G = H × Γ is a direct product.

(ii) The Schur index s_{χ} is not divisible by p.

(iii) There is a maximal 𝔬_{χ}-order Δ_{χ} contained in Λ̃_{χ}^{𝔬}(G) such that Λ^{𝔬_{χ}}(Γ′_{χ}) ⊗_{𝔬_{χ}} Δ_{χ} = Λ̃_{χ}^{𝔬}(G).

Remark 3.6. Fix a p-adic Lie-group G of dimension 1. Then (ii) and (iii) hold for all χ whenever K is sufficiently large. This follows from Theorem 1.11 and Lemma 1.14.

Remark 3.7. If p does not divide the order of the commutator subgroup of G, then Λ^{𝔬}(G) is a direct sum of matrix rings over commutative rings by [JN13, Proposition 4.5]. In particular, no skewfields occur in the Wedderburn decomposition of Q^{K}(G) and thus (ii) holds for every χ ∈ Irr (G).

Proof of Theorem 3.5. Let us define

δ_{χ}^{−1} := p^{n}𝔇^{−1}(Λ^{𝔬_{χ}}(Γ′_{χ})/R) = w_{χ} · 𝔇^{−1}(𝔬_{χ}/𝔬)Λ^{𝔬_{χ}}(Γ′_{χ}),

where the second equality follows from Lemma 2.4 and Remark 1.8.

Lemma 3.8. We have an equality D_{χ}(Λ̃^{𝔬}(G)) = D(Λ̃_{χ}^{𝔬}(G)/Λ^{𝔬_{χ}}(Γ′_{χ})) · δ_{χ}^{−1}.

Proof. This is a special case of Lemma 2.1.

By Theorem 2.8 and the definition of the central conductor we obtain

F_{χ}(Λ̃^{𝔬}(G)/Λ^{𝔬}(G)) = Λ^{𝔬_{χ}}(Γ′_{χ}) ∩ (\frac{|H|}{χ(1)} · D_{χ}(Λ̃^{𝔬}(G)))

for each χ ∈ Irr (G). Hence it must be shown that

Λ^{𝔬_{χ}}(Γ′_{χ}) ∩ (\frac{|H|}{χ(1)} · D_{χ}(Λ̃^{𝔬}(G))) = \frac{|H|}{χ(1)}δ_{χ}^{−1}d_{χ}  (13)

for each character χ. We note that

\frac{|H|}{χ(1)}δ_{χ}^{−1}d_{χ} ⊆ \frac{|H|}{χ(1)}D_{χ}(Λ̃^{𝔬}(G)) ⊆ Λ̃^{𝔬}(G);

so each element of \frac{|H|}{χ(1)}δ_{χ}^{−1}d_{χ} is integral over R, and thus lies in Λ^{𝔬_{χ}}(Γ′_{χ}). This gives one inclusion in (13). Now let y ∈ Λ^{𝔬_{χ}}(Γ′_{χ}). Then by Lemma 3.8 we have

y ∈ \frac{|H|}{χ(1)} ·D_{χ}(Λ̃^{𝔬}(G)) ⟺ yδ_{χ} ⊆ \frac{|H|}{χ(1)}D(Λ̃_{χ}^{𝔬}(G)/Λ^{𝔬_{χ}}(Γ′_{χ})) ⟺ yδ_{χ} ⊆ \frac{|H|}{χ(1)}d_{χ}

giving the reverse inclusion in (13). Lemma 3.3 and Lemma 3.4 imply the remaining assertions apart from the claim that r_{χ} = 0 in case (i) and (iii). Let us first assume that G = H × Γ is a direct product. Choose a maximal 𝔬-order 𝔐(H) containing 𝔬[H]. Let χ ∈ Irr (G). Then res _{H}^{G}χ = η is an irreducible character of H and there is a character ρ of type W such that χ ⊗ ρ is trivial on Γ. As χ ∼_{K} χ ⊗ ρ, we may henceforth assume that ρ = 1. We then have w_{χ} = 1, K_{χ} = K(η), e_{χ} = e(η) and γ_{χ} = γe_{χ} = γ′_{χ}. Then 𝔐(H)ε_{χ} is a maximal 𝔬_{χ}-order such that Λ̃_{χ}^{𝔬}(G) := Λ^{𝔬_{χ}}(Γ) ⊗_{𝔬_{χ}} 𝔐(H)ε_{χ} is a maximal R-order in A_{χ}. It follows that (i) is a special case of (iii). Now assume that (iii) holds and let y ∈ (d_{χ})_{𝔭_{χ}} be arbitrary. We have to show that y ∈ Λ^{𝔬_{χ}}(Γ′_{χ})_{𝔭_{χ}}. As the reduced trace is Q^{K_{χ}}(Γ′_{χ})-linear, we may alter y by a unit in Λ^{𝔬_{χ}}(Γ′_{χ})_{𝔭_{χ}} so that we may assume that y = π_{χ}^{k} for an appropriate integer k; in particular, we then have y ∈ K_{χ}. By hypothesis there is a maximal 𝔬_{χ}-order Δ_{χ} contained in Λ^{𝔬_{χ}}(Γ′_{χ}) such that Q^{K_{χ}}(Γ′_{χ}) ⊗_{𝔬_{χ}} Δ_{χ} = A_{χ}. Hence an 𝔬_{χ}-basis of Δ_{χ} is also a Q^{K_{χ}}(Γ′_{χ})-basis of A_{χ} which we may use to compute the reduced trace. Hence y ∈ K_{χ} has

tr_{K_{χ}⊗_{𝔬χ} Δ_{χ}/K_{χ}}(yΔ_{χ}) ⊆ Λ^{𝔬_{χ}}(Γ′_{χ})_{𝔭_{χ}} ∩ K_{χ} = 𝔬_{χ}

and we have to show that y ∈ 𝔬_{χ}; in other words we shall show that K_{χ} ∩ 𝔇^{−1}(Δ_{χ}/𝔬_{χ}) = 𝔬_{χ}. Suppose that this is not true. Then 𝔬_{χ} is properly contained in K_{χ} ∩ 𝔇^{−1}(Δ_{χ}/𝔬_{χ}). Then 𝔭_{χ}^{−1} ⊆ 𝔇^{−1}(Δ_{χ}/𝔬_{χ}) and thus 𝔭_{χ}Δ_{χ} contains 𝔇(Δ_{χ}/𝔬_{χ}). However, if 𝔓_{χ} denotes the radical of Δ_{χ}, then 𝔭_{χ}Δ_{χ} = 𝔓_{χ}^{e} for some positive integer e and 𝔇(Δ_{χ}/𝔬_{χ}) = 𝔓_{χ}^{e−1} by [Re03, Theorems 20.3 and 14.3], a contradiction.

Example. Assume that G = H × Γ is a direct product and thus

Λ^{𝔬}(G) = Λ^{𝔬}(Γ)[H] = Λ^{𝔬}(Γ) ⊗_{𝔬} 𝔬[H].  (14)

Let χ ∈ Irr (G). As we have seen in the proof of Theorem 3.5 we then have w_{χ} = 1, K_{χ} = K(η) and γ′_{χ} = γe_{χ}. Hence

F(Λ^{𝔬}(G)) = ⨁_{χ∈Irr (G)/∼_{K}} \frac{|H|}{χ(1)} · 𝔇^{−1}(𝔬_{χ}/𝔬)Λ^{𝔬_{χ}}(Γ) = Λ^{𝔬}(Γ) ⊗_{𝔬} F(𝔬[H])

which can be shown more directly using (14) and Jacobinski’s central conductor formula (1). So the main obstacle to derive Theorem 3.5 directly from Jacobinski’s result is the fact that in general G = H ⋊ Γ is only semi-direct.

We finally determine the integers r_{χ} whenever G is a pro-p-group. For simplicity we will also assume that K = ℚ_{p}. Then K_{χ} = ℚ_{p,χ} and 𝔬_{χ} =: ℤ_{p,χ} denotes the ring of integers in ℚ_{p,χ}. Let 𝔔_{η} and 𝔔_{χ} be the maximal ideals in ℤ_{p}[η] and ℤ_{p,χ}, respectively. If x is a rational number, we let ⌊x⌋ ∈ ℤ denote the largest integer such that ⌊x⌋ ≤ x.

Theorem 3.9. Assume that G is a pro-p-group and a p-adic Lie group of dimension 1. Fix χ ∈ Irr (G) and let A_{χ} ≃ (D_{χ})_{n_{χ}×n_{χ}} be the corresponding simple component of Q(G). Write 𝔇(ℤ_{p}[η]/ℤ_{p,χ}) = 𝔔_{η}^{k_{χ}} for some integer k_{χ} ≥ 0. Then D_{χ} is a cyclic skewfield with Schur index s_{χ} = [ℚ_{p}(η) : ℚ_{p,χ}] and r_{χ} = −⌊\frac{k_{χ}}{s_{χ}} ⌋.

Proof. By [La12, Theorem 1] we know that D_{χ} is a cyclic skewfield with Schur index s_{χ} = [ℚ_{p}(η) : ℚ_{p,χ}]. More precisely, the extension ℚ_{p}(η)/ℚ_{p,χ} is cyclic and totally ramified, as G is pro-p and thus ℚ_{p}(η) is a subfield of ℚ_{p}(ζ_{p^{m}}) for some positive integer m. Let σ be a generator of the Galois group Gal(ℚ_{p}(η)/ℚ_{p,χ}). Then by [La12, Theorem 1] the skewfield D_{χ} is given by

D_{χ} ≃ ⨁_{i=0}^{s_{χ}−1} (Q^{ℚ_{p}(η)}(Γ^{s_{χ}})) γ^{i},  (15)

where γ^{s_{χ}} is a topological generator of Γ^{s_{χ}} ≃ ℤ_{p} and for x ∈ Q^{ℚ_{p}(η)}(Γ^{s_{χ}}) we have γx = σ(x)γ (note that this is not the same Γ as in G = H ⋊ Γ; it is just a topological group isomorphic to ℤ_{p} which we again denote by Γ). Then D_{χ} has center Q^{ℚ_{p,χ}}(Γ^{s_{χ}}) which is isomorphic to Q^{ℚ_{p,χ}}(Γ′_{χ}) under the isomorphism (15). Now let d = ∑_{i=0}^{s_{χ}−1} x_{i}γ^{i} ∈ D_{χ} be arbitrary, x_{i} ∈ Q^{ℚ_{p}(η)}(Γ^{s_{χ}}), 0 ≤ i < s_{χ}. We claim that

tr_{D_{χ}/Q^{ℚp,χ} (Γ^{sχ})}(d) = Tr_{Q^{ℚp(η)}(Γ^{sχ})/Q^{ℚp,χ} (Γ^{sχ})}(x_{0}).  (16)

For this one only has to check that we have an isomorphism

ℚ_{p}(η) ⊗_{ℚ_{p,χ}} D_{χ} ≃ (Q^{ℚ_{p}(η)}(Γ))_{s_{χ}×s_{χ}}

x_{0} ↦ ( \begin{matrix}x_{0} \\ & σ(x_{0}) \\ & & . & . & . \\ & & & & & σ^{s_{χ}−1}(x_{0}) \end{matrix} )

γ ↦ ( \begin{matrix}0 & & & γ \\ γ & ... & & 0 \\ & ... & 0 & ⋮ \\ & & γ & 0 \end{matrix} )

From this one can easily compute the reduced trace in (16). Now put

Σ_{χ} := ⨁_{i=0}^{s_{χ}−1} (Λ^{ℤ_{p}[η]}(Γ^{s_{χ}})) γ^{i} ⊆ D_{χ}.

Then Σ_{χ} is a Λ^{ℤ_{p,χ}}(Γ^{s_{χ}})-order in D_{χ}. Moreover, 𝔔_{η}Σ_{χ} is a two-sided ideal of Σ_{χ}, since γ𝔔_{η} = σ(𝔔_{η})γ = 𝔔_{η}γ. We consider the Λ^{ℤ_{p,χ}}(Γ^{s_{χ}})_{𝔭_{χ}}-order Σ_{𝔭_{χ}} := (Σ_{χ})_{𝔭_{χ}} in D_{χ}. Then 𝔔_{η}Σ_{𝔭_{χ}} is again two-sided and we have

(𝔔_{η}Σ_{𝔭_{χ}})^{s_{χ}} = 𝔔_{χ}Σ_{𝔭_{χ}} = 𝔭_{χ}Σ_{𝔭_{χ}} ⊆ rad(Σ_{𝔭_{χ}}).

Hence also 𝔔_{η}Σ_{𝔭_{χ}} ⊆ rad(Σ_{𝔭_{χ}}) by [Re03, §6, Exercise 3]. We now observe that γ ↦ 1 + T induces an isomorphism

Σ_{χ}/𝔔_{η}Σ_{χ} ≃ 𝔽_{p}[[T]].

Moreover, this isomorphism induces

Σ_{𝔭_{χ}}/𝔔_{η}Σ_{𝔭_{χ}} ≃ 𝔽_{p}((T))

and thus the inclusion 𝔔_{η}Σ_{𝔭_{χ}} ⊆ rad(Σ_{𝔭_{χ}}) is an equality. Moreover, the order Σ_{𝔭_{χ}} is quasi-local. As 𝔔_{η} is a principal ideal, there is a short exact sequence

0 → Σ_{𝔭_{χ}} → Σ_{𝔭_{χ}} → 𝔽_{p}((T)) → 0

and thus the projective dimension of 𝔽_{p}((T)) considered as left Σ_{𝔭_{χ}}-module equals 1. Now [Ra69, Corollary 1.3] gives that also gl.dim (Σ_{𝔭_{χ}}) = 1. It then follows from [AG60, Theorem 2.3] that Σ_{𝔭_{χ}} is a maximal Λ^{ℤ_{p,χ}}(Γ^{s_{χ}})_{𝔭_{χ}}-order. By (16) we find that

tr_{D_{χ}/Q^{ℚp,χ} (Γ^{sχ})}(Σ_{𝔭_{χ}}) = Tr_{Q^{ℚp(η)}(Γ^{sχ})/Q^{ℚp,χ} (Γ^{sχ})}(Λ^{ℤ_{p}[η]}(Γ^{s_{χ}})_{𝔭_{χ}})
= Tr_{ℚ_{p}(η)/ℚ_{p,χ}}(ℤ_{p}[η])Λ^{ℤ_{p,χ}}(Γ^{s_{χ}})_{𝔭_{χ}}
= 𝔔_{χ}^{r′_{χ}}Λ^{ℤ_{p,χ}}(Γ^{s_{χ}})_{𝔭_{χ}},

where r′_{χ} ∈ ℤ is maximal such that

ℤ_{p}[η] ⊆ 𝔔_{χ}^{r′_{χ}}𝔇(ℤ_{p}[η]/ℤ_{p,χ})^{−1} = 𝔔_{η}^{s_{χ}r′_{χ}−k_{χ}}

and thus r′_{χ} = ⌊\frac{k_{χ}}{s_{χ}} ⌋. Now let x ∈ Q^{ℚ_{p,χ}}(Γ^{s_{χ}}). Then x belongs to (d_{χ})_{𝔭_{χ}} if and only if

tr_{D_{χ}/Q^{ℚp,χ} (Γ^{sχ})}(x · Σ_{𝔭_{χ}}) = x · tr_{D_{χ}/Q^{ℚp,χ} (Γ^{sχ})}(Σ_{𝔭_{χ}}) ⊆ Λ^{ℤ_{p,χ}}(Γ^{s_{χ}})_{𝔭_{χ}}

and thus if and only if x ∈ 𝔭_{χ}^{−r′_{χ}}. Therefore we have r_{χ} = −r′_{χ} = −⌊\frac{k_{χ}}{s_{χ}} ⌋ as desired.

Example. We continue the example in [La12, p. 1233]. For this let p = 3 and let H be the cyclic group of order 9. Choose a generator h of H. We put G = H ⋊ Γ, where the action of Γ on H is determined by γhγ^{−1} = h^{4}. Let η be the irreducible character of H with η(h) = ζ_{9}. Then St(η) = H × ⟨γ^{3}⟩ and χ = ind _{St(η)}^{G}χ′ with χ′(h) = η(h) and χ′(γ^{3}) = 1 is an irreducible character of G with open kernel. We have χ(1) = w_{χ} = 3 and ℚ_{p}(η) = ℚ_{p}(ζ_{9}), ℚ_{p,χ} = ℚ_{p}(ζ_{3}). We see that s_{χ} = [ℚ_{p}(ζ_{9}) : ℚ_{p}(ζ_{3})] = 3. Moreover, the different 𝔇(ℤ_{p}[ζ_{9}]/ℤ_{p}[ζ_{3}]) is easily computed and we find that k_{χ} = 3. Thus r_{χ} = −⌊\frac{k_{χ}}{s_{χ}} ⌋ = −1 is non-trivial and (12) is a proper inclusion in this case.

4 Consequences of the central conductor formula

In this section we derive several consequences of our main Theorem 3.5.

4.1 Extensions of lattices

Corollary 4.1. Let M be a Λ^{𝔬}(G)-lattice and let N be a finitely generated Λ^{𝔬}(G)-module. Then

( ⨁_{χ∈Irr (G)/∼_{K}} \frac{|H|w_{χ}}{χ(1)} · 𝔇^{−1}(𝔬_{χ}/𝔬)d_{χ}) · Ext_{Λ^{𝔬}(G)}^{i}(M, N)

is finite for all integers i ≥ 1.

Proof. An R-module M is finite if and only if it is pseudo-null, i.e. M_{𝔭} = 0 for every height one prime ideal 𝔭 of R. We do induction on the integer i. As R_{𝔭} is a Dedekind domain for every height one prime ideal 𝔭, the case i = 1 is now immediate from Theorem 3.5 and [CR81, Theorem 29.4]. For k sufficiently large, there is an exact sequence

M′ ↣ Λ^{𝔬}(G)^{k} ↠ M.

Then M′ also is a Λ^{𝔬}(G)-lattice. Applying Hom_{Λ^{𝔬}(G)}(−, N) to the above exact sequence gives isomorphisms

Ext_{Λ^{𝔬}(G)}^{j}(M′, N) ≃ Ext_{Λ^{𝔬}(G)}^{j+1}(M, N)

for all integers j ≥ 1. The case j = i − 1 gives the induction step.

Example. The stronger statement that the central conductor annihilates Ext_{Λ^{𝔬}(G)}^{i}(M, N) is not true. Assume that G = Γ and K = ℚ_{p}. Then Corollary 4.1 simply asserts that Ext_{Λ(Γ)}^{i}(M, N) is finite for every finitely generated torsionfree Λ(Γ)-module M and every finitely generated Λ(Γ)-module N. Suppose that Ext_{Λ(Γ)}^{i}(M, N) = 0 for every finitely generated Λ(Γ)-module N; then M is projective and thus free as Λ(Γ)-module. However, there are many examples of torsionfree Λ(Γ)-modules which are not free (take for example the maximal ideal of Λ(Γ)).

For χ ∈ Irr (G) we put 𝔞_{χ}^{−1} := (𝔇^{−1}(𝔬_{χ}/𝔬) ∩ K) · R ⊇ R.

Corollary 4.2. We have

⋂_{χ∈Irr (G)/∼_{K}} \frac{|H|w_{χ}}{χ(1)} · 𝔞_{χ}^{−1} ⊆ R ∩ F(Λ^{𝔬}(G)).

In fact, if r_{χ} = 0 for all χ ∈ Irr (G), then a proof similar to that of [CR81, Theorem 27.13(ii)] shows that the inclusion in Corollary 4.2 is an equality. For every Λ^{𝔬}(G)-module M let Υ(M) := {e_{χ} | e_{χ} · Q^{K}(G) ⊗_{Λ^{𝔬}(G)} M = 0}.

Corollary 4.3. Let M and N be Λ^{𝔬}(G)-lattices and let i ≥ 1 be an integer. Then

( ⋂_{e_{χ}∉Υ(M)} \frac{|H|w_{χ}}{χ(1)} · 𝔞_{χ}^{−1}) Ext_{Λ^{𝔬}(G)}^{i}(M, N)

is finite. In particular, (⋂_{χ∈Irr (G)/∼_{K}} \frac{|H|w_{χ}}{χ(1)} · 𝔞_{χ}^{−1}) · Ext_{Λ^{𝔬}(G)}^{i}(M, N) is finite for every Λ^{𝔬}(G)-lattices M and N and every integer i ≥ 1.

Proof. The last assertion is an immediate consequence of Corollary 4.1 and Corollary 4.2. The first assertion is also easy and is shown exactly in the same way as [CR81, Theorem 29.9].

Corollary 4.4. Let M and N be Λ^{𝔬}(G)-lattices and assume that Q^{K}(G) ⊗_{Λ^{𝔬}(G)} M is absolutely simple. Then there is a unique idempotent e_{χ} ∉ Υ(M) and for every integer i ≥ 1 we have that \frac{|H|w_{χ}}{χ(1)} · Ext_{Λ^{𝔬}(G)}^{i}(M, N) is finite.

Example. Assume that p divides |H| and that K = ℚ_{p}. We consider Λ(Γ) as Λ(G)-module in the natural way; then Q(G) ⊗_{Λ(G)} Λ(Γ) is absolutely simple and e_{χ_{0}} is the unique idempotent not contained in Υ(Λ(Γ)), where χ_{0} denotes the trivial character. Moreover, we have χ_{0}(1) = 1 and w_{χ_{0}} = 1. Let Δ(G, Γ) be the kernel of the augmentation map aug : Λ(G) ↠ Λ(Γ) which sends each h ∈ H to 1. We then have an exact sequence

Hom_{Λ(G)}(Λ(Γ), Δ(G, Γ)) ↣ Hom_{Λ(G)}(Λ(Γ), Λ(G)) ⟶^{aug_{∗}} Hom_{Λ(G)}(Λ(Γ), Λ(Γ))

↠ Ext_{Λ(G)}^{1}(Λ(Γ), Δ(G, Γ)).

We have a canonical isomorphism Hom_{Λ(G)}(Λ(Γ), Λ(Γ)) ≃ Λ(Γ), f ↦ f(1) and the image of aug_{∗} in Λ(Γ) under this identification is given by aug ((∑_{h∈H} h)Λ(G)) = |H|Λ(Γ). We therefore have a canonical isomorphism

Ext_{Λ(G)}^{1}(Λ(Γ), Δ(G, Γ)) ≃ Λ(Γ)/|H|Λ(Γ);

in particular, this module is not finite. However, |H|Ext_{Λ(G)}^{1}(Λ(Γ), Δ(G, Γ)) = 0 is finite.

4.2 Non-commutative Fitting invariants

For the following we refer the reader to [Ni10]. Let A be a separable L-algebra and Λ be an R-order in A, finitely generated as R-module, where R is an integrally closed complete commutative noetherian local domain with field of quotients L. Let N and M be two ζ(Λ)-submodules of an R-torsionfree ζ(Λ)- module. Then N and M are called nr(Λ)-equivalent if there exists an integer n and a matrix U ∈ Gl_{n}(Λ) such that N = nr(U) · M, where nr : A → ζ(A) denotes the reduced norm map which extends to matrix rings over A in the obvious way. We denote the corresponding equivalence class by [N]_{nr(Λ)}. We say that N is nr(Λ)-contained in M (and write [N]_{nr(Λ)} ⊆ [M]_{nr(Λ)}) if for all N′ ∈ [N]_{nr(Λ)} there exists M′ ∈ [M]_{nr(Λ)} such that N′ ⊆ M′. We will say that x is contained in [N]_{nr(Λ)} (and write x ∈ [N]_{nr(Λ)}) if there is N_{0} ∈ [N]_{nr(Λ)} such that x ∈ N_{0}. Now let M be a finitely presented (left) Λ-module and let

Λ^{a} ⟶^{h} Λ^{b} ↠ M  (17)

be a finite presentation of M. We identify the homomorphism h with the corresponding matrix in M_{a×b}(Λ) and define S(h) = S_{b}(h) to be the set of all b × b submatrices of h if a ≥ b. The Fitting invariant of h over Λ is defined to be

Fitt_{Λ}(h) = \{ \begin{matrix}[0]_{nr(Λ)} & if & a < b \\ [⟨nr(H) | H ∈ S(h)⟩_{ζ(Λ)}]_{nr(Λ)} & if & a ≥ b. \end{matrix}

We call Fitt_{Λ}(h) a Fitting invariant of M over Λ. One defines Fitt_{Λ}^{max}(M) to be the unique Fitting invariant of M over Λ which is maximal among all Fitting invariants of M with respect to the partial order “⊆”. We now specialize to the situation in this article, where Λ is Λ^{𝔬}(G). Then Theorem 3.5 and [JN13, Corollary 6.2] imply the following result.

Corollary 4.5. Let M be a finitely presented Λ^{𝔬}(G)-module. Then

( ⨁_{χ∈Irr (G)/∼_{K}} \frac{|H|w_{χ}}{χ(1)} · 𝔇^{−1}(𝔬_{χ}/𝔬)d_{χ}) · Fitt_{Λ^{𝔬}(G)}^{max}(M) ⊆ Ann_{Λ^{𝔬}(G)}(M).

Together with Corollary 4.2 and [Ni10, Lemma 3.4] this yields the following corollary.

Corollary 4.6. Let M be a finitely presented Λ^{𝔬}(G)-module. Then

( ⋂_{e_{χ}∉Υ(M)} \frac{|H|w_{χ}}{χ(1)} · 𝔞_{χ}^{−1}) · Fitt_{Λ^{𝔬}(G)}^{max}(M) ⊆ Ann_{Λ^{𝔬}(G)}(M).

Remark 4.7. Note that in fact H(Λ^{𝔬}(G)) · Fitt_{Λ^{𝔬}(G)}^{max}(M) ⊆ Ann_{Λ^{𝔬}(G)}(M), where H(Λ^{𝔬}(G)) is a certain ideal of ζ(Λ^{𝔬}(G)) which always contains the central conductor. In general, however, this containment is not an equality. Though the ideal H(Λ^{𝔬}(G)) is hard to determine in general, considerable progress is made in [JN13]; in particular, by [JN13, Proposition 4.5] one knows that H(Λ^{𝔬}(G)) equals ζ(Λ^{𝔬}(G)) (to wit: is best possible) if and only if p does not divide the order of the commutator subgroup of G (which is finite).

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Andreas Nickel Universität Bielefeld Fakultät für Mathematik Postfach 100131 33501 Bielefeld Germany anickel3@math.unibielefeld.de 628

Documenta Mathematica 19 (2014)