A conductor formula for completed group algebras

Mathematik

A conductor formula for completed group algebras

Andreas Nickel

Preprint Nr. 06/2012

Andreas Nickel^∗

Abstract

Letobe the ring of integers in a nite extension ofQp. IfGis a nite group andΓ is a maximal order containing the group ringoG, Jacobinski's conductor formula gives a complete description of the central conductor ofΓintooGin terms of characters of G. We prove a similar result for completed group algebraso[[G]], whereGis ap-adic Lie group of dimension1. We will also discuss several consequences of this result.

Introduction

Let^obe the ring of integers in a number eldK and consider the group ring^oGof a nite group G over ^o. The central conductor F(oG) consists of all elements x in the center of oGsuch thatxΓ⊂^oG, whereΓ is a chosen maximal order containing^oG, i.e.

F(oG) ={x∈ζ(oG)|xΓ⊂^oG}.

Here, for any ring Λ, we write ζ(Λ) for the center ofΛ. 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 ofG. More precisely, we have

F(oG) =⊕

χ

|G|

χ(1)D⁻¹(o[χ]/o),

where the sum runs through all irreducible characters of G modulo Galois action, and D⁻¹(o[χ]/o)denotes the inverse dierent of^o[χ], the ring of integers inK(χ) :=K(χ(g)|g∈ G), with respect to ^o. Jacobinski's main interest was in determining annihilators ofExt; in fact, he showed that

F(oG)·Ext¹_oG(M, N) = 0

for all ^oG-lattices M and ^oG-modulesN. For instance, this implies that|G|/χ(1)annihi- lates Ext¹_oG(Mχ, N) if Mχ is an ^oG-lattice such that K ⊗oMχ is absolutely simple with characterχ. Later, Roggenkamp [Ro71] showed that the annihilators achieved in this way are in fact 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 nite extension K of Qp and G is a p-adic Lie group of dimension 1, i.e.

∗I acknowledge nancial support provided by the DFG

2010 Mathematics Subject Classication: 16H10, 16H20, 11R23

Keywords: central conductor, completed group algebras, extensions of lattices, Fitting invariants

1

G can be written as a semi-direct product H_oΓ with nite H and a cyclic pro-p-group Γ, isomorphic to Zp. 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]]. But it turns out that the results provided by Ritter and Weiss are not sucient for our purposes such that we have to determine the structure of Q^K(G) in more detail, thereby generalizing results of Lau [La] (where K = Qp and G is pro-p). The main result of this rst section is that there is always a nite Galois extension E ofK such thatQ^E(G) splits. In section2 and 3we provide the necessary preparations for our main result which will be stated and proved in section 4. More precisely, if we dene the central conductor in complete analogy to the group ring case, then we have an equality

F(o[[G]]) =⊕

χ/∼

|H|w_χ

χ(1) · D⁻¹(o_χ/o)o_χ[[Γ_χ]],

where the sum runs through all irreducible characters ofGwith open kernel up to a certain explicit equivalence relation. Moreover, w_χ is the index of a certain subgroup (depending on χ) in Gand ^oχ 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.

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. And even if we localize at a height one prime ideal, the residue eld will not be nite. Hence we do not have the well elaborated theory of maximal orders over discrete valuation rings with nite residue eld at our disposal. We will overcome this problem by replacing our chosen maximal ^o[[T]]-order by a suitable maximal ^o-order.

Finally, we draw some consequences in section 5. Especially, we obtain annihilation results for the corresponding Ext-groups in complete analogy to the group ring case. We further apply our main result to the theory of non-commutative Fitting invariants intro- duced by the author [Ni10]. In fact, this theory may be applied to^o[[G]]-modules even ifG is non-abelian. But in contrast to the commutative case, the Fitting invariant of a nitely presented ^o[[G]]-module M might not be contained in the annihilator of M. To achieve annihilators one has to multiply by a certain ideal of ζ(o[[G]]) which is hard to determine in general. But it is easily seen that this ideal always contains the central conductor such that our main theorem provides a method to compute explicit annihilators of a nitely presented ^o[[G]]-module, at least, if we are able to compute its Fitting invariant.

1 On the total ring of fractions of a completed group algebra

Let p be an odd prime and let G be a pronite group which contains a nite normal subgroup H such that G/H ≃ Γ for a pro-p-group Γ, isomorphic to Zp; thus G can be written as a semi-direct productHoΓand is ap-adic Lie group of dimension1. We denote the completed group algebra Zp[[G]] by Λ(G). If K is a nite eld extension of Qp with

ring of integers^o, we putΛ^o(G) :=o⊗_ZpΛ(G) =o[[G]]. We x a topological generatorγ of Γand choose a natural numbernsuch thatγ^pn is central inG. Since alsoΓ^pn ≃Zp, there is an isomorphism ^o[[Γ^pn]] ≃ ^o[[T]] induced by γ^pn 7→ 1 +T. Here, R := o[[T]] denotes the power series ring in one variable over ^o. If we viewΛ^o(G) as an R-module, there is a decomposition

Λ^o(G) =

p⊕ⁿ−1 i=0

Rγⁱ[H].

Hence Λ^o(G) is nitely generated as an R-module and an R-order in the separable L :=

Quot(R)-algebraQ^K(G) :=⊕

iLγⁱ[H]. Note that Q^K(G) is obtained fromΛ^o(G) by in- verting all regular elements andQ^K(G) =K⊗QpQ(G), whereQ(G) :=Q^Qp(G).

LetQ^c_p be an algebraic closure of Qp and x an irreducibleQ^c_p-valued characterχofG with open kernel. Choose a nite Galois extension E ofQp such that the characterχhas a realizationV_χ over E. Let η be an irreducible constituent ofres^G_Hχ and set

St(η) :={g∈G:η^g =η}, e_η = η(1)

|H|

∑

h∈H

η(h⁻¹)h, e_χ= ∑

η|res^G_Hχ

e_η.

By [RW04, Corollary to Proposition 6]e_χ is a primitive central idempotent ofQ^E(G). In fact, any primitive central idempotent of Q^c(G) := Q^cp⊗_Qp 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^G_Hρ = 1). Since the occurring irreducible constituents ofres^G_Hχ are precisely the Galois conjugates ofη by [CR81, Proposition 11.4], we have an equality

eχ=

w∑χ−1 i=0

e_ηγi, (1)

where w_χ = [G : St(η)]. By [RW04, Proposition 5] there is a distinguished element γ_χ ∈ ζ(Q^E(G)e_χ) which generates a procyclic p-subgroup Γ_χ of (Q^E(G)e_χ)^× and acts trivially onV_χ. Moreover,γ_χinduces an isomorphismQ^E(Γ_χ)−→^≃ ζ(Q^E(G)e_χ)by [RW04, Proposition 6]. Note that we may write γχ =γ^wχ ·c =c·γ^wχ, where c ∈(E[H]eχ)^× by [RW04, Proposition 5 and its proof].

Proposition 1.1. Let Gbe ap-adic Lie group of dimension 1 and letK be a nite exten- sion of Qp. For any irreducible character χ of G with open kernel put K_χ:=K(χ(h)|h∈ H). Then there is an isomorphism

ζ(Q^K(G))≃⊕

χ/∼

Q^Kχ(Γχ),

where the sum runs through all irreducible characters of G with open kernel up to the equivalence relation: χ ∼χ^′ if and only if there isσ ∈Gal(Kχ/K) such that (res^G_Hχ)^σ = res^G_Hχ^′.

Proof. Since there are only nitely many central primitive idempotents e_χ of Q^c(G), we may choose a nite Galois extension E of Qp such that E[H]contains each eχ. We may also assume that K is a subeld of E. Now let σ be an element of Gal(E/K). Then σ acts on Q^E(G) and e^σ_χ = e_χ^σ. Moreover by the above, e_χ^σ = e_χ if and only if there is a character ρ of type W such that χ=χ^σ ⊗ρ, thus if and only if res^G_Hχ= res^G_Hχ^σ. Since the center of Q^K(G) coincides with the Gal(E/K)-invariants of ζ(Q^E(G)), we have an equality

ζ(Q^K(G)) =⊕

χ/∼

ζ(Q^K(G)εχ), εχ= ∑

σ∈Gal(Kχ/K)

eχ^σ.

Note that χ^σ depends on a chosen lift ofσ, but eχ^σ does not. Now let β = (βσ)σ ∈ ⊕

σ∈Gal(Kχ/K)

ζ(Q^E(G)eχ^σ) = ⊕

σ∈Gal(Kχ/K)

Q^E(Γχ^σ)

be invariant under Gal(E/K). The uniqueness of γχ implies that γχ^σ = γ^σ_χ; thus β is determined by β₁ andβ₁ lies inQ^E(Γ_χ)^Gal(E/Kχ)=Q^Kχ(Γ_χ) as desired.

Remark 1.2. In the special case, where K =Qp and G is a pro-p-group, this was shown by Lau [La] using a dierent method. The same is true for Corollary 1.5 below.

Corollary 1.3. Let K be a nite extension of Qp with ring of integeres ^o. Choose a maximal order Λ˜^o(G) containing Λ^o(G). Then

ζ( ˜Λ^o(G))≃⊕

χ/∼

Λ^oχ(Γχ),

where ^oχ denotes the ring of integers in K_χ.

Remark 1.4. Here, Λ˜^o(G) is an order over R = o[[T]], where we have identied 1 +T with γ^pn for a chosen largen. Now letχ be an irreducible character ofGwith open kernel.

If n is suciently large, thenγ^pn acts trivially on Vχ and henceγχ^pn = (γ^pn)^wχeχ. As γ^pn is central in G, the integer wχ divides pⁿ. We have shown that the inclusion

R=o[[T]]Λ^oχ(Γχ) is induced by 1 +T 7→γχ^pn/wχ.

Corollary 1.5. The algebra Q^K(G) has Wedderburn decomposition Q^K(G)≃⊕

χ/∼

(D_χ)_nχ×nχ,

where nχ ∈ N and Dχ is a skeweld with center Q^Kχ(Γχ). If sχ denotes the Schur index of Dχ, then we have an equality χ(1) =nχsχ.

Proof. All assertions are immediate from Proposition 1.1 apart from the last equality. Let us denote the simple component (D_χ)_nχ×nχ by A_χ. With E as in Proposition 1.1 we

compute

(n_χs_χ)² = dim_QKχ(Γχ)(A_χ)

= [Kχ:K]⁻¹·dim_Q^K_(Γχ)(Aχ)

= [K_χ:K]⁻¹·dim_QE(Γχ)(E⊗KA_χ)

(1)

= [Kχ:K]⁻¹· ∑

σ∈Gal(Kχ/K)

dim_Q^E_(Γχ)(Q^E(G)eχ^σ)

(2)

= [K_χ:K]⁻¹· ∑

σ∈Gal(Kχ/K)

χ^σ(1)²

= χ(1)²

Here, (1) is implied by the isomorphism E⊗KA_χ ≃ ⊕

σ∈Gal(Kχ/K)Q^E(G)e_χ^σ and (2) is shown in the proof of [RW04, Proposition 6].

Theorem 1.6. There is a nite Galois extension E of K such that Q^E(G) splits.

Proof. We choose E as in Proposition 1.1 and let L^′:=E⊗KL=Q^E(Γ^pn). As L^′-vector space, we have a decomposition

Q^E(G) =

p⊕ⁿ−1 i=0

L^′[H]γⁱ.

Now letχ be an irreducible character ofGwith open kernel. Enlarging E if necessary, we may assume that the group ring E[H]splits. Since E is a subeld ofL^′, we obtain

Q^E(G)eχ =

p⊕ⁿ−1 i=0

L^′[H]eχγⁱ

=

p⊕ⁿ−1 i=0

w⊕χ−1 j=0

L^′[H]e_ηγjγⁱ

=

p⊕ⁿ−1 i=0

w⊕χ−1 j=0

L^′_η(1)×η(1)γⁱ,

where we have used equation (1). We now choose an indecomposable idempotentfη =fηeη

of L^′[H]e_η = L^′_η(1)×η(1). Observe that for any other indecomposable idempotent f_η^′ of L^′[H]e_η we have an isomorphism f_ηL^′[H]f_η^′ ≃ L^′. As Q^E(G)e_χ is a simple algebra over its center Q^E(Γ_χ) by Corollary 1.5, and f_η is also an idempotent in Q^E(G)e_χ, it suces to show that fηQ^E(G)eχfη is a eld, namelyQ^E(Γχ). For this, we rst observe that for any 0 ≤ i < pⁿ also fη,i := γⁱfηγ⁻ⁱ is an indecomposable idempotent which belongs to L^′[H]e_ηγi such that

fηL^′[H]fη,i≃

{ L^′ ifwχ|i 0 otherwise.

This implies that we have an isomorphism

f_ηQ^E(G)e_χf_η ≃

p⊕ⁿ−1

i=0 wχ|i

L^′γⁱ =

w⁻χ⊕¹pⁿ−1 i=0

L^′γ^wχi=

w⁻χ⊕¹pⁿ−1 i=0

L^′γ_χⁱ.

Here, the last equality holds, as we may write γ_χ = γ^wχ ·c = c·γ^wχ, where c lies in (E[H]e_χ)^×. Finally,L^′ =Q^E(Γ^pn)andγ^pn identies withγχ^pn/wχ by Remark 1.4 such that

w⁻χ⊕¹pⁿ−1 i=0

L^′γ_χⁱ =Q^E(Γ_χ)

as desired.

Corollary 1.7. The semi-simple algebra Q^c(G) splits.

Remark 1.8. If Aχ = (Dχ)nχ×nχ is a simple component of Q^K(G), then the arguments in the proof of Theorem 1.6 can be rened to show that K(η)⊗KχAχ splits, whereK(η) = K(η(h)|h∈H). See [La, Theorem 1] for details in the caseK=Qp andG a pro-p-group.

In fact, in this special case one can determine the occurring skewelds very explicitly.

2 Traces and conductors

Recall that L=Quot(R)and denote by Tr the ordinary trace map from Q^K(G) toL. Lemma 2.1. The elements γⁱh, 0≤i < pⁿ, h∈H form anL-basis of Q^K(G) such that

Tr (γⁱh) =

{ pⁿ|H| ifγⁱh= 1 0 otherwise.

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

Proof. Let0≤i, j < pⁿ andh, h^′ ∈H such that γⁱhγ^jh^′ =γ^i+jh_jh^′ with h_j :=γ^−jhγ^j ∈ H. Assume that γⁱhγ^jh^′ =x·γ^jh^′, wherex∈L. To prove the desired formula forTr we have to show that i = 0 and h = 1. In fact, we see that hjh^′ = h^′; hence h = hj = 1. Moreover, we have γ^i+j = x·γ^j which implies that x = γⁱ. But x belongs to L which does not contain γⁱ for 0 < i < pⁿ; hence also i = 0. It is now easily checked that (pⁿ|H|)⁻¹h⁻¹γ⁻ⁱ,0≤i < pⁿ,h∈H is the dual basis.

Lemma 2.2. Let K^′ be a nite eld extension of K with ring of integers ^o′. Consider Λ^o′(Γ) as R-order via the embedding

RΛ^o′(Γ), 1 +T 7→γ^pn. Then the inverse dierent D⁻¹(Λ^o′(Γ)/R) :=

{

x∈ Q^K′(Γ)|Tr_QK′(Γ)/L(xΛ^o′(Γ))⊂R } is given by

D⁻¹(Λ^o′(Γ)/R) =p⁻ⁿD⁻¹(o^′/o)Λ^o′(Γ),

where D⁻¹(o^′/o) denotes the usual inverse dierent of ^o′ with respect to ^o.

Proof. If K^′ =K, the result follows from Lemma 2.1 with G= Γ. Hence we may assume that n= 0. But if x1, . . . , xk form an^o-basis of^o′, then x1, . . . , xk are also an ^o[[T]]-basis of ^o′[[T]] which is isomorphic toΛ^o′(Γ)via 1 +T 7→ γ. Hence its dual basis with respect to the ordinary traceTr_K′/K of elds, is also a dual basis with respect toTr_QK′(Γ)/L.

By Corollary 1.5 we may write Q^K(G) =⊕

χ/∼A_χ, where A_χ = (D_χ)_nχ×nχ, n_χ ∈N and Dχ is a skeweld with Schur index sχ and center Q^Kχ(Γχ). By the same Corollary, we have χ(1) =s_χn_χ such that the ordinary trace may be written as

Tr =∑

χ/∼

χ(1)tr_χ, (2)

wheretrχ denotes the reduced trace fromAχ to L. Moreover, we have tr_χ= Tr_QKχ(Γχ)/L◦tr_A

χ/Q^Kχ(Γχ), (3)

where Tr_QKχ(Γχ)/L denotes the ordinary trace of elds and tr_A

χ/Q^Kχ(Γχ) denotes the re- duced trace from A_χ into its center. Recall from Remark 1.4 that we have chosen a su- ciently large integern≥0such thatR=o[[T]]embeds intoΛ^oχ(Γ_χ) via 1 +T 7→γ_χ^pn/wχ. Now Lemma 2.2 implies that the following denition does not depend on n.

Denition 2.3. Choose a maximal R-order Λ˜^o(G) containing Λ^o(G). We have a de- composition Λ˜^o(G) =⊕

χ/∼Λ˜^o_χ(G), where each Λ˜^o_χ(G) is a maximal R-order in A_χ. For suciently large n we call the two-sided Λ˜^o_χ(G)-lattice

D⁻χ¹( ˜Λ^o(G)) =Dnorm⁻¹ ( ˜Λ^o_χ(G)/R) :=pⁿ·{

x∈Aχ|trχ(xΛ˜^o_χ(G))⊂R }

the normalized inverse dierent.

We point out that this is abuse of notation, since in generalD⁻_χ¹( ˜Λ^o(G))might not be invertible.

Denition 2.4. Let Λ⊂Λ˜ be a pair of rings. Then ( ˜Λ : Λ)_l:=

{

x∈Λ˜|xΛ˜ ⊂Λ }

is called the left conductor of Λ˜ into Λ. Similarly, ( ˜Λ : Λ)r:=

{

x∈Λ|˜ Λx˜ ⊂Λ }

is called the right conductor of Λ˜ into Λ.

Using Lemma 2.1 and equation (2), we can adjust the proof of Jacobinski's conductor formula given in [CR81, Theorem 27.8] to show the following result.

Theorem 2.5. Let Λ˜^o(G) be a maximalR-order containing Λ^o(G). Then ( ˜Λ^o(G) : Λ^o(G))_l= ( ˜Λ^o(G) : Λ^o(G))_r =⊕

χ/∼

|H|

χ(1)D⁻χ¹( ˜Λ^o(G)).

3 Some further preliminaries

Letπ be a prime element in ^o. For anyR-moduleM we write M_(π) for the localization of M at the prime (π). In particular, if Λ is anR-order in the L-algebra A, then Λ_(π) is an R_(π)-order in A.

Lemma 3.1. Let A be a separable L-algebra of nite dimension over L and let Λ be an R-order inA. IfA is split, then there is an ^o-order ∆in Λ_(π) such that L⊗^o∆ =A. Proof. Choose a maximal R-order Λ˜ in A containing Λ. Then Λ˜_(π) is a maximal R_(π)- order containing Λ_(π) by [Re75, Theorem 11.1]. Note that we have π^NΛ˜_(π) ⊂ Λ_(π) if N is suciently large. There are natural numbers k > 0 and n_i, 1 ≤ i ≤ k such that A≃⊕_k

i=1Lni×ni. We putΛ˜0 :=⊕_k

i=1Rni×ni and observe that this is a maximalR-order inA by [Re75, Theorem 8.7]. The global dimension ofΛ˜₀ is given by

gl.dim ˜Λ0= gl.dim R= 2<∞.

Here, the second equality follows from [Ei95, Corollary 19.6], and the rst equality holds, since the global dimension is invariant under Morita equivalence (cf. [Ra69, Corollary, p. 476]). Recall that a noetherian ring Γ is called quasi-local if Γ/rad(Γ) is a simple artinian ring, where rad(Γ) denotes the Jacobson radical of Γ. As R/rad(R) is a eld, any component Rni×ni of Λ˜0 is quasi-local. As R is a regular local ring of dimension 2, these observations permit us to apply [Ra69, Theorem 5.4] componentwise which implies the existence of an invertible element a∈A such thatΛ =˜ a⁻¹Λ˜₀a.

For 1 ≤ i ≤ k and 1 ≤ j, l ≤ ni let eⁱ_jl ∈ Λ˜0 be the element which is zero everywhere except for thei-th component, where it is equal to the matrix with1in position (j, l) and 0everywhere else. Then the elements1anda⁻¹eⁱ_jlawith(i, j, l)̸= (1,1,1)form anR-basis of Λ˜. ForN as above, the free ^o-module

∆ :=o⊕ ⊕

(i,j,l)̸=(1,1,1)

π^Na⁻¹eⁱ_jla·^o⊂Λ_(π)

is closed under multiplication, thus is an ^o-order in the separable K-algebraK⊗^o∆. As the^o-rank of∆equals theL-dimension ofA, we haveL⊗o∆ =Aas desired.

Lemma 3.2. Let A⊂ Q^K(G) be a semi-simple component of Q^K(G), i.e. A is the direct sum of some, but maybe not all A_χ. If Λ is an R-order in A, then there is an ^o-order ∆ in Λ_(π) such that L⊗o∆ =A.

Proof. By Theorem 1.6 there is a nite Galois extensionK^′ ofK such thatA^′ :=K^′⊗KA splits. Let^o′ be the ring of integers inK^′ with prime elementπ^′ and R^′ =o^′⊗oR=o^′[[T]]

with eld of fractionsL^′=K^′⊗KL. ThenΛ^′ :=o^′⊗oΛis anR^′-order inA^′. By Lemma 3.1 there is an^o′-order∆^′inΛ^′_(π′)such thatL^′⊗o^′∆^′ =A^′. We put∆ := (∆^′)^Gal(K′/K). Then∆ is contained in(Λ^′_(π′))^Gal(K′/K) = Λ_(π)and a ring containing^o. Note that, by construction, multiplication in ∆is induced by multiplication in A. As∆is an ^o-submodule of∆^′ and

∆^′ is nitely generated and free over ^o′ (and thus over^o), also ∆is nitely generated and free over^o. Hence ∆is an^o-order. Finally,∆contains aK^′-basis ofK^′⊗o^′∆^′ by Hilbert's Theorem 90. ThusL⊗o∆ =A, as both sides have the same dimension overL.

Corollary 3.3. There exists a maximal oderΛ˜ in A which contains a maximal ^o-order∆˜ such that L⊗^o∆ =˜ A.

Proof. Choose any ^o-order ∆ as in Lemma 3.2 and a maximal ^o-order ∆˜ in K ⊗o ∆ containing ∆. Then also L⊗^o∆ =˜ A. Moreover, R⊗^o∆˜ is an R-order in A and hence contained in a maximal orderΛ˜. Obviously,∆˜ is contained inΛ˜.

Corollary 3.4. Let Rd_(π) be the completion of R_(π) with respect to the prime (π) and Lˆ = Quot(Rd_(π)). Then any maximal Rd_(π)-order Λˆ in Aˆ := ˆL⊗LA contains a maximal o-order∆ˆ such that Lˆ⊗o∆ = ˆˆ A.

Proof. Take a maximal orderΛ˜ as in Corollary 3.3. Then clearly Lˆ⊗o∆ = ˆ˜ A. Moreover,

∆˜ is contained in the (π)-adic completion Λ˜d_(π) of Λ˜_(π) which is a maximal Rd_(π)-order in Aˆ. Now let Λˆ be an arbitrary maximal Rd_(π)-order inAˆ. SinceRd_(π) is a complete discrete valuation ring, it follows from [Re75, Theorem 17.3] that there is an a ∈ Aˆ^× such that Λ =ˆ aΛ˜d_(π)a⁻¹. Then ∆ :=ˆ a∆a˜ ⁻¹ has the desired properties.

4 A formula for the central conductor

Denition 4.1. Let Λ˜^o(G) be a maximal order containing Λ^o(G). Then the central con- ductor of Λ˜^o(G) intoΛ^o(G) is dened to be

F(Λ^o(G)) =F( ˜Λ^o(G)/Λ^o(G)) := ζ(Λ^o(G))∩( ˜Λ^o(G) : Λ^o(G))l

= ζ(Λ^o(G))∩( ˜Λ^o(G) : Λ^o(G))r

= {

x∈ζ(Λ^o(G))|xΛ˜^o(G)⊂Λ^o(G) }

.

Remark 4.2. The theorem below shows that the central conductor only depends onΛ^o(G) and not on the maximal order containing Λ^o(G). Hence the notationF(Λ^o(G))is justied.

Theorem 4.3. Let Λ˜^o(G) be a maximal order containing Λ^o(G). Then the central con- ductor of Λ˜^o(G) intoΛ^o(G) is given by

F(Λ^o(G)) =⊕

χ/∼

|H|w_χ

χ(1) · D⁻¹(o_χ/o)Λ^oχ(Γ_χ).

In particular, the central conductorF(Λ^o(G))does not depend on the maximal orderΛ˜^o(G). Proof. According to Corollary 1.5 we writeQ^K(G) =⊕

χ/∼A_χ, where eachA_χ≃(D_χ)_nχ×nχ is simple. Similarly,Λ˜^o(G) decomposes into⊕

χ/∼Λ˜^o_χ(G), where each Λ˜^o_χ(G)is a maximal R-order in Aχ with centerΛ^oχ(Γχ). We dene

d⁻_χ¹ :=

{

x∈A_χ|tr_A

χ/Q^Kχ(Γχ)(xΛ˜^o_χ(G))⊂Λ^oχ(Γ_χ) } δ⁻_χ¹ := pⁿD⁻¹(Λ^oχ(Γχ)/R) =wχ· D⁻¹(o_χ/o)Λ^oχ(Γχ), where the last equality follows from Lemma 2.2 and Remark 1.4.

Lemma 4.4. We have an equality D⁻χ¹( ˜Λ^o_χ(G)) =d⁻_χ¹·δ_χ⁻¹.

Proof. This follows easily from the denitions using equation (3). In fact, the proof is similar to the corresponding statement in the proof of [CR81, Theorem 27.13]; note that only δ_χ⁻¹ has to be invertible.

By Theorem 2.5 and the denition of the central conductor we obtain F(Λ^o(G)) =⊕

χ/∼

Λ^oχ(Γ_χ)∩ ( |H|

χ(1) · D⁻χ¹( ˜Λ^o_χ(G)) )

.

Hence it must be shown that Λ^oχ(Γχ)∩

(|H|

χ(1)· D⁻χ¹( ˜Λ^o_χ(G)) )

= |H|

χ(1)δ⁻_χ¹ (4) for each irreducible characterχ. We note that

|H|

χ(1)δ⁻_χ¹ ⊂ |H|

χ(1)D⁻χ¹( ˜Λ^o_χ(G))⊂Λ˜^o(G);

so each element of _χ(1)^|H|δ_χ⁻¹ is integral over R, and thus lies in Λ^oχ(Γχ). This gives one inclusion in (4). For the reverse inclusion let y∈Λ^oχ(Γ_χ). Then by Lemma 4.4 we have

y∈ |H|

χ(1) · D⁻_χ¹( ˜Λ^o_χ(G)) ⇐⇒ yδχ ⊂ |H|

χ(1)d⁻_χ¹ ⇐⇒ yδχ⊂ |H|

χ(1)(Q^Kχ(Γχ)∩d⁻_χ¹).

Hence the theorem follows from the lemma below.

Lemma 4.5. We have Q^Kχ(Γχ)∩d⁻_χ¹ = Λ^oχ(Γχ).

Proof. To show the non-trivial inclusion let y∈ Q^Kχ(Γχ)∩d⁻_χ¹. Then in particular tr_Aχ/QKχ(Γχ)(y) =y·χ(1)∈Λ^oχ(Γχ).

Letπ_χbe a prime element in ^oχ. Asχ(1)is an integer, the above equation shows that we may localize and even complete at the prime (πχ). More precisely, let Λˆχ and Rˆχ be the completions of Λ˜^o_χ(G) andΛ^oχ(Γχ) at the prime(πχ), respectively. ThenΛˆχ is a maximal Rˆ_χ-order inAˆ_χ:= ˆL_χ⊗_Q^Kχ_(Γχ)A_χ, whereLˆ_χdenotes the quotient eld ofRˆ_χ. Nowy∈Lˆ_χ is such that

tr_Aˆ

χ/Lˆχ(yΛˆχ)⊂Rˆχ

and we wish to show thaty belongs toRˆχ. As the reduced trace isLˆχ-linear, we may alter y by a unit in Rˆ_χ such that we may assume thaty=π^k_χ for an appropriate integerk. By Corollary 3.4 there is a maximal^oχ-order ∆_χ contained inΛˆ_χ such thatLˆ_χ⊗oχ∆_χ= ˆA_χ. Hence an^oχ-basis of∆_χis also aLˆ_χ-basis ofAˆ_χwhich we may use to compute the reduced trace. Hence y∈K_χ is such that

tr_Kχ⊗o

χ∆χ/Kχ(y∆_χ)⊂Rˆ_χ∩K_χ=o_χ.

But ^oχ is a complete discrete valuation ring with nite residue eld such that we may conclude as in the proof of [CR81, Theorem 27.13] that y∈^oχ as desired.

5 Consequences of the central conductor formula

In this section we derive several consequences of our main Theorem 4.3.

Corollary 5.1. Let χ be an irreducible character of Gwith open kernel. Then

|H|wχ

χ(1) ∈Zp.

In particular, nχ and the Schur index sχ are divisors of |H|wχ in Zp.

Proof. By Theorem 4.3 the quotient ^|H_χ(1)^|wχ ∈Qpbelongs to the central conductorF(Λ^o(G))⊂ ζ(Λ^o(G)), so it is integral. The second assertion is clear, since χ(1) =nχsχ by Corollary 1.5.

5.1 Annihilation of Ext

Corollary 5.2. Let M be a Λ^o(G)-lattice and let N be a Λ^o(G)-module. Then



⊕

χ/∼

|H|w_χ

χ(1) · D⁻¹(o_χ/o)Λ^oχ(Γχ)



·Extⁱ_Λo(G)(M, N) = 0

for all integers i≥1.

Proof. We do induction on the integer i. The case i= 1 is immediate from Theorem 4.3 and [CR81, Theorem 29.4]. Fork suciently large, there is an exact sequence

M^′Λ^o(G)^k M.

As M and Λ^o(G) are projective (in fact free) as R-modules, so is M^′, that is M^′ is a Λ^o(G)-lattice. Applying Hom_Λ^o_(G)(_, N) to the above exact sequence gives isomorphisms

Ext^j_Λo(G)(M^′, N)≃Ext^j+1_Λo(G)(M, N) for all integers j≥1. The casej=i−1 gives the induction step.

Now we put^d−_χ¹ := (D⁻¹(o_χ/o)∩K)·R ⊇R. Then a proof similar to that of [CR81, Theorem 27.13 (ii)] shows the following result.

Corollary 5.3. We have

R∩ F(Λ^o(G)) = ∩

χ/∼

|H|wχ

χ(1) ·^d−χ¹. For aΛ^o(G)-latticeM let Υ(M) :={

eχ|eχ· Q^K(G)⊗M = 0} . Corollary 5.4. Let M and N be Λ^o(G)-lattices. Then ∩

eχ̸∈Υ(M)(|H|wχ/χ(1))d⁻_χ¹ anni- hilates Extⁱ_Λo(G)(M, N). In particular,



∩

χ/∼

|H|w_χ χ(1) ·^d−_χ¹



·Extⁱ_Λo(G)(M, N) = 0 for any Λ^o(G)-lattices M and N and any integer i≥1.

Proof. The last assertion is an immediate consequence of Corollary 5.2 and Corollary 5.3.

The rst assertion is also easy and is shown exactly in the same way as [CR81, Theorem 29.9].

Corollary 5.5. LetM andN beΛ^o(G)-lattices and assume thatQ^K(G)⊗M is absolutely simple. Then there is a unique idempotent eχ̸∈Υ(M) and for any integer i≥1 we have

|H|w_χ

χ(1) ·Extⁱ_Λo(G)(M, N) = 0.

In fact, the annihilation results above are in some sense best possible. A proof along the lines of the proof [CR81, Theorem 29.22] gives the following analogue of a result of Roggenkamp [Ro71].

Corollary 5.6. Let x∈R. Then

1. The element x annihilates Ext¹_Λo(G)(M, N) for all Λ^o(G)-lattices M and N if and only if x∈∩

χ/∼ |H|wχ

χ(1) ·^d−χ¹.

2. For each central idempotent e of Q^K(G) there exists a Λ^o(G)-lattice M (namely M = ˜Λ^o(G)e) such that eM =M and

x·Ext¹_Λo(G)(M, N) = 0 for all N ⇐⇒ x∈ ∩

eχ|e

|H|wχ

χ(1) ^d

−1 χ .

5.2 Non-commutative Fitting invariants

For the following we refer the reader to [Ni10]. Let A be a separable L-algebra andΛ be an R-order in A, nitely generated asR-module, whereR is an integrally closed complete commutative noetherian local domain with eld of quotientsL. LetN andMbe twoζ(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 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 N0 ∈[N]_nr(Λ) such thatx∈N0. Now letM be a nitely presented (left)Λ-module and let

Λ^a−→^h Λ^b M (5)

be a nite presentation of M. We identify the homomorphism h with the corresponding matrix in M_a×b(Λ) and dene S(h) =S_b(h) to be the set of all b×b submatrices of h if a≥b. The Fitting invariant of hover Λ is dened to be

FittΛ(h) =

{ [0][ _nr(Λ) if a < b

⟨nr(H)|H∈S(h)⟩ζ(Λ)

]

nr(Λ) if a≥b.

We callFitt_Λ(h)a Fitting invariant ofM overΛ. One denesFitt^max_Λ (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 4.3 and [Ni10, Lemma 4.1 and Theorem 4.2] imply the following result.

Corollary 5.7. Let M be a nitely presented Λ^o(G)-module. Then



⊕

χ/∼

|H|wχ

χ(1) · D⁻¹(o_χ/o)Λ^oχ(Γχ)



·Fitt^max_Λo(G)(M)⊂Ann_Λ^o_(G)(M).

Together with Corollary 5.3 and [Ni10, Lemma 3.4] this yields the following corollary.

Corollary 5.8. Let M be a nitely presented Λ^o(G)-module. Then



 ∩

eχ̸∈Υ(M)

|H|w_χ χ(1) ·^d−_χ¹



·Fitt^max_Λo(G)(M)⊂Ann_Λ^o_(G)(M).

Remark 5.9. Note that in factH(Λ^o(G))·Fitt^max_Λo(G)(M)⊂Ann_Λo(G)(M), whereH(Λ^o(G)) is a certain ideal ofζ(Λ^o(G))which always contains the central conductor. In general, how- ever, this containment is not an equality. Though the idealH(Λ^o(G))is hard to determine in general, considerable progress is made in [JN]; in particular, by [JN, Corollary 4.15]

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 nite).

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