Decomposition of the Witt–Burnside ring and Burnside ring of an abelian profinite group

Advances in Mathematics 222 (2009) 485–526 www.elsevier.com/locate/aim

Decomposition of the Witt–Burnside ring and Burnside ring of an abelian profinite group ✩ Young-Tak Oh Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea Received 19 August 2008; accepted 6 May 2009 Available online 2 June 2009 Communicated by David J. Benson

Abstract Let G, H be abelian profinite groups whose orders are coprime and assume that q ranges over the set q q q of integers. The aim of this paper is to establish an isomorphism of functors WG ◦ WH ∼ = WG×H , where q WG denotes the q-deformed Witt–Burnside ring functor of G introduced in [Y.-T. Oh, q-Deformation of Witt–Burnside rings, Math. Z. 207 (1) (2007) 151–191]. To do this, we first establish an isomorphism of q q q q functors BG ◦ BH ∼ = BG×H , where BG denotes the q-deformed Burnside ring functor of G which was also introduced in [Y.-T. Oh, q-Deformation of Witt–Burnside rings, Math. Z. 207 (1) (2007) 151–191]. As an application, we derive a pseudo-multiplicative property of the q-Möbius function associated to the lattice of open subgroups of the direct sum of G and H . © 2009 Elsevier Inc. All rights reserved. MSC: 05A99; 05E99; 13K05; 19A22; 20E15; 20E18 Keywords: Witt–Burnside ring; Burnside ring; Profinite group; Witt-vector; Möbius function; Special λ ring

1. Introduction Let G be a profinite group and p a prime. The Witt–Burnside ring of G was first introduced by Dress and Siebeneicher [6] as a group-theoretical generalization of the ring of p-typical Witt vectors and the universal ring of Witt vectors. The theory of p-typical Witt vectors, contrived by Witt [22], has deep connections with the structure of complete discrete valuation rings A of ✩

This research was supported by KRF Grant #2007-314-C00005 and KOSEF Grant 2009-0070299. E-mail address: [email protected].

0001-8708/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.aim.2009.05.004

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characteristic 0, in which p is a uniformizing element and the residue field has characteristic p and is supposed to be perfect. To be more precise, he introduced a covariant functor, denoted by Wp , from the category of commutative rings into itself which assigns A/pA to A. Later he together with Lang [14] found a more general covariant functor W not depending on p. Roughly, Wp can be viewed as a truncated version of W when one restricts the index from the set of positive integers to {1, p, p 2 , . . .}. In the most literature Wp is called as the ring of p-typical Witt vectors and W the universal ring of Witt vectors. For more information, see [1,11–13]. Despite of many remarkable properties, however, they had remained as a mystery due to their very unusual nature for a long period. Even the structure of Wp (Z) and W(Z) had not been fully understood until the end of nineteen-seventies. The first combinatorial description of W(Z) was presented by Metropolis and Rota [15]. They proved that W(Z) is isomorphic to the necklace ring over Z which was created based on the combinatorics of necklaces. Then Dress and Siebeneicher [7] noticed that the necklace ring over Z is isomorphic to the Burnside ring of almost finite cyclic sets, that is, C-sets with no infinite cycles and with only finitely many cycles of length n for every integer n. Here C denotes the multiplicative infinite cyclic group. Motivated by this simple but remarkable observation, they came to introduce a functor WG from the category of commutative rings into itself satisfying (1)

WCˆ p = Wp ,

(2)

WCˆ = W,

and (3)

1 ˆ WG (Z) ∼ = Ω(G).

ˆ Here Cˆ denotes the profinite completion of C, Cˆ p the pro p-completion of C, and Ω(G) the Burnside ring of G. We call WG the Witt–Burnside ring functor of G. As in the case with Witt vectors, the nature of Witt–Burnside rings seems to be very bizarre. Intensive efforts have been made to realize them more naturally. We refer the reader to [3,4,8,10]. The Burnside ring of G has intimate relations with WG . Among its numerous properties we notice that it induces a covariant functor, denoted by BG ,2 on the category of commutative rings since its ring operations are completely determined by universal polynomials, that is, polynomiˆ als with integer coefficients. By construction it is obvious that BG (Z) is isomorphic to Ω(G). It is quite noteworthy that, restricted to the subcategory of binomial rings, it turn out to be naturally isomorphic to WG [17], where binomial rings mean special λ-rings whose Adams operations are the identity for all n  1. One of the most striking features of WG and BG might be that they have a q-deformation respectively when q ranges over the set of integers. It was shown in [18] that, for any integer q, q q there exist functors WG , BG on the category of commutative rings, satisfying that (1)

W1G = WG ,

(2)

B1G = BG ,

and (3)

q

W0G = B0G . q

In particular, the structure of WG at q = 0 is extremely simple. However the structure of WG q and BG for general q is still mysterious. For instance, although 20 years have passed since Witt–Burnside rings were introduced by Dress and Siebeneicher, the question if their functorial composition is compatible with taking direct sum remains open. Very recently, motivated by [2,19,20], the following conjecture was proposed in [21]. 1 This isomorphism was extended to the class of special λ-rings in [16,17]. 2 The author has used  NrG to denote BG in the former papers such as [18,21].

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Conjecture 1.1. (See [21].) Let q be any integer. If G, H are profinite groups whose orders are coprime, then q q q WG ◦ WH ∼ = WG×H

(1)

and (2)

q q q BG ◦ BH ∼ = BG×H .

The aim of the present paper is to prove this conjecture in case where G, H are abelian profinite groups whose orders are coprime. It should be noted that it is straightforward in case of q = 1, but is no longer obvious for general q. A central role is played in the proof of Conjecture 1.1 by Claim 3.5 and the formula on the number of subgroups in finite abelian p-groups (see formula (6.3)). The latter isomorphism in Conjecture 1.1 has a significant combinatorial interpretation associated with the q-Möbius function of the lattice, denoted by O(G), of open subgroups of G. q Define μG : O(G) × O(G) → Q[q] recursively via q

μG (U, U ) = 1,

for all U ∈ O(G),  q q (V :W )−1 μG (U, W ),

q

μG (U, V ) = −

for all U  V in O(G).

U ⊆W V W ∈O (G)

Note that q is being considered as an indeterminate, not a specific integer. When q = 1, it coq incides with the Möbius function, denoted by μG , of O(G). In contrast with μG , μG does not have a multiplicative property unless q = 1, 0, −1. Instead it possesses the following pseudomultiplicative property: Theorem 1.1. Suppose that G, H are profinite groups whose orders are coprime. For every U, S ∈ O(G) and V , T ∈ O(H ) with U ⊆ S and V ⊆ T , we have q

μG×H (U × V , S × T ) q

q

= μG (U, S)μH (V , T ) +



q

q

(W : U )(Z : V )fW,Z (q)μG (W, S)μH (Z, T ),

U ⊆W ⊆S V ⊆Z⊆T U ×V =S×T

where fW,Z (q) ∈ Q[q]’s are subject to the conditions: (1) fW,Z (q) are integer-valued polynomials in q, and (2) If S = U and T = V , then fW,Z (q) = 0 for every W, Z with U ⊆ W ⊆ S, V ⊆ Z ⊆ T if and only if q = 1, 0, −1. This paper is organized as follows: In Section 2, we review the basic definitions and notation q q on WG and BG . In Section 3, we establish a functorial isomorphism q q q BG ◦ BH ∼ = BG×H

(1.1)

where G, H are abelian profinite groups whose orders are coprime and q ranges over the set of integers (Theorem 3.3). The proof follows immediately from Lemma 3.4, and the proof of

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Lemma 3.4 largely depends on Claim 3.5. But, its being too long and complicated, we place its proof at the end of the paper. In Section 4, a functorial isomorphism q q q WG ◦ WH ∼ = WG×H

will be dealt with under the same condition as Section 3 (Theorem 4.2). As an application, we interpret the functorial isomorphism (1.1) as a pseudo-multiplicative property of the q-Möbius function associated with the lattice of open subgroups of G × H in Section 5. 2. Definitions and notation q

q

In this section, we review the basic definitions and notation on WG and BG very briefly. For more details, refer to [18,21]. In addition, the familiarity with profinite groups is assumed to some extent. Let G be a profinite group. Given open subgroups U and V of G, we say that U is subconjugate to V if U is a subgroup of some conjugates of V . This gives rise to a partial ordering on the set of the conjugacy classes of open subgroups of G, and will be denoted by [V ]  [U ]. Denote this poset by O(G). Tacitly we will always fix an enumeration of O(G) subject to the following condition throughout this paper: If [V ]  [U ], then [V ] precedes [U ]. For instance, if G is abelian, then O(G) is just the set of open subgroups of G equipped with the ordering V  U if and only if U ⊆ V . For any G-space X and any subgroup U of G define φU (X) to be the cardinality of the set X U of U -invariant elements of X and let G/U denote the G-space of left cosets of U in G. q q With the above notation, let us introduce covariant functors WG , BG , which were recently introduced in [18]. To do this, we first introduce the ghost ring functor of G, denoted by ghG , which is a unique functor from the category of commutative rings into itself satisfying that (1) as a set, ghG (A) coincides with the set AO(G) of all maps from the set O(G) of open subgroups of G into the ring A which are constant on conjugacy classes, (2) addition and multiplication are defined componentwise, and (3) for every ring homomorphism f : A → B and every α ∈ ghG (A) one has ghG (f )(α) = f ◦α. Theorem 2.1. (See [18].) Let G be any profinite group and let q be any integer. Then there exists a unique covariant functor, denoted by WG , from the category of commutative rings into itself subject to the following conditions: (1) as a set, WG (A) = AO(G) , q q (2) for every ring homomorphism f : A → B and every α ∈ WG (A) one has WG (f )(α) = f ◦α, and q q (3) the map ΦG : WG (A) → ghG (A), defined by q

  q ΦG (α) [U ] =



 (V :U ) φU (G/V )q (V :U )−1 α [V ] ,

[G][V ][U ]

is a ring homomorphism. Here (V : U ) represents (G : U )/(G : V ).

∀[U ] ∈ O(G),

Y.-T. Oh / Advances in Mathematics 222 (2009) 485–526 q

q

489 q

In fact, ΦG : WG → ghG is a natural transformation. Thus the map ΦG appearing in (3) should q be written as (ΦG )A . We, however, conventionally omit the suffix A in case where no confusion happens. Theorem 2.2. (See [18].) Let G be any profinite group and let q any an integer. Then there is a q unique functor, denoted by BG , from the category of commutative rings into itself subject to the following conditions: (1) as a set, BG (A) = AO(G) , q q (2) for every ring homomorphism f : A → B and every α ∈ BG (A) one has BG (f )(α) = f ◦ α, and q q (3) the map ϕG : BG (A) → ghG (A), defined by q



  q ϕG (α) [U ] =

  φU (G/V )q (V :U )−1 α [V ] ,

∀[U ] ∈ O(G),

[G][V ][U ]

is a ring homomorphism. q

When q = 1, WG coincides with the Witt–Burnside ring functor WG introduced by Dress q and Siebeneicher [6]. Similarly, when q = 1, BG coincides with the Burnside ring functor BG . It 0 might be worthwhile to remark that (i) WG = B0G and (ii) it has quite simple structure since 0 0 ΦG (α) = ϕG (α) =

   NG (U ) : U α(U ) U ∈O(G) ,

where NG (U ) denotes the normalizer of U in G. q rqG Remark 2.3. It should be remarked that the functor BG (resp., BG ) has been denoted by N rG ) in [18]. However, we will adopt the former notation here for simplicity of notation. (resp., N 1 Denote by ZG the ring Z[ (NG (U ):U ) : U is an open subgroup of G] and denote by D(G) the set {(NG (U ) : U ): U is an open subgroup of G}.

Lemma 2.4. For a commutative ring A the followings hold. q

q

(a) A is D(G)-torsion free ⇐⇒ ΦG is injective ⇐⇒ ϕG is injective. q q (b) A is a ZG -algebra ⇐⇒ ΦG is surjective ⇐⇒ ϕG is surjective. Proof. We will prove only (a) since (b) can be proved in the same way as (a). Assume that A is q q D(G)-torsion free. If ΦG (α) = ΦG (β), then for every [U ] ∈ O(G) 

 (V :U ) φU (G/V )q (V :U )−1 α [V ]

[G][V ][U ]

=



[G][V ][U ]

 (V :U ) φU (G/V )q (V :U )−1 β [V ] .

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Since α([G]) = β([G]), we can show by induction that     φU (G/U )α [U ] = φU (G/U )β [U ] . Thus we can conclude that α([U ]) = β([U ]) since A is D(G)-torsion free and φU (G/U ) = (NG (U ) : U ) ∈ D(G). For the converse, assume that Φ q is injective. If A is not D(G)-torsion free, there is a nonzero element a ∈ A and [U ] ∈ O(G) such that (NG (U ) : U )a = 0. Now, let α be the element in q q WG (A) defined by α([U ]) = a and α([W ]) = 0 if [W ] = [U ]. We will show that α ∈ ker ΦG . q q Unless [U ]  [W ], then ΦG (α)([W ]) = 0 by definition of ΦG . While, if [U ]  [W ], then 

  q ΦG (α) [W ] =

 (S:W ) φW (G/S)q (S:W )−1 α [S]

[G][S][W ]

 (U :W ) = ϕW (G/U )q (U :W )−1 α [U ] = ϕW (G/U )q (U :W )−1 a (U :W ) . Note that ϕU (G/U ) divides ϕW (G/U ) for every [U ]  [W ] since the group Aut(G/U ) is acting freely on the set of G-morphisms from G/W to G/U and the number of elements of this q q set equals ϕW (G/U ) (see [6]). This show that ΦG (α)([W ]) = 0. As a consequence, ΦG has a q nontrivial kernel, but it contradicts our assumption that Φ is injective. Therefore A should be D(G)-torsion free. In the exactly same way as above, we can show that A is not D(G)-torsion q free if and only if ϕG is surjective. This completes the proof. 2 q

q

As in the case q = 1, BG has close connection with WG . For the precise definition of binomial rings, see [9,17]. Proposition 2.5. (See [18].) Let A be a binomial ring and q be an integer. Then there exists a q q q ring isomorphism τG : WG (A) → BG (A) which makes the following diagram τq

q

WG (A)  q ΦG

 Bq (A) G





       ghG (A)

q ϕG

commutative. It should be remarked that the converse of Proposition 2.5 does not hold (see Remark 4.4). q q Structure of WG (resp. BG ) is more or less different from that of WG (resp. BG ). For instance, q q WG (Z) and BG (Z) are no longer unital unless q = ±1. Nevertheless, many properties at q = 1 are still available for any integer q. Particularly, for each open subgroup U of G, we have natural transformations [18]:

Y.-T. Oh / Advances in Mathematics 222 (2009) 485–526 q

q

q

q

vU : WU → WG , q

q

491

q

fU : WG → WU , q

q-IndG U : BU → BG ,

q

q

q-ResG U : BG → BU .

Restricting the domain to the category of binomial rings, one can derive the following formulae: q

q τ q ◦ fU = q-ResG U ◦τ ,

q

q τ q ◦ vU = q-IndG U ◦τ .

Convention 2.1. (a) Throughout this paper, we assume that G is an abelian profinite group. (b) Throughout this paper, the terminology “subgroup” will denote an open subgroup. (c) If there is no danger of confusion, we usually use the notation U ⊆ S ⊆ G instead of G  S  U . Tacitly U, S represent open subgroups of G. 3. Decomposition of q-deformed Burnside rings Let G and H be abelian profinite groups whose orders are coprime. In this section, we will study the decomposition of the q-deformed Burnside ring of the direct sum of G and H . This decomposition property was first proposed as a conjecture in [21, Section 4] (see also Conjecture 1.1(2)). 3.1. Background In this subsection, we provide all prerequisites indispensable to understanding and clarification of the decomposition of the q-deformed Burnside ring. In the theory of profinite groups it is well known that the order of a profinite group can be expressed as a Steinitz number (or super natural number) whose definition is given below. Definition 3.1.  (a) A Steinitz number is a formal infinite product p k(p) over all primes p, in which k(p) is a nonnegative integer or infinity. When k(p) is 0 for all primes p, the associated number is assumed to be 1.    (b) Two Steinitz numbers p k(p) and p k (p) are said to be coprime if there is no prime p with k(p), k  (p)  1. We require the condition that the orders of G and H are coprime because every open subgroup of G × H arises uniquely as U × V for some U ∈ O(G) and V ∈ O(H ) under this assumption. It follows that O(G × H ) can be identified with O(G) × O(H ) as a poset under the correspondence U × V → (U, V ) if we endow O(G) × O(H ) with the following ordering (U, V )  (U  , V  )

⇐⇒

U  ⊆ U and V  ⊆ V .

Note that U × V  U  × V  if and only if (U, V )  (U  , V  ). Now, given a commutative ring A, let ιA : ghG ◦ ghH (A) → ghG×H (A),

f → ιA (f ),

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where ιA (f )(U × V ) = f (U )(V ) for all U ∈ O(G), V ∈ O(H ). Due to the above identification ιA turns out to be a ring isomorphism. Furthermore, for a ring homomorphism g : A → B, we have the following commutative diagram ιA

ghG ◦ ghH (A) −−−−→ ghG×H (A) ⏐ ⏐ ⏐ ⏐gh ghG ◦ghH (g)   G×H (g) ιB

ghG ◦ ghH (B) −−−−→ ghG×H (B). Here the notation “◦” means the composite of functors. The commutativity can be verified as follows: For any f ∈ ghG ◦ ghH (A) and any U ∈ O(G), V ∈ O(H ),     ιB ◦ ghG ◦ ghH (g) (f )(U × V ) = ghG ◦ ghH (g)(f ) (U )(V )   = g f (U )(V )   = g ιA (f )(U × V )   = ghG×H (g) ◦ ιA (f )(U × V ). Consequently we can establish a functorial isomorphism ι : ghG ◦ ghH → ghG×H . In a similar way we can also establish a functorial isomorphism BG ◦ BH ∼ = BG×H . q Quite surprisingly, the same phenomenon happens for BG×H for an arbitrary integer q. However, the proof is no longer obvious contrary to the case q = 1. To explain the decomposition of q q q q q BG×H in more detail, let us first investigate BG ◦ BH , the composition of BG and BH . By the q q functoriality of ϕG and ϕH we have the following diagram q

q

q

ϕG ◦Id

q

BG ◦ BH −−−−→ ghG ◦ BH ⏐ ⏐ ⏐ q q ⏐  BG ◦ϕH  ghG ◦ϕHq q

q

ϕG ◦Id

BG ◦ ghH −−−−→ ghG ◦ ghH , where Id represents the identity transformation. The commutativity of the above diagram can q q be verified as follows: Let A be a commutative ring. For any f ∈ BG ◦ BH (A) and any U ∈ O(G), V ∈ O(H ), we have   q  q q  ϕG ◦ Id ◦ BG ◦ ϕH (f )(U )(V )   q q  = (G : S)q (S:U )−1 BG ◦ ϕH (f )(S)(V ) U ⊆S⊆G

=



(G : S)q (S:U )−1

U ⊆S⊆G

=



U ⊆S⊆G V ⊆T ⊆H



(H : T )q (T :V )−1 f (S)(T )

V ⊆T ⊆H

(G : S)(H : T )q (S:U )+(T :V )−2 f (S)(T ).

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In the same manner, we have   q   q ghG ◦ ϕH ◦ ϕG ◦ Id (f )(U )(V )   q  = (H : T )q (T :V )−1 ϕG ◦ Id (f )(U )(T ) V ⊆T ⊆H

=



(H : T )q (T :V )−1

V ⊆T ⊆H

=





(G : S)q (S:U )−1 f (S)(T )

U ⊆S⊆G

(G : S)(H : T )q (S:U )+(T :V )−2 f (S)(T ).

U ⊆S⊆G V ⊆T ⊆H q

q

Remark 3.2. Very often we use the vector notation to denote an element of BG ◦ BH (A). This is possible due to the identification O(G) × O(H ) with O(G × H ). More precisely, q q q q (XU,V )U ∈O(G), V ∈O(H ) ∈ BG ◦ BH (A) represents the function f ∈ BG ◦ BH (A) such that f (U )(V ) = XU,V . q

q

q

Define ϕG,H : BG ◦ BH → ghG×H by   q   q ι ◦ ghG ◦ ϕH ◦ ϕG ◦ Id . With this preparation, we can state the main theorem of this section as follows: Theorem 3.3. Let G, H be abelian profinite groups whose orders are coprime and q any integer. Then there is a unique functorial isomorphism, q

q

q

q

bG,H : BG ◦ BH → BG×H which makes the following diagram q

q

bG,H

q

BG ◦ BH  q ϕG,H

 

       ghG×H

 Bq G×H  q ϕG×H

commutative. The key point in proving Theorem 3.3 is to show the integrality of certain polynomials with rational coefficients, that is, those polynomials mentioned indeed have integer coefficients. Our proof, however, is more or less lengthy and complicated, we will devote the subsequent subsection fully to it.

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3.2. Key lemma q

In this subsection, let us investigate the decomposition of BG×H (R) where R is the polynomial ring

Z XU,V : U ∈ O(G), V ∈ O(H ) . Indeed the following lemma plays a crucial role in proving Theorem 3.3. Lemma 3.4. Let R denote the polynomial ring Z[XU,V : U ∈ O(G), V ∈ O(H )] and let q be any integer. And assume that G, H are abelian profinite groups whose orders are coprime. Then there is a unique ring isomorphism,  q  q q q bG,H R : BG ◦ BH (R) → BG×H (R) which makes the following diagram q

q

(bG,H )R

q

BG ◦ BH (R)  q ϕG,H





 Bq G×H (R)  q  ϕG×H 

(3.1)

      ghG×H (R)

commutative. Proof. Let us first show the uniqueness of (bG,H )R . Assume that (bG,H )R and (b G,H )R satisfy the commutativity of the diagram (3.1), that is, q

q

q

 q   q  q q q ϕG,H = ϕG×H ◦ bG,H R = ϕG×H ◦ b G,H R . It implies that  q   q   q ϕG×H ◦ bG,H R − b G,H R = 0. q

Now, the uniqueness follows from the injectivity of ϕG×H since R is torsion free (see Lemma 2.4). Therefore, it suffices to show the existence of such a ring isomorphism satisfying the comq q mutativity of the diagram (3.1). Let X = (XU,V )U ∈O(G), V ∈O(H ) ∈ BG ◦ BH (R). Recall that q q ϕG : BG (A) → gh(A) is invertible when A is a Q-algebra. Denote by Z = (ZU,V )U ∈O(G), V ∈O(H ) the vector  q −1 q q ϕG×H ◦ ϕG,H (X) ∈ BG×H (Q ⊗ R). In an inductive way, one can show easily that

ZU,V − XU,V ∈ Q XS,T : U ⊆ S ⊆ G, V ⊆ T ⊆ H, (S, T ) = (U, V )

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for all U ∈ O(G) and V ∈ O(H ). We, however, claim that its coefficients are all integers, that is,

ZU,V − XU,V ∈ Z XS,T : U ⊆ S ⊆ G, V ⊆ T ⊆ H, (S, T ) = (U, V )

(3.2)

for all U ∈ O(G) and V ∈ O(H ). If this is true, the vector Z will give rise to a desired ring q isomorphism. To be more precise, the existence of bG,H satisfying (3.1) follows from defining  q  q q q bG,H R : BG ◦ BH (R) → BG×H (R),   (aU,V ) U ∈O(G) → ZU,V (aS,T ) U ∈O(G) . V ∈O (H )

V ∈O (H )

Here ZU,V (aS,T ) denotes the value obtained by the specialization of XS,T ’s into aS,T ’s. In the following, we will prove (3.2). Mathematical induction on index of open subgroups will be used. As a first step, we notice the identity ZG,H = XG,H , which can be verified immediately by comparing the (G, H )th coordinate on q

ϕG,H (X)

q

and ϕG×H (Z).

Next, we choose (and then fix) any U ∈ O(G) and V ∈ O(H ), and assume that ZS,T ∈ R for all S ∈ O(G), T ∈ O(H ) satisfying that U ⊆ S ⊆ G, V ⊆ T ⊆ H, and (S × T : U × V ) > 1. Under q the above induction hypothesis, we claim ZU,V ∈ R. First, we note the equality ϕG,H (X) = q ϕG×H (Z), which follows from the definition of Z. Equivalently, for every U ∈ O(G) and V ∈ O(H ),  U ⊆S⊆G

=

(G : S)q 

(S:U )−1

 

(H : T )q

(T :V )−1

XS,T

V ⊆T ⊆H

(G × H : S × T )q (S×T :U ×V )−1 ZS,T .

(3.3)

U ⊆S⊆G V ⊆T ⊆H

Now, let us consider the case where U = U, V = V . From our induction hypothesis it follows that (G : U )(H : V )ZU,V ∈ R. For a prime p dividing (G : U )(H : V ), let   ν := ordp (G : U )(H : V ) . Here, for any nonzero integer a, ordp (a) means the p-valuation of a, the exponent of p in the highest power of p dividing a. Since p is an arbitrary prime dividing (G : S)(H : T ), the proof will be complete if we can show that p ν |(G : U )(H : V )ZU,V . Since the orders of G, H are coprime, p must divide only one among (G : U ) and (H : V ). For simplicity, we assume that p|(G : U ). Then our induction hypothesis implies

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(G : U )(H : V )ZU,V  ≡ (G × H : S × T )q (S×T :U ×V )−1 ZS,T

  mod p ν

U ⊆S⊆G V ⊆T ⊆H ordp ((G:S))=ν



=

(G : S)(H : T )q (S:U )(T :V )−1 ZS,T

U ⊆S⊆G V ⊆T ⊆H

−



(G : S)(H : T )q (S:U )(T :V )−1 ZS,T .

(3.4)

U ⊆S⊆G V ⊆T ⊆H ordp ((G:S))<ν

Due to Eq. (3.3) the right-hand side of Eq. (3.4) can be rewritten as 

(H : T )q

V ⊆T ⊆H

−

(T :V )−1

 

(G : S)q

(S:U )−1

XS,T

U ⊆S⊆G



(G : S)(H : T )q (S:U )(T :V )−1 ZS,T .

(3.5)

U ⊆S⊆G V ⊆T ⊆H ordp ((G:U ))<ν

Consequently we arrive at the following congruence of polynomial values at q: (G : U )(H : V )ZU,V   (T :V )−1 ≡ (H : T )q V ⊆T ⊆H

−





(G : S)q

(S:U )−1

XS,T

U ⊆S⊆G ordp ((G:U ))<ν

(G : S)(H : T )q (S:U )(T :V )−1 ZS,T

  mod p ν .

(3.6)

U ⊆S⊆G V ⊆T ⊆H ordp ((G:U ))<ν

Hence it is enough to show that the right-hand side is 0 (mod p ν ). To do this let us split the values of q into two cases. Case 1. Firstly, we assume that q is divisible by p. Note that q (S:U )−1 ≡ 0 (mod p ordp ((S:U )) ). It then implies   (G : S)q (S:U )−1 ≡ 0 mod p ν since (G : S) ≡ 0 (mod p ordp ((G:S)) ). In the same manner, (T :V ) (T :V )−1    q ≡ 0 mod p ν . (G : S)q (S:U )(T :V )−1 ≡ (G : S) q (S:U )−1 Thus, from Eq. (3.6) it follows that (G : U )(H : V )ZU,V ≡ 0 (mod p ν ).

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497

Case 2. Secondly, we assume that q is not divisible by p, that is, (q, p) = 1. (S:U )

One can see that if ordp ((G : S)) < ν then q (S:U ) ≡ q p (mod p ordp ((S:U )) ), as p does not divide q. On the other hand, the condition (q, p) = 1 implies that q is a unit in Z/p ν Z. Therefore we have   mod p ν (S:U )   (T :V ) · q −1 mod p ν ≡ (G : S)q p (S:U )  (T :V )−1  mod p ν . ≡ (G : S)q p

(G : S)q (S:U )(T :V )−1 ≡ (G : S)q (S:U )(T :V ) · q −1

Applying these congruences to Eq. (3.6) we have (G : U )(H : V )ZU,V ≡



(H : T )q

(T :V )−1



V ⊆T ⊆H

(G : S)q

(S:U ) p −1

XS,T

U ⊆S⊆G ordp ((G:S))<ν



−



(G : S)(H : T )q

(S:U ) p (T :V )−1

ZS,T

  mod p ν . (3.7)

U ⊆S⊆G V ⊆T ⊆H ordp ((G:U ))<ν (S:U )

−1

(S:U )

(T :V )−1

From now on, we will investigate the modular behavior of q p and q p in the righthand side of Eq. (3.7). In fact, this modular behavior turns out to be crucial in the proof. To do this, let us view q as an indeterminate for the time being. To an open subgroup W of G with U ⊆ W ⊆ G let us associate a polynomial CW (q) ∈ Z[q] recursively via the following formula

0 if W = U, (W :U ) CW (q) =  −1 q p − U ⊆KW CK (q)q (W :K)−1 otherwise. This definition makes sense because the summation is over finite number of K’s. Indeed W/U is a finite abelian group and there is an index-preserving bijection   π : K ∈ O(W ): U ⊆ K ⊆ W → O(W/U ), K → K/U. (3.8) Here index-preserving means that (W : K) = (W/U : K/U ). By definition q

(W :U ) p −1

=



CK (q)q (W :K)−1

U ⊆K⊆W

for U  W ⊆ G. For instance, one can see in a recursive way that ⎧ ⎨0 CW (q) = 1 ⎩ p−1 q (1 − α)

if W = U, if (W : U ) = p, if (W : U ) = p 2 ,

where α denotes the number of subgroups K satisfying U ⊆ K ⊆ W , (K : U ) = p.

(3.9)

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Now, assume that S is an open subgroup of G satisfying U ⊆ S ⊆ G and ordp ((G : S)) < ν (equivalently, (S : U ) is divisible by p). Note that S/U is a finite abelian group. According to the fundamental structure theorem of finite abelian groups we can write (up to isomorphism) S/U as a direct product M × N where M is the Sylow p-subgroup of S/U , and N is a finite abelian group whose order is not divisible by p. Thus we have q

(S:U ) p −1

=q

|M||N| p −1

  |M| −1 = q |N |−1 q |N | p .

(3.10)

In order to investigate the properties of CS (q) we will introduce another polynomial which is essentially the same as CS (q). To every finite abelian p-group G we assign cG (q) ∈ Z[q] determined by c{0} = 0, and for |G| > 1, by the following recursive formula cG (q) = −



cS (q)q (G:S)−1 + q

|G| p −1

.

(3.11)

{0}⊆SG

Due to the bijection (3.8) one can see that CS (q) = cS/U (q) if S/U is a p-group. For instance, if G is a finite cyclic p-group, then it is easy to show that  cG (q) =

1 0

if |G| = p, otherwise.

The following lemma plays a key role in the remaining part. Claim 3.5. Let G be a finite abelian p-group and k a positive integer. If q is an integer coprime to p, then we have the following congruence of polynomial values at q: 

cG (q) ≡ −

 (G:S)−1  k  |G| −1 cS (q) q k + q p

  mod |G| .

{0}⊆SG

The proof of Claim 3.5 can be found in Section 6. Let us return to the proof again. It was shown in Eq. (3.10) that q

(S:U ) p −1

  |M| −1 = q |N |−1 q |N | p .

Since M is a finite abelian p-group, we can apply Claim 3.5. Let k = |N |. Then 

q |N |

 |M| −1 p

≡



(M:L)−1  cL (q) q |N |

  mod |M| .

{0}⊆L⊆M

Notice that there is an index-preserving bijection π −1 |O(M) : O(M) → {W : U ⊆ W ⊆ S and W/U is a p-group}. Thus the right-hand side of Eq. (3.12) equals

(3.12)

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CW (q)q (S:W )−1

499

  mod p ordp ((S:U ))

U ⊆W ⊆S W/U is a p -group

(recall that CW (q) = cW/U (q)). Consequently we can establish the following congruence of polynomial values at q q

(S:U ) p −1



≡

CW (q)q (S:W )−1

  mod p ordp ((S:U )) .

(3.13)

U ⊆W ⊆S W/U is a p -group

Therefore we have 

(H : T )q (T :V )−1

V ⊆T ⊆H



(H : T )q (T :V )−1

V ⊆T ⊆H

CW (q)

U ⊆W ⊆G W/U is a p -group



≡

XS,T

 

(G : S)q

(S:U ) p −1

XS,T

  mod p ν

CW (q)





(G : S)(H : T )q (S:W )+(T :V )−2 XS,T

  mod p ν

W ⊆S⊆G V ⊆T ⊆H

U ⊆W ⊆G W/U is a p -group

≡

(S:U ) p −1

U ⊆S⊆G



≡

(G : S)q

U ⊆S⊆G ordp ((G:S))<ν



≡





(G : S)(H : T )q (S:W )(T :V )−1 ZS,T

  mod p ν

W ⊆S⊆G V ⊆T ⊆H



CW (q)

U ⊆W ⊆G W/U is a p -group

(G : S)(H : T )q (S:W )(T :V )−1 ZS,T

  mod p ν .

W ⊆S⊆G ordp ((G:S))<ν V ⊆T ⊆H

Note that the second congruence follows from Eq. (3.13) and the third one from Eq. (3.3). Plugging this result into Eq. (3.7) again yields the following congruence of polynomial values at q: (G : U )(H : V )ZU,V   (S:U ) (T :V )−1 ≡− (G : S)(H : T ) q p − U ⊆S⊆G ordp ((G:S))<ν V ⊆T ⊆H



CW (q)q (S:W )(T :V )−1 ZS,T

U ⊆W ⊆S W/U is a p -group

  mod p ν .

For our purpose (that is, (G : U )(H : V )ZU,V ≡ 0 (mod p ν )) it suffices to show q

(S:U ) p (T :V )−1

−

 U ⊆W ⊆S W/U is a p -group

CW (q)q (S:W )(T :V )−1 ≡ 0

  mod p ordp ((S:U ))

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for every open subgroup S of G such that U ⊆ S ⊆ G and ordp ((G : S)) < ν. But, it can be verified immediately by virtue of Claim 3.5. More precisely, q

(S:U ) p (T :V )−1

−



CW (q)q (S:W )(T :V )−1

U ⊆W ⊆S W/U is a p -group

= q (T :V )−1

  (T :V )  (S:U ) −1 p q −



(S:W )−1  CW (q) q (T :V )



U ⊆W ⊆S W/U is a p -group

≡0

  mod p ordp ((S:U )) (by Claim 3.5 and Eq. (3.12)).

This complete the proof. 3.3. Proof of Theorem 3.3 and its generalization We close this section by providing a proof of Theorem 3.3. A generalization of Theorem 3.3 will be also given (refer to Corollary 3.6). Proof of Theorem 3.3. Let R = Z[XU,V : U ∈ O(G), V ∈ O(H )], and consider q

q

X = (XU,V ) U ∈O(G) ∈ BG ◦ BH (R). V ∈O (H )

Denote by Z = (ZU,V )U ∈O(G), V ∈O(H ) the vector −1  q q q ◦ ϕG,H (X) ∈ BG×H (Q ⊗ R). ϕG×H We have shown in the proof of Lemma 3.4 that

ZU,V − XU,V ∈ Z XS,T : U ⊆ S ⊆ G, V ⊆ T ⊆ H, (S, T ) = (U, V ) for all U ∈ O(G) and V ∈ O(H ). Now, for any commutative ring A, let us define  q  q q q bG,H A : BG ◦ BH (A) → BG×H (A),   (aU,V ) U ∈O(G) → ZU,V (aS,T ) U ∈O(G) . V ∈O (H )

V ∈O (H )

Here ZU,V (aS,T ) means the value obtained by the specialization of XS,T ’s into aS,T ’s. To be more precise, let π : R → A be a ring homomorphism assigning XU,V to aU,V for all U ∈ O(G) and V ∈ O(H ). Then ZU,V (aS,T ) = π(ZU,V ). It is not difficult to show that this assignment q induces a natural transformation bG,H on the category of commutative rings and satisfies the commutativity  q  q q ϕG,H = ϕG×H ◦ bG,H A since Z is a universal object in the category of commutative rings.

Y.-T. Oh / Advances in Mathematics 222 (2009) 485–526

501

For the uniqueness, assume that b G,H is also a natural isomorphism satisfying the commutativity of the diagram (3.3). Let A and π be as above. Then, for any a = (aU,V )U ∈O(G), V ∈O(H ) in q q q q q BG ◦ BH (A), we obtain that BG ◦ BH (π)(X) = a. By the uniqueness of (bG,H )R in Lemma 3.4 we obtain the following commutative diagram q

q

q

q

BG ◦BH (π)

q

q

q

BG ◦ BH (R) −−−−−−→ BG ◦ BH (A) ⏐ ⏐ ⏐(b q ) ⏐ q q (b G,H )R (=(bG,H )R )  G,H A q

q

BG×H (R)

BG×H (π)

−−−−−→

q

BG×H (A).

It forces that (b G,H )A = (bG,H )A since q

q



So, we are done.

b G,H q

 A

 q   q  q (a) = b G,H A ◦ BG ◦ BH (π) (X)   q   q q = BG ◦ BH (π) ◦ bG,H R (X)   q   q q = bG,H A ◦ BG ◦ BH (π) (X)  q  = b G,H A (a).

2

As before, let q be any integer. Assume that Gi ’s (1  i  k)  are abelian profinite groups whose orders are mutually coprime. Every open subgroup of iG i , in this case, arises  U for some U ∈ O(G ), 1  i  k. It follows that O( uniquely as i i i i i Gi ) can be identified   O(G ) as a poset under the correspondence U → (U , U , . . . , Uk ) if we endow with i i 1 2 i i  O(G ) with the following ordering i i   (U1 , U2 , . . . , Uk )  U1 , U2 , . . . , Uk

⇐⇒

Ui ⊆ Ui

for 1  i  k.

  Note that i Ui  i Ui if and only if (U1 , U2 , . . . , Uk )  (U1 , U2 , . . . , Uk ). We will define a natural transformation q

q

q

q

ϕG1 ,...,Gk : BG1 ◦ BG2 ◦ · · · ◦ BGk → ghk

i=1 Gi

in the following way: Given a commutative ring A, let us define   q q q q ϕG1 ,...,Gk A : BG1 ◦ BG2 ◦ · · · ◦ BGk (A) → ghk

i=1 Gi

(A)

by sending X = (XU1 ,...,Uk )Ui ∈O(Gi ), 1ik to Y = (YU1 ,...,Uk )Ui ∈O(Gi ), 1ik . Here YU1 ,...,Uk =





1ik Ui ⊆Si ⊆Gi



 k  (Si :Ui )−1 (Gi : Si )q XS1 ,...,Sk . i=1

With this notation, we can obtain a generalization of Theorem 3.3.

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Y.-T. Oh / Advances in Mathematics 222 (2009) 485–526

Corollary 3.6. Let q be any integer and assume that Gi ’s are abelian profinite groups whose orders are coprime. Then there is a unique natural isomorphism ∼

= → Bk bG1 ,...,Gk : BG1 ◦ BG2 ◦ · · · ◦ BGk − q

q

q

q

q

i=1 Gi

satisfying q

q

ϕG1 ,...,Gk = ϕk

i=1 Gi

q

◦ bG1 ,...,Gk .

(3.14)

Proof. Let R be the polynomial ring

Z XU1 ,...,Uk : Ui ∈ O(Gi ), 1  i  k , and let q

q

q

X = (XU1 ,...,Uk )U ∈O(Gi ), 1ik ∈ BG1 ◦ BG2 ◦ · · · ◦ BGk (R). Denote by Z = (ZU1 ,...,Uk )U ∈O(Gi ), 1ik the vector  q ϕk

−1

i=1 Gi

q

q

◦ ϕG1 ,...,Gk (X) ∈ Bk

i=1 Gi

(Q ⊗ R).

One can show immediately that ZU1 ,...,Uk − XU1 ,...,Uk

∈ Q XS1 ,...,Sk : Ui ⊆ Si ⊆ Gi , but (Ui )1ik = (Si )1ik , 1  i  k for all Ui ∈ O(Gi ) and 1  i  k. However, we claim that ZU1 ,...,Uk − XU1 ,...,Uk

∈ Z XS1 ,...,Sk : Ui ⊆ Si ⊆ Gi , but (Ui )1ik = (Si )1ik , 1  i  k

(3.15)

for all Ui ∈ O(Gi ) and 1  i  k. As in Theorem 3.3, it will give rise to a unique natural isomorphism satisfying Eq. (3.14). To justify (3.15) we will use mathematical induction on the number k of Gi ’s. (3.15) is obvious when k = 1. Now, assume that (3.15) is true for k − 1 (k  2). It induces a unique natural isomorphism ∼

= bG2 ,...,Gk : BG2 ◦ BG2 ◦ · · · ◦ BGk − → Bk q

q

q

q

q

i=2 Gi

q

q

satisfying ϕG2 ,...,Gk = ϕk

i=2 Gi

q

◦ bG2 ,...,Gk , and which gives the following commutative diagram:

Y.-T. Oh / Advances in Mathematics 222 (2009) 485–526

q

q BG1

q ◦ BG2

q ◦ · · · ◦ BGk

q

BG (bG 1

2 ,...,Gk

)

b q BG1

q ◦ Bk

q  G1 , ki=2 Gi

i=2 Gi

503

q

Bk

i=1 Gi

q ϕ k G1 , i=2 Gi

q ϕG ,...,G k 1

Id

ghk

i=1 Gi

Id

ghk

i=1 Gi

q

ϕk

i=1 Gi

ghk

i=1 Gi

.

q

Letting bG1 ,...,Gk be b

q  G1 , ki=2 Gi

 q  q ◦ BG1 bG2 ,...,Gk ,

it will be a desired natural isomorphism satisfying our requirement. To show this, let    q  q Y = (YU1 ,...,Uk ) U ∈O(Gi ) = BG1 bG2 ,...,Gk R (X) 1ik

q

q

q

where X = (XU1 ,...,Uk ) Ui ∈O(Gi ) ∈ BG1 ◦ BG2 ◦ · · · ◦ BGk (R). Then our induction hypothesis im1ik

plies that, given all U1 ∈ O(G1 ), we have YU1 ,...,Uk − XU1 ,...,Uk

∈ Z XU1 ,S2 ,S3 ,...,Sk : Ui ⊆ Si ⊆ Gi , but (Ui )2ik = (Si )2ik , 2  i  k

(3.16)

for all Ui ∈ O(Gi ) and 2  i  k. To be more precise, YU1 ,...,Uk = Y(U1 )(U2 ) · · · (Uk )  q  q = BG1 bG2 ,...,Gk R (X)(U1 )(U2 ) · · · (Uk )   q = bG2 ,...,Gk R ◦ X(U1 )(U2 ) · · · (Uk )     q = bG2 ,...,Gk R X(U1 ) (U2 ) · · · (Uk ). It should be remarked once more that we are using two notations to denote elements of our concerned rings. For instance, the vector (YU1 ,...,Uk )U ∈O(Gi ), 1ik means a function Y ∈ q q BG1 (bG2 ,...,Gk )R such that Y(U1 )(U2 ) · · · (Uk ) = YU1 ,...,Uk . Now, according to our induction q hypothesis, (bG2 ,...,Gk )R (X(U1 ))(U2 ) · · · (Uk ) should be a polynomial in X(U1 )(S2 ) · · · (Sk ) = XU1 ,S2 ,...,Sk ’s with integer coefficients for all Ui ⊆ Si ⊆ Gi (2  i  k). Note that U1 is being fixed. On the other hand, Theorem 3.3 implies ZU1 ,...,Uk − YU1 ,...,Uk

∈ Z YS1 ,...,Sk : Ui ⊆ Si ⊆ Gi , but (Ui )1ik = (Si )1ik , 1  i  k for all Ui ∈ O(Gi ) and 1  i  k.

(3.17)

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Finally, by combining (3.16) with (3.17), we can conclude that ZU1 ,...,Uk − XU1 ,...,Uk

∈ Z XS1 ,...,Sk : Ui ⊆ Si ⊆ Gi , but (Ui )1ik = (Si )1ik , 1  i  k . q

Now, the existence and the uniqueness of bG1 ,...,Gk can be shown as in the proof of Theorem 3.3. 2 4. Decomposition of q-deformed Witt–Burnside rings As before, assume that G and H are abelian profinite groups whose orders are coprime. The goal of this section is to establish a functorial isomorphism q q q WG ◦ WH ∼ = WG×H . q

q

By virtue of the functoriality of ΦG and ΦH , we have the following diagram q

q

q

ΦG ◦Id

q

WG ◦ WH −−−−→ ghG ◦ WH ⏐ ⏐ ⏐ q q ⏐ WG ◦ΦH  ghG ◦ΦHq q

ΦG ◦Id

q

WG ◦ ghH −−−−→ ghG ◦ ghH . q

In general, this diagram is not commutative. Let A be a commutative ring A. For any f ∈ WG ◦ q WH (A) and any U ∈ O(G), V ∈ O(H ), we have  q   q q ΦG ◦ Id ◦ WG ◦ ΦH (f )(U )(V )   q (S:U ) q (G : S)q (S:U )−1 WG ◦ ΦH (f )(S) (V ) = U ⊆S⊆G

=



(G : S)q

(S:U )−1

 

U ⊆S⊆G

(H : T )q

(T :V )−1

(T :V )

(S:U )

f (S)(T )

.

V ⊆T ⊆H

But,   q  q ghG ◦ ΦH ◦ ΦG ◦ Id (f )(U )(V )   q  (T :V ) (H : T )q (T :V )−1 ΦG ◦ Id (f )(U ) (T ) = V ⊆T ⊆H

=

 V ⊆T ⊆H

(H : T )q (T :V )−1

 

(G : S)q (S:U )−1 f (S)(T )(S:U )

(T :V ) .

U ⊆S⊆G q

q

q

In this section, we will use the natural transformation ΦG,H : WG ◦ WH → ghG×H defined by   q  q ι ◦ ghG ◦ ΦH ◦ ΦH ◦ id .

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505

Given a commutative ring A, q

q

q

ΦG,H (A) : WG ◦ WH (A) → ghG×H (A) maps X = (XU,V )U ∈O(G), V ∈O(H ) to Y = (YU,V )U ∈O(G), V ∈O(H ) , where YU,V =



(G : S)q (S:U )−1

U ⊆S⊆G

 

(T :V )

(H : T )q (T :V )−1 XS,T

(S:U ) .

V ⊆T ⊆H

First, we will investigate a functorial property of Witt–Burnside rings, i.e., the case where q = 1. Lemma 4.1. Suppose that G, H are any abelian profinite groups whose orders are coprime. Then there is a unique functorial isomorphism, ωG,H : WG ◦ WH → WG×H satisfying ΦG,H = ΦG×H ◦ ωG,H .

(4.1)

Proof. Let R be the polynomial ring

Z XU,V : U ∈ O(G), V ∈ O(H ) , and consider X = (XU,V )U ∈O(G), V ∈O(H ) ∈ WG ◦ WH (R). Recall that ΦG : WG (A) → gh(A) is invertible if A is a Q-algebra. Denote by Z = (ZU,V )U ∈O(G), V ∈O(H ) the vector (ΦG×H )−1 ◦ ΦG,H (X) ∈ WG×H (Q ⊗ R). It is not difficult to show that

ZU,V − XU,V ∈ Q XS,T : U ⊆ S ⊆ G, V ⊆ T ⊆ H, (S, T ) = (U, V ) for all U ∈ O(G), V ∈ O(H ). We, however, claim that ZU,V − XU,V is a polynomial with integer coefficients. That is,

ZU,V − XU,V ∈ Z XS,T : U ⊆ S ⊆ G, V ⊆ T ⊆ H, (S, T ) = (U, V )

(4.2)

for all U ∈ O(G), V ∈ O(H ). As in Theorem 3.3, it will induce a unique natural isomorphism, say ωG,H , satisfying Eq. (4.1). Note that ZG,H = XG,H . It can be shown by comparing the (G, H )th component on either side of ΦG,H (X) = (ΦG×H )(Z).

(4.3)

To prove (4.2), we will utilize mathematical induction on index. Choose (and then fix) any U ∈ O(G) and V ∈ O(H ). Assume that ZS,T ∈ R for all S ∈ O(G), T ∈ O(H ) satisfying that U ⊆

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S ⊆ G, V ⊆ T ⊆ H, and (S ×T : U ×V ) > 1. With this hypothesis, we will show that ZU,V ∈ R. For this purpose it is quite useful to rewrite Eq. (4.3) in the following way: For every U ∈ O(G) and V ∈ O(H ), (S:U )     (T :V ) (S×T :U ×V ) (G : S) (H : T )XS,T = (G × H : S × T )ZS,T . (4.4) U ⊆S⊆G

V ⊆T ⊆H

U ⊆S⊆G V ⊆T ⊆H

Applying the induction hypothesis to the case U = U , V = V , we obtain (G : U )(H : V )ZU,V ∈ R. For any prime p dividing (G : U )(H : V ), let ν be ordp ((G : U )(H : V )). Since p is an arbitrary prime dividing (G : U )(H : V ), our claim will be verified if we can show that p ν divides (G : U )(H : V )ZU,V . Case 1. Firstly, we assume that p divides (G : U ). From our induction hypothesis it follows that (G : U )(H : V )ZU,V  (S×T :U ×V ) ≡ (G × H : S × T )ZS,T U ⊆S⊆G V ⊆T ⊆H ordp ((G:S))=ν

=



(S:U )(T :V )

(G : S)(H : T )ZS,T

U ⊆S⊆G V ⊆T ⊆H



−

(S:U )(T :V )

(G : S)(H : T )ZS,T

U ⊆S⊆G V ⊆T ⊆H ordp ((G:S))<ν



= ΦG×H (Z)U,V −

  mod p ν

(S:U )(T :V )

(G : S)(H : T )ZS,T

.

U ⊆S⊆G V ⊆T ⊆H ordp ((G:S))<ν

Replacing ΦG×H (Z)U,V by ΦG,H (X)U,V yields that (G : U )(H : V )ZU,V (S:U )    (T :V ) ≡ (G : S) (H : T )XS,T U ⊆S⊆G

−

V ⊆T ⊆H



(S:U )(T :V )

(G : S)(H : T )ZS,T

  mod p ν

U ⊆S⊆G V ⊆T ⊆H ordp ((G:S))<ν



≡

(S:U )   (T :V ) (G : S) (H : T )XS,T

U ⊆S⊆G ordp ((G:S))<ν

−



U ⊆S⊆G V ⊆T ⊆H ordp ((G:S))<ν

V ⊆T ⊆H (S:U )(T :V )

(G : S)(H : T )ZS,T

  mod p ν .

(4.5)

Y.-T. Oh / Advances in Mathematics 222 (2009) 485–526

507

The second congruence was obtained by ignoring the terms containing (G : S) with ordp ((G : S)) = ν since they vanish. Let us recall Eq. (3.13). Specializing q into 1 gives rise to the identity: 

1≡

CW (1)

  mod p ordp ((S:U )) .

(4.6)

U ⊆W ⊆S W/U is a p -group

By Eq. (4.6) we can deduce the following:   (S:U ) (T :V ) (G : S) (H : T )XS,T

 U ⊆S⊆G

V ⊆T ⊆H



≡

CW (1)

U ⊆W ⊆G W/U is a p -group

  (S:U ) (T :V ) (G : S) (H : T )XS,T

 W ⊆S⊆G

  mod p ν .

(4.7)

V ⊆T ⊆H

In the same manner, we have 

(S:U )(T :V )

(G : S)(H : T )ZS,T

U ⊆S⊆G V ⊆T ⊆H ordp ((G:U ))<ν

≡



CW (1)

U ⊆W ⊆G W/U is a p -group

≡

 U ⊆W ⊆G W/U is a p -group



(S:U )(T :V )

(G : S)(H : T )ZS,T

  mod p ν

W ⊆S⊆G V ⊆T ⊆H

CW (1)



 (W :U ) (S:W )(T :V ) (G : S)(H : T ) ZS,T

  mod p ν .

(4.8)

W ⊆S⊆G V ⊆T ⊆H

The second congruence follows from the relation (S : U ) = (S : W )(W : U ). Recall that our induction hypothesis says that

ZS,T − XS,T ∈ Z XS  ,T  : S ⊆ S  ⊆ G, T ⊆ T  ⊆ H, (S  , T  ) = (S, T )

(4.9)

for all S ∈ O(G), T ∈ O(H ) satisfying that U ⊆ S ⊆ G, V ⊆ T ⊆ H, and (S × T : U × V ) > 1. From now on, let us view XS,T as a polynomial in ZS  ,T  ’s. Since ZG,H = XG,H , the assertion (4.9) implies by induction that

XS,T − ZS,T ∈ Z ZS  ,T  : S ⊆ S  ⊆ G, T ⊆ T  ⊆ H, (S  , T  ) = (S, T ) for all S ∈ O(G), T ∈ O(H ) satisfying that U ⊆ S ⊆ G, V ⊆ T ⊆ H, and (S × T : U × V ) > 1. To emphasize this viewpoint, that is, XS,T is a polynomial in ZS  ,T  ’s, we will use XS,T (Z) instead of XS,T . Applying this notation to Eq. (4.8) we obtain the relation:

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Y.-T. Oh / Advances in Mathematics 222 (2009) 485–526



(S:U )(T :V )

(G : S)(H : T )ZS,T

U ⊆S⊆G V ⊆T ⊆H ordp ((G:U ))<ν



≡



CW (1)

 (W :U ) (S:W )(T :V ) (G : S)(H : T ) ZS,T

U ⊆W ⊆G W/U is a p -group

W ⊆S⊆G V ⊆T ⊆H





≡

CW (1)

U ⊆W ⊆G W/U is a p -group

  mod p ν

   (W :U ) (T :V ) (S:W ) (G : S) (H : T )XS,T Z

W ⊆S⊆G

V ⊆T ⊆H

  mod p ν .

(4.10)

The second congruence follows from Eq. (4.4). And, the notation XS,T (Z (W :U ) ) appearing in the third summation means the polynomial obtained from XS,T (ZS  ,T  ) by the substitution of (W :U ) ZS  ,T  ’s with ZS  ,T  ’s for all S ⊆ S  ⊆ G, T ⊆ T  ⊆ H . Applying Eqs. (4.7) and (4.10) to Eq. (4.5) we arrive at the following congruence: (G : U )(H : V )ZU,V 

≡

CW (1)

U ⊆W ⊆G W/U is a p -group

−

 



(S:U )   (T :V ) (G : S) (H : T )XS,T (Z)

W ⊆S⊆G

V ⊆T ⊆H

 (T :V ) (H : T )XS,T Z (W :U )

(S:W )

  mod p ν .

V ⊆T ⊆H

Recall that CW (q) ≡ 0 (mod p ordp ((W :U ))−1 ) for any integer q with (q, p) = 1 (refer to Eq. (6.10)). Thus we have CW (1)(G : S) ≡ 0

  mod p ordp ((G:U ))−ordp ((S:W ))−1 .

As a consequence, our claim that (G : U )(H : V )ZU,V ≡ 0 (mod p ν ) will turn out to be true if one can only show  

(T :V )

(S:U )

(H : T )XS,T (Z)

V ⊆T ⊆H

≡

 

 (T :V ) (H : T )XS,T Z (W :U )

(S:W )

  mod p ordp ((S:W ))+1 .

(4.11)

V ⊆T ⊆H

As in the proof of Lemma 3.4 (more precisely, refer to Eq. (3.10)) write (up to isomorphism) S/U as M × N where M is the Sylow p-subgroup of S/U and N is an abelian group whose order is not divisible by p. Denote W/U by K. Then K is a subgroup of M since W/U is a p-group. Let |K| = p i and |M| = p n+i (n  0). Using this notation, let us rewrite Eq. (4.11) as

Y.-T. Oh / Advances in Mathematics 222 (2009) 485–526

 

(T :V )

(H : T )XS,T (Z)

pi pn |N |

V ⊆T ⊆H

≡

 

509

 i (T :V ) (H : T )XS,T Z p

pn |N |

  mod p n+1 .

(4.12)

V ⊆T ⊆H

In the following, we will see the that Eq. (4.12) holds. It follows immediately from the congruence  

(T :V )

p i p n

(H : T )XS,T (Z)

V ⊆T ⊆H

≡

 

 i (T :V ) (H : T )XS,T Z p

pn

  mod p n+1 ,

(4.13)

 i (T :V ) (H : T )XS,T Z p

(mod p) (4.14)

V ⊆T ⊆H

and (4.13) follows from the congruence  

(T :V )

pi

(H : T )XS,T (Z)



≡

V ⊆T ⊆H

V ⊆T ⊆H k

k

due to the fact that if a ≡ b (mod p) then a p ≡ bp (mod p k+1 ) for all k  1. Finally, Eq. (4.14) can be obtained from the combination of  i pi XS,T (Z) ≡ XS,T Z p

(4.15)

(mod p)

with the congruence  

(T :V )

p i

(H : T )XS,T (Z)



≡

V ⊆T ⊆H

pi

(H : T )XS,T (Z)(T :V )

(mod p).

V ⊆T ⊆H

Case 2. Secondly, we assume that p divides (H : V ). In this case, we have (G : U )(H : V )ZU,V ≡

 U ⊆S⊆G V ⊆T ⊆H

−

(S:U )(T :V )

(G : S)(H : T )ZS,T 

(S:U )(T :V )

(G : S)(H : T )ZS,T

U ⊆S⊆G V ⊆T ⊆H ordp ((H :T ))<ν

By Eq. (4.4), the first summation in the right-hand side can be written as

  mod p ν . (4.16)

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Y.-T. Oh / Advances in Mathematics 222 (2009) 485–526



 (G : S)

U ⊆S⊆G



(H

(T :V ) : T )XS,T

(S:U )

  mod p ν .

(4.17)

V ⊆T ⊆H ordp ((H :T ))<ν

From Eq. (4.6) (see also Eq. (3.13)) it follows that 

1≡

  mod p ordp ((T :V )) .

CW (1)

V ⊆W ⊆T W/V is a p -group

Following the method used in Case 1, we can do the following transformations: 

(S:U )(T :V )

(G : S)(H : T )ZS,T

U ⊆S⊆G V ⊆T ⊆H ordp ((H :T ))<ν



≡

V ⊆W ⊆H W/V is a p -group





CW (1)

V ⊆W ⊆H W/V is a p -group

≡

 pi (S:U )(T :W ) (G : S)(H : T ) ZS,T

U ⊆S⊆G W ⊆T ⊆H



≡



CW (1)

 (G : S)

U ⊆S⊆G

U ⊆S⊆G

V ⊆W ⊆H W/V is a p -group

  mod p ν .

 i (T :W ) (H : T )XS,T Z p

(S:U )

W ⊆T ⊆H

 CW (1)



(G : S)





 i (T :W ) (H : T )XS,T Z p

(S:U )

W ⊆T ⊆H

(4.18)

Here (W : V ) = p i . From now on, view XS,T as a polynomial in ZS  ,T  ’s as in Case 1. Exploiting (4.17) together with Eq. (4.18) we can reduce Eq. (4.16) to the following congruence: (G : U )(H : V )ZU,V   ≡ (G : S) U ⊆S⊆G

−





(H : T )XS,T (Z)(T :V )

(S:U )

V ⊆T ⊆H ordp ((H :T ))<ν

(G : S)

U ⊆S⊆G

 CW (1)

 V ⊆W ⊆H W/V is a p -group

  mod p ν .



 i (T :W ) (H : T )XS,T Z p

(S:U )

W ⊆T ⊆H

For our purpose it suffices to show that the difference between 

 V ⊆T ⊆H ordp ((H :T ))<ν

(T :V )

(H : T )XS,T (Z)

(S:U ) (4.19)

Y.-T. Oh / Advances in Mathematics 222 (2009) 485–526

511

and  CW (1)

 V ⊆W ⊆H W/V is a p -group



 i (T :W ) (H : T )XS,T Z p

(S:U ) (4.20)

W ⊆T ⊆H

vanishes modulo (mod p ν ). This can be verified in the following way: Note that (4.19) can be expanded as 

d(T1 , . . . , Tr : a1 , . . . , ar )

r 

XS,Tj (Z)aj (Tj :V ) ,

(4.21)

j =1

(T1 ,...,Tr ) (a1 ,...,ar ) 1r(S:U )

where the sum is over pairs (T1 , T2 , . . . , Tr ) and (a1 , a2 , . . . , ar ) satisfying the following conditions: (1) Tj ’s (1  j  r) are distinct open subgroups of H such that V ⊆ Tj ⊆ H and ordp ((H : Tj )) < ν, and (2) aj  1 for all j = 1, 2, . . . , r and a1 + · · · + ar = (S : U ). And, the coefficient d(T1 , . . . , Tr : a1 , . . . , ar ) is given by   r  ar−1 + ar  (S : U ) (S : U ) − a1 ··· (H : Tj )aj . a2 ar−1 a1 j =1

Using the fact that

r

i=1 (H

: Ti ) is a multiple of (H :

 1ir

Ti ), we can prove that

   (4.22) d(T1 , . . . , Tr : a1 , . . . , ar ) ≡ 0 mod p ordp ((H : 1ir Ti )) .  r to ri=1 Ti . It follows that Indeed, r i=1 H /Ti isequal r the kernel of the product map H → r H /r i=1 Ti is isomorphic to a subgroup of i=1 H /Ti , so i=1 (H : Ti ) is a multiple of (H : i=1 Ti ). In the same way as above, we can show that (4.20) has the expansion  r     i a (T :W ) . (4.23) d(T1 , . . . , Tr : a1 , . . . , ar ) CW (1) XS,Tj Z p j j V ⊆W ⊆H W/V is a p -group W ⊆Tj , ∀j =1,...,r

(T1 ,...,Tr ) (a1 ,...,ar ) 1r(S:U )

j =1

As a consequence, (4.19)–(4.20) can be expressed as  d(T1 , . . . , Tr : a1 , . . . , ar ) (T1 ,...,Tr ) (a1 ,...,ar ) 1r(S:U )



aj (Tj :V )

× XS,Tj (Z)

−

 V ⊆W ⊆H W/V is a p -group W ⊆Tj , ∀j =1,...,r

CW (1)

r  j =1

 pi aj (Tj :W ) . XS,Tj Z

(4.24)

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Y.-T. Oh / Advances in Mathematics 222 (2009) 485–526

 In case where ( 1j r Tj : V ) is not divisible by p, d(T1 , . . . , Tr : a1 , . . . , ar ) ≡ 0

  mod p ordp ((H :V ))

 by (4.22). Now, let us consider the case where ( 1j r Tj : V ) is divisible by p. Due to the fact k

k

that if a ≡ b (mod p) then a p ≡ bp (mod p k+1 ) for all k  1 and (4.15)  i pordp (Tj :W )  i p ordp (Tj :W ) XS,Tj Z p ≡ XS,Tj (Z)p = XS,Tj (Z)p

ordp (Tj :V )

modulo (mod p ordp (Tj :W )+1 ). Therefore r 

 i a (T :W ) XS,Tj Z p j j = XS,Tj (Z)aj (Tj :V ) + fW (XS,Tj : 1  j  r),

j =1

coefficients divisible by where fW (XS,Tj : 1  j  r) is a polynomial in XS,Tj ’s whose  (mod p ordp (Tj :W )+1 ) for some j and thus divisible by (mod p ordp ( this result with CW (1) ≡ 0 (mod p ordp ((W :U ))−1 ) together with 

CW (1) ≡ 1

1j r

Tj :W )+1

). Combining

   mod p ordp (( 1j r Tj :V ))

V ⊆W ⊆H (W :V)=p i , i1 W ⊆ 1j r Tj

(refer to Eq. (4.6)) yields that aj (Tj :V )

XS,Tj (Z)



−

r 

CW (1)

V ⊆W ⊆H W/V is a p -group W ⊆Tj , ∀j =1,...,r

 i a (T :W ) XS,Tj Z p j j

j =1



vanishes modulo (mod p ordp ( 1j r Tj :V ) ). As a consequence, we can conclude that (4.24) vanishes modulo (mod p ordp (H :V ) ). This completes the proof. 2 Now, we are ready to show the functorial property of q-deformed Witt–Burnside rings. Theorem 4.2. Let G, H be abelian profinite groups whose orders are relatively prime. Then there is a unique functorial isomorphism, q

q

q

q

ωG,H : WG ◦ WH → WG×H q

q

q

satisfying ΦG,H = ΦG×H ◦ ωG,H .

Y.-T. Oh / Advances in Mathematics 222 (2009) 485–526

513

Proof. As in the proof of Lemma 4.1 let

R = Z XU,V : U ∈ O(G), V ∈ O(H ) , and let X = (XU,V )U ∈O(G), V ∈O(H ) . Also, we denote by Z = (ZU,V )U ∈O(G), V ∈O(H ) the vector  q −1 q q ◦ ΦG,H (X) ∈ WG×H (Q ⊗ R). ΦG×H For our purpose it suffices to show

ZU,V − XU,V ∈ Z XS,T : U ⊆ S ⊆ G, V ⊆ T ⊆ H, (S, T ) = (U, V ) for all U ∈ O(G), V ∈ O(H ). From the equality q

q

ΦG,H (X) = ΦG×H (Z) we have 

(G : S)q

U ⊆S⊆G

=

(S:U )−1

 

(H : T )q

(T :V )−1

(T :V ) XS,T

(S:U )

V ⊆T ⊆H



(S×T :U ×V )

(G × H : S × T )q (S×T :U ×V )−1 ZS,T

(4.25)

U ⊆S⊆G V ⊆T ⊆H

for every U ∈ O(G) and V ∈ O(H ). Multiplying q to both sides of Eq. (4.25) yields (S:U )   (G : S) (H : T )(qXS,T )(T :V )

 U ⊆S⊆G

=



V ⊆T ⊆H

(G × H : S × T )(qZS,T )(S×T :U ×V ) .

U ⊆S⊆G V ⊆T ⊆H

But, it has already been shown in the middle of proving Lemma 4.1 that

qZU,V − qXU,V ∈ Z qXS,T : U ⊆ S ⊆ G, V ⊆ T ⊆ H, (S, T ) = (U, V ) .

(4.26)

Since qZU,V − qXU,V has no constant term, (4.26) implies that

ZU,V − XU,V ∈ Z XS,T : U ⊆ S ⊆ G, V ⊆ T ⊆ H, (S, T ) = (U, V ) . This completes the proof.

2 q

q

Define the exchange transformation ex : WG×H → WH ×G and ex : ghG×H → ghH ×G so that exA (f )(V )(U ) = f (U )(V ) for any commutative ring A. Then one can have the following commutative diagram:

514

Y.-T. Oh / Advances in Mathematics 222 (2009) 485–526 q

q WG

q

ωG,H

q ◦ WH

q

ex

q WG×H q

ΦG,H

ωH,G

q

q

ΦH,G

ΦH ×G ex

ghG×H

q

WH ◦ WG

q

ΦG×H

ghG×H

q WH ×G

ghH ×G

ghH ×G .

In the remaining part, we deal with a generalization of Theorem 4.2. Let q be any integer and assume that Gi ’s are abelian profinite groups whose orders are coprime. Let us define a natural transformation q

q

q

q

ΦG1 ,...,Gk : WG1 ◦ WG2 ◦ · · · ◦ WGk → ghk

i=1 Gi

in the following way: Given a commutative ring A, we define q

q

q

q

ΦG1 ,...,Gk (A) : WG1 ◦ WG2 ◦ · · · ◦ WGk (A) → ghk

i=1 Gi

(A)

by assigning X = (XU1 ,...,Uk )Ui ∈O(Gi ), 1ik to Y = (YU1 ,...,Uk )Ui ∈O(Gi ), 1ik . Here YU1 ,...,Uk is defined recursively as below: YU  ,...,U  ,U = 1

k

k−1

YU  ,...,U  ,U 1

k−2

 Uk ⊆Sk ⊆Gk

k−1 ,Uk

.. . YU 1 ,U2 ,...,Uk =

=



U2 ⊆S2 ⊆G2



U1 ⊆S1 ⊆G1

:U

(S

Uk−1 ⊆Sk−1 ⊆Gk−1



YU1 ,U2 ,...,Uk =

(S :U )

(Gk : Sk )XU1k,U2k,...,Uk k−1 (Gk−1 : Sk−1 )Y k−1 

)

U1 ,...,Uk−1 ,Uk

(S :U )

2 2 (G2 : S2 )YU  ,U  ,U

3 ,...,Uk

(S1 :U1 ) (G1 : S1 )YU  ,U ,U

3 ,...,Uk

1

1

2

2

.

We can establish the following functorial isomorphism in the way as Corollary 3.6 was proven. Corollary 4.3. Let q be any integer and assume that Gi ’s are abelian profinite groups whose orders are mutually coprime. Then there is a unique natural isomorphism ∼

= → W ωG1 ,...,Gk : WG1 ◦ WG2 ◦ · · · ◦ WGk − q

q

q

q

satisfying ΦG1 ,...,Gk = Φk

i=1 Gi

q

q

q

i∈I

Gi

q

◦ ωG1 ,...,Gk .

By the trivial modification of the proof of Corollary 3.6 one can deduce that  q q q  q ◦ WG1 ωG2 ,...,Gk ωG1 ,...,Gk = ω k G1 ,

and the following diagram is commutative:

i=2 Gi

Y.-T. Oh / Advances in Mathematics 222 (2009) 485–526

q

q WG1

q ◦ WG2

q ◦ · · · ◦ WGk

q

WG (ωG 1

2 ,...,Gk

)

ω q WG1

q ◦ Wk

i=2 Gi

q

Wk

i=1 Gi

q ω k G1 , i=2 Gi

q ΦG ,...,G k 1

Id

ghG×H

q  G1 , ki=2 Gi

515

q

Φk

i=1 Gi

Id

ghG×H

ghG×H .

Remark 4.4. The converse of Proposition 2.5 is not true. In view of the following diagram q

WG×H (Z) ∼ =

∼ = (by Th. 4.2)

q

q

WG ◦ WH (Z)

∼ = (by Prop. 2.5)

q

q

q

q

WG ◦ BH (Z)

(by Prop. 2.5) ∼ = (by Th. 3.3)

q

BG×H (Z)

q

q

BG ◦ BH (Z)

BG ◦ BH (Z)

we can deduce that q q q q WG ◦ BH (Z) ∼ = BG ◦ BH (Z) q

although BH (Z) is not a binomial ring unless q = ±1. Similarly, q q q q BG ◦ WH (Z) ∼ = BH ◦ WH (Z) q

although WH (Z) is not a binomial ring unless q = ±1. 5. Pseudo-multiplicativeness of q-Möbius function In this section, we investigate a combinatorial interpretation of the functorial isomorphism q provided in Theorem 3.3. To do this, we define an O(G) × O(G) matrix λG by  q

λG (V , W ) =

(G : W )q (W :V )−1 0

if V ⊆ W, otherwise.

When q = 1 and G is a finite group, the matrix λG is called the table of marks3 of G. One can q q q see that the natural isomorphism ϕG : BG → ghG is given by the left multiplication by λG . Let q q ψG be the inverse of λG . −1 Convention 5.1. It should be remarked that λ−1 G , ψG do not denote the inverse of λG , ψG , respectively. To denote its inverse we will use the notation (λG )−1 , (ψG )−1 .

Theorem 5.1. Let q be an indeterminate and let G, H be abelian profinite groups whose orders are coprime. For every U, S ∈ O(G) and V , T ∈ O(H ) with U ⊆ S and V ⊆ T , we have 3 The concept of a table of marks of a finite group was first introduced by William Burnside in the second edition of his classical book “Theory of groups of finite order”.

516

Y.-T. Oh / Advances in Mathematics 222 (2009) 485–526 q

q

q

ψG×H (U × V , S × T ) = ψG (U, S)ψH (V , T ) +



q

q

fW,Z (q)ψG (W, S)ψH (Z, T ),

U ⊆W ⊆S V ⊆Z⊆T U ×V =S×T

where fW,Z (q) ∈ Q[q] satisfies the following conditions: (1) fW,Z (q) are integer-valued polynomials in q, and (2) If S = U and T = V , then fW,Z (q) = 0 for every W, Z with U ⊆ W ⊆ S, V ⊆ Z ⊆ T if and only if q = 1, 0, −1. Proof. Let R denote the polynomial ring

Z XU,V : U ∈ O(G), V ∈ O(H ) , and let ⎛ . ⎞ .. ⎜ ⎟ X = ⎝ XV ⎠ .. .

⎛ ,

V ∈O (H )

⎞

.. .

⎜ ⎟ where XV = ⎝ XU,V ⎠ .. .

U ∈O (G)

for every V ∈ O(H ). ⎛ . ⎞ .. ⎜ ⎟ Y = ⎝ YV ⎠ .. .

⎛ q

= ϕG,H (X), V ∈O (H )

.. .

⎞

⎜ ⎟ where YV = ⎝ YU,V ⎠ .. .

. U ∈O (G)

Multiply both sides of ⎛ . ⎞ .. ⎞ .. . ⎟ ⎜ ⎟ q ⎜ q X λ λH ⎝ G V ⎠ = ⎝ YV ⎠ .. .. . V ∈O(H ) . V ∈O (H ) ⎛

q

by the inverse of λH to deduce ⎛

⎛ . ⎞ .. ⎞ .. . ⎜ λq X ⎟ ⎟ q ⎜ = ψH ⎝ YV ⎠ . ⎝ G V⎠ .. .. . V ∈O(H ) . V ∈O (H )

(5.1)

Compare the V th component on either side of Eq. (5.1) to derive the relation q

λG X V =



q

(5.2)

ψH (V , T )YT .

V ⊆T q

Again, let us multiply either side of Eq. (5.2) by the inverse of λH to deduce

Y.-T. Oh / Advances in Mathematics 222 (2009) 485–526



q

XV = ψG

517

q ψH (V , T )YT .

V ⊆T

Its U th component is given by XU,V =



q

q

ψG (U, S)ψH (V , T )YS,T

U ⊆S V ⊆T

for all U ∈ O(G), V ∈ O(H ). q Next, let us consider bG,H (X), which is equal to ⎛

.. .

⎞

⎜ ⎟ q ψG×H ⎝ YU ×V ⎠ . .. U ∈O (G) . V ∈O (H )

Whenever necessary, we will use the identification YU,V = YU ×V . The U × V th component of the above vector is given by 

q

ψG×H (U × V , S × T )YS,T .

U ⊆S V ⊆T

By Theorem 3.3 



q

ψG×H (U × V , S × T )YS,T −

U ⊆S V ⊆T

q

q

ψG (U, S)ψH (V , T )YS,T

(5.3)

U ⊆S V ⊆T

belongs to

Z XS  ,T  : U ⊆ S  ⊆ G, V ⊆ T  ⊆ H, (S  , T  ) = (U, V ) . Here XS  ,T  is given by  S  ⊆S T  ⊆T

ψG (S  , S)ψH (T  , T )YS,T . q

q

Write this polynomial in the form 

fS  ,T  (q)XS  ,T  .

(5.4)

S  ,T 

Then the first assertion (1) for fW,Z (q) can be confirmed by picking out the coefficient of YS,T in the polynomials (5.3) and (5.4). The “if”-part of the second property (2) has already been shown in [21]. For the converse, let us assume that fW,Z (q) = 0 for every W , Z with U ⊆ W ⊆ S, V ⊆ Z ⊆ T , but not U × V =

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Y.-T. Oh / Advances in Mathematics 222 (2009) 485–526

S × T . If q = 0, then Eq. (3.3) implies XU,V = ZU,V for all U ∈ O(G), V ∈ O(H ). This means that b0G,H is nothing but the identity transformation. Hence 

0 0 ψG (U, S)ψH (V , T )YS,T =

U ⊆S V ⊆T



0 ψG×H (U × V , S × T )YS,T ,

U ⊆S V ⊆T

and which justifies our assertion at q = 0. Now, assume that q is not zero. Choose U ⊂ W0 ⊂ S and V ⊂ Z0 ⊂ T such that p := (W0 : U ) and p  := (Z0 : V ) are distinct primes. Put Z = q (ZU,V )U ∈O(G), V ∈O(H ) = bG,H (X). In view of Theorem 3.3 and Eq. (3.3) we can show that ZU,V =



gM,N (q)XM,N

U ⊆M⊆G V ⊆N ⊆H

where gU,V (q) = 1, gW0 ,Z0 (q) = −

gW0 ,V (q) = 0, q

pp  −1

−q pp 

p+p  −2

gU,Z0 (q) = 0, =−

q

(p−1)(p  −1)

pp 

and

−1

.

Hence  U ⊆S V ⊆T

=

q

ψG×H (U × V , S × T )YS,T 

q

q

ψG (U, S)ψH (V , T )YS,T + gW0 ,Z0 (q)

U ⊆S V ⊆T

q

q

ψG (W0 , S)ψH (Z0 , T )YS,T

W0 ⊆S Z0 ⊆T



+



gM,N (q)

U ⊆M⊆G V ⊆N ⊆H (M,N )=(U,V ),(U,Z0 ),(W0 ,V ),(W0 ,Z0 )



q

q

ψG (M, S)ψH (N, T )YS,T .

M⊆S N ⊆T

Picking out the coefficient of YS,T on either side of the above identity yields q

ψG×H (U × V , S × T ) q

q

q

q

= ψG (U, S)μ¯ H (V , T ) + gW0 ,Z0 (q)ψG (W0 , S)ψH (Z0 , T )  q q + gM,N (q)ψG (M, S)ψH (N, T ). U ⊆M⊆S V ⊆N ⊆T (M,N )=(U,V ),(U,Z0 ),(W0 ,V ),(W0 ,Z0 ) 

By assumption gW0 ,Z0 (q) = q (p−1)(p −1) − 1 = 0, which forces q = 1, −1.

2

Y.-T. Oh / Advances in Mathematics 222 (2009) 485–526

519

q

Define a O(G) × O(G) matrix ζG by  q

ζG (V , W ) = q

q (W :V )−1 0

if V ⊆ W, otherwise.

def

q

Let μG be the inverse of ζG . Since μG ( = μ1G ) denotes the Möbius function of the lattice of q q q open subgroups of G at q = 1, we may regard μG as a q-deformation of μG . From μG ζG = δ it follows that q

μG (U, U ) = 1,

for all U ∈ O(G),  q q (V :W )−1 μG (U, W ),

q

μG (U, V ) = −

for all U  V .

(5.5)

U ⊆W V

Convention 5.2. It should be remarked that ζG−1 , μ−1 G do not denote the inverse of ζG , μG , respectively. To denote its inverse we will use the notation (ζG )−1 , (μG )−1 . Corollary 5.2. Suppose that G, H are profinite groups whose orders are coprime. For every U, S ∈ O(G) and V , T ∈ O(H ) with U ⊆ S and V ⊆ T , we have q

μG×H (U × V , S × T ) q



q

= μG (U, S)μH (V , T ) +

q

q

(W : U )(Z : V )fW,Z (q)μG (W, S)μH (Z, T ).

U ⊆W ⊆S V ⊆Z⊆T U ×V =S×T

Proof. The desired result follows from the identity q

q

ψG (U, V ) = (G : U )μG (U, V ).

2

(5.6)

q

In view of Eq. (5.5) it is obvious that μG (U, V ) depends on U and V , but not on G. In particular, q

q

q

μG (U, V ) = μV (0, V /U ) = μV /U (0, V /U ). Definition 5.3. Given a finite abelian H , we define μq (H ) by q

μH (0, H ). q

It is obvious that μq (H ) = μG (U, V ) if V /U isomorphic to H . Furthermore, in view of Eq. (5.5), one has the following recursive form: μq (0) = 1, μq (H ) = −

 W H

q (H :W )−1 μq (W ),

for all W  H.

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Y.-T. Oh / Advances in Mathematics 222 (2009) 485–526

Note that q

q

ψH (0, H ) = |H | · μH (0, H ) = |H |μq (H )

(by Eq. (5.6)),

where |H | represents the order of H . Corollary 5.4. Suppose that G, H are finite abelian groups which are coprime. Then we have μq (G × H ) = μq (G)μq (H ) +



|W ||Z|fW,Z (q)μq (G/W )μq (H /Z).

W ⊆G Z⊆H

Proof. It follows from Corollary 5.2.

2

6. Proof of Claim 3.5 This section is fully devoted to the proof of Claim 3.5 which played a key role in the proof of Lemma 3.4. To begin with, let us recall Claim 3.5 for the readers’ convenience. Claim 3.5. Let G be a finite abelian p-group and k a positive integer. If q is an integer coprime to p, then we have the following congruence of polynomial values at q: 

cG (q) ≡ −

 (G:S)−1  k  |G| −1 cS (q) q k + q p

  mod |G| .

(6.1)

{0}⊆SG

Before starting the proof, let us reformulate Claim 3.5 into a simpler form. By definition cG (q) turns out to be identical to cG (q) if G and G are isomorphic. It is well known that the isomorphism class of every finite abelian p-group can be identified with a partition called its type. For instance, the type of Z/p λ1 × Z/p λ2 × · · · × Z/p λl Z with λ1  λ2  · · ·  λl  1 is given by the partition (λ1 , . . . , λl ). From now on we will denote cG (q) by c (type of G). Let P be the set of partitions. In view of Eq. (3.11) we have c(λ) = −



c(μ)aλ (μ; p)q p

|λ|−|μ| −1

+ qp

|λ|−1 −1

,

with c(∅) = 0

(6.2)

μλ

for every λ ∈ P. Here aλ (μ; p) denotes the number of subgroups of type μ in a finite abelian p-group of type λ. Indeed, it is well known (for example, see [5]) that aλ (μ; p) =

 j 1

p

μj +1 (λj −μj )



λj − μj +1

μj − μj +1

(6.3)

, p

where λ , μ are partitions which are conjugate to λ, μ, respectively, and binomial coefficient, that is  n (1 − p n )(1 − p n−1 ) · · · (1 − p n−k+1 ) . = k p (1 − p)(1 − p 2 ) · · · (1 − p k )

·

· p

denotes the p-

Y.-T. Oh / Advances in Mathematics 222 (2009) 485–526

521

Due to the above discussion, Eq. (6.1) can be restated follows: c(λ) ≡ −



 p|λ|−|μ| −1  k p|λ|−1 −1 c(μ)aλ (μ; p) q k + q

  mod p |λ| ,

(6.4)

μλ

where |λ| stands for the weight of λ, i.e., |λ| = λ1 + λ2 + · · · . Plugging Eq. (6.2) into Eq. (6.4) yields 

 p|λ|−|μ| −1 |λ|−|μ| −1  c(μ)aλ (μ; p) q k − qp

μλ

 p|λ|−1 −1 |λ|−1 −1 ≡ qk − qp

  mod p |λ| .

(6.5)

When |λ| = 1, there is no proper subpartition of λ. Thus each side of Eq. (6.5) vanishes. So, we will assume |λ|  2 from now on. Put

Aλ (i) =

c(1)aλ (1; p) − 1 if i = 1,  μ⊂λ c(μ)aλ (μ; p) if i = 2, . . . , |λ| − 1.

(6.6)

|μ|=i

Using this notation we can rewrite Eq. (6.5) as 

    p|λ|−i −1 |λ|−i Aλ (i) q k − q p −1 ≡ 0 mod p |λ| .

(6.7)

1i|λ|−1

As a consequence, Claim 3.5 reduces to the problem proving the congruence (6.7). Proof of Claim 3.5. Assume that λ is a partition with |λ|  2. For all 1  i  |λ| − 1 let us suppose that Aλ (1) + Aλ (2) + · · · + Aλ (i) ≡ 0

  mod p i

which will be verified later. By dint of Eq. (6.8), for all 1  i  |λ| − 1, we have 

 p|λ|−j −1 |λ|−j −1  Aλ (j ) q k − qp

1j i

≡

 

  p|λ|−i −1 |λ|−i Aλ (j ) q k − q p −1

  mod p |λ|−(i−1) .

1j i

This is because, for all j  i, we have  k p|λ|−j −1  p|λ|−i −1  |λ|−j −1  |λ|−i q ≡ qk − qp − q p −1 by Fermat’s Little Theorem. In particular, if i = |λ| − 1, then

  mod p |λ|−(i−1)

(6.8)

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Y.-T. Oh / Advances in Mathematics 222 (2009) 485–526



 p|λ|−j −1 |λ|−j −1  Aλ (j ) q k − qp

1j |λ|−1



  p−1 Aλ (j ) q k − q p−1



≡

1j |λ|−1

≡0



mod p |λ|



due to Eq. (6.8) together with the congruence (q k )p−1 − q p−1 ≡ 0 (mod p). In conclusion, the proof will be completed if we can show Eq. (6.8). In the following, we will prove Eq. (6.8) using mathematical induction on |λ|. First, we assume that |λ| = 2. Since the only proper subpartition of λ is μ = (1), it follows that   λ Aλ (1) = c(1)aλ (1; p) − 1 = 1 · 1 − 1 ≡ 0 (mod p). 1 p

(6.9)

Now, let us assume that our assertion holds for all partitions λ with the weight 2  |λ|  n − 1, where n  3. Now, let |λ| = n. The case where i = 1 is trivial since Eq. (6.9) is still true. So, we assume that 1 < i  n − 1. Our assertion will be verified in five steps. Step 1. Firstly, we will show that if a subpartition μ of λ has the weight i, then c(μ) ≡ 0

  mod p i−1 .

(6.10)

By Eq. (6.2) we have c(μ) = −



c(η)aμ (η; p)q p

i−j −1

c(η)aμ (η; p)q p

i−j −1

η⊂μ |η|=j 1j i−1

=−



+ qp

i−1 −1

  i−1 − c(1)aλ (1; p) − 1 q p −1

η⊂μ |η|=j 2j i−1

=−



Aμ (j )q p

i−j −1

by (6.6).

(6.11)

1j i−1

Note that Aλ (1) was defined to be c(1)aλ (1; p) − 1. Combining our induction hypothesis with Fermat’s Little Theorem we can derive the congruence:   i−j Aμ (1) + Aμ (2) + · · · + Aμ (j ) q p −1   i−j −1 −1 ≡ Aμ (1) + Aμ (2) + · · · + Aμ (j ) q p

  mod p i

for 1  j  i − 1. Applying this formula repeatedly for j = 1, 2, . . . , i − 1 to Eq. (6.11) yields   c(μ) ≡ − Aμ (1) + Aμ (2) + · · · + Aμ (i − 1) q p−1

  mod p i .

Y.-T. Oh / Advances in Mathematics 222 (2009) 485–526

523

So, we can conclude c(μ) ≡ 0

  mod p i−1 .

Step 2. Secondly, we will show that Aλ (i) ≡ c(λ(i)) (mod p i ), where λ(i) denotes a unique subpartition of λ satisfying (i) λ1 + λ2 + · · · + λr−1 < i  λ1 + λ2 + · · · + λr , (ii) λ(i)j = λj for j = 1, 2, . . . , r − 1, and (iii) λ(i)r = i − (λ1 + λ2 + · · · + λr−1 ). For instance, if λ = (5, 3, 3, 1), then λ(9) = (3, 3, 2, 1). To prove our assertion, we will show that if a subpartition μ of λ has the weight i and is different from λ(i), then aλ (μ; p) is always divisible by p. It can be proved as follows: the formula (6.3) implies that aλ (μ; p) is divisible by 

p

   j 1 μj +1 (λj −μj )

.

But, it can be easily verified that the exponent is always positive unless μ = λ(i). Thus, from Step 1 it follows that c(μ)aλ (μ; p) is divided by p i unless μ = λ(i). As a result,      c(μ)aλ (μ; p) ≡ c λ(i) mod p i Aλ (i) = μ⊂λ |μ|=i

since aλ (λ(i); p) ≡ 1 (mod p) by the formula (6.3). Step 3. By Step 1 again, we have   c λ(i) = −



Aλ(i) (j )q p

i−j −1

1j i−1

    ≡ − Aλ(i) (1) + Aλ(i) (2) + · · · + Aλ(i) (i − 1) q p−1 mod p i     ≡ − Aλ(i) (1) + Aλ(i) (2) + · · · + Aλ(i) (i − 1) mod p i since Aλ(i) (1) + Aλ(i) (2) + · · · + Aλ(i) (i − 1) ≡ 0 (mod p i−1 ) and q p−1 ≡ 1 (mod p). Step 4. Finally, we will show that Aλ (j ) ≡ Aλ(i) (j ) (mod p i ) for all 1  j  i − 1. Let μ be a subpartition of λ of weight j . Then the maximum p-power dividing aλ (μ; p) is given by 

p

   l1 μl+1 (λl −μl )

,

which follows from the formula (6.3). Thus, if the inequality 

  μl+1 λl − μl  i − j + 1

l1

holds, then c(μ)aλ (μ; p) ≡ 0 (mod p i ) since p j −1 |c(μ) (see Step 1).

(6.12)

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Firstly, assume that 2  j  i − 1. In case where μ is not contained in λ(i), the inequality (6.12) always turns out to be true. Indeed, there exists an integer s > r satisfying μ1  μ2  · · ·  μr+1  · · ·  μs > μs+1 = 0, where r is the largest positive integer such that λ(i)r is not zero. Hence we have 

   λl − μl

  μl+1 λl − μl 

1ls−1

l1

     λ1 + λ2 + · · · + λr − μ1 + μ2 + · · · + μs−1    i − j − μs  i − j + 1.

As a result, we can establish the following congruence: 

Aλ (j ) ≡

  mod p i .

c(μ)aλ (μ; p)

(6.13)

μ⊂λ(i) |μ|=j μ does not satisfy inequality (6.12)

On the other hand, for μ ⊂ λ(i) of weight j , it holds that 

      μl+1 λl − μl = μl+1 λ(i)l − μl .

l1

l1

Since λ(i)l = λl for 1  l  r − 1 (refer to Step 2). It implies that 

Aλ(i) (j ) ≡

c(μ)aλ(i) (μ; p)

  mod p i .

(6.14)

μ⊂λ(i) |μ|=j μ does not satisfy inequality (6.12)

Secondly, assume that j = 1. In this case, we already know that Aλ (1) ≡ aλ (1; p) − 1 (mod p i ) (resp. Aλ(i) (1) ≡ aλ(i) (1; p) − 1 (mod p i )), since actually Aλ (1) = aλ (1; p) − 1 (resp. Aλ (1) = aλ (1; p) − 1). If μ ⊂ λ(i) of weight j does not satisfy the inequality (6.12), then N := i − j + 1 −



  μl+1 λl − μl > 0.

l1

Letting the length of μ be s, i.e., μs > 0 and μs+1 = 0, aλ (μ; p) − aλ(i) (μ; p) =p



   l1 μl+1 (λl −μl )

·

s−1   λ − μ l l+1  − μ μ l l+1 p l=1

 ·

λs μs



 − p

λ(i)s μs

. p

Y.-T. Oh / Advances in Mathematics 222 (2009) 485–526

525

In case where s < r, aλ (μ; p) − aλ(i) (μ; p) = 0 since λ(i)l = λl for 1  l  r − 1. Let s = r. In     r is divisible by p N . Note that this case, we claim that μλr − λ(i) μ r



λr μr

− p

p

 λ(i)r μr p 

=

p

r











(1 − p λ(i)r ) · · · (1 − p λ(i)r −μr +1 ) (1 − p λr ) · · · (1 − p λr −μr +1 ) − .   (1 − p)(1 − p 2 ) · · · (1 − p μr ) (1 − p)(1 − p 2 ) · · · (1 − p μr )

For our purpose, it suffices to show that N  λ(i)r − μr + 1. Indeed, this is the case since λ(i)r − μr + 1 − N = λ(i)r − μr + 1 +



  μl+1 λl − μl

l1

    − λ1 + · · · + λr−1 + λ(i)r + μ1 + · · · + μr−1 + μr       =1+ λl − μl μl+1 λl − μl − 1lr−1

 1 (since

μ1

1lr−1

 · · ·  μr

 1).

This shows that aλ (μ; p) − aλ(i) (μ; p) is divisible by p i−j +1 , and thus     c(μ) aλ (μ; p) − aλ(i) (μ; p) ≡ 0 mod p i . So, in view of Eqs. (6.13) and (6.14) we obtain Aλ (j ) ≡ Aλ(i) (j )

  mod p i for 1  j  i − 1.

Step 5. Putting Step 1 throughout Step 4 together, we deduce Aλ (1) + Aλ (2) + · · · + Aλ (i)

    mod p i (by Step 2) ≡ Aλ (1) + Aλ (2) + · · · + Aλ (i − 1) + c λ(i)       ≡ Aλ (1) + · · · + Aλ (i − 1) − Aλ(i) (1) + · · · + Aλ(i) (i − 1) mod p i     = Aλ (1) − Aλ(i) (1) + · · · + Aλ (i − 1) + · · · + Aλ(i) (i − 1)   ≡ 0 mod p i (by Step 4).

This completes the proof.

(by Step 3)

2

Acknowledgment The author would like to express his sincere gratitude to the referee for his/her correction of many errors in the previous version and valuable advices.

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