generalized burnside–grothendieck ring functor and aperiodic ring functor associated with...

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Journal of Algebra 291 (2005) 607–648

www.elsevier.com/locate/jalgebr

Generalized Burnside–Grothendieck ring functoand aperiodic ring functor associated with

profinite groups

Young-Tak Oh1

Korea Institute for Advanced Study, 207-43 Cheongryangri-dong, Dongdaemun-gu, Seoul 130-722Republic of Korea

Received 15 September 2004

Available online 25 March 2005

Communicated by Jean-Yves Thibon

Abstract

For every profinite groupG, we construct two covariant functors∆G andAPG which are equiv-alent to the functorWG introduced in [A. Dress, C. Siebeneicher, The Burnside ring of profigroups and the Witt vectors construction, Adv. Math. 70 (1988) 87–132]. We call∆G the general-ized Burnside–Grothendieck ring functor andAPG the aperiodic ring functor (associated withG).In caseG is abelian, we also construct another functor ApG from the category of commutativrings with identity to itself as a generalization of the functor Ap introduced in [K. VaradarK. Wehrhahn, Aperiodic rings, necklace rings, and Witt vectors, Adv. Math. 81 (1990) 1–29nally, it is shown that there existq-analogues of these functors (i.e.,WG,∆G,APG, and ApG) incaseG is the profinite completion of the multiplicative infinite cyclic groupC. 2005 Elsevier Inc. All rights reserved.

Keywords:Necklace ring; Witt–Burnside ring; Burnside–Grothendieck ring

E-mail address:[email protected].
1 This research was supported by KOSEF Grant # R01-2003-000-10012-0.
0021-8693/$ – see front matter 2005 Elsevier Inc. All rights reserved.doi:10.1016/j.jalgebra.2004.12.022

608 Y.-T. Oh / Journal of Algebra 291 (2005) 607–648

ctedete val-ions,

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1. Introduction

Since E. Witt introduced it in [8,16], the universal ring of Witt vectors has attramany mathematicians for their remarkable connections to other areas such as discruation rings, arithmetic functions, formal group laws, Witt groups, symmetric functand so on. The universal ring of Witt vectorsW(A) associated with a commutative ringAwith 1 = 0 can be characterized by the following properties:

(W1) As a set, it isAN.

(W2) For any ring homomorphismf :A → B, the mapW(f ) : a → (f (an))n≥1 is a ringhomomorphism fora = (an)n≥1.

(W3) The mapswm :W(A) → A defined by

a →∑d|m

dam/dd for a = (an)n≥1

are ring homomorphisms.

In [10] Metropolis and Rota gave a new interpretation of the ring of Witt vectors uthe concept of so called ‘string’, and using it they found an algebraic structure isomorpto W(A) for integral domainsA of characteristic zero. They called this algebraic structhenecklace ring Nr(A) over A. The underlying set ofNr(A) is AN. Addition is definedcomponent-wise and multiplication is defined by

(bc)n =∑

[i,j ]=n

(i, j)bicj

for b = (bn)n≥1, c = (cn)n≥1 ∈ Nr(A). Here,[i, j ] represents the least common multipand (i, j) the greatest common divisor ofi andj . In particular, in [4] the necklace rinNr(Z) over Z was interpreted as the Burnside–Grothendieck ringΩ(C) of isomorphismclasses of almost finite cyclic sets. Here,C denotes the multiplicative infinite cyclic grou

On the other hand, in [3] Dress and Siebeneicher showed that the classical constof Witt vectors can be explained as a special case of the construction of a certaintor using group-theoretical language. In detail, they proved that there exists a covfunctor WG from the category of commutative rings into itself associated with an atrary profinite groupG, and that the classical construction of Witt vectors can be recovby considering the caseG = C, the profinite completion ofC. Furthermore, they alsproved that ifA = Z the corresponding Witt–Burnside ringWG(Z) is isomorphic to theBurnside–Grothendieck ringΩ(G) of isomorphism classes of almost finiteG-spaces. Inview of Dress and Siebeneicher’s work it would be quite natural to question whetherexists a functor equivalent toWG which coincides withΩ(G) whenA = Z. In caseG isfinite, M. Brun [2] constructed such a functor which is indeed the left adjoint of thegebraic functor from the category ofG-Tambara functors to the category of commutat
rings with an action ofG. Another construction was made by the author [11–13] whenWG

Y.-T. Oh / Journal of Algebra 291 (2005) 607–648 609

e

nner.

[9]

is

itionsunctorss

iesnduc-shownd

s withn then we

df

is viewed as a functor from the category of specialλ-rings to the category of commutativrings (see Section 2).

In this paper we remove such restrictions. We construct two functors∆G andAPG

from the category of commutative rings with identity to itself in a purely algebraic maActually these two functors will turn out to be equivalent toWG.

Surprisingly these functors haveq-analogues in some cases. More precisely, inC. Lenart showed that there existq-deformations ofWG(R), NrG(R), andAPG(R) whenG = C for any integerq and for any commutative torsion-free ringR with identity. Weobserve that indeed his results work for every commutative ringR with identity. Based onthis observation we constructq-analoguesWq

G(R), NrqG(R), ∆qG(R), andAPq

G(R) when

G = C for every commutative ringR with identity, and also show that their constructionfunctorial.

This paper is organized as follows. In Section 2, we introduce the basic definand notation needed throughout this paper. In Sections 3 and 4 we construct the f∆G, APG and show that they are equivalent toWG. One of many important propertieof WG(R), ∆G(R), andAPG(R) is that they always come equipped with two familof additive homomorphisms. One is inductions and the other is restrictions. In fact itions are additive, whereas restrictions are ring homomorphisms. In Section 5 it isthat natural equivalences amongWG, ∆G, andAPG are compatible with inductions anrestrictions. Finally, in Section 6 we deal withq-analogues of these functors in caseG = C.

2. Preliminaries

2.1. Witt Burnside rings and necklace rings

Throughout this paper the rings we consider will be commutative associative ring1 = 0, and the subrings will have 1. We begin with the basic definitions and notation ocovariant functorWG introduced by Dress and Siebeneicher. For complete informatiorefer to [3]. LetG be an arbitrary profinite group. For anyG-spaceX and any subgroupU of G defineϕU(X) to be the cardinality of the setXU of U -invariant elements ofXand letG/U denote theG-space of left cosets ofU in G. With this notation Dress anSiebeneicher showed that there exists a unique covariant functorWG from the category ocommutative rings into itself satisfying the following three conditions:

(WG1) For any commutative ringA the ringWG(A) coincides, as a set, with the setAO(G)

of all maps from the setO(G) of open subgroups ofG into the ringA which areconstant on conjugacy classes.

(WG2) For every ring homomorphismh :A → B and everyα ∈ WG(A) one hasWG(h)(α) = h α.

(WG3) For any open subgroupU of G the family of mapsφAU :WG(A) → A defined by

α →∑′

ϕU(G/V ) · α(V )(V :U)

UV G


,

asslled

npfted

,

r. Tos

eit

provides a natural transformation from the functorWG into the identity. HereU V means that the open subgroupU of G is subconjugate toV , i.e., thereexists someg ∈ G with U gVg−1, (V : U) means the index ofU in gVg−1 andthe symbol “

∑′” (and “∏′”, too) is meant to indicate that for each conjugacy cl

of subgroupsV with U V exactly one summand has to be taken. They caWG(A) theWitt–Burnside ringof G overA.

Remark 2.1. Let G be a profinite group andX be an almost finiteG-space (see [3]). Thethe stabilizerGx of x ∈ X in G is closed of finite index inG. Hence it is an open subgrouof G. This is why Dress and Siebeneicher used profinite groups in the construction oWG.However, there is no need forG be restricted to profinite groups. In [5] Graham construca functorFG for every groupG with properties similar toWG, and used it to constructWG.

Much effort has been made to construct a functor which is equivalent toWG (see, e.g.[2,5,11,13,15]). Here, we introduce a functor from the category of specialλ-rings to thecategory of commutative rings which is equivalent toWG if WG is also viewed as a functofrom the category of specialλ-rings to the category of commutative rings (see [11–13])do this, let us review the definitions and notations of specialλ-rings and Adams operationvery briefly. For further information one can refer to [1,7]. Aλ-ring R is a commuta-tive ring with unity with operationsλn :R → R, n = 0,1,2, . . . such that (1)λ0(x) = 1,(2) λ1(x) = x, and (3)λn(x + y) = ∑n

r=0 λr(x)λn−r (y). If t is an indeterminate, then wdefineλt (x) = ∑

n0 λn(x)tn for x ∈ R. By the third condition in the above definition,is straightforward that

λt (x + y) = λt (x)λt (y).

It is well known that for any commutative ringA with unity, we are able to give aλ-ringstructure onΛ(A) := 1+ a1t + a2t

2 + · · · | ai ∈ A.

Definition 2.2 [1,7]. A λ-ring R is said to bespecialif λt :R → Λ(R) is aλ-ring homo-morphism, that is, a ring homomorphism commuting with theλ-operations.

Grothendieck showed that for any commutative ringA with unity Λ(A) becomes aspecialλ-ring for theλ-ring structure mentioned above. LetR be aλ-ring, and we definethenth Adams operationΨ n :R → R by

d

dtlogλt (x) =

∞∑n=0

(−1)nΨ n+1(x)tn for all x ∈ R.

With this notation, let us introduce a functorNrG from the category of specialλ-rings
to the category of commutative rings as follows:


d so

is-

ps

(NrG1) As a set,

NrG(R) =∏′

UG, open

R.

(NrG2) For every specialλ-ring homomorphismf :R → S, one has

NrG(f )(x) = (f (xU )

)′UG

for all x = (xU )′UG

∈ NrG(R).

(NrG3) For every specialλ-ring R, the map

ϕR =∏′

UG, open

ϕRU ,

where

ϕRU : NrG(R) → R, (xV )′V →

∑′

UV G

ϕU(G/V )Ψ (V :U)(xV ),

is a ring homomorphism.

For a specialλ-ring R, NrG(R) is called thenecklace ring ofG overR. We can checkthat the addition inNrG(R) is defined component-wise and the multiplication is definethat theU th component of the product of two sequencesx = (xV )′

V Gandy = (yW )′

WG

is given by

(x · y)U =(∑′

V,W

∑VgW⊆GZ(g,V,W)

Ψ (V :Z(g,V,W))(bV )Ψ (W :Z(g,V,W))(cW )

), (2.1)

where the sum is over

Z(g,V,W) := V ∩ gWg−1

which are conjugate toU in G (see [13]). It was also shown in [13] thatWG(R) is iso-morphic toNrG(R). In proving this fact two families of maps play a crucial role. OneexponentialmapsτU and the other is inductions IndG

U , whereU ranges over open subgroups ofG. More precisely, the mapτU :R → NrU(R) is defined by

r → (MU(r,V )

)′V U

,

where the factorMU(r,V ) is determined in the following way: First, we writer as a sum ofone-dimensional elements, sayr1+r2+· · ·+rm. Then, consider the set of continuous ma
from U to the topological spacer := r1, r2, . . . , rm with regard to the discrete topology


e

s

rlying

.

with trivial U -action. It is well known that this set becomes aU -space with regard to thcompact-open topology via the following standardU -action

(u · f )(x) = f(u−1 · x)

, u, x ∈ U.

Decompose this set into disjointU -orbits and consider its union, say⊔

h Uh, whereh

runs through a system of representatives of this decomposition. After writingU/Uh =⊔1≤i≤(U :Uh) wiUh, whereUh represents the isotropy subgroup ofh, we let

[h] :=(U :Uh)∏

i=1

h(wi).

Clearly, this is well defined sinceh is Uh-invariant. With this notation,MU(r,V ) is givenby

∑h[h], whereh is taken over the representatives such thatUh is isomorphic toU/V .

Note that theUh is isomorphic toU/V if and only if Uh is conjugate toV . IndeedU actson h freely moduloUh, that is,wi · h = wj · h ⇒ i = j . On the other hand, inductionIndG

U : NrU(R) → NrG(R) are defined by

(xV )′V U → (yW )′WG, (2.2)

whereyW is the sum ofxV ’s such thatV is conjugate toW in G. It is clear that IndGU isadditive.

Lemma 2.3 [13]. For every open subgroupU of G and every specialλ-ring R, τU ismultiplicative.

Using inductions IndGU and exponential mapsτU for all open subgroups ofG simulta-neously, the author constructed a mapτR as follows:

τR :WG(R) → NrG(R), α →∑′

U

IndGU τU

(α(U)

).

In [11,13] the mapτR has been called theR-Teichmüller map.

Remark 2.4. It should be noted that the subscript in a function representing the undering, for exampleR in τR andϕR , will be usually omitted if it causes no confusion.

Theorem 2.5 [13, Theorem 3.3].For every specialλ-ring R, τ is a ring isomorphismMoreover, the following diagram is commutative:

WG(R) NrG(R)

τ

ϕΦ whereΦ =∏′

UG, open

φRU .

RO(G)


efor the

ons

Finally, we recall inductions and restrictions associated withWG(R) and NrG(R)

briefly. In the construction of the functorWG in [3] they played a very important rol(see also [11–13]). For inductions for the necklace rings, see Eq. (2.2). Restrictionsnecklace rings are defined to be

ResGU : NrG(R) → NrU(R), (bV )′V G →(∑′

V

∑g

Ψ (V :Z(g,U,V ))(bV )

)′

WU

,

whereg ranges over a set of representatives ofU -orbits ofG/V satisfyingZ(g,U,V ) isconjugate toW in U . Via the isomorphismτ , one can also define inductions and restriction Witt–Burnside rings. More precisely, we have inductionvU :WU(R) → WG(R) givenby

τ−1 IndGU τ,

andfU :WG(R) → WU(R) be restrictions given by

τ−1 ResGU τ

for every open subgroupU of G (see [3,13]).

Example 2.6. Let G = C. ThenWG∼= W. Furthermore,

Φ(q1, q2, . . . , qn, . . .) =(∑

d|ndq

n/dd

)n∈N

,

ϕ(b1, b2, . . . , bn, . . .) =(∑

d|ndΨ n/d(bd)

)n∈N

.

The multiplication ofNrG(R) is given by

(b1, b2, . . . , bn, . . .) · (c1, c2, . . . , cn, . . .) =( ∑

[i,j ]=n

(i, j)Ψ n/i(bi)Ψn/j (cj )

)n∈N

and we have the following exponential map

τ C :R → RN, r → (M(r,n)

),

where

M(r,n) := M ˆ(r, Cn

) = 1 ∑µ(d)Ψ d(r)n/d , n 1.

C nd|n


wing

as a

ble

ulti-upns:

Here,µ represents the classical Möbius inversion function. Finally, we have the folloinductions

IndC

Cr:Nr

C(R) → Nr

C(R), (bn)n∈N → (bn/r )n∈N,

with bn/r := 0 if n/r /∈ N, and restrictions

fr :NrC(R) → Nr

C(R), (bn)n∈N →

( ∑[r,j ]=rn

(r, j)Ψ r/(r,j)(bj )

)n∈N

.

2.2. Binomial rings and necklace rings

A specialλ-ring in whichΨ n = id for all n 1 will be called abinomial ring.

Lemma 2.7 [13]. LetR be a commutative ring with unity which has noZ-torsion and suchthat ap = a modpR if p is a prime. ThenR has a unique specialλ-ring structure withΨ n = id for all n 1.

For example, the rings containingQ (as a subring),Z, andZ(r) satisfy the conditionin Lemma 2.7. Here,Z(r) means the ring of integers localized atr , that is,m/n ∈ Q |(n, r) = 1. Hence they are binomial rings.

Let R be a binomial ring. In this case, we can see that the multiplication (2.1) hsimple form. For two sequencesx = (xV )′V andy = (yW )′W , the U th component of theproduct is given by

(x · y)U :=∑′

V,W

pWV (U)xV · yW , (2.3)

wherepWV (U) represents the number ofVgW ’s in a system of representations of the dou

cosetsVgW ⊆ G such thatZ(g,V,W) is conjugate toU in G.In general, we can construct a new ring from an arbitrary commutative ring with M

plication (2.3). LetxU andyU be indeterminates whereU ranges over every open subgroof G. For every open subgroupU of G let us consider the following systems of equatio

∑′

UV G

ϕU(G/V )sV =∑′

UV G

ϕU(G/V )(xV + yV ),

∑′

UV G

ϕU(G/V )pV =∑′

UV G

ϕU(G/V )xV ·∑′

UV G

ϕU(G/V )yV .

It is clear thatsG = xG + yG andpG = xG · yG. SolvingsU andpU inductively, one canshow that for every open subgroupU of G,

sU = xU + yU ,

pU = ∑′V,WpW

V (U)xV · yW .(2.4)


ing

on 2.2,

ap. But,

Since all the coefficients insU andpU are integers, we obtain the following theorem:

Theorem 2.8. Associated with every profinite groupG there exists a unique functorNrGfrom the category of commutative rings with identity into itself satisfying the followconditions:

(NrG1) As a set

NrG(R) =∏′

UG, open

R.

(NrG2) For every ring homomorphismf :R → S, one has

NrG(f )(x) = (f (xU )

)′UG


∈ NrG(R).

(NrG3) For every ringR the map

ϕ =∏′

UG, open

ϕRU ,

where

ϕRU : NrG(R) → R, (xV )′V →

∑′

UV G

ϕU(G/V )xV ,


Note that the operations ofNrG(R) are given by Eq. (2.4). The functorNrG also comesequipped with inductions and restrictions. Inductions are defined same as in Sectiand restrictions are defined to be

ResGU : NrG(R) → NrU(R), (bV )′V G →(∑′

V

∑g

bV

)′

WU

,

whereg ranges over a set of representatives ofU -orbits ofG/V satisfyingZ(g,U,V ) isconjugate toW in U .

Remark 2.9. (a) If A is a binomial ring, thenNrG(R) coincides withNrG(R). In this case,we can also consider the maps such as the exponential maps and the Teichmüller min general, this is not the case sinceMU(r,V ) cannot be defined. For example, ifG = C,the factor

1 ∑µ(d)rn/d, n 1,

nd|n


ives

r

is a nonsense unlessµ(d)rn/d is divided byn. In this case, we have to considernM(r,n)

instead ofM(r,n), which explains why the notion of the aperiodic rings is needed.(b) If G = C, thenNrG(R) coincides with the necklace algebraNr(R) in [10].(c) If R containsQ as a subring thenϕ is an isomorphism, and ifR is a torsion-free

ring which does not containQ as a subring then it is just injective. However,ϕ is neitherinjective nor surjective in general.

Let us investigateϕ (in the above theorem) in more detail. Since it isR-linear, it can beexpressed as a matrix form. LetP be a partially ordered set consisting of (representatof conjugacy classes of) open subgroups ofG with the partial order such that

V W ⇔ W V.

Consider (and fix) an enumerationVi | 1≤ i, Vi ∈ P of P satisfying the condition

Vi Vj ⇒ i j.

For this enumeration we define the matrixζP by

ζP (V,W) := ϕW(G/V ).

Using the fact thatϕW(G/V ) = 0 unlessV W , we know thatζP is a upper-triangulamatrix with the diagonal elements(NG(Vi) : Vi), the index ofVi in NG(Vi). With thisnotation

ϕ(x) = ζPtx, wherex =

xV1

xV2...

.

As a easy consequences of this expression, we obtain thatϕR is invertible if and only iffor everyU ∈ P the index(NG(V ) : V ) is a unit. When invertible, the inverse ofϕR canbe described as follows: First, letP(U) be a subset ofP consisting of open subgroupsVof G such thatV U . Now, we set

µP(U) = ζ−1P(U),

whereζP (U) is the matrix obtained fromζP by restricting the index toP(U). Then, fora ∈ RO(G)

ϕ−1(a)G

...

ϕ−1(a)U

= µt

P (U)

aG

...

aU

.

That is,

ϕ−1(a)U =∑

µP(U)(V ,U)aV . (2.5)
V U


ns of

lied to

lent

fine

If G is abelian andR is torsion-free, the inverse ofϕ gets much simpler. For an opesubgroupU of G, we letP(U) be a partially ordered set consisting of open subgroupG containingU with the partial order such that

V W ⇔ W V.

Applying Möbius inversion formula for arbitrary posets yields that

ϕ−1(a)U = 1

(G : U)

∑V U

µabP(U)(V ,U)aV

for a = (aU )UG ∈ RO(G) if a belongs to the image ofϕ. Here,µabP(U) is given by the

inverse of the matrixζ abP(U) given by

ζ abP(U)(V ,W) :=

1, if V W,

0, otherwise.

Let R containQ as a subring. Then, Theorem 2.8 implies that fora,b ∈ RO(G)

ϕ−1(ab)U =∑

W,W ′GW∩W ′=U

pW ′W (U)ϕ−1(a)W ϕ−1(b)W ′ . (2.6)

Especially, identity (2.6) comes to us as an interesting and simple formula when appthe groupC. For simplicity, letE(a, n) := ϕ−1(a)

Cn for a ∈ RN, n ∈ N.

Proposition 2.10 (cf. Theorem 3.2 in [10]).Let R be a commutative ring containingQ asa subring. Then fora = (an)n1,b = (bn)n1 ∈ RN the following equation holds:

E(ab, n) =∑

[i,j ]=n

(i, i)E(a, i)E(b, j), whereE(a, n) = 1

n

∑d|n

µ(d)an.

3. Generalized Burnside–Grothendieck ring functor ∆G

In this section, given a profinite groupG, we shall introduce a covariant functor∆G

from the category of commutative rings to itself. Indeed, it will turn out to be equivawith the functorWG in [3]. Let us define it in the following steps:

Case 1. Suppose thatR containsQ (or a field of characteristic zero) as a subring. We de∆G(R) by NrG(R). Theorem 2.5 implies that the map

τ :WG(R) → ∆G(R), α →∑′

IndGU τU

(α(U)

),

UG


g,

ith in

is a ring isomorphism. Here, the mapτU :R → ∆U(R) is defined by

r → (MU(r,U)

)′UG

,

whereMU(r,U) is given associated with the binomial ring structure ofR. CombiningTheorem 2.5 with Eq. (2.5), we obtain that

MG(r,U) =∑V U

µP(U)(V ,U)r(G:V ).

We can also obtain inductions, restrictions, andϕ as in Section 2.2.

Case 2. Suppose thatR does not containQ (or a field of characteristic zero) as a subrinbut it is torsion-free. We denote byRQ the rationalization ofR, i.e., RQ := R ⊗Z Q.Under the natural embedding ofR into RQ we obtain a mapτU |R :R → ∆U(RQ) fromτU :RQ → ∆U(RQ). By Lemma 2.3 this map is multiplicative. CombiningτU |R with theinduction map

IndGU :∆U(RQ) → ∆G(RQ),

we obtain a bijective map fromWG(R) to Im(τRQ|WG(R))

τ (= τR) :=∑′

UG

IndGU τU

∣∣R

.

Letting

∆G(R) := τR

(WG(R)

)it is clear that∆G(R) is a subring of∆G(RQ), and moreover it is isomorphic toWG(R).

Remark 3.1. Very often it is very important to know how∆G(R) looks explicitly. Cer-tainly, it is not just a product of copies ofR. However, as mentioned in Section 2.2,

∆G(R) = NrG(R)

if R has a binomial ring structure.

Next, let us discuss exponential maps and inductions. Restrictions will be dealt wSection 5.

Lemma 3.2. Suppose thatR does not containQ (or a field of characteristic zero) as asubring, but it is torsion-free. Then

(a) Im(τU |R) ⊂ ∆U(R).G
(b) Im(IndU |∆U (R)) ⊂ ∆G(R).


e

on

Proof. (a) The desired result follows from the definition of∆U(R).(b) By [13, Section 3.2], fora = (aW )′

WU∈ WU(RQ),

(τRQ)−1 IndGU τRQ(a) = (pV )′V G,

wherepV is a polynomial with integral coefficients in thoseaW (W an open subgroupUto whichW is sub-conjugate inG). In view of the definition of∆G(R), this implies that ifa = (aW )′

WU∈ WU(R) thenτRQ((pV )′

V G) should be in∆G(R). Thus we complete th

proof. By virtue of Lemma 3.2, we are able to get the following maps

τU :R → ∆U(R) and IndGU :∆U(R) → ∆G(R)

from the mapτU |R and IndGU |∆U (R). The following proposition would be a generalizatiof Theorem 3.2 in [10] corresponding to a profinite groupG.

Proposition 3.3. Suppose thatR is torsion-free. InRQ, for an open subgroupV of aprofinite groupG, andr, s ∈ R, we have

MG(rs,V )∑′

W,W ′pW ′

W (V )MG(r,W)MG

(s,W ′). (3.1)

If G is abelian, formula(3.1) reduces to

MG(rs,V ) =∑W,W ′

W∩W ′=V

(G : W + W ′)MG(r,W)MG

(s,W ′). (3.2)

In particular, if G = C, for every positive integern andr, s ∈ R, we have

M(rs,n) =∑

[i,j ]=n

(i, j)M(r, i)M(s, j), (3.3)

where[i, j ] is the least common multiple and(i, j) the greatest common divisor ofi andj .

Proof. From the multiplicativity ofτG formula (3.1) follows. In caseG is abelian, thenumber of all elements in a system of representations of the double cosetsWgW ′ ⊆ G

where is given by(G : W + W ′). Thus we have formula (3.2). Finally, ifG = C,

iZ ∩ jZ = [i, j ]Z, iZ + jZ = (i, j)Z.

This yields formula (3.3).


-

ing

Finally, from ϕRQ|∆G(R) let us obtain the map

ϕ (= ϕR) :∆G(R) → RO(G).

Indeed this map is well defined since Im(ϕRQ|∆G(R)) ⊂ RO(G), which follows from thefact

ImΦR = ImΦRQ|WG(R) and ΦRQ = ϕRQ τRQ.

Consequently we arrive at the following commutative diagram:

WG(R) ∆G(R) ⊂ ∆G(RQ)

RO(G) ⊂ RQO(G)

τ∼=ϕΦ

ϕRQ

Note thatΦRQ, ϕRQ are ring-isomorphisms, whereasΦ, ϕ are injections but not surjections in general.

Case 3. Finally, we suppose thatR is not torsion-free. In this case, we start by choosa surjective ring homomorphismρ :R′ → R from a torsion free ringR′. For example, wemay takeR′ = Z[R] andρ :Z[R] → R defined by

∑nr · er →

∑nrr,

whereer is the basis element ofZ[R] corresponding tor ∈ R. By the functoriality ofWG

there exists a surjective ring homomorphismWG(ρ) :WG(R′) → WG(R) defined by

(aU )′UG → (ρ(aU )

)′UG

.

Thus it holds

kerWG(ρ) = WG(kerρ),

and which implies that the map

τR′ :WG

(R′)/WG(kerρ) → ∆G

(R′)/∆G(kerρ),

which is induced fromτR′ , is a ring isomorphism. Set

( ′)
∆G(R) := ∆G R /∆G(kerρ) and τ (= τR) := τR′ .


ideatonbetter]

s of

e

g

Remark 3.4. Note that∆G(R′) and ∆G(kerρ) are considered as the images (ins∆G(R′Q)) of WG(R′) andWG(kerρ) . The regrettable point for this construction is thwe cannot use the combinatorics of∆G(R′) to describe the abelian group structure∆G(R) in general. (M. Brun indicated this to me.) But, in some cases there are a fewconstructions. For example, see Theorem 2.5 in caseR is a specialλ ring. And also see [2whenG is a finite group andR is any commutative ring.

By definition∆G(R) andτ are well defined, i.e., they are independent of the choice(R′, ρ) (up to isomorphism).

Let us define exponential maps and inductions. For an open subgroupU of G it is clearthat the induced map

τ U :R′/kerρ → ∆U

(R′)/∆U(kerρ)

from the mapτU :R′ → ∆U(R′) is well defined sinceτU (kerρ) ⊂ ∆U(kerρ). Hence, byabuse of notation, we can regardτU as a map fromR to ∆U(R). In the same manner wcan get inductions

IndGU :∆U(R) → ∆G(R).

By constructionτU is multiplicative, IndGU is additive, and

τ =∑′

UG

IndGU τU .

We can also define a map

ϕ :∆G(R′)/∆G(kerρ) → R′O(G)/(kerρ)O(G)

from ϕR′ :∆G(R′) → R′O(G). RegardingϕR as a map from∆G(R) to RO(G), we cancheck easily thatΦ = ϕ τ . It should be noted that in this caseΦ andϕ is neither injectivenor surjective in general.

Until now we have described∆G(R) for a given commutative ringR. We shall nowdescribe morphisms. Given a ring homomorphismf :A → B, we can define a natural rinhomomorphism

∆G(f ) :∆G(A) → ∆G(B)

via τ , i.e., by

∑′IndG

U τU (xU ) →∑′

IndGU τU

(f (xU )

)(3.4)

UG UG


i-ve

for x = (xU )′UG

∈ WG(A). Actually, if A andB are torsion free, then

∆G(f )(x) = (f (xU )

)′UG

,

which can be verified as follows: First, recall in [13] that

τ(α) =( ∑′

UG

∑′

V U

MU

(α(U),V

))′

WG

,

whereV ranges over open subgroups which are conjugate toW in G. Applying the defin-ition (3.4), we have

f

( ∑′

UG

∑′

V U

MU

(α(U),V

)) =∑′

UG

∑′

V U

MU

(f

(α(U)

),V

)

sincef can be viewed as a ring homomorphism fromAQ to BQ. This says that the defintion (3.4) coincides with the definition (3.4). IfA or B is not torsion free, choose surjectihomomorphismsρ :A′ → A andρ′ :B ′ → B for torsion free ringsA′ andB ′ respectively.With this situation, we have

Lemma 3.5. For x ∈ ∆G(A′)

∆G(f )(x + ∆G(kerρ)

) = y + ∆G

(kerρ′),

wherey ∈ ∆G(B ′) is any element satisfying the conditionf (xV + kerρ) = yV + kerρ′ forevery open subgroupV of G.

Proof. Write

x + ∆G(kerρ) = τA′(z) + ∆G(kerρ)

for somez = (zU )′U ∈ WG(A′).

τB ′ WG(f )((zU + kerρ)UG

) = τB ′((

f (zU + kerρ))UG

) = τB ′(z′) + ∆G

(kerρ′),

wherez′ = (z′U)′U ∈ WG(B ′) is any element satisfying

z′ + ∆G(kerρ) = (f (zV + kerρ)

)V G

.

So, we may take

yV =∑′ ∑′

MU

(z′U ,V ′),

UG V ′U


ate

e

ion.be

ring.g

profi-

for every open subgroupV of G. Here,V ′ ranges over open subgroups which are conjugto V in G. Thus, we have

f (xV + kerρ) = f

( ∑′

UG

∑′

V ′U

MU

(zU ,V ′) + kerρ

)

=∑′

UG

∑′

V ′U

MU

(f (zU + kerρ),V ′)

=∑′

UG

∑′

V ′U

MU

(z′U + kerρ,V ′)

=∑′

UG

∑′

V ′U

MU

(z′U ,V ′) + kerρ

= yV + kerρ. By definition of ∆G(f ), ∆G(f ) = τB WG(f ) τA

−1. Consequently, we have thfollowing theorem:

Theorem 3.6. The functor∆G is equivalent to the functorWG.

Remark 3.7. The functor∆G provides answer to the question proposed in IntroductHowever, our construction of it does depend onWG (see in Cases 2 and 3). It wouldvery worthwhile to find a definition not depending onWG.

4. Aperiodic ring functor APG

In [15] Varadarajan and Wehrhahn introduced the notion ofthe aperiodic ringand in-vestigated its properties and relations with the ring of Witt vectors over a torsion-freeThe aperiodic ring Ap(R) over a commutative ringR can be characterized by the followinproperties:

(Ap1) As a set, it isRN.(Ap2) For any ring homomorphismf :R → S, the map Ap(f ) : a → (f (an))n1 is a ring

homomorphism fora = (an)n1 ∈ Ap(R).(Ap3) The mapsϕm : Ap(R) → R defined by

a →∑d|m

ad for a = (an)n1

are ring homomorphisms.

In this section we generalize the construction of the aperiodic ring to an arbitrary
nite group. To begin with, we deal with the case whereG is an abelian profinite group.


f

.s

Associated with an abelian groupG, let us introduce a functor ApG from the category ocommutative rings to itself as follows:

(ApG1) As a set, it is ∏′

UG, open

R.

(ApG2) For every ring homomorphismf :R → S, one has

ApG(f )(x) = (f (xU )

)′UG


∈ ApG(R).

(ApG3) For every ringR the map

ϕ =∏′

UG, open

ϕRU ,

where

ϕRU : ApG(R) → R, (xV )′V →

∑′

UV G

xV ,


It is not difficult to show that ApG is a functor. Indeed the addition of ApG(R) is definedcomponent-wise, and its multiplication is defined so that forx = (xV )′V andy = (yW )′W inApG(A), theU th component ofx · y is given by

(x · y)U :=∑′

V,WG

∑VgW⊆GZ(g,V,W)

xV yW ,

whereg runs through a system of representations of the double cosetsUgV ⊆ G, andZ(g,V,W) runs overG-conjugates toU . In particular, by considering the caseG = C,we can recover that classical aperiodic ring.

Now, we remove the condition thatG is abelian. LetG be an arbitrary profinite groupAssociated withG, we introduce a functorAPG from the category of commutative ringto itself, which will turn out to be equivalent toWG. Given a ringA, we callAPG(A) theaperiodic ring ofG overR. DefineAPG(R) in the following steps:

Case 1. Suppose thatR is an arbitrary commutative ring containingQ (or a field of char-acteristic zero) as a subring. In this case,APG(R) is defined as follows:

(APG1) As a set, it is ∏′R.

UG, open


ee need

ve

(APG2) Addition is defined component-wise.(APG3) Multiplication is defined so that forx = (xV )′V andy = (yW )′W in APG(R), the

U th component ofx · y is given by

(x · y)U :=∑′

V,WG

∑VgW⊆GZ(g,V,W)

cVU (g)xV yW , (4.1)

whereg runs through a system of representations of the double cosetsUgV ⊆ G,andZ(g,V,W) runs overG-conjugates toU , and the coefficientcV

U (g) is givenby

cVU (g) = (G : Z(g,U,V ))

(G : U)(G : V ).

Remark 4.1. Observe that ifG is abelian, then in Eq. (4.1)

∑VgW⊆GZ(g,V,W)

cVU (g) = (G : U ∩ V )(G : U + V )

(G : U)(G : V )= 1. (4.2)

For example, letG = C. Then the multiplication ofAPG(R) is given by

(ab)n =∑

[i,j ]=n

aibj for a,b ∈APG(R).

If follows thatAPG(R) exactly coincides with ApG(R). But, for non-abelian groups, thcoefficients in Eq. (4.1) do not always have integral values. This is the reason why wthe conditionR containsQ.

Rather than verify thatAPG(R) is indeed a commutative ring directly, we shall proit indirectly. To do this, for every conjugacy class of open subgroupsU of G, let us definea mapϕR

U :APG(R) → R by

x →∑′

UV G

1

(G : V )ϕU(G/V )xV

for x = (xU )′U ∈ APG(R). Now, we set

ϕ (= ϕR) =∏′

U

ϕRU :∆G(R) → RO(G)

by ϕ(x)(U) = ϕRU(x) for all x ∈ ∆G(R).


-

Proposition 4.2 (cf. [15]). For every commutative ringR containingQ (or a field of char-acteristic zero) as a subring, we have

(a) ϕ is an isomorphism of the additive groupAPG(R) ontoRO(G).(b) For anyx,y ∈ APG(R),

ϕ(xy) = ϕ(x)ϕ(y).

Proof. (a) It is clear thatϕ is a homomorphism of additive groups. Letx = (xU )′U ∈APG(R) satisfyϕ(x) = 0, where0 is the zero element ofRO(G). Then for every conjugacy class of open subgroupsU of G, ϕR

U(x) = 0. If U = G, thenxG = 0. Now, assumethatxV = 0 for all V such that(G : V ) < (G : U). Then, from

∑′

UV G

1

(G : V )ϕU(G/V )xV = 0

we can getxU = 0. Thusx = 0 ∈APG(R), and kerϕ = 0.Next, we will show thatϕ is surjective. For anya = (aU )′U ∈ RO(G) we want to find an

elementx = (xU )′U ∈ APG(R) satisfying

∑′

UV G

1

(G : V )ϕU(G/V )xV = aU

for every conjugacy class of open subgroupsU of G. If U = G, thenxG = aG. Let us usemathematical induction on the index. Assume that we have foundxV for all V ’s such that(G : V ) < (G : U). Then, since

1

(G : U)ϕU(G/U)xU = aU −

∑′

UV GU =V

1

(G : V )ϕU(G/V )xV ,

xU is determined by the assumption. This completes the proof of (a).(b) For anyx = (xU )′U ,y = (yV )′V ∈ APG(R),

ϕRU(x) · ϕR

U(y) =∑′

UV G

1

(G : V )ϕU(G/V )xV ·

∑′

UWG

1

(G : W)ϕU(G/W)yW

=∑′

UV G

UWG

xV yW

(G : V )(G : W)ϕU(G/V )ϕU(G/W).

Let eU ∈ RO(G) be the element given byeU = (δU,V )′V G

. Then,


two

ropo-

,

ong

ϕRU(xy) = ϕR

U

(∑′

V,W

∑VgW⊆G

cVU (g)xV yWeZ(g,V,W)

)

=∑′

V,W

∑VgW⊆G

UZ(g,V,W)

cVU (g)xV yWϕR

U(eZ(g,V,W))

=∑′

V,W

xV yW

∑VgW⊆G

UZ(g,V,W)

cVU (g)

(G : Z(g,V,W))ϕU

(G/Z(g,V,W)

)

=∑′

V,W

xV yW

(G : V )(G : W)

∑VgW⊆G

UZ(g,V,W)

ϕU

(G/Z(g,V,W)

).

Applying the fact thatϕRU is a ring homomorphism, it is immediate that

ϕRU(xy) = ϕR

U(x)ϕRU (y).

As easy but very important results of Proposition 4.2, we obtain the followingcorollaries:

Corollary 4.3. Suppose thatR is an arbitrary commutative ring containingQ (or a fieldof characteristic zero) as a subring. ThenAPG(R) is a commutative ring.

Corollary 4.4. If G is abelian, then for every commutative ringR,

ϕ : ApG(R) → RO(G)

is a ring isomorphism(see the remark containing the identity(4.2)).

Proof. SinceG is abelian, we haveϕU(G/U) = (G : U) for every open subgroupU of G.Using this fact, we can notice that no non-integer coefficients appear in the proof of Psition 4.2. Therefore we can conclude that Proposition 4.2 holds for ApG(R) over everycommutative ringR.

As ϕ−1 does,ϕ−1 :RO(G) → ApG(R) has a very simple form ifG is abelian. In detail

ϕ−1(a)U =∑V U

µabP(U)(V ,U)aV

for a = (aU )UG ∈ RO(G). This observation provides us many interesting relations amaU ,bV ’s when applied to the identity

ϕ−1(ab)U∑

′ϕ−1(a)Wϕ−1

R (b)W ′
W,W GW∩W ′=U


10.

w

aps.

for a,b ∈ RO(G). For example, takingG = C, we can get an analogue of Proposition 2.For simplicity, letF(a, n) := ϕ−1(a)

Cn for a ∈ RN, n ∈ N.

Proposition 4.5 (cf. [15]). For every commutative ringR, we have the formula

F(ab, n) =∑

[i,j ]=n

F (a, i)F (b, j),

wherea = (an)n1,b = (bn)n1 ∈ RN and

F(a, n) =∑d|n

µ(d)an.

To show that∆G(R) is isomorphic toAPG(R) let us introduce a map

θ (= θR) : ∆G(R) →APG(R)

defined by

(xU )′UG → ((G : U)xU

)′UG


.

Proposition 4.6. Suppose thatR is an arbitrary commutative ring containingQ (or a fieldof characteristic zero) as a subring. Then the mapθ is a ring isomorphism andϕ = θ ϕ.

Proof. Sinceϕ andϕ are ring isomorphismsθ is a ring isomorphism. Now, let us shoϕ = ϕ θ . For anyx = (xU )′

UG∈ ∆G(R), we get

ϕRU θ(x) =

∑′

UV G

1

(G : V )ϕU(G/V )(G : V )xV =

∑′

UV G

ϕU(G/V )xV = ϕRU (x).

This completes the proof.As ∆G(R) does,APG(R) comes equipped with exponential maps and induction m

First, let us investigate exponential maps. AnexponentialmapΥ U :R → APU(R) is de-fined by

r → (SU(r,V )

)′V U

,

whereSU(r,V ) := (U : V )MU(r,V ) for all open subgroupsV of U .

Proposition 4.7. Suppose thatR is an arbitrary commutative ring containingQ (or a field
of characteristic zero) as a subring. ThenΥ U is multiplicative.


Proof. Observe that

Υ U = θ τU . (4.3)

So, the desired result follows from the fact thatθ andτU are multiplicative. For every conjugacy class of open subgroupsU of G, inductions

IndGU :APU(R) →APG(R)

are defined by

(xV )′V U → (yW )′WG,

whereyW is the sum of(G : U)xV ’s such thatV is conjugate toW in G.

Lemma 4.8. Suppose thatR is an arbitrary commutative ring containingQ (or a field ofcharacteristic zero) as a subring. Then,

θ IndGU = IndG

U θ. (4.4)

Proof. For anyx = (xV )′V U

∈ ∆U(R),

θ IndGU(R)(x) = (yW )′WG,

whereyW = ∑′V (G : W)xV . On the other hand,

IndGU θ(x) = IndG

U

(((U : V )xV

)′V U

) =(∑′

V

(G : U)(U : V )xV

)′

WG

=(∑′

V

(G : W)xV

)′

WG

= (yW )′WG.

This completes the proof.Composingτ with θ , we get a ring homomorphism fromWG(R) to APG(R). Let

γ (= γR) := θ τ . Note that its explicit form is

γ =∑′

UG

IndGU Υ U

since


g,

.

is

-

,

θ ∑′

UG

IndGU τU =

∑′

UG

IndGU θ τU (by Lemma 4.8)

=∑′

UG

IndGU Υ U (by (4.3)).

Case 2. Suppose thatR does not containQ (or a field of characteristic zero) as a subrinbut it is torsion-free. In this case, we defineAPG(R) as follows:

Letting

APG(R) := γRQ

(WG(R)

),

then clearlyAPG(R) is a subring ofAPG(RQ). As in Case 1, let us define

θ (= θR) :∆G(R) → APG(R) by (aU )′UG → ((G : U)aU

)′UG

.

That is,θ = θRQ|∆G(R). In view of Eq. (4.3) and identity (4.4),θ is clearly well definedThat is,

θRQ

(∆G(R)

) ⊂ APG(R).

Proposition 4.9. If R does not containQ (or a field of characteristic zero) as a subring,but it is torsion-free, the mapθ is a ring isomorphism. Moreover, the following diagramcommutative:

∆G(R) APG(R) ⊂ APG(RQ)

RO(G) ⊂ RQO(G)

θ∼=

ϕRQ|APG(R)ϕ

ϕRQ

Proof. In view of the definition ofAPG(R) it is clear thatθ is a ring isomorphism. Moreover, we already know thatϕ comes fromϕRQ|∆G(R) and

ϕRQ = ϕRQ θRQ

(by Proposition 4.6). Hence

ϕRQ

(APG(R)

) ⊂ RO(G),

and the commutativity is immediate.DenoteϕRQ|APG(R) by ϕ (= ϕR) :APG(R) → RO(G). Similarly, by abuse of notation

define exponential maps and inductions by restriction:


-

the

Υ U = θ τU , IndGU :APU(R) →APG(R).

Lemma 4.10. Suppose thatR does not containQ (or a field of characteristic zero) as asubring, but it is torsion-free. ThenIndG

U is well defined, i.e.,

Im(IndG

U |APG(R)

) ⊂ APG(R).

Proof. Note that

IndGU θRQ

(∆U(R)

) = IndGU

(APU(R)

)(by Proposition 4.9)

= θRQ IndGU

(∆U(R)

)(by Lemma 4.8)

⊂ θRQ

(∆G(R)

)(by Lemma 3.2)

⊂ APG(R). From the multiplicativity ofΥ U we have an analogue of Proposition 3.3.

Proposition 4.11. Assume thatR is torsion-free. InRQ, for r, s ∈ R and every open subgroupV of G, we have

SG(rs,V ) =∑′

W,W ′

∑WgW ′⊆G

Z(g,W,W ′)

cW ′W (g)SG(r,W)SG

(s,W ′) (4.5)

where the sum is overZ(g,W,W ′)’s which are conjugate toV . If G is abelian, then theidentity(4.5) reduces to

SG(rs,V ) =∑

W,W ′GW∩W ′=V

SG(r,W)SG

(s,W ′). (4.6)

In particular, if G = C, then(4.6) reduces to the following simple form

SG(rs, n) =∑

[i,j ]=n

SG(r, i)SG(s, j) for all n ∈ N. (4.7)

Proof. Formula (4.5) follows from the multiplicativity ofΥ U . If G is abelian, by applyingformula (4.2) to formula (4.5) we get formula (4.6). Finally, formula (4.7) follows fromfact iZ ∩ jZ = [i, j ]Z.

Finally, settingγ (= γR) := θ τ , then it is an isomorphism fromWG(R) to APG(R).


sm

toin

Case 3. Finally, we suppose thatR is not torsion-free. For a surjective ring homomorphiρ :R′ → R from a torsion free ringR′, let us consider the following diagram

∆G(R′)θR′∼= APG(R′)

∆G(R) = ∆G(R′)/∆G(kerρ)θR′∼= APG(R′)/APG(kerρ).

Let

APG(R) := APG

(R′)/APG(kerρ) and θ (= θR) := θR′ .

Since∆G(R) is well defined,APG(R) is well defined, too. Finally, let us discuss howobtain exponential maps and inductions, and the mapϕ. In this case, one way to obtaexponential mapsΥ G is to consider the map

Υ G : R′/kerρ → APG

(R′)/APG(kerρ),

which is induced fromΥ G :R′ → APG(R′), and the other is to let

Υ U := θU τU . (4.8)

Of course they coincide with each other. Similarly, as for inductions, the mapIndGU(R′):

APU(R′) → APG(R′) induces the map

IndGU

(R′) :APU

(R′)/APU(kerρ) → APG

(R′)/APG(kerρ).

By abuse of notation, let

IndGU := IndG

U(R) = IndGU

(R′).

As a final step, from the mapϕR′ :APG(R′) → R′O(G), let us derive a map

ϕR′ :APG(R)/APG(kerρ) → R′O(G)/(kerρ)O(G).

Let ϕ (= ϕR) := ϕR′ . Regardingϕ as a map fromAPG(R) to RO(G), then ϕ = ϕ θ ,which indeed follows from the definition ofϕ. However, in this case,ϕ andϕ are neitherinjective nor surjective in general.

Lemma 4.12. LetR be any commutative ring with identity. Then we have

θ IndGU = IndG

U θ (4.9)


s

from

s

of

and

θ τ =∑′

UG

IndGU Υ U . (4.10)

Proof. First, let us prove the identity (4.9). IfR is torsion-free, the identity (4.4) justifiethe desired result. Now, assume thatR is not torsion-free. Then, for anyx = (xV )′

V U∈

∆U(R′),

θ IndGU

(x + ∆U(kerρ)

) = θ(IndG

U(x) + ∆G(kerρ)) = θR′ IndG

U(x) +APG(kerρ).

On the other hand,

IndGU θR

(x + ∆U(kerρ)

) = IndGU

(θR′(x) +APU(kerρ)

)= IndG

U θR′(x) +APG(kerρ).

By (4.4)

θR′ IndGU(x) = IndG

U θR′(x).

Thus we proved the identity (4.9). On the other hand, the identity (4.10) followsEqs. (4.8) and (4.9).

Finally, setγ := θ τ .Now, we are ready to describe morphisms. Given a ring homomorphismf :A → B, we

can define a ring homomorphism

APG(f ) :APG(A) → APG(B)

via γ , i.e., by

∑′

UG

IndGU Υ U(xU ) →

∑′

UG

IndGU Υ U

(f (xU )

)

for x = (xU )′UG ∈ WG(R). As shown in Section 3, ifA andB are torsion free, it holdthat

APG(f )(x) = (f (xU )

)′UG

.

If A orB is not torsion free, choose surjective homomorphismsρ :A′ → A andρ′ :B ′ → B

for torsion free ringsA′ andB ′ respectively. With this situation, we have an analogue
Lemma 3.5.


t

Lemma 4.13. For x ∈ APG(A′)

APG(f )(x +APG(kerρ)

) = y +APG

(kerρ′),

wherey ∈APG(B ′) is any element satisfying

f (xV + kerρ) = yV + kerρ′

for every open subgroupV of G.

By definition of∆G(f ) we haveAPG(f ) = θB ∆G(f ) θA−1. In other words, the

following diagram is commutative:

∆G(A)

θA

∆G(f )∆G(B)

θB

APG(A)APG(f )

APG(B).

Summarizing our discussion until now, we can state the following result:

Theorem 4.14. The functorAPG is equivalent to the functor∆G. Hence, it is equivalento the functorWG.

5. Inductions and restrictions

In the previous sections we introduced inductions

vU :WU(R) → WG(R),

IndGU :∆U(R) → ∆G(R),

IndGU :APU(R) → APG(R).

We also introduced restrictions

fU :WG(R) → WU(R)

in Section 2. In this section we define restrictions

ResGU :∆G(R) → ∆U(R), ResGU :APG(R) → APU(R),


nsfor-on

ictions

e

g,

], for

and then show that these restrictions and inductions are compatible with natural tramationsτ , θ , andγ . Finally, we remark that there exist inductions and restrictionsRO(G):

νU :RO(U) → RO(G) and FU :RO(G) → RO(U),

and that they are compatible with natural transformations, too. First, we define restrResGU :∆G(R) → ∆U(R) in the following steps:

Case 1. Suppose thatR containsQ (or a field of characteristic zero) as a subring. Defin

ResGU :∆G(R) → ∆U(R) by (bV )′V G →(∑′

V

∑g

bV

)′

WU

,

whereg ranges over a set of representatives ofU -orbits ofG/V satisfyingZ(g,U,V ) isconjugate toW in U (see Section 2).

Case 2. Suppose thatR does not containQ (or a field of characteristic zero) as a subrinbut it is torsion free. From the map ResG

U(RQ)|∆G(R) let us obtain the map

ResGU := ResGU(R) :∆G(R) → ∆U(R).

Actually the following lemma implies that this map is well defined.


Im(ResGU |∆G(R)

) ⊂ ∆U(R).

Proof. In essence the proof is same as that of Lemma 3.2(b). By [13, Section 3.2a = (aW )′

WU∈ WG(RQ)

(τRQ)−1 ResGU τRQ(a) = (qV )′V U ,

whereqV is a polynomial with integral coefficients in thoseaWi1≤ i ≤ k (where,Wi : 1≤

i ≤ k is a system of subgroups ofG containing a conjugate ofU ). In view of the definitionof ∆U(R), this implies that ifa = (aW )′

WU∈ WG(R) thenτRQ((qV )′

V U) should be in

∆U(R). Thus we complete the proof.Case 3. Suppose thatR is not torsion free. For a surjective ring homomorphismρ :R′ → R

from a torsion free ringR′, the map ResGU(R′) :∆G(R′) → ∆U(R′) induces

ResGU(R′) :∆G

(R′)/∆G(kerρ) → ∆U

(R′)/∆U(kerρ).

Let


e

g,

ResGU := ResGU(R) = ResGU(R′).

Next, we define restrictions associated with the functorAPG

ResGU :APG(R) → APU(R)

as follows:

Case 1. Suppose thatR containsQ (or a field of characteristic zero) as a subring. Defin

ResGU :APG(R) → APU(R) by (bV )′V G →

(∑′

V

∑g

(U : W)

(G : V )bV

)′

WU

,

whereg ranges over a set of representatives ofU -orbits ofG/V satisfyingZ(g,U,V ) isconjugate toW in U .

Lemma 5.2. Suppose thatR is an arbitrary commutative ring containingQ (or a field ofcharacteristic zero) as a subring. Then

θ ResGU = ResGU θ.

Proof. For anyx = (xV )′V G

∈ ∆G(R),

θ ResGU(x) =(

(U : W)∑′

V

∑g

xV

)′

WU

,

whereg ranges over a set of representatives ofU -orbits ofG/V such thatZ(g,U,V ) isU -conjugate toW . On the other hand,

ResGU θ(x) = ResG

U

(((G : V )xV

)′V G

)=

(∑′

V

∑g

(U : W)

(G : V )(G : V )xV

)′

WU

=(

(U : W)∑′

V

∑g

xV

)′

WU

.

This completes the proof.Case 2. Suppose thatR does not containQ (or a field of characteristic zero) as a subrinor it is torsion free. The following lemma implies that the map

ResGU :APG(R) → APU(R)

is well defined.


sm

n inlity (c)

andbtained


Im(ResG

U |∆G(R)

) ⊂ APU(R).

Proof. By virtue of Lemma 5.2 we can apply the method of the proof of Lemma 4.10.Case 3. Finally, suppose thatR is not torsion free. For a surjective ring homomorphiρ :R′ → R from a torsion free ringR′, the mapResG

U(R′) :APG(R′) → APU(R′) in-duces

ResGU(R′) :APG(R′)/APG(kerρ) → APU

(R′)/APU(kerρ).

Let

ResGU := ResG

U(R) = ResGU(R′).

Proposition 5.4. With this notation, we have

(a) ResGU τ = τ fU .(b) ResG

U θ = θ ResGU .(c) ResG

U γ = γ fU .

Proof. The proof is similar with Proposition 5.5. By definition ofAPG, ResGU , andResGU

we may assume thatR containsQ as a subring. In this case, equality (a) was prove[13, Section 2.3] and equality (b) was proven Lemma 5.2. On the other hand, equafollows from (a) and (b) (by definition ofγ ). Proposition 5.5. For every commutative ringR with identity, the mapsτ, θ andγ preserveinduction maps. That is, for every open subgroupU of G we have

(a) IndGU τ = τ vU .

(b) IndGU θ = θ IndG

U .(c) IndG

U γ = γ vU .

Proof. We may assume thatR containsQ as a subring since∆G(R), IndGU , andIndG

U areconstructed fromRQ if R is torsion free and from a torsion free ringR′ if R is not torsionfree. So, assume thatR ⊃ Q. In this case, assertion (a) was proven in [13, Section 2.3]assertion (b) was proven in Lemma 4.8. On the other hand, assertion (c) can be oby combining (a) with (b) (by definition ofγ ).

Finally, let us consider inductions and restrictions onRO(G)

νU :RO(U) → RO(G), FU :RO(G) → RO(U).


at

pen

tssis

Note that ifR is torsion-free, thenϕ is one-to-one. Therefore in this case we can defineνU

andFU via ϕ on the image ofϕ, i.e.,

νU := ϕ IndGU ϕ−1, FU := ϕ ResGU ϕ−1.

Note that this method is no more valid unlessR is torsion-free. However, we claim ththese maps are still valid even in caseR has torsion.

Lemma 5.6. Let R be a commutative ring containingQ (or a field of characteristic zero)as a subring. For anyb = (bV )′

V G∈ RO(G), let us write

ϕ−1(b) = (aV )′V G.

Then for every open subgroupV of G, aV can be written as

1

ϕV (G/V )

∑′

V WG

cWbW

for somecW ∈ Z.

Proof. In order to prove this we shall use the mathematical induction on index(G : V ).First, note thataG = bG. Now, we assume that the desired assertion holds for all osubgroupsW of G such that(G : W) < (G : V ). From

∑′

V WG

ϕV (G/W)aW = bV ,

we know that

aV = 1

ϕV (G/V )

(bV −

∑′

V WGV W

ϕV (G/W)aW

).

Note thatϕW(G/W) dividesϕV (G/W) for everyV W since the group Aut(G/W) isacting freely on the set ofG-morphisms fromG/V to G/W and the number of elemenof this set equalsϕV (G/W) (see [3]). Combining these facts with induction hypotheyields our assertion. Proposition 5.7. Let R be a commutative ring containingQ (or a field of characteristiczero) as a subring. For anyb = (bV )′

V U∈ RO(U) we let

ϕ IndGU ϕ−1(b) = (tV )′V G.


at

l-

here

Then for every open subgroupV of U , tV is a polynomial inbW : V W (in G) andW U

with integer coefficients. UnlessV is not an open subgroup ofU , thentV is zero.

Proof. Writing

ϕ−1(b) = (aV ′)′V ′U ,

then

IndGU ϕ−1(b) = (αV )′V G,

whereαV is the sum ofaV ′ ’s such thatV ′ is conjugate toV in G. Therefore

tV =∑′

V WG

ϕV (G/W)αW . (5.1)

Now, we claim thatϕW(U/W) divides ϕV (G/W). To show this we observe thϕW(G/W) is (NG(W) : W), the index ofW in the normalizer ofW in G, andNU(W) =NG(W) ∩ U . This implies thatϕW(U/W) dividesϕW(G/W). On the other hand, we aready knows thatϕW(G/W) dividesϕV (G/W). Hence we can conclude thatϕW(U/W)

dividesϕV (G/W). Now, apply Lemma 5.6 to Eq. (5.1) to get our assertion. In case wV is not an open subgroup ofU , αW = 0 for all V W . ThereforetV is zero.

Proposition 5.7 has an amusing consequence that we can define inductionsνU for arbi-trary commutative rings using polynomialstV ’s.

Remark 5.8. In [14] the explicit form ofνU was computed. Actually it is given by

(bV )′V U → (cW )′WG, wherecW =∑′

V UV is conjugate toW in G

[NG(W) : NU(V )

]bV .

By definition ofνU it is straightforward that

νU ϕ = ϕ IndGU .

On the other hand, we defineFU :RO(G) → RO(U) by

(bV )′V G → (cW )′WU wherecW :=

bV if W is conjugate toV in G,

0 otherwise.

Indeed ifR containsQ as a subring, then we can verify that

FU = ϕ ResGU ϕ−1.


t

func-

iebung

und inp

thout

s on

g

Hence we can conclude that (see [11]):

FU ϕ = ϕ ResGU .

6. q-deformation of the functor W and its equivalent functors

Let R be a commutative ring with unity. In this section we show there exisq-deformations ofW(R), Nr(R), andAP(R) (the subscriptG = C will be omitted) whereqranges over the integers. Furthermore, we also show that their constructions will betorial and they are equivalent. More precisely, we prove the following theorem:

Theorem 6.1. The functorsWq , C(Fq, ·), ∆q , andAPq are equivalent for allq ∈ Z. Andeach natural equivalence among them is compatible with the Frobenius and Verschoperators.

For a torsion-free ringR, we letWF (R) be the group of Witt-vectors overR associatedwith the formal group lawF andC(F,R) be the group of curves inF . It is well knownthatWF (R) andC(F,R) are values of functors from formal group laws overR to groups,and that the Artin–Hasse exponential map

HF :WF (R) → C(F,R), α →∑n1

Fαnt

n

gives a natural equivalence of functors (see [6,9]). On the other hand, C. Lenart fo[9] that it is possible to endowWF (R) with multiplicative structure for some formal groulaws, more precisely for

Fq(X,Y ) := X + Y − qXY, q ∈ Z.

To avoid confusion we shall adapt and use all notations and definitions in [9] wichanges (and without any explanations). Fix a formal group lawFq for someq ∈ Z, anddefine a mapλ :Gh(R) → Gh(R) by λ(x) = (nxn)n1 for all x = (xn)n1. With thisnotation, C. Lenart proved the following facts on the multiplicative structure ofWq(R)

(the superscriptq will be used instead ofFq ).

Lemma 6.2 (Lenart [9]).

(a) There is a ring structure onWq(Z) such that the restriction ofλ wq is a ring homo-morphism. The mapHq provides an isomorphic ring structure onC(Fq,Z).

(b) Let R be a torsion free commutative ring with identity. There are ring structureNrq(R), Wq(R), andC(Fq,R) such that the restrictions ofλ gq andλ wq are ringhomomorphism, and the restriction ofHq is a ring isomorphism. The correspondin
Frobenius and the Verschiebung operators are endomorphisms of these rings.


re,

a:

(c) If R is aQ-algebra orR = Z or R = Z(r), there are restrictions of the mapsT q andcq

to the ringsNrq(R),Wq(R), andC(Fq,R), and they are isomorphisms. Furthermothe Verschiebung and Frobenius operators commutes with the mapsT q andcq .

(d) The statements in(b) also holds ifq is viewed as an indeterminate, andR is an algebraover the ring of numerical polynomials inq (in particular, if it coincides with this ring).

Actually Lemma 6.2 implies a little more. LetA = Z[a1, a2, . . . ;b1, b2, . . .], and leta = (an)n1 andb = (bn)n1. Define

Φq :Wq(A) → Gh(A), x →(∑

d|ndqn/d−1x

n/dd

)n1

.

Let us solve the following equations

Φq(sq) = Φq(a) + Φq(b),

Φq(pq) = Φq(a) · Φq(b),

Φq(ιq) = −Φq(a),

where sq = (sqn )n1, pq = (p

qn)n1, and ιq = (ι

qn)n1. Applying Lemma 6.2(a) and

M. Hopkins’s proof (see [9, p. 727]) to these equations, we obtain the following lemm

Lemma 6.3 (Lenart [9]). Fix q ∈ Z. Thensqn ,p

qn are polynomials inad, bd ’s for d | n

with integral coefficients. Similarly,ιqn is a polynomial inad ’s for 1 ≤ d ≤ n with integralcoefficients for everyn 1.

Lemma 6.3 has an amusing consequence that we can defineWq as a functor from thecategory of commutative rings with identity to the category of commutative rings.

Remark 6.4. In general,Wq(R) does not have the identity unlessR containsQ as a sub-ring. Indeed, if exists, the identity can be determined inductively by letting∑

d|ndqn/d−1a

n/dd = 1 for all n.

Forq ∈ Z, let us defineWq(R) for any commutative ringR with identity as follows:

(Wq1) As a set, it isRN.(Wq2) For any ring homomorphismf :R → S, the mapWq(f ) : a → (f (an))n1 is a

ring homomorphism fora = (an)n1.(Wq3) The mapΦq :Wq(R) → Gh(R) defined by

a →(∑

d|ndqn/d−1a

n/dd

)n1

for a = (an)n1



ls

a-

dtor-

ame

[9]):

On the other hand, since Artin–Hasse mapHq is determined by universal polynomiawith integer coefficients

Hq(x) = x1t + x2t2 + (x3 − qx1x2)t

3 + (x4 − qx1x3)t4 + · · · ,

we can endowC(Fq,R) with the ring structure viaHq . Consequently for every commuttive ring R with identity we get a ring isomorphismHq :Wq(R) → C(Fq,R). Frobeniusand Verschiebung operators can be also definedWq(R) andC(Fq,R), which are preserveby Hq since they are also given by universal polynomials with integer coefficients insion free cases.

As we did in Sections 3 and 4, we can construct functors∆q andAPq which areequivalent toWq (so toC(Fq, ·)). Indeed the process of their construction is exactly sas that of∆G andAPG. First, we define∆q(R) as follows:

Case 1. If R containsQ as a subring, then we let

∆q(R) = Nrq(R).

The ringNrq(R) is defined by the following conditions (for the completeness, refer to

(Nrq1) As a set, it isRN.(Nrq2) Addition is defined component-wise.(Nrq3) Multiplication is defined so that forx = (xn)n1 andy = (yn)n1 in Nrq(R), the

nth component ofx · y is given by∑[i,j ]|n

(i, j)Pn,i,j (q)xiyj ,

wherePn,i,j (q), [i, j ] | n are numerical polynomials inQ[q] given by

j

(i, j)q

∑d|n/[i,j ]

τq

(n

[i, j ]d ,n

i

)S(q[i,j ]/j , d

),

and the notationτq(i, n) denotes the quantity∑

d|i µq(1, d)ζ q(d,n) (for the defi-nition of µq andζ q , see below).

Introduce theq-exponential mapMq :R → Nrq(R) defined by

x → (Mq(x,n)

)n1, whereMq(x, n) =

∑d|n

µq(d,n)qd−1

dxd.

Here,µq(d,n) is the(d,n)th entry of the inverse of the matrixζ q defined on the latticeD(n) of divisors given by

q

d1 qd2/d1−1 if d1 | d2,
ζ (d1, d2) := d2
0 otherwise.


op-fineds

) has

Note that unlessq = 1, Mq is not multiplicative. Denote Frobenius and Verschiebungerators byV q

r andfqr for r 1, respectively. Note that Verschiebung operators are de

regardless ofq, that is,V qr = V 1

r = Vr , whereasf qr , which is a ring homomorphism, i

defined by

x →(

r∑d|rn

τ q

(rn

[r, d] ,rn

d

)xd

)n1

. (6.1)

From [9] it follows that theq-Teichmüller map

T q :Wq(R) → Nrq(R), x →∞∑

n=1

VnMq(xn), x = (xn)n1

is a ring isomorphism.Finally, we letϕq = λ gq . More precisely,ϕq :∆q(R) → Gh(R) is defined by

x →(∑

d|ndqn/d−1xd

)n1

for x = (xn)n1 ∈ ∆q(R). It is well known (see [6,9]) that

Φq = ϕq T q,

and moreover Lemma 6.2 says thatϕq is a ring homomorphism.

Case 2. If R does not containQ as a subring, but it is torsion free, then we let

∆q(R) := T q(Wq(R)

).

By definition ∆q(RQ) is isomorphic to Wq(R) and C(Fq,R). It is easy to ver-ify that there exists a restriction of the mapsVr :∆q(RQ) → ∆q(RQ) (respectivelyf

qr :∆q(RQ) → ∆q(RQ), Mq :RQ → Nrq(RQ), andϕ

q

RQ:∆q(RQ) → Gh(RQ)) to the

map Vr :∆q(R) → ∆q(R) (respectivelyf qr :∆q(R) → ∆q(R), Mq :R → Nrq(R), and

ϕqR :∆q(R) → Gh(R)). For example, for Frobenius operators observe that Eq. (6.1

only integer coefficients, i.e.,

rτ q

(rn

[r, d] ,rn

d

)∈ Z

for all q ∈ Z. This implies thatf qr (∆q(R)) ⊂ ∆q(R).

Proposition 6.5. If R is a torsion-free commutative ring with unity such thatap =a modpR if p is a prime, then

q q
∆ (R) = Nr (R).


t

en

e

Proof. By [9] we know that

Mq(x,n) =∑d|n

(dτq

(n

d,n

))M(x,d),

anddτq(n/d,n) ∈ Q[q] are numerical polynomials for all positive integersd,n with d | n.On the other hand, by hypothesis onR, it has a unique specialλ-ring structure withΨ n = idfor all n 1 (see [13, Section 2]]). It implies thatM(x,d) ∈ R for x ∈ R. Therefore we geMq(x,n) ∈ R, and which means that∆q(R) ⊂ Nrq(R). Now, to show∆q(R) ⊃ Nrq(R)

choose an arbitrary elementx = (xn)n1 ∈ Nrq(R). As in the classical case (i.e.,q = 1),we can findan’s in R inductively such thatT q((an)n1) = x. This completes the proof.

In caseR is torsion-free,ϕq is an injective ring homomorphism and its inverse is givby

(ϕq

)−1(x) =

∑d|n

µq(d,n)xd

d

if x belongs to the image ofϕq . Thus, we can state aq-analogue of Proposition 2.10.

Proposition 6.6. Let R be a torsion-free ring. Assume that the elementa = (an)n1 andb = (bn)n1 are in Im ϕq . Then the following equation holds(in RQ):

Eq(ab, n) =∑

[i,j ]=n

(i, i)Pn,i,j (q)Eq(a, i)Eq(b, j), whereEq(a, n) :=∑d|n

µq(d,n)ad

d.

In particular, by considering the casea = (qn−1xn)n1,b = (qn−1yn)n1, we can re-cover Proposition 5.15 in [9]

Mq(qxy) = qMq(x) · Mq(y).

Case 3. Finally, if R is not torsion-free, for a surjective ring homomorphismρ :R′ → R

from a torsion free ringR′, we define

∆q(R) := ∆q(R′)/∆q(kerρ).

Let us obtainT q :Wq(R) → ∆q(R), Vr , fqr (r 1), Mq , andϕq in the same way as w

did in Section 3.


iebunge

for

Consequently we arrive at the following commutative diagram:

Wq(R) ∆q(R)

Gh(R)

Hq T q−1

C(Fq,R)

λ−1 Eq

T q

∼= ∼=ϕqΦq

Note that all maps appearing in this diagram preserve each Frobenius and Verschoperator since they do in caseR = Z. As for morphisms of∆q , they can be obtained in thsame way as done in Section 3.

Finally, we are going to introduceq-aperiodic ring functorAPq . To begin with, wedefine Apq(R) as follows:

(Apq1) As a set, it isRN.(Apq2) Addition is defined component-wise.(Apq3) Multiplication is defined so that forx = (xn)n andy = (yn)n in Apq(R), thenth

component ofx · y is given by

∑[i,j ]|n

n

[i, j ]Pn,i,j (q)xiyj .

Let us define a mapϕq : Apq(R) → Gh(R) by

x →(∑

d|nqn/d−1xd

)n1

for x = (xn)n1 ∈ Apq(R).

Proposition 6.7. For every commutative ringR wit identity, we have

(a) ϕq is an isomorphism of the additive groupApq(R) ontoGh(R).(b) For anyx,y ∈ Apq(R),

ϕq(xy) = ϕq(x)ϕq(y).

Proof. (a) This proof is identical to that of Proposition 4.2(a).(b) Let x = (xn)n1 andx = (xn)n1. To prove our assertion we have to show that

everyn 1 it holds

∑qn/d−1

( ∑Pd,i,j (q)xiyj

)=

(∑qn/e−1xe

)(∑qn/f −1yf

).

d|n [i,j ]|d e|n f |n


he

ne

).t the

Equivalently, we have only to show that for everyn 1 andi, j | n∑

[i,j ]|dd|n

d

[i, j ]qn/d−1Pd,i,j (q) = qn/i+n/j−2. (6.2)

This follows from the fact thatϕq is a ring homomorphism. In fact, by computing tcoefficient ofxiyj in ϕq(x · y) andϕq(x)ϕq(y), we obtain

∑d|n

dqn/d−1( ∑

[i,j ]|d(i, j)Pd,i,j (q)

)= ijqn/i+n/j−2,

and which is clearly equivalent to identity (6.2).Note that the inverse ofϕq is given by

(ϕq

)−1(x) =

∑d|n

µq(d,n)n

dxd.

Thus we have aq-analogue of Proposition 4.5.

Proposition 6.8. Let R be a commutative ring. Then fora = (an)n1,b = (bn)n1 ∈ RN

the following equation holds:

Fq(ab, n) =∑

[i,j ]=n

Pn,i,j (q)F q(a, i)F q(b, j), whereFq(a, n) :=∑d|n

µq(d,n)n

dad.

According to Proposition 6.7 Apq(R) is a commutative ring. As in Section 4, we defi

Vr : Apq(R) → Apq(R) by (xn)n1 → (rxn/r )n1 with xn/r := 0 ifn

r/∈ N,

and define

fqr : Apq(R) → Apq(R) by x →(

r∑d|rn

τ q

(rn

[r, d] ,rn

d

)n

dxd

)n1

.

Actually, these operators are defined via the isomorphismθq (see Case 1 in the belowThis means thatθq is compatible with these operators. Now, we are ready to construcfunctorAPq . Let us defineAPq(R) in the following steps.

Case 1. Suppose thatR is an arbitrary commutative ring containingQ (or a field of char-acteristic zero) as a subring. In this case, we let

APq(R) := Apq(R).


t

.at the

From Proposition 6.7 it follows that the map

θq :∆q(R) → APq(R)

given by

x → (nxn)n1

for all x = (xn)n1 is a ring isomorphism, andϕq = ϕq θq . We already mentioned thaθq preserves the Frobenius and Verschiebung operators. Now, composingθq with T q weget the isomorphism

∞∑n=1

Vn Sq :Wq(R) → APq(R).

Here, theq-exponential mapSq :R → APq(R) is given by

x → (Sq(x, n)

)n1, whereSq(x, n) := nMq(x, n).

Case 2. If R does not containQ as a subring, but it is torsion free, then we let

APq(R) := θq

RQ

(∆q(R)

).

Let us obtainθq :∆q(R) → APq(R), Vr , fqr (r 1), Sq , andϕq in the same way viarestrictions as we did in Section 4.

Case 3. Finally, if R is not torsion-free, for a surjective ring homomorphismρ :R′ → R

from a torsion free ringR′, we define

APq(R) := APq(R′)/APq(kerρ).

By construction∆q(R) is isomorphic toAPq(R) for every commutative ringR withidentity, and the process to obtain the Frobenius and Verschiebung operators andϕq seemsto be routine. So we shall skip it. Similarly, it can be shown that the following diagram

∆q(R) APq(R)

Gh(R)

θq

∼=ϕqϕq

is commutative. The set of morphisms ofAPq is same with that ofAP (see Section 4)Summing up our arguments until now, we can establish Theorem 6.1 suggested
beginning of this section.


f many

–

, Adv.

cklace

ath.,

2005)

[Adv.

1–29.6

Acknowledgments

The author expresses his sincere gratitude to the referees for their corrections oerrors in the previous version and valuable advices.

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generalized burnside–grothendieck ring functor and aperiodic ring functor associated with...

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