Moduli space of pairings on complex roots of unity

Moduli space of pairings on complex roots of unity

Laurent Poinsot

LIPN - UMR CNRS 7030

Université Paris XIII, Sorbonne Paris Cité - Institut Galilée

Joint-work with Nadia El Mrabet - Université Paris 8

14th International Conference on Arithmetic, Geometry, Cryptography and Coding Theory Marseille

Table of contents

1 Introduction

2 Category theoretic definitions and notations

3 Categories of bilinear maps

4 Pairings as a Rees quotient

5 Length of a pairing and local finiteness

6 Classification of pairings on R/Z

Pairings

Let A,B,C be three modules over some commutative ring R with a unit.

A pairingis a non-degeneratebilinear mapf :A×B→C. Non-degeneracy means that γ_f:a∈A7→f(a,L) and δ_f:b∈B7→f(L,b) are bothone-to-one.

Examples

• Let 1→A→G →B →1 be a short exact sequence of groups, where A,B are Abelian, andAlies inZ(G). Thecommutator [·,·]ofG factors to a bilinear map [·,·] :B×B→Awhich is non-degenerate if, and only if, A=Z(G) (R. Baer, 1938).

• Leth· | ·i:A×Ab →R/Zdefined byha|χi=χ(a).

• Weil, Tate pairings and their recent generalizations to Abelian varieties.

• LetR be any field, andX be any set. Let us denote by K^(X) the vector space of finitely supported maps (i.e., the vector space with basis X). Let h· | ·i:K^X ×K^(X)→Kbe given by hf |gi=P

x∈X f(x)g(x) is a pairing.

Cryptographic applications

• MOV attack to solve discrete logarithm problem by transport from an elliptic curve to a finite field.

• A. Joux’s one-round key exchange tri-partite Diffie-Hellman protocol.

• Identity-based cryptography.

Objective of this talk

• Provide aclassification of pairings – under a suitable equivalence relation – from finite Abelian groups to the complex unit circle.

• Show that the set of equivalence classes of pairings is almost amoduli space: it is actually a subset of rational points of some (pro-)affine algebraic variety.

Warning: The classification from this talk is of course different from C.T.C Wall’s classification of skew or symmetric non-singular bilinear forms on finite Abelian groups (1964) because the equivalence relations under consideration are not the same. My equivalence relation is of categorical origin since it is the relation of isomorphism in a suitable category.

Table of contents

1 Introduction

2 Category theoretic definitions and notations

3 Categories of bilinear maps

4 Pairings as a Rees quotient

5 Length of a pairing and local finiteness

6 Classification of pairings on R/Z

Table of contents

1 Introduction

2 Category theoretic definitions and notations

3 Categories of bilinear maps

4 Pairings as a Rees quotient

5 Length of a pairing and local finiteness

6 Classification of pairings on R/Z

Notations

Let C be a category.

The class of objects of C is denoted byOb(C).

Let A,B be objects of C. The class of arrows from Ato B is denoted by C(A,B). Of course, f ∈C(A,B) is also denoted by f:A→B.

Names of categories

• LetC be a category.

- Cmono is the subcategory of C consisting of the objects of C and with arrows the monomorphisms (left cancellable arrows).

- Ciso is the core ofC,i.e., the groupoid with objects those of C, and arrows the isomorphisms inC.

- More generally, ifS is a class of arrows containing for each object its identity morphism, and closed under composition, then CS is the subcategory of C defined in the obvious way.

• Usual categories of setsSet,Ab andAbfin of Abelian and finite Abelian groups.

• LetR be a commutative ring with a unit (R 6=0). Let R-Mod be the category of R-modules, and letR-Freefin be its full subcategory of free R-modules of finite rank.

Monoidal category

A monoidal categoryis a categoryC with a bifunctor ⊗:C×C→C, an object I, and three natural isomorphisms

- α:L⊗(L⊗L)∼= (L⊗L)⊗L(associativity constraint or associator), - λ:I ⊗L∼=id_C (left unit),

- ρ:L⊗I ∼=id_C (right unit).

that satisfy somecoherence axioms.

Coherence for associativity

Mac Lane - Stasheff’s Pentagon

For all A,B,C,D∈Ob(C), the following diagram commutes.

A⊗(B⊗(C ⊗D))

idA⊗αB,C,D

tt

αA,B,(C⊗D)

**

A⊗((B⊗C)⊗D)

αA,(B⊗C),D

(A⊗B)⊗(C ⊗D)

α(A⊗B),C,D

(A⊗(B⊗C))⊗D

αA,B,C⊗id_D //((A⊗B)⊗C)⊗D

Coherence for units

For all A,B∈Ob(C), the following diagram commutes.

A⊗(I ⊗B) ^{αA,I,B //}

idA⊗λB &&

(A⊗I)⊗B

ρA⊗idB

xxA⊗B

Symmetric monoidal category

Let (C,⊗,I, α, λ, ρ)be a monoidal category. A braidingis a natural isomorphism σ:L⊗L∼= (L⊗L)◦τ whereτ:C×C→C×C is the usual flip τ(A,B) = (B,A) such that σB,A◦σA,B =id_A⊗B and

ρA =λA◦γA,I for every objectsA,B.

A symmetric monoidal category is a monoidal category with a coherent braiding,i.e., for all A,B,C ∈Ob(C) the following diagram commutes.

A⊗(B⊗C)^αA,B,C//

idA⊗σB,C

(A⊗B)⊗C^{σ(A⊗B),C//}C ⊗(A⊗B)

αC,A,B

A⊗(C ⊗B)_α

A,C,B//(A⊗C)⊗B

σA,C⊗id_B//(C ⊗A)⊗B

Monoidal closed category

A closed categoryis a symmetric monoidal category C in which for each object B the functor L⊗B:C →C has a specified right adjoint (L)^B:C →C (which is referred to as theinternal hom functor or

exponential), i.e., for every objectsA,C, there is a natural isomorphism (in the category of sets)

Curry_A,B,C:C(A⊗B,C)∼=C(A,C^B) .

Examples: every Cartesian closed category (Set, Kelley spaces, category of all small categories, ...), Ab,Abfin,R-Mod,R-Freefin, commutative Hopf algebras, ...

Notations concerning coproducts

Let C be a category with a binary coproduct ⊕. In what follows, for every objects A,B ofC,q_A:A→A⊕B,q_B:B →A⊕B denote the natural injections (while they are not required to be injective!).

Let α∈C(A,A⁰) andβ∈C(B,B⁰). Then,[α, β]∈C(A⊕B,A⁰⊕B⁰) denotes the unique arrowγ such thatγ◦q_A=q_A⁰◦αandγ◦q_B =q_B⁰◦β.

Slice category

Let B,C be categories, letF:B→C be a functor, and let C be a fixed object ofC. The slice categoryF/C over C has

- objects all pairs(f,A) where A∈Ob(B)andf ∈C(F(A),C),

- for each f ∈C(F(A),C) andg ∈C(F(B),C), an arrowα:f →g is a member of B(A,B) such that the following diagram commutes.

F(A)

f !!

F(α) //F(B)

}} g

C

Table of contents

1 Introduction

2 Category theoretic definitions and notations

3 Categories of bilinear maps

4 Pairings as a Rees quotient

5 Length of a pairing and local finiteness

6 Classification of pairings on R/Z

The category of bilinear maps

Let us assume that C is a closed category, and letD be a full subcategory of C (it may beC itself). (These data will be implicitly assumed.)

Let C be a given objects of C. The category of bilinear maps onC is the full subcategory BilD(C) of the slice category⊗/C. Objects: (f,(A,B)), A,B objects ofD andf ∈C(A⊗B,C). (It is equivalently defined as the slice category⊗ ◦(j_D×j_D), wherej_D:D,→C is the full inclusion functor.)

In what follows if(f,(A,B))is such an object, then it is identified with f itself and I use the following notations: L_f =AandR_f =B.

Thus an arrow α in BilD(C) fromf ∈C(L_f ⊗R_f,C) to

g ∈C(L_g ⊗R_g,C),L_f,R_f,L_g,R_g objects of D,i.e.,α∈BilD(C)(f,g), is a pair (αl, αr) with αl ∈D(L_f,L_g), and αr ∈D(Rf,R_g)such that the following diagram commutes.

L ⊗R ^{αl⊗αr //}L ⊗R

Currying

SinceC is assumed closed, for each f ∈C(L_f ⊗R_f,C), we may define its adjointγ_f ∈C(L_f,C^Rf)as Curry_Lf,Rf,C(f).

Because C is assumed to be symmetric with braiding say σ, we may define another adjoint δ_f ∈C(R_f,C^Lf)as Curry_Rf,Lf,C(σL_f,R_f(f)).

Category of pairings on C

Apairing onC is a bilinear mapf ∈C(A⊗B,C)such that both arrows γf

andβ_f are monomorphisms inC.

This is the translation of non-degeneracyin this setting.

Thecategory of pairings on C is the full subcategory PairD(C) ofBilD(C) with objects all the pairings on C.

Perfect pairing

A pairing f on C is said to beperfectwhenever γ_f andδ_f are actually isomorphisms in C.

Thecategory of perfect pairings on C is the full subcategoryPerf_D(C) of PairD(C) (and obviously also ofBilD(C)) with objects all the perfect pairings on C.

In brief: Perf_D(C),→PairD(C),→BilD(C),→ ⊗/C (where the arrows denote the full inclusion functors).

Isomorphisms in these categories

Let f,g be two objects ofBilD(C), and letα= (αl, αr) :f →g be an arrow.

The arrow α∈BilD(C)_iso(f,g) (respectively, BilD(C)_mono(f,g))if, and only if,αl ∈Diso(L_f,L_g) (resp.,Dmono(L_f,L_g)) and αr ∈Diso(R_f,R_g) (resp.,Dmono(R_f,R_g)).

Because Perf_D(C) andPairD(C) are full subcategories ofBilD(C), their isomorphisms are the same as those of BilD(C).

Remark

Let f,g ∈Ob(BilD(C))such that f ∼=g (as bilinear maps). We observe that f is a pairing (respectively, a perfect pairing) if, and only if,g also is.

Similarly, let us assume thatf,g ∈Ob(PairD(C)). Then, f is a perfect pairing if, and only if, g is so.

Remarks

Let us denote by Cat the category of all (large) categories.

We may define a functor BilD(L) :C →Cat as follows.

Let C,C⁰ be two objects of C and let φ∈C(C,C⁰). Then, BilD(φ) :BilD(C)→BilD(C⁰)is the functor defined for an object f ∈C(L_f ⊗R_f,C) by BilD(φ)(f) =φ◦f ∈C(A⊗B,C⁰) and which acts as the identityon arrows.

By restriction, we may define two other functors:

• PairD(L) : Cmono →Catfull emb (where “full emb” stands for “full embedding”, i.e., a particular class of functorial monomorphisms: full functors, injective on arrows). So for every monomorphism φ∈C(C,C⁰), PairD(C) may be seen as a full subcategory ofPairD(C⁰).

• Perf_D(L) :Ciso →Catiso.

∼ ⁰

Remark

Let us assume that E ,→D ,→C =R-Mod are full embeddings of

categories (i.e.,E is a full subcategory ofD which a full subcategory ofC).

We have the following commutative diagram of full inclusions for every C ∈Ob(C).

Perf_D(C) ^{ //}PairD(C) ^{ //}BilD(C)

Perf_E^? (C)

OO

 //PairE^? (C)

OO

 //BilE^? (C)

OO

Adjoints related to perfect pairings for C = R - Mod

Let D be a full subcategory ofR-Mod, let C be a givenR-module, and let f ∈Ob(Perf_D(C)).

For every φ∈R-Mod(D_f,D_f), there is aunique ^†φ∈R-Mod(R_f,R_f) such that f ◦(φ⊗id_Rf) =f ◦(id_Lf ⊗^†φ). (Define

†φ(y) =δ⁻¹_f (f(φ(·)⊗y))∈R_f for eachy ∈R_f.)

For every ψ∈R-Mod(R_f,R_f), there is aunique ψ^†∈R-Mod(L_f,L_f) such that f ◦(id_Lf ⊗ψ) =f ◦(ψ^†⊗id_Rf) (Defineψ^†(x) =γ_f⁻¹(f(x ⊗ψ(·))).) Actually, ^†(L) :R-Mod(L_f,L_f)→R-Mod(R_f,R_f)^op and

(L)^†:R-Mod(R_f,R_f)^op →R-Mod(L_f,L_f) areisomorphisms ofR-algebras, inverse one from the other.

In particular when D =R-Freefin, if f:L_f ⊗R_f →C is a perfect pairing,

Example

• LetR =Zso thatC =Ab and letD =Abfin. Let Abe a finite Abelian group, and let us consider the natural pairingh· | ·i_A:A×Ab→R/Q defined by the evaluation ha|χi_A =χ(a), where Ab=Ab(A,R/Q).

Let φ∈Abfin(A,A), then^†φ∈Abfin(A,b A)b is given by^†φ(χ) =χ◦φ.

• Similarly, letD =R-Freefin. We also consider the natural pairing h· | ·i_n:Rⁿ×(Rⁿ)^∗→R given by the evaluation hv |`i<sub>n</sub>=`(v).

Let φ∈R-Freefin(Rⁿ,Rⁿ), then ^†φ∈R-Freefin((Rⁿ)^∗,(Rⁿ)^∗) is given by

φ(`) =`◦φ.

Table of contents

1 Introduction

2 Category theoretic definitions and notations

3 Categories of bilinear maps

4 Pairings as a Rees quotient

5 Length of a pairing and local finiteness

6 Classification of pairings on R/Z

Rees quotient

Let S be a semigroup, andI ⊆S. The set I is a (two-sided) idealof S if

SI ⊆S ⊇IS .

An idealI is said to beprime when xy ∈I implies that either x ∈I or y ∈I.

Given an ideal I, we may form theRees quotient S/I of S by I: it is the set (S\I)t { ∞ } with the following operation: letx,y ∈S, thenx ×y =xy if xy 6∈I,x×y =∞ otherwise, andz× ∞=∞=∞ ×z for every z ∈(S\I)t { ∞ }.

It is equivalently defined as the quotient semigroup S/∼= by the

Remark

If I is a prime ideal ofS, thenS/I is zero-divisor free, therefore it is a usual semigroup, sayT, with a two-sided absorbing element∞ freely adjoined T^∞=T t { ∞ }.

A symmetric monoidal structure on Bil

(C )

From now on, C =R-Mod, and D is cocartesian for ⊕.

Let A,B,C,D∈Ob(D). Since

(A⊕B)⊗(C ⊗D)∼= (A⊗C)⊕(A⊗D)⊕(B⊗C)⊕(B⊗D), (A⊕B)⊗(C ⊗D)has acoproduct presentation:

A⊗C

q_A⊗q_C

((

A⊗D

q_A⊗q_D

vv

(A⊕B)⊗(C ⊕D)

B⊗C

qB⊗qC

66

B⊗D

qB⊗qD

hh

A symmetric monoidal structure on Bil

(C )

Let f,g ∈Ob(BilD(C)).

We define f⊥g: (L_f ⊕L_g)⊗(R_f ⊕R_g)→C by (f⊥g)◦q_Lf ⊗q_Rf =f,

(f⊥g)◦q_Lg ⊗q_Rg =g, (f⊥g)◦q_Lf ⊗q_Rg =0Lf⊗Rg,C, (f⊥g)◦q_Lg ⊗q_Rf =0Lg⊗R_f,C

where 0A,B is thezero arrowfromAto B.

Because D is assumed cocartesian for⊕,f⊥g ∈Ob(BilD(C)).

A remark

Let q_f ∈BilD(C)(f,f⊥g)be defined by q_f = (q_Lf,q_Rf), and similarly, let qg ∈BilD(C)(g,f⊥g)be given by qg = (q_Lg,q_Rg).

It is not true that(f⊥g,q_f,q_g)forms a coproduct for f,g.

Nevertheless it satisfies the following universal property: let h∈BilD(C), α∈BilD(C)(f,h) andβ ∈BilD(C)(g,h) such that

h◦(α_l ⊗α_r) =0Lf⊗R_f,C andh◦(β_l⊗β_r) =0Lg⊗Rg,C, then there is a unique arrow γ ∈BilD(C) such that γ◦q_f =α andγ◦q_g =β, namely γ = ([α_l, β_l],[αr, βr]).

Symmetric monoidal structure at the level of arrows

Let f,f⁰,g,g⁰ ∈BilD(C), andα ∈BilD(C)(f,f⁰),β ∈BilD(C)(g,g⁰), then α⊥β ∈BilD(C)(f⊥g,f⁰⊥g⁰) is defined by

α⊥β= (α_l⊕β_l, αr ⊕βr) .

Theunit of ⊥is00⊗0,C and the coherent braiding is given by

σ_f,g = (σ_Lf,Lg, σ_Rf,Rg) : f⊥g →g⊥f where σ_A,B:A⊕B ∼=B⊕Ais the natural twist of C.

An essential property of ⊥

Theorem

Let f,g ∈BilD(C).

The bilinear map f⊥g is a pairing (respectively, a perfect pairing) if, and only if,f andg are pairings (respectively, perfect pairings) themselves.

As a corollary, the set of equivalent classes of pairings Pair_D(C) is a

sub-monoid of the (commutative) monoidBil_D(C)(under ⊥) of equivalent classes of bilinear maps, and the set of equivalent classes of perfect pairings PerfD(C) is a sub-monoid of Pair_D(C).

Table of contents

1 Introduction

2 Category theoretic definitions and notations

3 Categories of bilinear maps

4 Pairings as a Rees quotient

5 Length of a pairing and local finiteness

6 Classification of pairings on R/Z

Locally finite monoids

Let S be a semigroup, andx ∈S. Adecomposition ofx of lengthn is a n-tuple (x1,· · ·,xn)∈Sⁿ such that x =x1· · ·xn. Adecomposition ofx is then a decomposition of x of length n for somen. A semigroup is said to be locally finiteif all its members admit onlyfinitely many decompositions.

If M is a monoid (with identity 1), and x ∈M, then a decomposition (x₁,· · ·,x_n) of x is said to benon-trivial if nox_i’s are equal to 1. A monoid is said to belocally finite if all its members admit onlyfinitely many non-trivial decompositions.

Let S be any semigroup (or monoid), andx ∈S. We denote byD_n(x)the set of all (non trivial) decompositions of x of lengthn.

Remark

Length

Any locally finite semigroup (respectively, monoid) S may be equipped with a lengthfunction defined as follows: forx ∈S,

`(x) =max{n∈N:D_n(x)6=∅ }.

In particular for a locally finite monoid M,`(x) =0 if, and only if,x =1.

A member x ofM is said to beindecomposable if`(x) =1. It is clear that such indecomposable elements generate the monoid M.

Finite decomposition monoids

A monoid M is said to be afinite decompositionmonoid if for everyx ∈M, {(y,z)∈M²:x =yz}

is finite. Clearly, any finite monoid is a finite decomposition monoid. So is any locally finite monoid. A non trivial finite group is a finite

decomposition but not locally finite monoid.

Let R be a commutative ring with a unit, and letAbe a commutative R-algebra with a unit. Let M be a finite decomposition monoid. Then, A^M admits a structure of R-algebra with a unit given by

(fg)(x) =P

yz=xf(y)g(z) (convolution product). This algebra is denoted by A[[M]] and called thelarge algebra of M (for instance,C[[N^∗]]is the algebra of Dirichlet’s series).

Remark: Möbius inversion

Finite decomposition and algebraic monoids

Let M be a finite decomposition monoid. LetR[x_a:a∈M]be the polynomial algebra in the indeterminate x_a, for each a∈M (i.e., the R-algebra of the free commutative monoid generated by the set M). Then, the functor (L)^M:R-CAlg →Set (defined at the object level byA7→A^M) is a representable functorwith representing object R[x_a:a∈M]

(coordinate ring) sinceR-CAlg(R[x_a:a∈M],A)∼=A^M.

Moreover, the underlying multiplicative monoid structure of A[[M]] is natural in the algebraA of coefficients. The multiplication

m: (L)[[M]]×(L)[[M]]→(L)[[M]] and the unit e:∗ →(L)[[M]] are natural transformations between representable functors. According to Yoneda’s lemma they give rise to a structure of bialgebra on R[x_a:a∈M], so that(L)[[M]] becomes a(pro-affine) algebraic monoid.

Finally, the monoidM is a sub-monoid of the R-rational pointsR[[M]] of the pro-affine monoid scheme (L)[[M]]. The constant functorR-CAlg→Set with

Remark: Zêta function

Let M be a locally finite monoid. For every algebraA, letM_A be the augmentation ideal ofA[[M]],i.e., the set all functionsf inA^M vanishing at the identity of M.

Then,1+M_(L) is a pro-affine algebraic group scheme (this is the reason why in this case the zêta function ζA is invertible).

Back to pairings

From now, one we assume that D =R-Freefin or D =Abfin ifR =Z. We denote by [f]the class of equivalence under isomorphism of a bilinear map f ∈BilD(C).

• LetC =Ab, andD =Abfin. Then, there exists a homomorphism of monoids c:Bil_Abfin(C)→N^∗×N^∗ given byc([f]) = (|L_f|,|R_f|) (well-defined on equivalence classes).

• LetC =R-Mod, and D =R-Freefin. Then, there exists a homomorphism of monoids d:Bil_R-Freefin(C)→N×Ngiven by

d([f]) = (rank(L_f),rank(R_f))(well-defined on equivalence classes).

Back to pairings

From the existence of the morphisms c andd, it may be deduced that Bil_D(C),Pair_D(C) andPerf

D(C) are alllocally finite monoids.

So they are sub-monoids of rational points of some (pro-affine) algebraic variety.

Table of contents

1 Introduction

2 Category theoretic definitions and notations

3 Categories of bilinear maps

4 Pairings as a Rees quotient

5 Length of a pairing and local finiteness

6 Classification of pairings on R/Z

Pairings are perfect

From now on, C =Z-Mod =Ab,D =Abfin andC =R/Z.

Let Abe any finite Abelian group, and let us denote byAb=Ab(A,R/Z) the group of characters(ordual) ofA. It is well-known thatA∼=Ab (non natural isomorphism).

Let f ∈Ob(PairAb_fin(R/Z)). Then, L_f ,→Rb_f ∼=R_f ,→bL_f ∼=L_f, so that L_f ∼=R_f, and γ_f, δ_f are isomorphisms. Thus,f is a perfect pairing,i.e., PairAbfin(R/Z) =Perf_Abfin(R/Z).

Natural pairing

Let Abe a finite Abelian group. Thenatural pairing is defined by h· | ·i_A:A×bA→R/Zby

ha|χi_A =χ(a) .

Theorem

Let f ∈Ob(PairAbfin(R/Z)), thenf ∼=h· | ·i_Lf.

Sketch of the proof: Let α:R_f ∼=L_f, and defineg:L_f ×L_f →R/Z by g(a,b) =f(a, α⁻¹(b))so thatg ∼=f. Since δg:L_f →bL_f is an

isomorphism, h=g◦(id_Lf ⊗δ⁻¹_g )∼=g. Moreover,

h(a, χ) =g(a, δ_g⁻¹(χ)) =δg(δ_g⁻¹(χ))(a) =χ(a) =ha|χi_Lf.

Equivalence classes of pairings

As a corollary, equivalence classes of pairings and isomorphic classes of finite Abelian groups are in one-one correspondence.

SinceAb(A⊕B,R/Z)∼=Ab(A,R/Z)×Ab(B,R/Z), it follows that A\⊕B ∼=Ab⊕Bb (isomorphic as groups).

Moreover,

h· | ·i_A⊕B ∼=h· | ·i_A⊥h· | ·i_B . In conclusion, Pair_Abfin(R/Z) =Perf

Abfin

(R/Z)∼=L

p∈PMp, where P is the set of all prime numbers, and M_p is the free commutative monoid

generated by prime powers pⁱ’s, i ∈N^∗. In particular,Pair_Abfin(R/Z)is a free commutative monoid.

Remark

Remark

Let f:A×B →Zn be a pairing. Then, it may be seen as aR/Z-valued pairing (since PairAbfin(Zn)is a full subcategory of PairAbfin(R/Z)) and as such it is a perfect pairing (thus A∼=B) andf is isomorphic to h· | ·i_A. Remark

• Letf :Z^mn ⊗Zn→Z^mn be given by

f((x_i modn)^m_i=1,y) = (x_iy modn)ⁿ_i=1. It is a pairing (obviously non perfect wheneverm>1).

• SinceC^∗ ∼=R/Zare isomorphic groups, PairAbfin(C^∗)∼=PairAbfin(R/Z).

Moreover, PairAbfin(Q/Z)∼=PairAbfin(R/Z) because of the torsion in finite Abelian groups.

Remark

Let p be a prime number, and let Z(p^∞)be the Prüfer p-group, i.e., the direct limit of 0,→Zp ,→Zp² ,→ · · · (certainly,Z(p^∞) is a sub-group of Q/Z).

Let p-Abfin be the full subcategory ofAbfin of all finite Abelian p-groups.

The category Pairp-Ab_fin(Z(p^∞))is a full subcategory ofPairAb_fin(Q/Z), therefore eachf ∈Ob(Pairp-Abfin(Z(p^∞))may be seen as a perfect pairing (so thatL_f ∼=R_f) with values inQ/Z, and as so it is isomorphic toh· | ·i_Lf. It becomes easy to see that Pair_p-Abfin(Z(p^∞))is the free commutative monoidM_p generated prime powerspⁱ,i ≥1 in such a way that

Pair_Abfin(Q/Z)∼=M

p∈

Pair_p-Abfin(Z(p^∞))

The zêta function of Pair

( R / Q )

According to Pierre Cartier and Dominique Foata, for any algebra A, the inverse µA of the zêta functionζA of the free monoid PairAb_fin(R/Q) is given by

µ_A([h· | ·i_Z

pi]) =−1_A for all prime numbersp and integers i >0,

µ_A(0) =1_A , and for all [f]∈PairAbfin(R/Q)with `([f])≥2,

µA([f]) =0.

(Because PairAbfin(R/Q)is a free commutative monoid, its length ` corresponds to the number of generators in a term.)