3 Morita theorems.

The Hermitian Morita Theorems

Los Teoremas Herm´ıticos de Morita P. Verhaege, A. Verschoren

University of Antwerp, RUCA

Department of Mathematics and Computer Science Antwerp, Belgium

Abstract

Similar to the Morita theorems proved in [1] and the relative version given by Van Oystaeyen and Verschoren in [9], we will prove in this note a (relative) hermitian version of the Morita theorems, i.e., we will describe which equivalences of the cate- gory of (relative) sesquilinear, resp. hermitian, modules are de- termined by a single object and viceversa. A first approach was made in [5], which includes some partial version of the Morita theorems in the hermitian context. As we will show in this note, the techniques developed in [6] permit us to present a complete solution to the problem of generalizing the Morita theorems to the hermitian case.

Key words and phrases: Morita theorems, hermitian forms.

Resumen

En esta nota probamos una versi´on herm´ıtica (relativa) de los teoremas de Morita, an´aloga a la dada en [1] y a la versi´on relativa dada por Van Oystaeyen y Verschoren en [9]. Esto es, describimos qu´e equivalencias de la categor´ıa de m´odulos sesqui- lineales, resp. herm´ıticos (relativos), est´an determinadas por un objeto ´unico y viceversa. Una primera aproximaci´on a la soluci´on de este problema aparece ya en [5], en donde se incluye una ver- si´on parcial de los Teoremas de Morita en el contexto herm´ıtico.

Como demostramos en esta nota, las t´ecnicas desarrolladas en [6]

nos han permitido presentar una soluci´on completa al problema

de la generalizaci´on, al contexto herm´ıtico, de los Teoremas de Morita.

Palabras y frases clave: teoremas de Morita, formas herm´ıti- cas.

1 Generalities.

Throughout this paper, R is a commutative ring with unit and all rings are unitaryR-algebras; the lettersA,A⁰,. . . , will denote suchR-algebras. Let us denote the category of left (resp. right)A-modules byA-mod(resp. mod-A) and the corresponding sets of morphisms by_A[M, N] (resp [M, N]_A). Bimod- ules will always be defined overR.

Analgebra with involutionis a couple (A, α), whereAis anR-algebra and α:A→AanR-linear map satisfyingα²= 1_Aandα(a₁a₂) =α(a₂)α(a₁) for everya₁,a₂∈A. We may define with respect to αa construction similar to the usual “restriction of scalars”. However, since we use an involution instead of algebra morphisms, we have to switch sides. So, if M is a left (resp. right) A-module, then αinduces a right (resp. left) A-module structure on M by putting m·a=α(a)m (resp. a·m =mα(a)) for everym∈M and a∈A.

We denote this module by ^αM (respM^α). If (A⁰, α⁰) is a second R-algebra with involution and if M is an (A, A⁰)-bimodule then the (A⁰, A)-bimodule

α⁰M^α is defined by putting a⁰·m·a=α(a)mα⁰(a⁰) for anya ∈A, a⁰ ∈A⁰ andm∈M. IfM is anA-bimodule, then we writeM_α=^αM^α.

Any left linear map f ∈ A[M, N] yields an obvious right linear map f^α ∈ [M^α, N^α]_A. Actually, (−)^α and ^α(−) define a category equivalence betweenA-modandmod-A.

(1.1) Let us briefly recollect some definitions and properties of abstract lo- calization. For a more detailed treatment, we refer to [2, 3, 4, 7, 8 et al]. We restrict to leftA-modules, rightA-modules being treated similarly.

A left exact subfunctorλof the identity inA-modsuch thatλ(M/λM) = 0 for anyM ∈A-modwill be called aradical. Any radical is completely deter- mined by the couple (Tλ,Fλ), where thetorsion classTλ(resp. thetorsionfree class Fλ)) consists ofλ-torsion (resp. λ-torsionfree) leftA-modules, i.e. left A-modulesM such thatλM = 0 (resp. λM =M). On the other hand, the radical λis also completely determined by the setLλ of leftA-idealsL such that A/Lisλ-torsion. We call this set theGabriel filter associated toλ. It is easy to see that m∈ λM if and only if there exists someL ∈ Lλ such that Lm= 0.

A leftA-moduleE is said to beλ-injective, if for anyλ-isomorphismf :M → N in A-mod, i.e., a morphism with both λ-torsion kernel and cokernel, and any morphismg:M →E there exists a morphismg :N →E extendingG, i.e., withg=g◦f. If this morphism is always unique as such, thenEis said to be λ-closed. This is also equivalent toE being λ-torsionfree andλ-injective.

The full subcategory ofA-modconsisting of theλ-closed leftA-modules will be denoted by (A, λ)-mod and it is well known that the inclusion functor

i_λ: (A, λ)-mod ,→A-mod possesses an exact adjoint

a_λ:A-mod→(A, λ)-mod

(the reflectorofA-modinto (A, λ)-mod). The left exact functor Q_λ=i_λ◦a_λ:A-mod→A-mod

is called thelocalization functor atλand may be described in many different ways. For instance letE be an injective hull ofM/λM, thenQ_λ(M) consists of those e ∈ E such that Le ⊆ M/λM for some L ∈ L_λ. So, for any left A-moduleM, there exists a canonicalλ-isomorphism

j_λ=j_j,M :M →Q_λ(M),

which is the composition of the canonical morphism M → M/λM and the inclusionM/λM ,→Q_λ(M). Ifλis a radical inA-mod, thenQ_λ(A) is canon- ically endowed with an R-algebra structure extending that ofA. Moreover, ifM is a leftA-module (resp. an (A, A⁰)-bimodule) thenQ_λ(M) possesses a natural left Q_λ(A)-module (resp. a (Q_λ(A), A⁰)-bimodule) structure.

(1.2)Let us fix radicalsλandλ⁰) inA-modandA⁰-modrespectively. Then we say that an (A, A⁰)-bimoduleP is (λ, λ⁰)-flatorrelatively flat (with respect to (λ, λ⁰)), if for any leftA⁰-linear mapf⁰ :M⁰ →N⁰ withλ⁰-torsion kernel, the left A-module Ker(P ⊗A⁰ f⁰) is λ-torsion. It is easy to see that P is (λ, λ⁰)-flat if and only ifQ_λ(P) is relatively flat, or equivalently if it satisfies each of the following conditions:

(1.2.1)for any injective left A⁰-linear map i⁰ :M⁰ ,→N⁰, the left A-module Ker(P ⊗A⁰i⁰) isλ-torsion.

(1.2.2) for anyλ⁰-torsion left A⁰-module T⁰, the left A-module P ⊗A⁰ T⁰ is λ-torsion.

The next (technical) result will play a key-role in all that follows:

(1.3) Lemma. [6,9]Let P be an (A, A⁰)-bimodule andM⁰ a left A⁰-module, then:

(1.3.1)Qλ(P ⊗_A⁰M⁰=Qλ(Qλ(P) ⊗_A⁰M⁰;

(1.3.2)if P is relatively flat, thenQλ(P ⊗A⁰M⁰=Qλ(P ⊗A⁰Q_λ0(M⁰));

(1.3.3)ifP is also relatively flat andλ-closed, then it has a canonical(Q_λ(A), Q_λ0(A⁰))-bimodule structure and for any left Q_λ0(A⁰)-module M⁰, the left A- modules P ⊗A⁰M⁰,P ⊗_Qλ0(A⁰₎M⁰ andP ⊗_Qλ0(A⁰₎Q_λ0(M⁰)sreλ-isomorphic.

LetM be an (A, A⁰)-bimodule, M” a left A⁰-module, then we will write M⊗b_A⁰M⁰ forQ_λ(M ⊗_A⁰M⁰) andm⊗b_A⁰m⁰ forj_λ(m⊗b_A⁰m⁰) for anym∈M andm⁰ ∈M⁰, wherej_λ:M ⊗_A⁰M⁰ →M⊗b_A⁰M⁰ is the canonical localization map. So, the previous lemma allows us to write:

P⊗bA⁰M⁰⊗bA⁰⁰M⁰⁰= (P⊗bA⁰M⁰)⊗bA⁰⁰M⁰⁰=P⊗bA⁰(M⁰⊗bA⁰⁰M⁰⁰) wheneverP is relatively flat.

(1.4) A λ-closed and (λ, λ⁰)-flat (A, A⁰)-bimodule P is said to be (λ, λ⁰)- invertible or relatively invertible (with respect to (λ, λ⁰)) , if there exists a λ⁰-closed and (λ⁰, λ)-flat (A⁰, A)-bimoduleQtogether withA-bimodule (resp.

A-bimodule) isomorphisms

ϕ:P⊗b_A⁰Q→Q_λ(A) resp. ψ:Q⊗_A P→Q_λ0(A⁰).

Moreover, cf. [9], we may always assume the above isomorphisms to fit into the following commutative diagrams:

P⊗b_A⁰Q⊗b_AP

ϕ⊗^bAP

P⊗^b_A0ψ

//Pb⊗A⁰Q_λ0(A⁰)

resp.

Q⊗b_AP⊗b_A⁰Q^Q⊗bAϕ//

ψ⊗^bA0Q

Q⊗bAQ_λ(A)

Q_λ(A)⊗bAP ^//P Q_λ0(A⁰)⊗bA⁰Q ^//Q

The module Q, which is obviously relatively invertible, is said to be an inverseforP, and is, as one easily verifies, isomorphic to_A[P, Q_λ(A)]. More- over, the evaluation map P⊗bA⁰(_A[P, Q_λ(A)]) :→ Q_λ(A), may then be used as an isomorphism.

This leads us to the relative version of the Morita theorems, cf. [9]:

(1.5) Theorem. Let λ (resp. λ⁰) be a radical in A-mod (resp. A⁰-mod).

Then there is a bijective correspondence between bimodule isomorphism classes of relatively invertible (A, A⁰)-bimodules and isomorphism classes of category equivalences between the categories (A, λ)-mod and(A⁰, λ⁰)-mod.

Note that the above correspondence is given by associating to any category equivalence F : (A, λ)-mod → (A⁰, λ⁰)-mod , the (λ⁰, λ)-invertible (A, A⁰)- bimoduleF(Q_λ(A)). Conversely, to any relatively invertible (A, A⁰)-bimodule Qwith inverseP, we associate the category equivalence

Q⊗b_A− ∼=_A[P,−] : (A, λ)-mod→(A⁰, λ⁰)-mod.

Let as point out thatQ_λ0(A⁰) and_A[P, P] are isomorphic as leftA⁰-bimodules.

(1.6) If λ is a radical in A-mod andα: A →A an R-involution, then one easily verifies the set {α(L) :L∈ Lλ} to be a Gabriel filter of rightA-ideals.

We will write α(λ) for the associated radical (in mod-A) andQ_α(λ) for the localization functor atα(λ) in mod-A. The functors (−)^αand^α(−) define a category equivalence between the categories (A, λ)-modandmod-(A, α(λ)).

Moreover, for any left A-module M, we have Q_λ(M)^α = Q_αλ(M^α) and if M is an (A, A⁰)-bimodule, then ^α0Q_λ(M)^α = Q_αλ(^α0M^α), where α⁰ is an R-involution on A⁰. In particular, if M is an A-bimodule, thenQ_λ(M)_α = Q_αλ(M_α).

(1.7)A triple (A, α, λ) is called atorsion triple, if (A, α) is anR-algebra with involution andλa radical inA-modwhich satisfies the equivalent conditions:

(1.7.1) the R-involution α: A → A extends (uniquely) to an R-involution b

α:Q_λ(A)→Qλ(A);

(1.7.2)theR-algebrasQ_λ(A) andQ_α(λ)(A) are isomorphic overA;

(1.7.3)there exists a (λ, λ⁰)-invertible (A, A⁰)-bimoduleP, for some algebra with involution (A⁰, α⁰) and radical λ⁰ in A⁰-mod], with the property that

P ∼=^α_A[P, Q_λ(A)]^α0 as (A, A⁰)-bimodules.

Note that these conditions are trivially fulfilled wheneverλis induced by a radical inR-mod; for other examples we refer to [6,10].

2 Hermitically invertible modules.

(2.1) Let us fix a torsion triple (A, α, λ) and aλ-closed leftA-moduleM. A map h:M×M →Q_λ(A) which is biadditive and satisfiesh(a₁m₁, a₂m₂) = a₁h(m₁, m₂)α(a₂) for everya₁, a₂∈A andm₁, m₂∈M is called a λ-sesqui- linear form. If, moreover, h(m₁, m₂) = α(h(mb ₂, m₁)), then h is called a λ-hermitian form. For anyλ-sesquilinear form h:M ×M →Q_λ(A), define h^a ∈ A[M,^α_A[M, Q_λ(A)]] by h^a(m₂)(m₁) =h(m₁, m₂) for any m₁, m₂ ∈M. This correspondence defines a bijection between theλ-sesquilinear forms onM and the leftA-linear maps fromM to^α_A[M, Q_λ(A)]. Ifh^a is an isomorphism, then his callednonsingular. If M is an (A, A⁰)-bimodule and h:M ×M → Q_λ(A) a λ-sesquilinear form satisfying h(m₁a⁰, m₂) = h(m₁, m₂α⁰(a⁰)), for any a⁰ ∈ A⁰ and m₁, m₂ ∈ M then h is said to be A⁰-compatible. So, an A⁰-compatible λ-sesquilinear morphismh:M ×M →Q_λ(A) is essentially a bimodule morphism M⊗b_A^0α0M^α→Q_λ(A). Note also that this is equivalent to requiring that the map h^a :M →^α_A[M, Q_λ(A)]^α0 is (A, A⁰)-linear.

IfM is aλ-closed leftA-module andh:M×M →Q_λ(A) aλ-sesquilinear form, then the couple (M, h) is called a λ-sesquilinear module or a relative sesquilinear module. Ifh is also λ=hermitian, then (M, h) is a λ-hermitian moduleorrelative hermitian module. It is said to beA⁰-compatible(resp. non- singular) wheneverhisA⁰-compatible (resp. nonsingular).

(2.2)Amorphismf : (M, h)→(N, k) betweenλ-sesquilinear leftA-modules is a left A-linear mapf :M →N such thath=k◦ (f ×f), or, equivalently such that the diagram

M ^{ha //}

f

αA[M, Q_λ(A)]

N ^{ka //α}_A[N, Q_λ(A)]

αA[f,Qλ(A)]

OO

commutes. We thus obtain categories S(A, α, λ), resp. H(A, α, λ), with ob- jects theλ-sesquilinear leftA-modules, resp. λ-hermitian leftA-modules, and with obvious morphisms.

(2.3) Fix some torsion triples (A, αλ) and (A, α⁰, λ⁰). A nonsingular λ- hermitian (A, A⁰)-bimodule (P, h) is calledhermitically(λ, λ⁰)-invertibleorrel- atively hermitically invertible, ifP is (λ, λ⁰)-invertible andhisA⁰-compatible.

As one easily verifies,his then alsoQ_λ0(A⁰)-compatible.

As an easy example, letp_Qλ(A):Q_λ(A)×Q_λ(A)→Q_λ(A) be defined by p_Qλ(A)(a₁, a₂) =a₁α(ab ₂),

for any a₁, a₂ ∈ Q_λ(A). Then (Q_λ(A), p_Qλ(A)) is a hermitically (λ, λ⁰)- invertible A-bimodule.

If (P, h) is a relatively hermitically invertible (A, A⁰)-bimodule, then we can makeQ=_A[P, Q_λ(A)] into a hermitically (λ⁰, λ)-invertible (A⁰, A)-bimod- ule by endowing it with the formk:Q×Q→Q_λ0(A⁰)∼=_A[P, P], defined by putting for any q₁, q₂∈Q:

k(q₁, q₂) :P→P:p7→k(q₁, q₂)(p) =h(p,(h^a)⁻¹(q₁))(h^a)⁻¹(q₂).

The module (Q, k) is usually referred to as an “inverse” of (P, h).

(2.4) Let (M, h) be a relatively flat A⁰-compatibleλ-sesquilinear (resp. λ- hermitian) (A, A⁰)-bimodule and (M⁰, h⁰) aλ-sesquilinear (resp. λ-hermitian) leftA⁰-module. Then we may define aλ-sesquilinear (resp. λ-hermitian) form

h ⊗_A⁰ h⁰:M ⊗_A⁰M⁰×M ⊗_A⁰M⁰→Q_λ(A) by

h ⊗_A⁰h⁰(m₁⊗_A⁰m⁰₁, m₂⊗_A⁰m⁰₂) = h(m₁h⁰(m⁰₁, m⁰₂), m₂)

= h(m₁, m₂h⁰(m⁰₁, m⁰₂)),

for anym₁, m₂∈M andm⁰₁, m⁰₂∈M⁰. One easily verifies the tensor product thus defined to be associative, and the form h⊗_A⁰h⁰ to be A⁰⁰-compatible, whenever (M⁰, h⁰) is.

SinceQ_λ(A) isα(λ)-closed and sincej_λ:M⊗bA⁰M⁰×M⊗bA⁰M⁰→Q_λ(A) is a λ-isomorphism, the formh ⊗A⁰h⁰ defines a unique λ-sesquilinear (resp.

λ-hermitian) form h⊗bA⁰h⁰ : M⊗bA⁰M⁰ ×M⊗bA⁰M⁰ → Q_λ(A) making the diagram

M⊗b_A⁰M⁰×M⊗b_A⁰M⁰

jλ×jλ

h⊗A0h⁰

**

U U U U U U U U U U U U U U U U U U

Q_λ(A)

M⊗bA⁰M⁰×M⊗bA⁰M⁰

h⊗^bA0h⁰

44

i

i

i

i

i

i

i

i

i

i

i

i

i

i

i

i

i

i

commutative, cf. [5]. It thus makes sense to define therelative tensor product (M, h)⊗b_A⁰(M⁰, h⁰) to be theλ-sesquilinear (resp. λ-hermitian) leftA-module (M⊗b_A⁰M⁰, h⊗b_A⁰h⁰). An easy unicity argument shows this tensor product to be associative, whenever it is defined.

3 Morita theorems.

(3.1) Fix torsion triples (A, α, λ) and (A⁰, α⁰, λ⁰). Recall from [5,6] that any relatively hermitically invertible (A, A⁰)-bimodule (P, h) determines an equiv- alence of categories

(P, h)⊗bA⁰−:S(A⁰, α⁰, λ⁰)→ S(A, α, λ) and an equivalence

(P, h)⊗bA⁰−:H(A⁰, α⁰, λ⁰)→ H(A, α, λ) Moreover, if (Q, k) is as in (2.3), then

(Q, k)⊗bA−:S(A, α, λ)→ S(A⁰, α⁰, λ⁰) resp.

(Q, k)⊗bA−:H(A, α, λ)→ H(A⁰, α⁰, λ⁰) is an inverse for (P, h)⊗bA⁰−.

(3.2)In order to establish the complete Morita theorems, we need a notion of

“good” category equivalence between categories of relative sesquilinear (resp.

relative hermitian) modules: a category equivalence F :S(A, α, λ)→ S(A⁰, α⁰, λ⁰)

resp.

F :H(A, α, λ)→ H(A⁰, α⁰, λ⁰)

is said to be decent, if it factorizes through a category equivalence F: (A, λ)-mod→(A⁰, λ⁰)-mod

note that we use the same simbol F, as no ambiguity may arise) i.e., if we have a commutative diagram of functors

S(A, α, λ)

F //S(A⁰, α⁰, λ⁰)

(A, λ)-mod ^{F //}(A⁰, λ⁰)-mod

where the vertical arrows are defined by forgetting the relative sesquilinear form (a similar condition holds for the category of relative hermitian modules) and if there exists an isomorphismη :F(^α_A[(−), Q_λ(A)])∼=^α_A⁰0[F(−), Q_λ0(A⁰)]

such that for everyλ-sesquilinear leftA-module (M, l), we have a commutative diagram

F(M)

F(l^a)

wwp p

p p

p p

p p

p p

p F(l)^a

''

O O O O O O O O O O O

F(^α_A[M, Q_λ(A)]) ^{ηM //α}_A⁰₀[F(M), Q_λ0(A⁰)]

If (M, l) is a λ-sesquilinear (A, A⁰⁰)-bimodule, then, by the naturality of η, we have that η_M is an (A⁰, A⁰⁰)-bimodule isomorphism. Moreover, we will only consider category equivalences between relatively sesquilinear modules which map relative hermitian modules to relative hermitian modules.

We will prove below that ifGis an inverse forF, thenGis decent as well.

Before we can show that the category equivalence induced by a relatively hermitically invertible bimodule is decent, we need the following lemma, whose proof is just a straightforward verification.

(3.3) Lemma. Let U be a right A⁰-module, V a left A-module and W an (A, A⁰)-bimodule, then the morphism

µ: [U,_A[V, W]]_A⁰ →A[V,[U, W]_A⁰]

defined by (µ(f)(v))(u) = f(u)(v), for every f ∈ [U,_A[V, W]]_A0, u∈ U and v ∈ V, is an isomorphism. If V is an (A, A⁰⁰)-bimodule, then µ is left A⁰⁰-

linear and if U is an (A⁰⁰, A⁰)-bimodule, then µis right A⁰⁰-linear.

(3.4) Proposition (Morita I).Fix torsion triples (A, α, λ)and(A⁰, α⁰, λ⁰).

Then any relatively hermitically invertible (A⁰, A)-bimodule (Q, k) defines a decent equivalence between the categoriesS(A, α, λ)andmathhcalS(A⁰, α⁰, λ⁰) and the categories H(A, α, λ)andmathcalH(A⁰, α⁰, λ⁰).

Proof. Let (P, h) be an inverse for (Q, k). Define for every λ-closed left A-moduleM the isomorphismη_M as the composition of the following isomor- phisms

Q⊗b_A^α_A[M, Q_λ(A)] ∼= _A[P,^α_A[M, Q_λ(A)]]∼=_A[P,[M^α, Q_λ(A)_α]_A]

∼= [M^α,_A[P, Q_λ(A)_α]]_A∼= [M^α,_A[P, Q_λ(A)]]_A

∼= [M^α, Q]_A∼=^α_A⁰[M, P]

∼= ^α_A⁰0[Q⊗bAM, Q⊗bAP]∼=^α_A⁰0[Q⊗bAM, Q_λ0(A⁰)].

An easy verification shows that

η_M(q⊗bAf)(q⁰⊗bAm⁰) =k(q⁰f(m⁰), q)

and that η :Q⊗b_A^α_A[(−), Q_λ(A)]∼=^α_A⁰0[Q⊗b_A(−), Q_λ⁰(A⁰)]. So, for every m∈ M andq∈Q, we have that

η_M(q⊗bAl^a(m)) = (k⊗bAl)^a(q⊗bAm),

i.e., (Q, k)⊗bA−is decent. By symmetry,P⊗bA⁰−is decent as well. ² Conversely,

(3.5) Proposition (Morita II). Let (A, α, λ) and (A⁰, α⁰, λ⁰) be torsion triples. Then every decent category equivalence betweenS(A, α, λ) andS(A⁰, α⁰, λ⁰)(resp. H(A, α, λ) andmathcalH(A⁰, α⁰, λ⁰)) is induced by a relatively hermitically invertible (A⁰, A)-bimodule.

Proof. Let F :H(A, α, λ)→ H(A⁰, α⁰, λ⁰) be a decent category equivalence, then (Q, k) = F(Q_λ(A), p_Qλ(A)) is a hermitically (λ⁰, λ)-invertible (A⁰, A)- bimodule. Let us now show that F(−) = (Q, k)⊗b_A−.

As F = Q⊗b_A− : (A, λ)-mod → (A⁰, λ⁰)-mod, we only have to verify that F(l) =k⊗b_Al, for everyλ-sesquilinear left A-module (M, l). Let

η:F(^α_A[(−), Q_λ(A)])→^{∼ α}_A⁰⁰[F(−), Q_λ0(A⁰)],

then

η_Qλ(A)=k^a:Q→^α_A⁰0[Q, Q_λ0(A⁰)].

Let (M, l) be a λ-sesquilinear left A-module, then for everym∈M we have a commutative diagram

Q

k^a

Q⊗^bAα

A[l^a(M),Qλ(A)]

//Qb⊗Aα

A[M, Q_λ(A)]

ηM

α⁰

A⁰[Qb⊗A, Q_λ0(A⁰)]

α0

A0[Q⊗^bAl^a(m),Q_λ0(A⁰)]

//α⁰

A⁰[Qb⊗AM, Q_λ0(A⁰)]

after identifying

Q=Qb⊗_A^α_A[Q_λ(A), Q_λ(A)]

and

α⁰

A⁰[Q⊗bA, Q_λ0(A⁰)] =^α_A⁰0[Qb⊗AQ_λ(A), Q_λ0(A⁰)].

So, since F(l)^a=η_M ◦(Q⊗b_Al^a), we have for everyq, q⁰∈Qandm, m⁰∈M (F(l)^a(q⊗bAm))(q⁰⊗bAm⁰) = ((η_M◦(Q⊗bAl^a))(q⊗bAm))(q⁰⊗bAm⁰)

= η_M(q⊗b_Al^a(m))(q⁰⊗b_Am⁰)

= ((η_M◦(Q⊗bAα

A[l^a(m), Q_λ(A)]))(q))(q⁰⊗bAm⁰)

= ((^α_A⁰0[Q⊗b_Al^a(m), Q_λ0(A⁰)]◦k^a)(q))(q⁰⊗b_Am⁰)

= (k^a(q)◦(Q⊗bAl^a(m))))(q⁰⊗bAm⁰)

= k^a(q)(q⁰l(m⁰, m))

= ((k⊗bAl)^a(q⊗bAm))(q⁰⊗bAm⁰),

henceF(l) =k⊗b_Al, as claimed. ²

(3.6) Corollary. With the same notations, if G is an inverse for F, then G= (P, h)⊗b_A⁰−, where (P, h) is an inverse for (Q, k).

In particular,Gis also decent, as claimed before.

References

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