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
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,A′,. . . , 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 byA[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 (respMα). If (A′, α′) is a second R-algebra
with involution and if M is an (A, A′)-bimodule then the (A′, A)-bimodule
α′
Mα is defined by putting a′·m·a=α(a)mα′(a′) for anya ∈A, a′ ∈A′
andm∈M. IfM is anA-bimodule, then we writeMα=αMα.
Any left linear map f ∈ A[M, N] yields an obvious right linear map
fα ∈ [Mα, Nα]
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
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
iλ: (A, λ)-mod ֒→A-mod
possesses an exact adjoint
aλ:A-mod→(A, λ)-mod
(the reflectorofA-modinto (A, λ)-mod). The left exact functor
Qλ=iλ◦aλ: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, thenQλ(M) consists
of those e ∈ E such that Le ⊆ M/λM for some L ∈ Lλ. So, for any left
A-moduleM, there exists a canonicalλ-isomorphism
jλ=jj,M :M →Qλ(M),
which is the composition of the canonical morphism M → M/λM and the inclusionM/λM ֒→Qλ(M). Ifλis a radical inA-mod, thenQλ(A) is
canon-ically endowed with an R-algebra structure extending that ofA. Moreover, ifM is a leftA-module (resp. an (A, A′)-bimodule) thenQ
λ(M) possesses a
natural leftQλ(A)-module (resp. a (Qλ(A), A′)-bimodule) structure.
(1.2)Let us fix radicalsλandλ′) inA-modandA′-modrespectively. Then
we say that an (A, A′)-bimoduleP is (λ, λ′)-flatorrelatively flat (with respect
to (λ, λ′)), if for any leftA′-linear mapf′ :M′ →N′ withλ′-torsion kernel,
the left A-module Ker(P ⊗A′ f′) is λ-torsion. It is easy to see that P is
(λ, λ′)-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 A′-linear map i′ :M′ ֒→N′, the left A-module
(1.2.2) for anyλ′-torsion left A′-module T′, the left A-module P ⊗
A′ T′ 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, A′)-bimodule andM′ a left A′-module,
map. So, the previous lemma allows us to write:
P⊗bA′M′⊗bA′′M′′= (P⊗bA′M′)⊗bA′′M′′=P⊗bA′(M′⊗bA′′M′′)
wheneverP is relatively flat.
(1.4) A λ-closed and (λ, λ′)-flat (A, A′)-bimodule P is said to be (λ, λ′
)-invertible or relatively invertible (with respect to (λ, λ′)) , if there exists a
λ′-closed and (λ′, λ)-flat (A′, A)-bimoduleQtogether withA-bimodule (resp.
A-bimodule) isomorphisms
ϕ:P⊗bA′Q→Qλ(A) resp. ψ:Q⊗A P→Qλ′(A′).
The module Q, which is obviously relatively invertible, is said to be an inverseforP, and is, as one easily verifies, isomorphic toA[P, Qλ(A)].
More-over, the evaluation map P⊗bA′(A[P, Qλ(A)]) :→ Qλ(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. λ′) be a radical in A-mod (resp. A′-mod).
Then there is a bijective correspondence between bimodule isomorphism classes of relatively invertible (A, A′)-bimodules and isomorphism classes of category
equivalences between the categories (A, λ)-mod and(A′, λ′)-mod.
Note that the above correspondence is given by associating to any category equivalence F : (A, λ)-mod → (A′, λ′)-mod , the (λ′, λ)-invertible (A, A′
)-bimoduleF(Qλ(A)). Conversely, to any relatively invertible (A, A′)-bimodule
Qwith inverseP, we associate the category equivalence
Q⊗bA− ∼=A[P,−] : (A, λ)-mod→(A′, λ′)-mod.
Let as point out thatQλ′(A′) andA[P, P] are isomorphic as leftA′-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) andQα(λ) for the
localization functor atα(λ) in mod-A. The functors (−)αandα(−) define a
category equivalence between the categories (A, λ)-modandmod-(A, α(λ)). Moreover, for any left A-module M, we have Qλ(M)α = Qαλ(Mα) and if
M is an (A, A′)-bimodule, then α′
Qλ(M)α = Qαλ(α
′
Mα), where α′ is an
R-involution on A′. In particular, if M is an A-bimodule, thenQ
λ(M)α =
Qαλ(Mα).
(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:Qλ(A)→Qλ(A);
(1.7.2)theR-algebrasQλ(A) andQα(λ)(A) are isomorphic overA;
(1.7.3)there exists a (λ, λ′)-invertible (A, A′)-bimoduleP, for some algebra
P ∼=α
A[P, Qλ(A)]α
′
as (A, A′)-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.
This correspondence defines a bijection between theλ-sesquilinear forms onM
and the leftA-linear maps fromM toα
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 beA′-compatible(resp.
non-singular) wheneverhisA′-compatible (resp. nonsingular).
(2.3) Fix some torsion triples (A, αλ) and (A, α′, λ′). A nonsingular λ
-hermitian (A, A′)-bimodule (P, h) is calledhermitically(λ, λ′)-invertibleor
rel-atively hermitically invertible, ifP is (λ, λ′)-invertible andhisA′-compatible.
As one easily verifies,his then alsoQλ′(A′)-compatible.
As an easy example, letpQλ(A):Qλ(A)×Qλ(A)→Qλ(A) be defined by
pQλ(A)(a1, a2) =a1αb(a2),
for any a1, a2 ∈ Qλ(A). Then (Qλ(A), pQλ(A)) is a hermitically (λ, λ
′
)-invertible A-bimodule.
If (P, h) is a relatively hermitically invertible (A, A′)-bimodule, then we
can makeQ=A[P, Qλ(A)] into a hermitically (λ′, λ)-invertible (A′, A
)-bimod-ule by endowing it with the formk:Q×Q→Qλ′(A′)∼=A[P, P], defined by
putting for any q1, q2∈Q:
k(q1, q2) :P→P:p7→k(q1, q2)(p) =h(p,(ha)−1(q1))(ha)−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 A′-compatibleλ-sesquilinear (resp. λ
-hermitian) (A, A′)-bimodule and (M′, h′) aλ-sesquilinear (resp. λ-hermitian)
leftA′-module. Then we may define aλ-sesquilinear (resp. λ-hermitian) form
h ⊗A′ h′:M ⊗A′M′×M ⊗A′M′→Qλ(A)
by
h ⊗A′h′(m1⊗A′m′1, m2⊗A′m′2) = h(m1h′(m′1, m′2), m2)
= h(m1, m2h′(m′
1, m′2)),
for anym1, m2∈M andm′1, m′
2∈M′. One easily verifies the tensor product
thus defined to be associative, and the form h⊗A′h′ to be A′′-compatible,
whenever (M′, h′) is.
SinceQλ(A) isα(λ)-closed and sincejλ:M⊗bA′M′×M⊗bA′M′→Qλ(A)
is a λ-isomorphism, the formh ⊗A′h′ defines a unique λ-sesquilinear (resp.
λ-hermitian) form h⊗bA′h′ : M⊗bA′M′ ×M⊗bA′M′ → Qλ(A) making the
M⊗bA′M′×M⊗bA′M′
commutative, cf. [5]. It thus makes sense to define therelative tensor product (M, h)⊗bA′(M′, h′) to be theλ-sesquilinear (resp. λ-hermitian) leftA-module
(M⊗bA′M′, h⊗bA′h′). 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 (A′, α′, λ′). Recall from [5,6] that any
relatively hermitically invertible (A, A′)-bimodule (P, h) determines an
equiv-alence of categories
(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(A′
, α′
, λ′
resp.
F :H(A, α, λ)→ H(A′, α′, λ′)
is said to be decent, if it factorizes through a category equivalence
F: (A, λ)-mod→(A′, λ′)-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)
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[(−), Qλ(A)])∼=α
′
A′[F(−), Qλ′(A′)]
such that for everyλ-sesquilinear leftA-module (M, l), we have a commutative diagram
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 A′-module, V a left A-module and W an
(A, A′)-bimodule, then the morphism
µ: [U,A[V, W]]A′ →A[V,[U, W]A′]
defined by (µ(f)(v))(u) = f(u)(v), for every f ∈ [U,A[V, W]]A′, u∈ U and
-linear and if U is an (A′′, A′)-bimodule, then µis right A′′-linear.
(3.4) Proposition (Morita I).Fix torsion triples (A, α, λ)and(A′, α′, λ′).
Then any relatively hermitically invertible (A′, A)-bimodule (Q, k) defines a
decent equivalence between the categoriesS(A, α, λ)andmathhcalS(A′, α′, λ′)
and the categories H(A, α, λ)andmathcalH(A′, α′, λ′).
Proof. Let (P, h) be an inverse for (Q, k). Define for every λ-closed left
A-moduleM the isomorphismηM as the composition of the following
isomor-phisms
Q⊗bAαA[M, Qλ(A)] ∼= A[P,αA[M, Qλ(A)]]∼=A[P,[Mα, Qλ(A)α]A]
∼
= [Mα,A[P, Qλ(A)α]]A∼= [Mα,A[P, Qλ(A)]]A
∼
= [Mα, Q] A∼=α
′
A[M, P]
∼
= αA′′[Q⊗bAM, Q⊗bAP]∼=α
′
A′[Q⊗bAM, Qλ′(A′)].
An easy verification shows that
ηM(q⊗bAf)(q′⊗bAm′) =k(q′f(m′), q)
and that η :Q⊗bAαA[(−), Qλ(A)]∼=α
′
A′[Q⊗bA(−), Qλ′(A′)]. So, for every m∈
M andq∈Q, we have that
ηM(q⊗bAla(m)) = (k⊗bAl)a(q⊗bAm),
i.e., (Q, k)⊗bA−is decent. By symmetry,P⊗bA′−is decent as well. 2 Conversely,
(3.5) Proposition (Morita II). Let (A, α, λ) and (A′, α′, λ′) be torsion
triples. Then every decent category equivalence betweenS(A, α, λ) andS(A′,
α′, λ′)(resp. H(A, α, λ) andmathcalH(A′, α′, λ′)) is induced by a relatively
hermitically invertible (A′, A)-bimodule.
Proof. Let F :H(A, α, λ)→ H(A′, α′, λ′) be a decent category equivalence,
then (Q, k) = F(Qλ(A), pQλ(A)) is a hermitically (λ
′, λ)-invertible (A′, A
)-bimodule. Let us now show that F(−) = (Q, k)⊗bA−.
As F = Q⊗bA− : (A, λ)-mod → (A′, λ′)-mod, we only have to verify
that F(l) =k⊗bAl, for everyλ-sesquilinear left A-module (M, l). Let
η:F(αA[(−), Qλ(A)])
∼
then
(3.6) Corollary. With the same notations, if G is an inverse for F, then
G= (P, h)⊗bA′−, where (P, h) is an inverse for (Q, k).
In particular,Gis also decent, as claimed before.
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