Injective Morita contexts (revisited)
Injective Morita Contexts (Revisited)
Dedicated to Prof. Robert Wisbauer
J. Y. Abuhlail ∗ S. K. Nauman
Department of Mathematics & Statistics Department of Mathematics
King Fahd University of Petroleum King AbdulAziz University
& Minerals, Box # 5046 P.O.Box 80203
31261 Dhahran (KSA) 21589 Jeddah (KSA)
abuhlail@kfupm.edu.sa synakhaled@hotmail.com
Abstract
This paper is an exposition of the so-called injective Morita contexts (in which
the connecting bimodule morphisms are injective) and Morita α-contexts (in which
the connecting bimodules enjoy some local projectivity in the sense of Zimmermann-
Huisgen). Motivated by situations in which only one trace ideal is in action, or
the compatibility between the bimodule morphisms is not needed, we introduce the
notions of Morita semi-contexts and Morita data, and investigate them. Injective
Morita data will be used (with the help of static and adstatic modules) to establish
equivalences between some intersecting subcategories related to subcategories of cat-
egories of modules that are localized or colocalized by trace ideals of a Morita datum.
We end up with applications of Morita α-contexts to ∗-modules and injective right
wide Morita contexts.
1 Introduction
Morita contexts, in general, and (semi-)strict Morita contexts (with surjective con-
necting bilinear morphisms), in particular, were extensively studied and developed expo-
nentially during the last few decades (e.g. [AGH-Z1997]). However, we sincerely feel that
there is a gap in the literature on injective Morita contexts (i.e. those with injective con-
necting bilinear morphisms). Apart from the results in [Nau1994-a], [Nau1994-b] (where
the second author initially explored this notion) and from an application to Grothendieck
groups in the recent paper ([Nau2004]), it seems that injective Morita contexts were not
studied systematically at all.
∗Corresponding Author
http://arxiv.org/abs/0704.0074v2
We noticed that in several results of ([Nau1993], [Nau1994-a] and [Nau1994-b]) that are
related to Morita contexts, only one trace ideal is used. Observing this fact, we introduce
the notions of Morita semi-contexts and Morita data and investigate them. Several results
are proved then for injective Morita semi contexts and/or injective Morita data.
Consider a Morita datum M = (T, S, P,Q,<,>T , <,>S), with not necessarily compat-
ible bimodule morphisms <,>T : P ⊗S Q→ T and <,>S: Q⊗T P → S. We say that M is
injective, iff <,>T and <,>S are injective, and to be a Morita α-datum, iff the associated
dual pairings Pl := (Q, TP ), Pr := (Q,PS), Ql := (P, SQ) and Qr := (P,QT ) satisfy
the α-condition (which is closely related to the notion of local projectivity in the sense of
Zimmermann-Huisgen [Z-H1976]). The α-condition was introduced in [AG-TL2001] and
further investigated by the first author in [Abu2005].
While (semi-)strict unital Morita contexts induce equivalences between the whole mod-
ule categories of the rings under consideration, we show in this paper how injective Morita
(semi-)contexts and injective Morita data play an important role in establishing equiva-
lences between suitable intersecting subcategories of module categories (e.g. intersections
of subcategories that are localized/colocalized by trace ideals of a Morita datum with sub-
categories of static/adstatic modules, etc.). Our main applications in addition to equiv-
alences related to the Kato-Ohtake-Müller localization-colocalization theory (developed in
[Kat1978], [KO1979] and [Mül1974]), will be to ∗-modules (introduced by Menini and Or-
satti [MO1989]) and to right wide Morita contexts (introduced by F. Castaño Iglesias and
J. Gómez-Torrecillas [C-IG-T1995]).
Most of our results will be stated for left modules, while deriving the “dual” versions for
right modules is left to the interested reader. Moreover, for Morita contexts, some results
are stated/proved for only one of the Morita semi-contexts, as the ones corresponding to
the second semi-context can be obtained analogously. For the convenience of the reader, we
tried to make the paper self-contained, so that it can serve as a reference on injective Morita
(semi-)contexts and their applications. In this respect, and for the sake of completeness, we
have included some previous results of the authors that are (in most cases) either provided
with new shorter proofs, or are obtained under weaker conditions.
This paper is organized as follows: After this brief introduction, we give in Section 2
some preliminaries including the basic properties of dual α-pairings, which play a central
role in rest of the work. The notions of Morita semi-contexts and Morita data are intro-
duced in Section 3, where we clarify their relations with the dual pairings and the so-called
elementary rngs. Injective Morita (semi-)contexts appear in Section 4, where we study
their interplay with dual α-pairings and provide some examples and a counter-example.
In Section 5 we include some observations regarding static and adstatic modules and use
them to obtain equivalences among suitable intersecting subcategories of modules related
to a Morita (semi-)context. In the last section, more applications are presented, mainly to
subcategories of modules that are localized or colocalized by a trace ideal of an injective
Morita (semi-)context, to ∗-modules and to injective right wide Morita contexts.
2 Preliminaries
Throughout, R denotes a commutative ring with 1R 6= 0R and A,A
′, B, B′ are unital
R-algebras. We have reserved the term “ring” for an associative ring with a multiplicative
unity, and we will use the term “rng” for a general associative ring (not necessarily with
unity). All modules over rings are assumed to be unitary, and ring morphisms are assumed
to respect multiplicative unities. If T and S are categories, then we write T ≤ S (T ≤ S)
to mean that T is a (full) subcategory of S, and T ≈ S to indicate that T and S are
equivalent.
Rngs and their modules
2.1. By an A-rng (T, µT ), we mean an (A,A)-bimodule T with an (A,A)-bilinear mor-
phism µT : T ⊗A T → T, such that µT ◦ (µT ⊗A idT ) = µT ◦ (idT ⊗A µT ). We call an A-rng
(T, µT ) an A-ring, iff there exists in addition an (A,A)-bilinear morphism ηT : A → T,
called the unity map, such that µT ◦ (ηT ⊗A idT ) = ϑ
T and µT ◦ (idT ⊗A ηT ) = ϑ
T (where
A⊗A T
≃ T and T ⊗A A
≃ T are the canonical isomorphisms). So, an A-ring is a unital
A-rng; and an A-rng is (roughly speaking) an A-ring not necessarily with unity.
2.2. A morphism of rngs (ψ : δ) : (T : A) → (T ′ : A′) consists of a morphism of R-algebras
δ : A→ A′ and an (A,A)-bilinear morphism ψ : T → T ′, such that µT ′◦χ
(A,A′)
(T ′,T ′)
◦(ψ⊗Aψ) =
ψ ◦µT (where χ
(A,A′)
(T ′,T ′)
: T ′ ⊗A T
′ → T ′ ⊗A′ T
′ is the canonical map induced by δ). By RNG
we denote the category of associative rngs with morphisms being rng morphisms, and
by URNG < RNG the (non-full) subcategory of unital rings with morphisms being the
morphisms in RNG which respect multiplicative unities.
2.3. Let (T, µT ) be an A-rng. By a left T -module we mean a left A-module N with a left
A-linear morphism φNT : T ⊗AN → N, such that φ
T ◦ (µT ⊗A idN) = φ
T ◦ (idT ⊗A φ
T ). For
left T -modulesM,N, we call a left A-linear morphism f :M → N a T -linear morphism,
iff f(tm) = tf(m) for all t ∈ T. The category of left T -modules and left T -linear morphisms
is denoted by TM. The category MT of right T -modules is defined analogously. Let (T : A)
and (T ′ : A′) be rngs. We call an (A,A′)-bimodule N a (T, T ′)-bimodule, iff (N, φNT ) is
a left T -module and (N, φNT ′) is a right T
′-module, such that φNT ′ ◦ (φ
T ⊗A′ idT ′) = φ
(idT ⊗Aφ
T ′). For (T, T
′)-bimodulesM,N, we call an (A,A′)-bilinear morphism f :M → N
(T, T ′)-bilinear, provided f is left T -linear and right T ′-linear. The category of (T, T ′)-
bimodules is denoted by TMT ′ . In particular, for any A-rng T, a left (right) T -module
M has a canonical structure of a unitary right (left) S-module, where S := End(TM)
(S := End(MT )); and moreover, with this structure M becomes a (T, S)-bimodule (an
(S, T )-bimodule).
Remark 2.4. Similarly, one can define rngs over arbitrary (not-necessarily unital) ground
rngs and rng morphisms between them. Moreover, one can define (bi)modules over such
rngs and (bi)linear morphisms between them.
Notation. Let T be an A-rng. We write TU (UT ) to denote that U is a left (right) T -
module. For a left (right) T -module TU, we consider the set
∗U := HomT−(U, T ) (U
Hom−T (U, T )) of all left (right) T -linear morphisms from U to T with the canonical right
(left) T -module structure.
Generators and cogenerators
Definition 2.5. Let T be an A-rng. For a left T -module TU consider the following sub-
classes of TM :
Gen(TU) := {TV | ∃ a set Λ and an exact sequence U
(Λ) → V → 0};
Cogen(TU) := {TW | ∃ a set Λ and an exact sequence 0 → W → U
Pres(TU) := {TV | ∃ sets Λ1,Λ2 and an exact sequence U
(Λ2) → U (Λ1) → V → 0};
Copres(TU) := {TW | ∃ sets Λ1,Λ2 and an exact sequence 0 →W → U
Λ1 → UΛ2};
A left T -module in Gen(TU) (respectively Cogen(TU), Pres(TU), Copres(TU)) is said to be
U-generated (respectively U-cogenerated, U-presented, U-copresented). Moreover,
we say that TU is a generator (respectively cogenerator, presentor, copresentor), iff
Gen(TU) = TM (respectively Cogen(TU) = TM, Pres(TU) = TM, Copres(TU) = TM).
Dual α-pairings
In what follows we recall the definition and properties of dual α-pairings introduced
in [AG-TL2001, Definition 2.3.] and studied further in [Abu2005].
2.6. Let T be an A-rng. A dual left T -pairing Pl = (V, TW ) consists of a left T -module
W and a right T -module V with a right T -linear morphism κPl : V →
∗W (equivalently
a left T -linear morphism χPl : W → V
∗). For dual left pairings Pl = (V, TW ), P
l = (V
′), a morphism of dual left pairings (ξ, θ) : (V ′,W ′) → (V,W ) consists of a triple
(ξ, θ : ς) : (V, TW ) → (V
′, T ′W
where ξ : V → V ′ and θ : W ′ → W are T -linear and ς : T → T ′ is a morphism of rngs,
such that considering the induced maps <,>T : V ×W → T and <,>T ′: V
′ ×W ′ → T ′ we
< ξ(v), w′ >T ′= ς(< v, θ(w
′) >T ) for all v ∈ V and w
′ ∈ W ′. (1)
The dual left pairings with the morphisms defined above build a category, which we denote
by Pl. With Pl(T ) ≤ Pl we denote the full subcategory of dual T -pairings. The category
Pr of dual right pairings and its full subcategory Pr(T ) ≤ Pr of dual right T -pairings are
defined analogously.
Remark 2.7. The reader should be warned that (in general) for a non-commutative rng T
and a dual left T -pairing Pl = (V, TW ), the following map induced by the right T -linear
morphism κPl : V →
<,>T : V ×W → T, < v, w >T := κPl(v)(w)
is not necessarily T -balanced, and so does not induce (in general) a map V ⊗T W → T. In
fact, for all v ∈ V, w ∈ W and t ∈ T we have
< vt, w > = κPl(vt)(w) = [κPl(v)t](w) = [κPl(v)(w)]t = < v,w >T t;
< v, tw > = κPl(v)(tw) = t[κPl(v)(w)] = t < v, w >T .
2.8. Let T be an A-rng, N,W be left T -modules and identify NW with the set of all
mappings fromW toN. Considering N with the discrete topology andNW with the product
topology, the induced relative topology on HomT−(W,N) →֒ N
W is a linear topology (called
the finite topology), for which the basis of neighborhoods of 0 is given by the set of
annihilator submodules:
Bf (0) := {F
⊥(HomT−(W,N)) | F = {w1, ..., wk} ⊂W is a finite subset},
where
F⊥(HomT−(W,N)) := {f ∈ HomT−(W,N)) | f(W ) = 0}.
2.9. Let T be an A-rng, Pl = (V, TW ) a dual left T -pairing and consider for every right
T -module UT the following canonical map
U : U ⊗T W → Hom−T (V, U),
ui ⊗T wi 7→ [v 7→
ui < v,wi >T ]. (2)
We say that Pl = (V, TW ) ∈ Pl(T ) satisfies the left α-condition (or is a dual left α-
pairing), iff α
U is injective for every right T -module UT . By P
l (T ) ≤ Pl(T ) we denote the
full subcategory of dual left T -pairings satisfying the left α-condition. The full subcategory
of dual right α-pairings Pαr (T ) ≤ Pr(T ) is defined analogously.
Definition 2.10. Let T be an A-rng, Pl = (V, TW ) be a dual left T -pairing and consider
κPl : V →
∗W and α
V : V ⊗T W → End(VT ).
We say Pl ∈ Pl(T ) is
dense, iff κPl(V ) ⊆
∗W is dense (w.r.t. the finite topology on ∗W →֒ TW );
injective (resp. semi-strict, strict), iff α
V is injective (resp. surjective, bijective);
non-degenerate, iff V
→֒ ∗W and W
→֒ V ∗ canonically.
2.11. Let T be an A-rng. We call a T -module W locally projective (in the sense of B.
Zimmermann-Huisgen [Z-H1976]), iff for every diagram of T -modules
0 // F
g′◦ι   
ι //W
// N // 0
with exact rows and finitely generated T -submodule F ⊆W : for every T -linear morphism
g : W → N, there exists a T -linear morphism g′ : W → L, such that g ◦ ι = π ◦ g′ ◦ ι.
For proofs of the following basic properties of locally projective modules and dual
α-pairings see [Abu2005] and [Z-H1976]:
Proposition 2.12. Let T be an A-ring and Pl = (V, TW ) ∈ Pl(T ).
1. The left T -module TW is locally projective if and only if (
∗W,W ) is an α-pairing.
2. The left T -module TW is locally projective, iff for any finite subset {w1, ..., wk} ⊆ W,
there exists {(fi, w̃i)}
i=1 ⊂
∗W ×W such that wj =
fi(wj)w̃i for all j = 1, ..., k.
3. If TW is locally projective, then TW is flat and T -cogenerated.
4. If Pl ∈ P
l (T ), then TW is locally projective.
5. If TW is locally projective and κP (V ) ⊆
∗W is dense, then Pl ∈ P
l (T ).
6. Assume TT is an injective cogenerator. Then Pl ∈ P
l (T ) if and only if TW is locally
projective and κPl(V ) ⊆
∗W is dense.
7. If T is a QF ring, then Pl ∈ P
l (T ) if and only if TW is projective and W
→֒ V ∗.
The following result completes the nice observation [BW2003, 42.13.] about locally
projective modules:
Proposition 2.13. Let T be a ring, TW a left T -module, S := End(TW )
op and consider
the canonical (S, S)-bilinear morphism
[, ]W :
∗W ⊗T W → End(TW ), f ⊗T w 7→ [w̃ 7→ f(w̃)w].
1. TW is finitely generated projective if and only if [, ]W is surjective.
2. TW is locally projective if and only if Im([, ]W ) ⊆ End(TW ) is dense.
Proof. 1. This follows by [Fai1981, 12.8.].
2. Assume TW is locally projective and consider for every left T -module N the canonical
mapping
[, ]WN :
∗ W ⊗T N → HomT (W,N), f ⊗T n 7→ [w̃ 7→ f(w̃)n].
It follows then by [BW2003, 42.13.], that Im([, ]WN ) ⊆ HomT (W,N) is dense. In
particular, setting N = W we conclude that Im([, ]W ) ⊆ End(TW ) is dense. On
the other hand, assume Im([, ]W ) ⊆ End(TW ) is dense. Then for every finite subset
{w1, ..., wk} ⊆ W, there exists
g̃i ⊗T w̃i ∈
∗W ⊗T W with
wj = idW (wj) = [, ]W (
g̃i ⊗T w̃i)(wj) =
g̃i(wj)w̃i for j = 1, ..., k.
It follows then by Proposition 2.12 “2” that TW is locally projective.�
3 Morita (Semi)contexts
We noticed, in the proofs of some results on equivalences between subcategories of
module categories associated to a given Morita context, that no use is made of the com-
patibility between the connecting bimodule morphisms (or even that only one trace ideal
is used and so only one of the two bilinear morphisms is really in action). Some results of
this type appeared, for example, in [Nau1993], [Nau1994-a] and [Nau1994-b]. Moreover,
in our considerations some Morita contexts will be formed for arbitrary associative rngs
(i.e. not necessarily unital rings). These considerations motivate us to make the following
general definitions:
3.1. By a Morita semi-context we mean a tuple
mT = ((T : A), (S : B), P, Q,<,>T , I), (3)
where T is an A-rng, S is a B-rng, P is a (T, S)-bimodule, Q is an (S, T )-bimodule,
<,>T : P ⊗S Q → T is a (T, T )-bilinear morphism and I := Im(<,>T ) ⊳ T (called the
trace ideal associated to mT ).We drop the ground rings A,B and the trace ideal I ⊳ T,
if they are not explicitly in action. If mT (3) is a Morita semi-context and T, S are unital
rings, then we call mT a unital Morita semi-context.
3.2. Let mT = ((T : A), (S : B), P, Q,<,>T ), mT ′ = ((T
′ : A′), (S ′ : B′), P ′, Q′, <,>T ′) be
Morita semi-contexts. By a morphism of Morita semi-contexts from mT to mT ′ we
mean a four fold set of morphisms
((β : δ), (γ : σ), φ, ψ) : ((T : A), (S : B), P, Q) → ((T ′ : A′), (S ′ : B′), P ′, Q′),
where (β : δ) : (T : A) → (T ′ : A′) and (γ : σ) : (S : B) → (S ′ : B′) are rng morphisms,
φ : P → P ′ is (T, S)-bilinear and ψ : Q→ Q′ is (S, T )-bilinear, such that
β(< p, q >T ) =< φ(p), ψ(q) >T ′ for all p ∈ P, q ∈ Q .
Notice that we consider P ′ as a (T, S)-bimodule and Q′ as an (S, T )-bimodule with actions
induced by the morphism of rngs (β : δ) and (γ : σ). By MSC we denote the category
of Morita semi-contexts with morphisms defined as above, and by UMSC < MSC the
(non-full) subcategory of unital Morita semi-contexts.
Morita semi-contexts are closely related to dual pairings in the sense of [Abu2005]:
3.3. Let (T, S, P,Q,<,>T ) ∈ MSC and consider the canonical isomorphisms of Abelian
groups
Hom(S,T )(Q,
≃ Hom(T,T )(P ⊗S Q, T )
≃ Hom(T,S)(P,Q
This means that we have two dual T -pairings Pl := (Q, TP ) ∈ Pl(T ) and Qr := (P,QT ) ∈
Pr(T ), induced by the canonical T -linear morphisms
κPl := ξ
−1(<,>T ) : Q→
∗P and κQr := ζ(<,>T ) : P → Q
On the other hand, let (S, T,Q, P,<,>S) ∈ MSC and consider the canonical isomorphisms
of Abelian groups
Hom(S,T )(Q,P
≃ Hom(S,S)(Q⊗T P, S)
≃ Hom(T,S)(P,
Then we have two dual S-pairings Pr := (Q,PS) ∈ Pr(S) and Ql := (P, SQ) ∈ Pl(S),
induced by the canonical morphisms
κPr := ξ
′−1(<,>S) : Q→ P
∗ and κQr := ζ
′(<,>S) : P →
3.4. By a Morita datum we mean a tuple
M = ((T : A), (S : B), P, Q,<,>T , <,>S, I, J), (4)
where the following are Morita semi-contexts.
MT := ((T : A), (S : B), P, Q,<,>T , I) and MS := ((S : B), (T : A), Q, P,<,>S, J) (5)
If, moreover, the bilinear morphisms <,>T : P ⊗S Q → T and < −, >S: Q ⊗T P → S are
compatible, in the sense that
< q, p >S q
′ = q < p, q′ >T and p < q, p
′ >S =< p, q >T p
′ ∀ p, p′ ∈ P, q, q′ ∈ Q, (6)
then we call M a Morita context. If T, S in a Morita datum (context) M are unital,
then we call M a unital Morita datum (context).
3.5. LetM = ((T : A), (S : B), P, Q,<,>T , <,>S) andM
′ = ((T ′ : A′), (S ′ : B′), P ′, Q′, <
,>T ′, <,>S′) be Morita contexts. Extending [Ami1971, Page 275], we mean by a mor-
phism of Morita contexts from M to M′ a four fold set of maps
((β : δ), (γ : σ), φ, ψ) : ((T : A), (S : B), P, Q) → ((T ′ : A′), (S ′ : B′), P ′, Q′),
where (β : δ) : (T : A) → (T ′ : A′), (γ : σ) : (S : B) → (S ′ : B′) are rng morphisms,
φ : P → P ′ is (T, S)-bilinear and ψ : Q→ Q′ is (S, T )-bilinear, such that
β(< p, q >T ) =< φ(p), ψ(q) >T ′ and γ(< q, p >S) =< ψ(q), φ(p) >S′ ∀ p ∈ P, q ∈ Q.
By MC we denote the category of Morita contexts with morphisms defined as above, and
by UMC <MC the (non-full) subcategory of unital Morita contexts.
Example 3.6. If R is commutative, then any Morita semi-context (R,R, P,Q,<,>R) yields
a Morita context (R,R, P,Q,<,>R, [, ]R), where [, ]R := Q⊗R P ≃ P ⊗R Q
−→ R.�
3.7. We call a Morita semi-context mT = (T, S, P,Q,<,>T ) semi-derived (derived),
iff S := End(TP )
op (and Q = ∗P ). We call a Morita datum, or a Morita context, M =
(T, S, P,Q,<,>T , <,>S) semi-derived (derived), iff S = End(TP )
op, or T = End(PS)
(S = End(TP )
op and Q = ∗P, or T = End(PS) and Q = P
Remark 3.8. Following [Cae1998, 1.2.] (however, dropping the condition that the bilinear
map <,>T : P ⊗SQ→ T is surjective), Morita semi-contexts (T, S, P,Q,<>T ) in our sense
were called dual pairs in [Ver2006]. However, we think the terminology we are using is more
informative and avoids confusion with other notions of dual pairings in the literature (e.g.
the ones studied by the first author in [Abu2005]). The reason for this specific terminology
(i.e. Morita semi-contexts) is that every Morita context contains two Morita semi-contexts
as clear from the definition; and that any Morita semi-context can be extended to a (not
necessarily unital) Morita context in a natural way as explained below.
Elementary rngs
In what follows we demonstrate how to build new Morita (semi-)contexts from a given
Morita semi-context. These constructions are inspired by the notion of elementary rngs in
[Cae1998, 1.2.] (and [Ver2006, Remark 3.8.]):
Lemma 3.9. Let mT := ((T : A), (S : B), P, Q,<,>T ) ∈ MSC.
1. The (T, T )-bimodule T := P ⊗S Q has a structure of a T -rng (A-rng) with multipli-
cation
(p⊗S q) ·T (p
′ ⊗S q
′) :=< p, q >T p
′ ⊗S q
′ ∀ p, p′ ∈ P, q, q′ ∈ Q,
such that <,>T : T → T is a morphism of A-rngs, P is a (T, S)-bimodule and Q is
an (S,T)-bimodule, where
(p⊗S q)⇀ p̃ :=< p, q >T p̃ and q̃ ↼ (p⊗S q) := q̃ < p, q >T .
Moreover, we have morphisms of T -rngs (A-rngs)
ψ : T → End(PS), p⊗S q 7→ [p̃ 7→< p, q >T p̃];
φ : T → End(SQ)
op, p⊗S q 7→ [q̃ 7→ q̃ < p, q >T ],
((T : A), (S : B), P, Q, idT) ∈ MSC and we have a morphism of Morita semi-contexts
(<,>T , idS, , idP , idQ) : (T, S, P,Q, idT) → (T, S, P,Q,<,>T ).
2. The (S, S)-bimodule S := Q⊗T P has a structure of an S-rng (B-rng) with multipli-
cation
(q⊗T p) ·S (q
′ ⊗T p
′) := q < p, q′ >T ⊗T p
′ = q⊗T < p, q
′ >T p
′ ∀ p, p′ ∈ P, q, q′ ∈ Q,
such that <,>S: S → S is a morphism of B-rngs, P is a (T,S)-bimodule and Q is
an (S, T )-bimodule, where
p̃ ↼ (q ⊗T p) :=< p̃, q >T p and (q ⊗T p)⇀ q̃ := q < p, q̃ >T .
Moreover, we have morphisms of S-rngs (B-rngs)
Ψ : S → End(TP )
op, q ⊗T p 7→ [p̃ 7→< p̃, q >T p],
Φ : S → End(QT ), q ⊗T p 7→ [q̃ 7→ q < p, q̃ >T ],
and M := ((T : A), (S : B), P, Q,<,>T , idS) is a Morita context.
Remarks 3.10. 1. Given ((S : B), (T : A), Q, P,<,>S) ∈ MSC, the (S, S)-bimodule
S := Q⊗T P becomes an S-rng with multiplication
(q ⊗T p) ·S (q
′ ⊗T p
′) :=< q, p >S q
′ ⊗T p
′ ∀ p, p′ ∈ P, q, q′ ∈ Q;
and the (T, T )-bimodule T := P ⊗S Q becomes a T -rng with multiplication
(p⊗S q) ·T (p
′⊗S q
′) := p < q, p′ >S ⊗S q
′ = p⊗S < q, p
′ >S q
′ ∀ p, p′ ∈ P, q, q′ ∈ Q.
Analogous results to those in Lemma 3.9 can be obtained for the S-rng S and the
T -rng T.
2. Given a Morita semi-context (T, S, P,Q,<,>T ) several equivalent conditions for the
T -rng T := P ⊗S Q to be unital and the modules TP, QT to be firm can be found in
[Ver2006, Theorem 3.3.]. Analogous results can be formulated for the S-rng Q⊗T P
and the S-modules PS, SQ corresponding to any (S, T,Q, P,<,>S) ∈ MSC.
Proposition 3.11. 1. Let mT = (T, S, P,Q,<,>T ) ∈ UMSC and assume the A-rng
T := P ⊗S Q to be unital. If <,>T : T → T respects unities (and mT is injective),
then <,>T is surjective (T
≃ T as A-rings).
2. Let mS = (S, T,Q, P,<,>S) ∈ UMSC and assume the B-rng S := Q ⊗S P to
be unital. If <,>S: S → S respects unities (and mS is injective), then <,>S is
surjective (S
≃ S as B-rings).
3. Let M = (T, S, P,Q,<,>T , <,>S) ∈ UMC and assume the rngs T := P ⊗S Q, T,
S := Q⊗S P to be unital. If <,>T : P ⊗S Q → T and <,>S: S → S respect unities,
then T
≃ T as A-ring, S
≃ S as B-rings and we have equivalences of categories
TM ≈ SM (and MT ≈ MS).
Proof. Assume T is unital with 1T =
pi ⊗S qi. If <,>T respects unities, then we
< pi, qi >T= 1T , and so for any t ∈ T we get t = t1T =
t < pi, qi >T=∑n
< tpi, qi >T∈ Im(<,>T ). One can prove “2” analogously. As for “3”, it is well
known that a unital Morita context with surjective connecting bimodule morphisms is
strict (e.g. [Fai1981, 12.7.]), hence T
≃ T, S
≃ S. The equivalences of categories
TM ≃ TM ≈ SM ≃ SM (and MT ≃ MT ≈ MS ≃ MS) follow then by classical Morita
Theory (e.g. [Fai1981, Chapter 12]).�
Definition 3.12. Let T be an A-rng, VT a right T -module and consider for every left
T -module TL the annihilator
ann⊗L(VT ) := {l ∈ L | V ⊗T l = 0}.
Following [AF1974, Exercises 19], we say VT is L-faithful, iff ann
L(VT ) = 0; and to be
completely faithful, iff VT is L-faithful for every left T -module SL. Similarly, we can
define completely faithful left T -modules.
Under suitable conditions, the following result characterizes the Morita data, which
are Morita contexts:
Proposition 3.13. Let M = (T, S, P,Q,<,>T , <,>S) be a Morita datum.
1. If M ∈ MC, then S
≃ S and T
≃ T as rngs.
2. Assume TP is Q-faithful and QT is P -faithful. Then M ∈ MC if and only if S
and T
≃ T as rngs.
Proof. 1. Obvious.
2. Assume S
≃ S and T
≃ T as rngs. If p ∈ P and q, q′ ∈ Q are arbitrary, then we have
for any p̃ ∈ P :
< q, p >S q
′ ⊗T p̃ = (q ⊗T p) ·S (q
′ ⊗T p̃) = (q ⊗T p) ·S (q
′ ⊗T p̃) = q < p, q
′ >T ⊗T p̃,
hence < q, p >S q
′ − q < p, q′ >T∈ annQ(P ) = 0 (since TP is Q-faithful), i.e.
< q, p >S q
′ = q < p, q′ >T for all p ∈ P and q, q
′ ∈ Q. Assuming QT is P -faithful,
one can prove analogously that < p, q >T p
′ = p < q, p′ >S for all p, p
′ ∈ P and
q ∈ Q. Consequently, M is a Morita context.�
4 Injective Morita (Semi-)Contexts
Definition 4.1. We call a Morita semi-context mT = (T, S, P,Q,<,>T , I) :
injective (resp. semi-strict, strict), iff <,>T : P ⊗S Q→ T is injective (resp. surjec-
tive, bijective);
non-degenerate, iff Q →֒ ∗P and P →֒ Q∗ canonically;
Morita α-semi-context, iff Pl := (Q, TP ) ∈ P
l (T ) and Qr := (P,QT ) ∈ P
r (T ).
Notation. By MSCα ≤ MSC (UMSCα ≤ UMSC) we denote the full subcategory of
(unital) Morita semi-contexts satisfying the α-condition. Moreover, we denote by IMSC ≤
MSC (IUMSC ≤ UMSC) the full subcategory of injective (unital) Morita semi-contexts.
Definition 4.2. We say a Morita datum (context) M = (T, S, P,Q,<,>T , <,>S, I, J) :
is injective (resp. semi-strict, strict), iff <,>T : P⊗SQ→ T and <,>S: Q⊗T P → S
are injective (resp. surjective, bijective);
is non-degenerate, iff Q →֒ ∗P, P →֒ Q∗, Q →֒ P ∗ and P →֒ ∗Q canonically;
satisfies the left α-condition, iffPl := (Q, TP ) ∈ P
l (T ) andQl := (P, SQ) ∈ P
l (S);
satisfies the right α-condition, iff Qr := (P,QT ) ∈ P
r (T ) and Pr := (Q,PS) ∈
Pαr (S);
satisfies the α-condition, or M is a Morita α-datum (Morita α-context), iff M
satisfies both the left and the right α-conditions.
Notation. By MCαl < MC (UMC
l < UMC) we denote the full subcategory of Morita
contexts satisfying the left α-condition, and by MCαr < MC (UMC
r < UMC) the full
subcategory of (unital) Morita contexts satisfying the right α-condition. Moreover, we set
α := MCαl ∩MC
r and UMC
α := UMCαl ∩ UMC
Lemma 4.3. Let M = (T, S, P,Q,<,>T , <,>S, I, J) ∈ MC. Consider the Morita semi-
context MS := (S, T,Q, P,<,>S), the dual pairings Pl := (Q, TP ) ∈ Pl(T ), Qr :=
(P,QT ) ∈ Pr(T ) and the canonical morphisms of rings
ρP : S → End(TP )
op and λQ : S → End(QT ).
1. If Qr is injective (semi-strict), then MS is injective (ρP : S → End(TP )
op is a
surjective morphism of B-rngs).
2. Assume PS is faithful and let Qr be semi-strict. Then S ≃ End(TP )
op (an isomor-
phism of unital B-rings) and MS is strict.
3. If Pl is injective (semi-strict), then MS is injective (λQ : S → End(QT ) is a surjec-
tive morphism of B-rngs).
4. Assume SQ is faithful and let Pl is semi-strict. Then S ≃ End(QT ) (an isomorphism
of unital B-rings) and MS is strict.
Proof. We prove only “1” and “2”, as “3” and “4” can be proved analogously.
Consider the following butterfly diagram with canonical morphisms
Q⊗T Q
∗P ⊗T P
Q⊗T P
idQ⊗TκQr
llYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY
⊗T idP
22eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee
uujjj
Hom−T (
∗P,Q)
(κPl ,Q)
Hom−T (Q
∗, P )
(κQr ,P )
ttjjjj
**TTT
End(QT ) End(TP )
qi ⊗T pi ∈ Q⊗T P be arbitrary. For every p̃ ∈ P we have
[(κQr , P ) ◦ α
qi ⊗T pi)](p̃) =
< p̃, qi >T pi
p̃ < qi, pi >S
= ρP (
< qi, pi >S)(p̃)
= (ρP◦ <,>S)(
qi ⊗T pi)(p̃),
i.e. α
P := (κQr , P ) ◦ α
P = ρP◦ <,>S; and
[, ]lP ◦ (κPl ⊗T idP ))(
qi ⊗T pi)](p̃) =
κPl(qi)(p̃)pi
< p̃, qi >T pi
p̃ < qi, pi >S
= ρP (
< qi, pi >S)(p̃)
= [(ρP◦ <,>S)(
qi ⊗T pi)](p̃),
i.e. [, ]lP ◦ (κPl ⊗T idP ) = ρP◦ <,>S . On the other hand, for every q̃ ∈ Q we have
((κPl , Q) ◦ α
qi ⊗T pi)(q̃) =
qi < pi, q̃ >T
< qi, pi >S)q̃
= λQ(
< qi, pi >S)(q̃)
= (λQ◦ <,>S)(
qi ⊗T pi),
i.e. α
Q := (κPl , Q) ◦ α
Q = λQ◦ <,>S and
([, ]rQ ◦ (idQ ⊗T κQr))(
qi ⊗T pi)](q̃) =
qiκQr(pi)(q̃)
qi < pi, q̃ >T
< qi, pi >S q̃
= λQ(
< qi, pi >S)(q̃)
= [(λQ◦ <,>S)(
qi ⊗T pi)](q̃),
i.e. [, ]rQ ◦ (idQ ⊗T κQr) = λQ◦ <,>S . Hence Diagram (7) is commutative.
(1) Follows directly from the assumptions and the equality α
P = ρP◦ <,>S .
(2) Let PS be faithful, so that the canonical left S-linear map ρP : S → End(TP )
is injective. Assume now that Qr is semi-strict. Then ρP is surjective by “1” , whence
bijective. Since rings of endomorphisms are unital, we conclude that S ≃ End(TP )
op is a
unital B-ring as well (with unity ρ−1P (idP )). Moreover, the surjectivity of α
P = ρP◦ <,>S
implies that <,>S is surjective (since ρP is injective), say 1S =
< q̃j, p̃j >S for some
{(q̃j , p̃j)}J ⊆ Q× P. For any
qi ⊗T pi ∈ Ker(<,>S), we have then
qi ⊗T pi = (
qi ⊗T pi) · 1S =
(qi ⊗T pi) · (
< q̃j, p̃j >S)
qi ⊗T pi < q̃j, p̃j >S =
qi⊗T < pi, q̃j >T p̃j
qi < pi, q̃j >T ⊗T p̃j =
< qi, pi >S q̃j ⊗T p̃j
< qi, pi >S)q̃j ⊗T p̃j = 0,
i.e. <,>S is injective, whence an isomorphism.�
The following result shows that Morita α-contexts are injective:
Corollary 4.4. MCαl ∪MC
r ≤ IMC.
Example 4.5. Let mT = (T, S, P,Q,<,>T ) be a non-degenerate Morita semi-context. If
T is a QF ring and the T -modules TP, QT are projective, then by Proposition 2.12 “7”
Pl := (Q, TP ) ∈ P
l (T ) and Qr := (P,QT ) ∈ P
r (T ) (i.e. mT is a Morita α-semi-
context, whence injective). On the other hand, let M = (T, S, P,Q,<,>T , <,>S) be a
non-degenerate Morita datum. If T, S are QF rings and the modules TP, QT , PS, SQ are
projective, then M is an Morita α-datum (whence injective).�
Every semi-strict unital Morita context is injective (whence strict, e.g. [Fai1981, 12.7.]).
The following example, which is a modification of [Lam1999, Example 18.30]), shows that
the converse is not necessarily true:
Example 4.6. Let T = M2(Z2) be the ring of 2× 2 matrices with entries in Z2. Notice that
∈ T is an idempotent, and that eTe ≃ Z2 as rings. Set
P := Te = {
| a′, c′ ∈ Z2} and Q := eT = {
| a, b ∈ Z2}.
Then P = Te is a (T, eTe)-bimodule and Q = eT is an (eTe, T )-bimodule. Moreover, we
have a Morita context
Me = (T, eTe, T e, , eT, <,>T , < . >eTe),
where the connecting bilinear maps are
<,>T : Te⊗eTe eT → T,
a′a a′b
c′a c′b
<,>eTe : eT ⊗T Te → eTe
aa′ + bc′ 0
Straightforward computations show that<,>T is injective but not surjective (as
Im(<,>T )) and that <,>eTe is in fact an isomorphism. This means that Me is an injective
Morita context that is not semi-strict (whence not strict).�
Definition 4.7. Let T be a rng and I ⊳ T an ideal. For every left T -module TV consider
the canonical T -linear map
ζI,V : V → HomT (I, V ), v 7→ [t 7→ tv].
We say T I is strongly V -faithful, iff annV (I) := Ker(ζI,V ) := 0. Moreover, we say I is
strongly faithful, if T I is V -faithful for every left T -module TV. Strong faithfulness of I
w.r.t. right T -modules can be defined analogously.
Remark 4.8. Let T be a rng, I ⊳ T an ideal and TU a left ideal. It’s clear that ann
U(IT ) ⊆
annU(I) := Ker(ζI,U). Hence, if T I is strongly U-faithful, then IT is U-faithful (which
justifies our terminology). In particular, if T I is strongly faithful, then IT is completely
faithful.
Morita α-contexts are injective by Corollary 4.4. The following result gives a
partial converse:
Lemma 4.9. Let M = (T, S, P,Q,<,>T , <,>S, I, J) ∈ MC and assume the Morita semi-
context MS := (S, T,Q, P,<,>S, J) is injective.
1. If SJ is strongly faithful, then Qr := (P,QT ) ∈ P
r (T ).
2. If JS is strongly faithful, then Pl := (Q, TP ) ∈ P
l (T ).
Proof. We prove only “1”, since “2” can be proved similarly. Assume MS is injective and
consider for every left T -module U the following diagram
Q⊗T U
ζJ,Q⊗T U ((QQ
HomT−(P, U)
ψQ,Uuukkk
HomS−(J,Q⊗T U)
where for all f ∈ HomT−(P, U) and
< qj , pj >S∈ J we define
ψQ,U(f)(
< qj , pj >S) :=
qj ⊗T f(pj).
Then we have for every
q̃i ⊗T ũi ∈ Q⊗T U and s =
< qj, pj >S∈ J :
(ψQ,U ◦ α
q̃i ⊗T ũi)(s) =
qj ⊗T [α
q̃i ⊗T ũi)](pj)
qj ⊗T
< pj , q̃i >T ũi]
qj⊗T < pj, q̃i >T ũi
qj < pj , q̃i >T ⊗T ũi
< qj , pj >S q̃i ⊗T ũi
= ζJ,Q⊗TU(
q̃i ⊗T ũi)(s),
i.e. diagram (8) is commutative. If SJ is strongly faithful, then Ker(ζJ,Q⊗TU) = annQ⊗TU(J) =
0, hence ζJ,Q⊗TU is injective and it follows then that α
U is injective.�
Proposition 4.10. Let M = (T, S, P,Q,<,>T , <,>S, I, J) ∈ IMC. If T I, IT , SJ and JS
are strongly faithful, then M ∈ MCα.
5 Equivalences of Categories
In this section we give some applications of injective Morita (semi-)contexts and in-
jective Morita data to equivalences between suitable subcategories of modules arising in
the Kato-Müller-Ohtake localization-colocalization theory (as developed in (e.g. [Kat1978],
[KO1979], [Mül1974]). All rings, hence all Morita (semi-)contexts and data, in this section
are unital.
Static and Adstatic Modules
5.1. ([C-IG-TW2003]) Let A and B be two complete cocomplete Abelian categories, R :
A → B an additive covariant functor with left adjoint L : B → A and let
ω : LR → 1A and η : 1B → RL
be the induced natural transformations (called the counit and the unit of the adjunction,
respectively). Related to the adjoint pair (L,R) are two full subcategories of A and B :
Stat(R) := {X ∈ A | LR(X)
≃ X} and Adstat(R) := {Y ∈ B | Y
≃ RL(Y )},
whose members are called R-static objects and R-adstatic objects, respectively. It is
evident (from definition) that we have equivalence of categories Stat(R) ≈ Adstat(R).
A typical situation, in which static and adstatic objects arise naturally is the
following:
5.2. Let T, S be rings, TUS a (T, S)-bimodule and consider the covariant functors
HlU := HomT (U,−) : TM → SM and T
U := U ⊗S − : SM → TM.
It is well-known that (TlU ,H
U) is an adjoint pair of covariant functors via the natural
isomorphisms
HomT (U ⊗S M,N) ≃ HomS(M,HomT (U,N)) for all M ∈ SM and N ∈ TM
and the natural transformations
ωlU : U ⊗S HomT (U,−) → 1TM and η
U : 1SM → HomT (U, U ⊗S −)
yield for every TK and SL the canonical morphisms
ωlU,K : U ⊗S HomT (U,K) → K and η
U,L : L→ HomT (U, U ⊗S L). (9)
We call the HlU-static modules U-static w.r.t. S and set
Statl(TUS) := Stat(H
U) = {TK | U ⊗S HomT−(U,K)
≃ K};
and the HlU-adstatic modules U-adstatic w.r.t. S and set
Adstatl(TUS) := Adstat(H
U) = {SL | L
≃ HomT−(U, U ⊗S L)}.
By [Nau1990a] and [Nau1990b], there are equivalences of categories
Statl(TUS) ≈ Adstat
l(TUS). (10)
On the other hand, one can define the full subcategories Statr(TUS) ≈ Adstat
r(TUS) :
Statr(TUS) := {KS | Hom−S(U,K)⊗T U ≃ K};
Adstatr(TUS) := {LT | L ≃ Hom−S(U, L⊗T U)}.
In particular, setting
Stat(TU) := Stat
l(TUEnd(TU)op); Adstat(TU) := Adstat
l(TUEnd(TU)op);
Stat(US) := Stat
r(End(SU)US); Adstat(US) := Adstat
r(End(SU)US),
there are equivalences of categories:
Stat(TU) ≃ Adstat(TU) and Stat(US) ≃ Adstat(US). (11)
Remark 5.3. The theory of static and adstatic modules was developed in a series of papers
by the second author (see the references). They were also considered by several other
authors (e.g. [Alp1990], [CF2004]). For other terminologies used by different authors, the
interested reader may refer to a comprehensive treatment of the subject by R. Wisbauer
in [Wis2000].
Intersecting subcategories
Several intersecting subcategories related to Morita contexts were introduced in
the literature (e.g. [Nau1993], [Nau1994-b]). In what follows we introduce more and we
show that many of these coincide, if one starts with an injective Morita semi-context.
Moreover, other results on equivalences between some intersecting subcategories related
to an injective Morita context will be reframed for arbitrary (not necessarily compatible)
injective Morita data.
Definition 5.4. 1. For a right T -module X, a T -submodule X ′ ⊆ X is called K-pure
for some left T -module TK, iff the following sequence of Abelian groups is exact
0 → X ′ ⊗T K → X ⊗T K → X/X
′ ⊗T K → 0;
2. For a left T -module Y, a T -submodule Y ′ ⊆ Y is called L-copure for some left
T -module TL, iff the following sequence of Abelian groups is exact
0 → HomT (Y/Y
′, L) → HomT (Y, L) → HomT (Y
′, L) → 0.
Definition 5.5. (Compare [KO1979, Theorems 1.3., 2.3.]) Let T be a ring, I ⊳ T an
ideal, U a left T -module and consider the canonical T -linear morphisms
ζI,U : U → HomT (I, U) and ξI,U : I ⊗T U → U.
1. We say TU is I-divisible, iff ξI,U is surjective (equivalently, iff IU = U).
2. We say TU is I-localized, iff U
≃ HomT (I, U) canonically (equivalently iff T I is
strongly U -faithful and T I ⊆ T is U -copure).
3. We say a left T -module U is I-colocalized, iff I⊗TU
≃ U canonically (equivalently,
iff TU is I-divisible and IT ⊆ T is U -pure).
Notation. For a ring T, an ideal I ⊳ T, and with morphisms being the canonical ones,
we set
ID := {TU | IU = U}; IF := {TU | U →֒ HomT−(I, U)};
IL := {TU | U ≃ HomT (I, U}; IC := {TU | I ⊗T U ≃ U};
DI := {UT | UI = U}; FI := {UT | U →֒ Hom−T (I, U)};
LI := {UT | U ≃ HomT (I, U}; CI := {UT | U ⊗T I ≃ U}; .
The following result is due to T. Kato, K. Ohtake and B. Müller (e.g. [Mül1974],
[Kat1978], [KO1979]):
Proposition 5.6. Let M = (T, S, P,Q,<,>T , <,>S, I, J) ∈ UMC. Then there are equiv-
alences of categories
IC ≈ JC, CI ≈ CJ , IL ≈ JL and LI ≈ LJ .
5.7. Let mT = (T, S, P,Q,<,>T , I) ∈ UMSC and consider the dual pairings Pl := (Q,
TP ) ∈ Pl(T ) and Qr := (P,QT ) ∈ Pr(T ). For every left (right) T -module U consider the
canonical S-linear morphism induced by <,>T :
U : Q⊗T U → HomT−(P, U) (α
U : U ⊗T P → Hom−T (Q,U)).
We define
Dl(mT ) := {TU | Q⊗T U
≃ HomT−(P, U)};
Dr(mT ) := {UT | U ⊗T P
≃ Hom−T (Q,U)}.
Moreover, set
Ul(mT ) := Stat
l(TPS) ∩Adstat
l(SQT ); Ur(mT ) := Stat
r(SQT ) ∩Adstat
r(TPS);
Vl(mT ) := Stat
l(TPS) ∩ Dl(mT ); Vr(mT ) := Stat
r(SQT ) ∩ Dr(mT );
Vl(mT ) := IC ∩ Dl(mT ); Vr(mT ) := CI ∩ Dr(mT );
V̂l(mT ) := Vl(mT )∩ IL; V̂r(mT ) := Vr(mT ) ∩ LI ;
Wl(mT ) := Adstat
l(SQT ) ∩ Dl(mT ); Wr(mT ) := Adstat
r(TPS) ∩ Dr(mT );
Wl(mT ) := IL ∩ Dl(mT ); Wr(mT ) := LI ∩ Dr(mT );
Ŵl(mT ) := Wl(mT )∩ IC; Ŵr(mT ) := Wr(mT ) ∩ CI ;
Xl(mT ) := Vl(mT ) ∩Wl(mT ); Xr(mT ) := Vr(mT ) ∩Wr(mT );
Xl(mT ) := Vl(mT ) ∩Wl(mT ); Xr(mT ) := Vr(mT ) ∩Wr(mT ).
X ∗l (mT ) := {S(Q⊗T U) | V ∈ Xl(mT )}; X
r (mT ) := {(U ⊗T P )S | V ∈ Xr(mT )};
l (mT ) := {S(Q⊗T U) | V ∈ Xl(mT )}; X
r(mT ) := {(U ⊗T P )S | V ∈ Xr(mT )}.
Given mS = (S, T,Q, P,<,>S, J) ∈ UMSC one can define analogously, the corresponding
intersecting subcategories of SM and MS.
As an immediate consequence of Proposition 5.6 we get
Corollary 5.8. Let M = (T, S, P,Q,<,>T , <,>S, I, J) ∈ IUMC and consider the asso-
ciated Morita semi-contexts MT and MS (5).
1. If IC ≤ Dl(MT ) and JC ≤ Dl(MS), then Vl(MT ) ≈ Vl(MS). Similarly, if CI ≤
Dr(MT ) and CJ ≤ Dr(MS), then Vr(MT ) ≈ Vr(MS).
2. If IL ≤ Dl(MT ) and JL ≤ Dl(MS), then Wl(MT ) ≈ Wl(MS). Similarly, if LI ≤
Dr(MT ) and LJ ≤ Dr(MS), then Wr(MT ) ≈ Wr(MS).
Starting with a Morita context, the following result was obtained in [Nau1993,
Theorem 3.2.]. We restate the result for an arbitrary (not necessarily compatible) Morita
datum and sketch its proof:
Lemma 5.9. Let M = (T, S, P,Q,<,>T , <,>S, I, J) be a unital Morita datum and con-
sider the associated Morita semi-contexts MT and MS in (5). Then there are equivalences
of categories
Xl(MT )
HomT−(P,−)
HomS−(Q,−)
Xl(MS) and Xr(MT )
Hom−T (Q,−)
Hom−S(P,−)
Xr(MS).
Proof. Let TV ∈ Xl(MT ). By the equivalence Stat
l(TPS)
HomT (P,−)
≈ Adstatl(TPS) in 5.2 we
have HomT−(P, V ) ∈ Adstat
l(TPS). Moreover, V ∈ Dl(M), hence HomT−(P, V ) ≃ Q⊗T V
canonically and it follows then from the equivalence Adstatl(SQT )
≈ Statl(SQT ) that
HomT−(P, V ) ∈ Stat
l(SQT ). Moreover, we have the following natural isomorphisms
P ⊗S HomT−(P, V ) ≃ V ≃ HomS−(Q,Q⊗T V ) ≃ HomS−(Q,HomT−(P, V )), (13)
i.e. HomT−(P, V ) ∈ Dl(MS). Consequently, HomT−(P, V ) ∈ Xl(MS). Moreover, (13)
yields a natural isomorphism V ≃ HomS−(Q,HomT−(P, V )). Analogously, one can show for
everyW ∈ Xl(MS) that HomS−(Q,W ) ∈ Xl(MT ) and thatW ≃ HomT−(P,HomS−(Q,W ))
naturally. Consequently, Xl(MT ) ≈ Xl(MS). The equivalences Xr(MT ) ≈ Xr(MS) can
be proved analogously.�
Proposition 5.10. Let M = (T, S, P,Q,<,>T , <,>S, I, J) be a unital injective Morita
datum and consider the associated Morita semi-contexts MT and MS in (5).
1. There are equivalences of categories
Statl(T IT ) ≈ Adstat
l(T IT ); Stat
l(SJS) ≈ Adstat
l(SJS);
Statr(T IT ) ≈ Adstat
r(T IT ); Stat
r(SJS) ≈ Adstat
r(SJS).
2. If Statl(T IT ) ≤ X
l (MS) and Stat
l(SJS) ≤ X
l (MT ), then there are equivalences of
categories
Statl(T IT ) ≈ Stat
l(SJS) and Adstat
l(T IT ) ≈ Adstat
l(SJS).
3. If Statr(T IT ) ≤ X
r (MS) and Stat
r(SJS) ≤ X
r (MT ), then there are equivalences of
categories
Statr(T IT ) ≈ Stat
r(SJS) and Adstat
r(T IT ) ≈ Adstat
r(SJS).
Proof. To prove “1”, notice that since M is an injective Morita datum, P ⊗S Q
and Q⊗T P
≃ J as bimodules and so the four equivalences of categories result from 5.2.
To prove “2”, one can use an argument similar to that in [Nau1994-b, Theorem 3.9.] to
show that the inclusion Statl(T IT ) = Stat
l(T (P ⊗S Q)T ) ≤ X
l (MS) implies Stat
l(T IT ) =
Statl(T (P ⊗S Q)T ) = Xl(MT ) and that the inclusion Stat
l(SJS) = Stat
l(S(Q ⊗T P )S) ≤
X ∗l (MT ) implies Stat
l(SJS) = Stat
l(S(Q ⊗T P )S) = Xl(MS). The result follows then by
Lemma 5.9. The proof of “3” is analogous to that of “2”.�
For injective Morita semi-contexts, several subcategories in (12) are shown in the
following result to be equal:
Theorem 5.11. Let mT = (T, S, P,Q,<,>T , I) ∈ IUMS. Then
1. Vl(mT ) = Vl(mT ), Wl(mT ) = Wl(mT ), whence
V̂l(mT ) = Ŵl(mT ) = Xl(mT ) = Xl(mT ) = IC∩Dl(mT )∩ IL and X
l (mT ) = X
l (mT ).
2. Vr(mT ) = Vr(mT ), Wr(mT ) = Wr(mT ), whence
V̂r(mT ) = Ŵr(mT ) = Xr(mT ) = Xr(mT ) = CI∩Dr(mT )∩LI and X
r (mT ) = X
r(mT ).
Proof. We prove only “1” as “2” can be proved analogously. Assume the Morita semi-
context mT = (T, S, P,Q,<,>T , I) is injective. By our assumption we have for every
V ∈ Dl(mT ) the commutative diagram
P ⊗S (Q⊗T V )
idP⊗S(α
(P ⊗S Q)⊗T V
<,>T⊗T idV≃
P ⊗S HomT−(P, V )
// V I ⊗T VξI,V
Then it becomes obvious that ωlP,V : P ⊗S HomT (P, V ) → V is an isomorphism if and only
if ξI,V : I ⊗T V → V is an isomorphism. Consequently
V(mT ) = Dl(mT ) ∩ Stat
l(TPS) = Dl(mT ) ∩ IC = V(mT ).
On the other hand, we have for every V ∈ Dl(mT ) the following commutative diagram
HomS−(Q,HomT−(P, V ))
// HomT−(P ⊗S Q, V )
HomS−(Q,Q⊗T V )
// HomT−(I, V )
(<,>T ,V )≃
It follows then that ηlP,L : V → HomS(Q,Q ⊗T P ) is an isomorphism if and only if
ζI,V : V → HomT (I, V ) is an isomorphism. Consequently,
W(mT ) = Dl(mT ) ∩ Adstat
l(TPS) = Dl(mT ) ∩I L = W(mT ).
Moreover, we have
V̂l(mT ) := Vl(mT ) ∩ IL = Vl(mT ) ∩ IL = IC ∩ Dl(mT )∩ IL
= IC ∩Wl(mT ) = IC ∩Wl(mT ) = Ŵl(mT ).
On the other hand, we have
Xl(mT ) = Vl(mT ) ∩Wl(mT ) = Vl(mT ) ∩Wl(mT ) = Xl(mT )
and so the equalities V̂l(mT ) = Ŵl(mT ) = Xl(mT ) = Xl(mT ) and X
l (mT ) = X
l (mT ) are
established.�
In addition to establishing several other equivalences of intersecting subcategories,
the following results reframe the equivalence of categories V̂ ≈ Ŵ in [Nau1994-b, Theorem
4.9.] for an arbitrary (not necessarily compatible) injective Morita datum:
Theorem 5.12. Let M = (T, S, P,Q,<,>T , <,>S, I, J) be an injective Morita datum and
consider the associated Morita semi-contexts MT and MS (5).
1. The following subcategories are mutually equivalent:
V̂l(MT ) = Ŵl(MT ) = Xl(MT ) = Xl(MT ) ≈ Xl(MS) = Xl(MS) = Ŵl(MS) = V̂l(MS).
2. If Vl(MT ) ≤ IL and Wl(MS) ≤ JC, then Vl(MT ) ≈ Wl(MS). If Wl(MT ) ≤ IC
and Vl(MS) ≤ JL, then Wl(MT ) ≈ Vl(MS).
3. The following subcategories are mutually equivalent:
V̂r(MT ) = Ŵr(MT ) = Xr(MT ) = Xr(MT ) ≈ Xr(MS) = Xr(MS) = Ŵr(MS) = V̂r(MS).
4. If Vr(MT ) ≤ LI and Wr(MT ) ≤ CJ , then Vr(MT ) ≈ Wr(MS). If Wr(MT ) ≤ CJ
and Vr(MS) ≤ LI , then Vr(MS) ≈ Wr(MT ).
Proof. By Lemma 5.9, Xl(MT ) ≈ Xl(MS) and so “1” follows by Theorem 5.11. If
Vl(MT ) ≤ IL and Wl(MS) ≤ JC, then we have
Vl(MT ) = Vl(MT ) ∩ IL = V̂l(MT ) ≈ Ŵl(MS) = Wl(MS) ∩ JC = Wl(MS).
On the other hand, if Wl(MT ) ≤ IL and Vl(MS) ≤ JC, then
Wl(MT ) = Wl(MT ) ∩ IC = Ŵl(MT ) ≈ V̂l(MS) = Vl(MS) ∩ JL = Vl(MS).
So we have established “2”. The results in “3” and “4” can be obtained analogously.�
6 More applications
In this final section we give more applications of Morita α-(semi-)contexts and
injective Morita (semi-)contexts. All rings in this section are unital, whence all Morita
(semi-)contexts are unital. Moreover, for any ring T we denote with TE an arbitrary, but
fixed, injective cogenerator in TM.
Notation. Let T be an A-ring. For any left T -module TV, we set
#V := HomT (V, TE). If
moreover, TVS is a (T, S)-bimodule for some B-ring S, then we consider
S V with the left
S-module structure induced by that of VS.
Lemma 6.1. (Compare [Col1990, Lemma 3.2.], [CF2004, Lemmas 2.1.2., 2.1.3.]) Let T be
an A-ring, S a B-ring and TVS a (T, S)-bimodule,
1. A left T -module TK is V -generated if and only if the canonical T -linear morphism
ωlV,K : V ⊗S HomT (V,K) → K (18)
is surjective. Moreover, V ⊗S W ⊆ Pres(TV ) ⊆ Gen(TV ) for every left S-module
2. A left S-module SL is
S V -cogenerated if and only if the canonical S-linear morphism
ηlV,L : L→ HomT (V, V ⊗S L) (19)
is injective. Moreover, HomT (V,M) ⊆ Copres(
S V ) ⊆ Cogen(
S V ) for every left
T -module TM.
Remark 6.2. Let T be an A-ring, S a B-ring and TVS a (T, S)-bimodule. Notice that for
any left S-module SL we have
ann⊗L(VS) := {l ∈ L | V ⊗S l = 0} = Ker(η
V,L),
whence (by Lemma 6.1 “2” ) VS is L-faithful if and only if SL is
S V -cogenerated. It follows
then that VS is completely faithful if and only if
S V is a cogenerator.
Localization and colocalization
In what follows we clarify the relations between static (adstatic) modules and subcate-
gories colocalized (localized) by a trace ideal of a Morita context satisfying the α-condition.
Recall that for any (T, S)-bimodule TPS we have by Lemma 6.1:
Statl(TPS) ⊆ Gen(TP ) and Adstat
l(TPS) ⊆ Cogen(
S P ). (20)
Theorem 6.3. Let M = (T, S, P,Q,<,>T , <,>S, I, J) ∈ UMC. Then we have
IC ⊆ ID ⊆ Gen(TP ). (21)
Assume Pr := (Q,PS) ∈ P
r (S). Then
1. Gen(TP ) = Stat
l(TPS) ⊆ IF.
2. If Gen(TP ) ⊆ IC, then IC = ID = Gen(TP ) = Stat
l(TPS).
3. If Qr := (P,QT ) ∈ P
r (T ), then T I ⊆ TT is pure and IC = ID.
Proof. For every left T -module TK, consider the following diagram with canonical mor-
phisms and let α2 := ζI,K ◦ ω
P,K. It is easy to see that both rectangles and the two right
triangles commutes:
P ⊗S Q⊗T K
idP⊗Sα
<,>T⊗T idK
P ⊗S HomT (P,K)
HomT (P,K)//
HomS(Q,HomT (P,K))
HomT (P ⊗S Q,K)
I ⊗T K
// HomT (I,K)
(<,>T ,K)
It follows directly from the definitions that IC ⊆ ID and Stat
l(TPS) ⊆ Gen(TP ). If TK
is I-divisible, then ξI,K◦ <,>T ⊗T idK = ω
P,K ◦ idP ⊗S α
K is surjective, whence ω
is surjective and we conclude that TK is P -generated by Lemma 6.1 “1”. Consequently,
ID ⊆ Gen(TP ).
Assume now that Pr ∈ P
r (S). Considering the canonical map ρQ : T → End(SQ)
the map ρQ◦ <,>T= α
Q is injective and so the bilinear map <,>T is injective (i.e.
P ⊗S Q
≃ I). Define α1 := (idP ⊗S α
K ) ◦ (<,>T ⊗T idK)
−1, so that the left triangles
commute. Notice that αPr
HomT (P,K)
is injective and the commutativity of the upper right
triangle in Diagram (22) implies that α2 is injective (whence ω
P,K is injective by the
commutativity of the lower right triangle).
1. If K ∈ Statl(TPS), then the commutativity of the lower right triangle (22) and the
injectivity of α2 show that ζI,K is injective; hence, Stat
l(TPS) ⊆ IF. On the other
hand, if TK is P -generated, then ω
P,K is surjective by Lemma 6.1 (1), thence bijective,
i.e. K ∈ Statl(TPS). Consequently, Gen(TP ) = Stat
l(TPS).
2. This follows directly from the inclusions in (21) and “1”.
3. Assume Qr := (P,QT ) ∈ P
r (T ). Since Pr ∈ P
r (S), it follows by analogy to Propo-
sition 2.12 “3” that PS is flat, hence idP ⊗S α
K is injective. The commutativity of
the upper left triangle in Diagram (22) implies then that α1 is injective, thence ξI,K
is injective by commutativity of the lower left triangle (i.e. T I ⊆ TT is K-pure). If
TK is divisible, then K ⊗T I
≃ K (i.e. K ∈ IC).�
Theorem 6.4. Let M = (T, S, P,Q,<,>T , <,>S, I, J) ∈ UMC. Then we have
JL ⊆ JF ⊆ Cogen(
S P ) and Adstat
l(TPS) ⊆ Cogen(
S P ).
Assume Qr := (P,QT ) ∈ P
r (T ). Then
1. JS ⊆ SS is pure and JC ⊆ Cogen(
S P ).
2. If Pr := (Q,PS) ∈ P
r (S), then JL ⊆ Adstat
l(TPS) ⊆ Cogen(
S P ) ⊆ JF.
3. If Pr ∈ P
r (S) and Cogen(
S P ) ⊆ JL, then JL = Cogen(
S P ) = Adstat
l(TPS).
Proof. For every right S-module L consider the commutative diagram with canonical
morphisms and let α3 be so defined, that the left triangles become commutative
J ⊗S L
ξJ,L //
ζJ,L // HomS(J, L)
(<,>S ,L)
HomS(Q⊗T P, L)
≃ can
Q⊗T P ⊗S L
(<,>S)⊗S idL
// HomT (P, P ⊗S L)
HomT (P,HomS(Q,L))
By definition JL ⊆ JF and Adstat
l(TPS) ⊆ Cogen(
S P ). If SL ∈ JF, then ζJ,L is injective
and it follows by commutativity of the right rectangle in Diagram (23) that ηlP,L is injective,
hence SL is
S P -cogenerated by Lemma 6.1 “2”. Consequently, JF ⊆ Cogen(
S P ).
Assume now that Qr ∈ P
r (T ). Then it follows from Lemma 4.3 that <,>S is injective
(hence Q⊗T P
≃ J) and so α4 := (can ◦ (<,>S, L))
−1 ◦ (P, αPrL ) is injective.
1. Since α3 is injective, ξJ,L is also injective for every SL, i.e. JS ⊆ SS is pure. If SL ∈
JC, then it follows from the commutativity of the left rectangle in Diagram (23) that
ηlP,L is injective, hence L ∈ Cogen(
S P ) by Lemma 6.1 (2).
2. Assume that Pr ∈ P
r (S), so that α4 is injective. If SL ∈ JL, then ζJ,L is an
isomorphism, thence ηlP,L is surjective (notice that α4 is injective). Consequently,
JL ⊆ Adstat
l(TPS).
3. This follows directly from the assumptions and “2”.�
∗-Modules
To the end of this section, we fix a unital ring T, a left T -module TP and set S :=
End(TP )
Definition 6.5. ([MO1989]) We call TP a ∗-module, iff Gen(TP ) ≈ Cogen(
S P ).
Remark 6.6. It was shown by J. Trlifaj [Trl1994] that all ∗-modules are finitely generated.
By definition, Statl(TPS) ≤ TM and Adstat
l(TPS) ≤ SM are the largest subcat-
egories between which the adjunction (P ⊗S −,HomT (P,−)) induces an equivalence. On
the other hand, Lemma 6.1 shows that Gen(TP ) ≤ TM and Cogen(
S P ) ≤ SM are the
largest such subcategories (see [Col1990, Section 3] for more details). This suggests the
following observation:
Proposition 6.7. ([Xin1999, Lemma 2.3.]) We have
TP is a ∗ -module ⇔ Stat(TP ) = Gen(TP ) and Adstat(TP ) = Cogen(
S P ).
Definition 6.8. A left T -module TU is said to be
semi-
-quasi-projective (abbr. s-
-quasi-projective), iff for any left T -module
TV ∈ Pres(TU) and any U-presentation
U (Λ) → U (Λ
′) → V → 0
of TV (if any), the following induced sequence is exact:
HomT (U, U
(Λ)) → HomT (U, U
(Λ′)) → HomT (U, V ) → 0;
weakly-
-quasi-projective (abbr. w-
-quasi-projective), iff for any left T -
module TV and any short exact sequence
0 → K → U (Λ
′) → V → 0
with K ∈ Gen(TU) (if any), the following induced sequence is exact:
0 → HomT (U,K) → HomT (U, U
(Λ′)) → HomT (U, V ) → 0;
self-tilting, iff TU is w-
-quasi-projective and Gen(TU) = Pres(TU);∑
-self-static, iff any direct sum U (Λ) is U -static.
(self)-small, iff HomT (U,−) commutes with direct sums (of TU);
Proposition 6.9. Assume M = (T, S, P,Q,<,>T , <,>S) is a unital Morita context.
1. If Pr := (Q,PS) ∈ P
r (S), then:
(a) Gen(TP ) = Stat
l(TPS);
(b) there is an equivalence of categories Gen(TP ) ≈ Cop(
S P );
(c) TP is
-self-static and Statl(TPS) is closed under factor modules.
(d) Gen(TP ) = Pres(TP );
2. If M ∈ UMCαr and Cogen(
S P ) ⊆ JL, then:
(a) Gen(TP ) = Stat
l(TPS) and Cogen(
S P ) = Adstat
l(TPS);
(b) there is an equivalence of categories Cogen(
S P ) ≈ Gen(TP );
(c) TP is a ∗-module;
(d) TP is self-tilting and self-small.
Proof. 1. If Pr ∈ P
r (S), then it follows by Theorem 6.3 that Gen(TP ) = Stat
l(TPS),
which is equivalent to each of “b” and “c” by [Wis2000, 4.4.] and to “d” by [Wis2000,
4.3.].
2. It follows by the assumptions, Theorems 6.3, 6.4 and 5.2 that Gen(TP ) = Stat
l(TPS) ≈
Adstatl(TPS) = Cogen(
S P ), whence Gen(TP ) ≈ Cogen(
S P ) (which is the definition
of ∗-modules). Hence “a” ⇔“b” ⇔“c”. The equivalence “a” ⇔ “d” is evident by
[Wis2000, Corollary 4.7.] and we are done.�
Wide Morita Contexts
Wide Morita contexts were introduced by F. Castaño Iglesias and J. Gómez-Torrecillas
[C-IG-T1995] and [C-IG-T1996] as an extension of classical Morita contexts to Abelian
categories.
Definition 6.10. Let A and B be Abelian categories. A right (left) wide Morita
context between A and B is a datum Wr = (G,A,B, F, η, ρ), where G : A ⇄ B : F are
right (left) exact covariant functors and η : F ◦ G −→ 1A, ρ : G ◦ F −→ 1B (η : 1A −→
F ◦ G, ρ : 1B −→ G ◦ F ) are natural transformations, such that for every pair of objects
(A,B) ∈ A× B the compatibility conditions G(ηA) = ρG(A) and F (ρB) = ηF (B) hold.
Definition 6.11. Let A and B be Abelian categories and W = (G,A,B, F, η, ρ) be a right
(left) wide Morita context. We call W injective (respectively semi-strict, strict), iff η
and ρ are monomorphisms (respectively epimorphisms, isomorphisms)
Remarks 6.12. Let W = (G,A,B, F, η, ρ) be a right (left) wide Morita context.
1. It follows by [CDN2005, Propositions 1.1., 1.4.] that if either η or ρ is an epimorphism
(monomorphism), then W is strict, whence A ≈ B.
2. The resemblance of injective left wide Morita contexts is with the Morita-Takeuchi
contexts for comodules of coalgebras, i.e. the so called pre-equivalence data for cate-
gories of comodules introduced in [Tak1977] (see [C-IG-T1998] for more details).
Injective Right wide Morita contexts
In a recent work [CDN2005, 5.1.], Chifan, et. al. clarified (for module categories) the
relation between classical Morita contexts and right wide Morita contexts. For the conve-
nience of the reader and for later reference, we include in what follows a brief description
of this relation.
6.13. Let T, S be rings, A := TM and B := SM. Associated to each Morita context
M = (T, S, P,Q,<,>T , <,>S) is a wide Morita context as follows: Define G : A ⇄ B : F
by G(−) = Q ⊗T − and F (−) = P ⊗S −. Then there are natural transformations η :
F ◦G −→ 1
and ρ : G ◦ F −→ 1
such that for each TV and WS :
ηV : P ⊗S (Q⊗T V ) → V,
pi ⊗S (qi ⊗T vi) 7→
< pi, qi >T vi,
ρW : Q⊗T (P ⊗S W ) → W,
qi ⊗T (pi ⊗S wi) 7→
< qi, pi >S wi.
Then the datum Wr(M) := (G, TM, SM, F, η, ρ) is a right wide Morita context.
Conversely, let T ′, S ′ be two rings and W ′r = (G
′, T ′M, S′M, F
′, η′, ρ′) be a right wide
Morita context between T ′M and S′M such that the right exact functors G
′ : T ′M ⇄
S′M : F
′ commute with direct sums. By Watts’ Theorems (e.g. [Gol1979]), there exists a
(T, S)-bimodule P ′ (e.g. F ′(S ′)) such that F ′ ≃ P ′⊗S′ −, an (S, T )-bimodule Q
′ such that
G′ ≃ Q′ ⊗T ′ − and there should exist two bilinear forms
<,>T ′: P
′ ⊗S′ Q
′ → T ′ and <,>S′: Q
′ ⊗T ′ P
′ → S ′,
such that the natural transformations η′ : F ′ ◦G′ → 1
, ρ : G′ ◦ F ′ → 1
are given by
η′V ′(p
′ ⊗S′ q
′ ⊗T ′ v
′) =< p′, q′ >T ′ v
′ and ρ′W ′(q
′ ⊗T p
′ ⊗S w
′) =< q′, p′ >S′ w
for all V ′ ∈ T ′M, W
′ ∈ S′M, p
′ ∈ P ′, q′ ∈ Q′, v′ ∈ V ′ and w′ ∈ W ′. It can be shown that
in this way one obtains a Morita context M′ = M′(W ′r) := (T
′, S ′, P ′, Q′, <,>T ′, <,>S′).
Moreover, it turns out that given a wide Morita context Wr, we have Wr ≃ Wr(M(Wr)).
The following result clarifies the relation between injective Morita contexts and
injective right wide Morita contexts.
Theorem 6.14. Let M = (T, S, P,Q,<,>T , <,>S) be a Morita context, A := TM, B :=
SM and consider the induced right wide Morita context Wr(M) := (G,A,B, F, η, ρ).
1. If Wr(M) is an injective right wide Morita context, then M is an injective Morita
context.
2. If M ∈ UMCαr , then Wr(M) is an injective right wide Morita context.
Proof. 1. Let Wr(M) be an injective right wide Morita context. Then in particular,
<,>T= ηT and <,>S= ρS are injective, i.e. M is an injective Morita context.
2. Assume that M satisfies the right α-condition. Suppose there exists some TV and∑
pi ⊗S (qi ⊗T vi) ∈ Ker(ηV ). Then for any q ∈ Q we have
0 = q ⊗T ηV (
(pi ⊗S qi)⊗T vi) =
q⊗T < pi, qi >T vi
q < pi, qi >T ⊗T vi =
< q, pi >S qi ⊗T vi
< q, pi >S (qi ⊗T vi) = α
pi ⊗S (qi ⊗T vi))(q).
Since Pr := (Q,PS) ∈ P
r (S), the morphism α
is injective and so
pi⊗S (qi⊗T
vi) = 0, i.e. ηV is injective. Analogously, suppose
qi ⊗T (pi ⊗S wi) ∈ Ker(ρW ).
Then for any p ∈ P we have
0 = p⊗S ρW (
qi ⊗T (pi ⊗S wi) =
p⊗S < qi, pi >S wi
p < qi, pi >S ⊗Swi =
< p, qi >T pi ⊗S wi
< p, qi >T (pi ⊗S wi) = α
qi ⊗T (pi ⊗S wi))(p).
Since Qr := (P,QT ) ∈ P
r (T ), the morphism α
is injective and so
qi ⊗T
(pi ⊗S wi) = 0, i.e. ρW is injective. Consequently, the induced right wide Morita
context Wr(M) is injective.�
Acknowledgement: The authors thank the referee for his/her careful reading of
the paper and for the fruitful suggestions, comments and corrections, which helped in
improving several parts of the paper. Moreover, they acknowledge the excellent research
facilities as well as the support of their respective institutions, King Fahd University of
Petroleum and Minerals and King AbdulAziz University.
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	Introduction
	Preliminaries
	Morita (Semi)contexts
	Injective Morita (Semi-)Contexts
	Equivalences of Categories
	More applications
