Monomorphism operator and perpendicular operator

0.19 MB·English

by  Keyan Song

MONOMORPHISM OPERATOR AND PERPENDICULAR OPERATOR
KEYANSONG†,PUZHANG∗
3
1
0
2 Department of Mathematics, Shanghai Jiao Tong University

n Shanghai 200240,PR China
a
J
4 Abstract. ForaquiverQ,ak-algebraA,andafullsubcategory X ofA-mod,themonomor-
1 phism category Mon(Q,X) is introduced. The main result says that if T is an A-module such
thatthereisanexactsequence0→Tm→···→T0→D(AA)→0witheachTi∈add(T),then
T] Mon(Q, ⊥T)= ⊥(kQ⊗kT);andifT iscotilting,thenkQ⊗kT isauniquecotiltingΛ-module,
R uptomultiplicitiesofindecomposabledirectsummands,suchthatMon(Q, ⊥T)= ⊥(kQ⊗kT).
. As applications, the category of the Gorenstein-projective (kQ⊗kA)-modules is character-
h ized as Mon(Q,GP(A)) if A is Gorenstein; the contravariantly finiteness of Mon(Q,X) can be
t
a described;andasufficientandnecessaryconditionforMon(Q,A)beingoffinitetypeisgiven.
m
[ 2010MathematicalSubjectClassification. 16G10, 16E65,16G50, 16G60.

1 Key words. monomorphism category, cotilting modules, derived category, contravariantly
v
finite, finite type
3
5
8 1. Introduction
2
.
1 1.1. WithaquiverQandak-algebraA,onecanassociatethemonomorphismcategoryMon(Q,A)
0 ([LZ]).IfQ=•→•itiscalledthesubmodulecategoryanddenotedbyS(A). IfQ=n•→···→•1
3
it is called the filtered chain category in D. Simson [S]; and it is denoted by S (A) in [Z].
1 n
:
v G.Birkhoff[B]initiatesthestudyofS(Z/hpti). C.M.RingelandM.Schmidmeier([RS1]-[RS3])
i
X
have extensively studied S(A). In particular, the Auslander-Reiten theory of S(A) is explicitly
r
a given ([RS2]). Since then the monomorphism category receives more attention. In [Z] relations
amongS (A)andthe Gorenstein-projectivemodulesandcotiltingtheoryaregiven. D.Kussin,H.
n
Lenzing,andH.Meltzer[KLM1]establishasurprisinglinkbetweenthestablesubmodulecategory
andthesingularitytheoryviaweightedprojectivelines(seealso[KLM2]). In[XZZ]theAuslander-
Reiten theory of S(A) is extended to S (A). For more related works we refer to [A], [RW], [SW],
n
[Mo], [C1], [C2], and [RZ].
1.2. LetX beafullsubcategoryofA-mod. WealsodefinethemonomorphismcategoryMon(Q,X).
For an A-module T, let ⊥T be the full subcategory of A-mod consisting of those modules X with
Exti (X,T)=0, ∀ i≥1. The main result of this paper gives a reciprocity of the monomorphism
A
operator Mon(Q,−) and the left perpendicular operator ⊥. Namely, if T is an A-module such
∗ Thecorrespondingauthor [email protected].
SupportedbytheNSFofChina(11271251), andDoctoralFundofMinistryofEducation(20120073110058).
† [email protected].
1
2 KEYANSONG,PUZHANG
that there is an exact sequence 0 → T → ··· → T → D(A ) → 0 with each T ∈ add(T), then
m 0 A i
Mon(Q, ⊥T) = ⊥(kQ⊗ T) (Theorem 3.1); and if T is a cotilting A-module, then kQ⊗ T is
k k
a unique cotilting Λ-module, up to multiplicities of indecomposable direct summands, such that
Mon(Q, ⊥T)= ⊥(kQ⊗ T) (Theorem 4.1).
k
Theorems 3.1 and 4.1 generalize [Z, Theorem 3.1(i) and (ii)] for Q = • → ··· → •. However,
the arguments in [Z] can not be generalized to the general case (cf. 3.1 and 4.1 below). Here we
adopt new treatments, in particular by using an adjoint pair (Coker ,S(i)⊗−) and Lemma 4.4.
i
1.3. Our main results have some applications, which generalize the corresponding results in [Z].
The category GP(A) of the Gorenstein-projective A-modules is Frobenius (cf. [AB], [AR],
[EJ]), and hence the corresponding stable category is triangulated ([H]). If A is Gorenstein (i.e.,
inj.dim A<∞andinj.dimA <∞),thenGP(A)= ⊥A([EJ,Corollary11.5.3]). TakingT = A
A A A
in Theorem 3.1 we have GP(Λ)=Mon(Q,GP(A)) if A is Gorenstein.
M. Auslander and I. Reiten [AR, Theorem 5.5(a)] have established a deep relation between
resolvingcontravariantlyfinite subcategoriesand cotilting theory, by assertingthat X is resolving
and contravariantly finite with X = A-mod if and only if X = ⊥T for some cotilting A-module
T, where X is the full subcategorby of A-mod consisting of those modules X, such that there is an
exact sequbence 0 → X → ··· → X → X → 0 with each X ∈ X. It is natural to ask when is
m 0 i
Mon(Q,X) contravariantlyfinite in Λ-mod? As anapplication of Theorem4.1 and [AR, Theorem
\
5.5(a)], we see that Mon(Q,X) is resolving and contravariantlyfinite with Mon(Q,X)=Λ-mod if
and only if X is resolving and contravariantly finite with X =A-mod (Theorem 5.1).
b
It is well-known that the representation type of Mon(Q,A) is different from the ones of A and
of Λ = kQ⊗ A. For example, k[x]/hxti is of finite type, while k(• → •)⊗ k[x]/hxti is of finite
k k
type if and only if t ≤ 3, and S (k[x]/hxti) is of finite type if and only if t ≤ 5. If t > 6 then
2
S (k[x]/hxti) is of “wild” type, while S (k[x]/hx6i) is of “tame” type ([S], Theorems 5.2 and 5.5).
2 2
AcompleteclassificationofindecomposableobjectsofS (k[x]/hx6i)isexhibitedin[RS3]. Inspired
2
by Auslander’s classical result: A is representation-finite if and only if there is an A-generator-
cogeneratorM suchthatgl.dimEnd (M)≤2([Au], ChapterIII), byusingTheorem4.1weprove
A
that Mon(Q,A) is of finite type if and only if there is a generator and relative cogenerator M of
Mon(Q,A) such that gl.dimEnd (M)≤2 (Theorem 6.1).
Λ
2. Preliminaries on monomorphism categories
In this section we fix notations, and give necessary definitions and facts.
2.1. Throughout this paper, k is a field, Q is a finite acyclic quiver (i.e., a finite quiver without
orientedcycles),andAisa finite-dimensionalk-algebra. Denote by kQthe pathalgebraofQover
k. PutΛ=kQ⊗ A,andD =Hom (−,k).LetP(i)(resp. I(i))betheindecomposableprojective
k k
(resp. injective) kQ-module, and S(i) the simple kQ-module, at i ∈ Q . By A-mod we denote
0
the category of finite-dimensional left A-modules. For an A-module T, let add(T) be the full the
subcategory of A-mod consisting of all the direct sums of indecomposable direct summands of T.
MONOMORPHISM OPERATOR AND PERPENDICULAR OPERATOR 3
2.2. Given a finite acyclic quiver Q = (Q ,Q ,s,e) with Q the set of vertices and Q the set
0 1 0 1
of arrows, we write the conjunction of a path p of Q from right to left, and let s(p) and e(p)
be respectively the starting and the ending point of p. The notion of representations of Q over
k can be extended as follows. By definition ([LZ]), a representation X of Q over A is a datum
X =(X , X , i∈Q , α∈Q ),orsimplyX =(X ,X ),whereeachX isanA-module,andeach
i α 0 1 i α i
X :X →X is anA-map. It isa finite-dimensional representation ifsois eachX . We call
α s(α) e(α) i
X the i-th branch of X. A morphism f from X to Y is a datum (f , i∈Q ), where f :X →Y
i i 0 i i i
is an A-map for i∈Q , such that for each arrow α:j →i the following diagram
0
fj //
X Y (2.1)
j j
Xα Yα
(cid:15)(cid:15) (cid:15)(cid:15)
X fi // Y
i i
commutes. Denote by Rep(Q,A) the category of finite-dimensional representations of Q over A.
f g
Note that a sequence of morphisms 0−→X −→Y −→Z −→0 in Rep(Q,A) is exact if and only
if 0−→X −f→i Y −g→i Z −→0 is exact in A-mod for each i∈Q .
i i i 0
Lemma 2.1. ([LZ, Lemma 2.1]) We have an equivalence Λ-mod∼=Rep(Q,A) of categories.
In the following we will identify a Λ-module with a representation of Q over A. If T ∈ A-mod
and M ∈ kQ-mod with M = (M ,i ∈ Q ,M ,α ∈ Q ) ∈ Rep(Q,k), then M ⊗ T ∈ Λ-mod with
i 0 α 1 k
M ⊗kT =(Mi⊗kT =TdimkMi, i∈Q0, Mα⊗kIdT, α∈Q1)∈Rep(Q,A).
2.3. Here is the central notion of this paper.
Definition 2.2. (i) ([LZ]) A representation X = (X ,X ,i ∈ Q ,α ∈ Q ) ∈ Rep(Q,A) is a
i α 0 1
monic representation of Q over A, or a monic Λ-module, if δ (X) is an injective A-map for each
i
i∈Q , where
0
δ (X)=(X ) : X −→X .
i α α∈Q1, e(α)=i M s(α) i
α∈Q1
e(α)=i
Denote by Mon(Q,A) the full subcategory of Rep(Q,A) consisting of all the monic representa-
tions of Q over A, which is called the monomorphism category of A over Q.
(ii) Let X be a full subcategory of A-mod. Denote by Mon(Q,X) the full subcategory of
Mon(Q,A) consisting of all the monic representations X = (X ,X ), such that X ∈ X and
i α i
Cokerδ (X)∈X for all i∈Q . We call Mon(Q,X) the monomorphism category of X over Q.
i 0
If X = A-mod then Mon(Q,X) = Mon(Q,A). For M ∈ kQ-mod and T ∈ A-mod, it is clear
that if M ∈Mon(Q,k) then M⊗ T ∈Mon(Q,A). In particular, P(i)⊗ T ∈Mon(Q,A) for each
k k
i∈Q .
0
Note that D(ΛΛ)∼=D(kQkQ)⊗kD(AA) as left Λ-modules. We need the following fact.
Lemma 2.3. ([LZ, Proposition 2.4]) Let IndP(A) (resp. IndI(A)) denote the set of pairwise
non-isomorphic indecomposable projective (resp. injective) A-modules. Then
4 KEYANSONG,PUZHANG
IndP(Λ)={P(i)⊗ P | i∈Q , P ∈IndP(A)}⊆Mon(Q,A),
k 0
and
IndI(Λ)={I(i)⊗ I | i∈Q , I ∈IndI(A)}.
k 0
In particular, for M ∈kQ-mod we have proj.dim(M ⊗ A)≤1, and inj.dim(M ⊗ D(A ))≤1.
k k A
2.4. Given X = (X ,X ) ∈ Λ-mod, for each i ∈ Q we have functors F and F+ from Λ-mod
j α 0 i i
to A-mod, respectively induced by F (X) = X and F+(X) := X (if i is a source then
i i i s(α)
αL∈Q1
e(α)=i
F+(X):=0).
i
We write Cokerδ (X) (cf. Definition 2.2 (i)) as Coker (X). Then we have a functor Coker :
i i i
Λ-mod −→ A-mod, explicitly given by Coker (X) := X / ImX (if i is a source then
i i α
αP∈Q1
e(α)=i
Coker (X) := X ). So we have an exact sequence of functors F+ −δ→i F −π→i Coker −→ 0,
i i i i i
i.e., we have the exact sequence of A-modules
X δ−i(→X)X π−i(→X)Coker (X)−→0
M s(α) i i
α∈Q1
e(α)=i
for each X ∈Λ-mod, where π (X) is the canonicalmap. It is clear that F+ and F are exact, and
i i i
Coker is right exact (by Snake Lemma). For i,j ∈Q and T ∈A-mod, we have
i 0
T, if j =i;
Coker (P(j)⊗ T)= (2.2)
i k 
0, if j 6=i.

Lemma 2.4. For each i∈Q , the restriction of functor Coker to Mon(Q,A) is exact.
0 i
Proof. Let 0 → (X ,X ) → (Y ,Y ) → (Z ,Z ) → 0 be an exact sequence in Mon(Q,A). Then
i α i α i α
we have the following commutative diagram with exact rows
// // // //
0 X Y Z 0
s(α) s(α) s(α)
L L L
δi(X) δi(Y) δi(Z)
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
// // // //
0 X Y Z 0.
i i i
Then the assertion follows from Snake Lemma since δ (Z) is injective. (cid:4)
i
Recallfrom[AR]thatX isresolvingifX containsalltheprojectiveA-modules,X isclosedunder
taking extensions, kernels of epimorphisms, and direct summands. Dually one has a coresolving
subcategory.
Lemma 2.5. Let X be a full subcategory of A-mod. Then
MONOMORPHISM OPERATOR AND PERPENDICULAR OPERATOR 5
(i) Mon(Q,X) is closed under taking extensions (resp. kernels of epimorphisms, direct sum-
mands) if and only if X is closed under taking extensions (resp. kernels of epimorphisms, direct
summands).
(ii) Mon(Q,X) is resolving if and only if X is resolving. In particular, Mon(Q,A) is resolving.
Proof. (i)canbesimilarlyprovedasLemma2.4. For(ii), byLemma2.3thebranchesofprojective
Λ-modules are projective A-modules. From this and (i) the assertion follows. (cid:4)
3. Reciprocity
3.1. This sectionis to provethe followingreciprocityofthe monomorphismoperatorandthe left
perpendicular operator.
Theorem 3.1. Let T be an A-module such that there is an exact sequence0→T →···→T →
m 0
D(A )→0 with each T ∈add(T), then Mon(Q, ⊥T)= ⊥(kQ⊗ T).
A j k
For Q = • → ··· → • this result has been obtained in [Z, Theorem 3.1(i)]. Since some adjoint
pairs in [Z, Lemma 1.2] are not available here, the arguments in [Z] can not be generalized to the
general case. Here we adopt the following adjoint pair (Coker ,S(i)⊗−).
i
3.2. The following observation will be used throughout this section.
Lemma 3.2. Let X = (X ,X ) ∈ Λ-mod and T ∈ A-mod. Then for each i ∈ Q we have an
i α 0
isomorphism of abelian groups which is natural in both positions
HomA(Cokeri(X),T)∼=HomΛ(X,S(i)⊗kT).
Proof. If we write S(i)⊗ T ∈ Λ-mod as (Y ,Y ), then Y = 0 for j 6= i and Y = T. Consider
k j α j i
the homomorphism Ψ:Hom (Coker (X),T)→Hom (X,S(i)⊗ T) given by
A i Λ k
f 7→Ψ(f)=(g ,j ∈Q ): X →S(i)⊗ T, ∀f ∈Hom (Coker (X),T),
j 0 k A i
where g = 0 for j 6= i, and g = f π (X) : X → T with the canonical map π (X) : X →
j i i i i i
Coker (X). By (2.1)itis clearthat Ψ(f)∈Hom (X,S(i)⊗ T)andΨ is surjective. Itis injective
i Λ k
since π (X) is surjective. (cid:4)
i
3.3. We need the following fact.
Lemma 3.3. Let T be an A-module. For each i∈Q we have ⊥(kQ⊗ T)= ⊥( (S(i)⊗ T)).
0 k k
i∈LQ0
Proof. Put S = S(i), and J to be the Jacobson radical of kQ with Jl = 0. Let X ∈
i∈LQ0
⊥(kQ⊗T). By the exact sequence 0 → J ⊗ T → kQ⊗ T → S ⊗ T → 0 we get the exact
k k k
sequence:
···→Extj(X,kQ⊗ T)→Extj(X,S⊗ T)→Extj+1(X,J ⊗ T)→··· .
Λ k Λ k Λ k
SincekQishereditary,J ∈add(kQ)andhenceExtj(X,J⊗ T)=0, ∀j ≥1. ThusX ∈ ⊥(S⊗ T).
Λ k k
6 KEYANSONG,PUZHANG
Conversely, let X ∈ ⊥(S ⊗ T). From the exact sequence 0 → Jl−1 ⊗ T → Jl−2 ⊗ T →
k k k
(Jl−2/Jl−1)⊗ T →0andbyJl−1⊗ T, Jl−2/Jl−1⊗ T ∈add(S⊗ T)weseeX ∈ ⊥(Jl−2⊗ T).
k k k k k
Continuing this process we finally see X ∈ ⊥(J0⊗ T)= ⊥(kQ⊗ T). (cid:4)
k k
Proposition 3.4. We have Mon(Q,A)= ⊥(kQ⊗ D(A )).
k A
Proof. By Lemma 3.3 it suffices to prove the following equality, for each i∈Q :
0
⊥(S(i)⊗ D(A ))={X =(X ,X )∈Rep(Q,A) | δ (X) is injective}.
k A j α i
Let P be the projective cover of X. Applying functor F+ and F to the exact sequence
X i i
0→Ω(X)→P →X →0 we get the following commutative diagram with exact rows
X
0 −−−−→ F+(Ω(X)) −−−−→ F+(P ) −−−−→ F+(X) −−−−→ 0
i i X i
δi(Ω(X)) δi(PX) δi(X)
  
y y y
0 −−−−→ F (Ω(X)) −−−−→ F (P ) −−−−→ F (X) −−−−→ 0.
i i X i
By Snake Lemma we have the exact sequence
0→Kerδ (X)→Coker (Ω(X))→Coker (P )→Coker (X)→0. (∗)
i i i X i
Assume that δ (X) is injective. Applying Hom (−,D(A )) to (∗) and by Lemma 3.2 we get
i A A
the following exact sequence (with Hom omitted)
0→(X,S(i)⊗ D(A ))→(P ,S(i)⊗ D(A ))→(Ω(X),S(i)⊗ D(A ))→0. (∗∗)
k A X k A k A
Applying Hom (−,S(i)⊗ D(A )) to 0→Ω(X)→P →X →0 we get the exact sequence
Λ k A X
0→(X,S(i)⊗D(A ))→(P ,S(i)⊗D(A))→(Ω(X),S(i)⊗D(A))→Ext1(X,S(i)⊗D(A))→0.
A X Λ
Comparingitwith(∗∗)weseeExt1(X,S(i)⊗ D(A ))=0. ByLemma2.3wehaveinj.dim(S(i)⊗
Λ k A k
D(A ))≤1, so X ∈ ⊥(S(i)⊗ D(A )).
A k A
Conversely,assumeX ∈ ⊥(S(i)⊗ D(A )). ApplyingHom (−,S(i)⊗ D(A ))to0→Ω(X)→
k A Λ k A
P →X →0 and using Lemma 3.2, we get the following exact sequence
X
0→(Coker (X),D(A ))→(Coker (P ),D(A ))→(Coker (Ω(X)),D(A ))→0,
i A i X A i A
i.e., 0 → Coker (X) → Coker (P(X)) → Coker (Ω(X)) → 0 is exact. Comparing it with (∗) we
i i i
see Kerδ (X)=0. (cid:4)
i
3.4. Replacing D(A ) in Proposition 3.4 by an arbitrary A-module T, we have
A
Proposition 3.5. Let T be an A-module. Then Mon(Q, ⊥T)= ⊥(kQ⊗ T) ∩ Mon(Q,A).
k
Proof. We first prove that for each i∈Q there holds the following equality
0
⊥(S(i)⊗ T) ∩ Mon(Q,A)={X =(X ,X )∈Rep(Q,A) | Coker (X)∈ ⊥T,
k j α i
δ (X) is injective for all j ∈Q }. (3.1)
j 0
MONOMORPHISM OPERATOR AND PERPENDICULAR OPERATOR 7
Let X ∈ Mon(Q,A) with a projective resolution ··· → P1 → P0 → X → 0. Since each Pi is
in Mon(Q,A) (cf. Lemma 2.3) and Mon(Q,A) is closed under taking the kernels of epimorphisms
(cf. Lemma 2.5), it follows from Lemma 2.4 that we have the exact sequence
···→Coker (P1)→Coker (P0)→Coker (X)→0.
i i i
We claim it is a projective resolution of Coker (X). In fact, by (2.2) we have
i
T, if j =i;
Coker (P(j)⊗ T)=
i k 
0, if j 6=i.

So Coker (kQ⊗ T) = T and Coker (kQ⊗ A) = A. Thus Coker (Pj) is a projective A-module
i k i k i
since Pj ∈add(kQ⊗ A).
k
ApplyingHom(−,S(i)⊗ T)to···→P1 →P0 →X →0,byLemma3.2wehavethefollowing
k
commutative diagram
0 −−−−→ (X,S(i)⊗ T) −−−−→ (P0,S(i)⊗ T) −−−−→ (P1,S(i)⊗ T) −−−−→ ···
k k k
≀ ≀ ≀
  
  
0 −−−−→ (Cokery(X),T) −−−−→ (Cokery(P0),T) −−−−→ (Cokery(P1),T) −−−−→ ···
i i i
Note that X ∈ ⊥(S(i)⊗ T) if and only if the upper row is exact, if and only if the lower one is
k
exact, if and only if Coker (X)∈ ⊥T. This proves (3.1).
i
Now, assume that X ∈ Mon(Q, ⊥T). By definition and (3.1) we know X ∈ ⊥(S(i) ⊗
k
T) ∩ Mon(Q,A) for each i ∈ Q . By Lemma 3.3 we know X ∈ ⊥(kQ ⊗ T) and hence
0 k
X ∈ ⊥(kQ⊗ T) ∩ Mon(Q,A).
k
Conversely,assume that X ∈ ⊥(kQ⊗ T) ∩ Mon(Q,A). By Lemma 3.3 X ∈ ⊥(S(i)⊗ T) ∩
k k
Mon(Q,A)foreachi∈Q . ToseeX ∈Mon(Q, ⊥T),by(3.1)itremainstoproveX ∈ ⊥T foreach
0 i
i∈Q . For each i∈Q , set l =0 if i is a source, and l =max{ l(p) |p is a path with e(p)=i}
0 0 i i
if otherwise, where l(p) is the length of p. We prove X ∈ ⊥T by using induction on l . If l =0,
i i i
then i is a source and X = Coker (X) ∈ ⊥T. Let l 6= 0. Then we have the exact sequence
i i i
0−→ X −→X −→Coker (X)−→0withCoker (X)∈ ⊥T. Sincel <l forα∈Q
s(α) i i i s(α) i 1
αL∈Q1
e(α)=i
and e(α)=i, by induction X ∈ ⊥T, and hence X ∈ ⊥T. This completes the proof. (cid:4)
s(α) i
3.5. Proof of Theorem 3.1. By Proposition3.5 it suffices to prove ⊥(kQ⊗ T)⊆Mon(Q, A).
k
ByProposition3.4itsufficesto prove⊥(kQ⊗ T)⊆ ⊥(kQ⊗ D(A )). LetX ∈ ⊥(kQ⊗ T). By
k k A k
assumptionwehaveanexactsequence0−→kQ⊗ T −→···−→kQ⊗ T −→kQ⊗ D(A )−→
k m k 0 k A
0 with each kQ⊗ T ∈add(kQ⊗ T). From this we see the assertion. (cid:4)
k j k
3.6. Let GP(A) denote the category of the Gorenstein-projective A-modules. If A is Gorenstein
(i.e., inj.dim A<∞ andinj.dimA <∞), then GP(A)= ⊥A ([EJ,Corollary11.5.3]). Note that
A A
if A is Gorenstein then so is Λ. Taking T = A in Theorem 3.1 we have
A
Corollary 3.6. Let A be a Gorenstein algebra. Then GP(Λ)=Mon(Q,GP(A)).
8 KEYANSONG,PUZHANG
4. Monomorphism categories and cotilting theory
4.1. The aim of this section is to prove the following
Theorem 4.1. Let T be a cotilting A-module. Then kQ⊗ T is a unique cotilting Λ-module, up
k
to multiplicities of indecomposable direct summands, such that Mon(Q, ⊥T)= ⊥(kQ⊗ T).
k
For Q = • →···→ • this result has been obtained in [Z, Theorem 3.1(ii)]. We stress that the
proof in [Z] can not be generalized to the general case. Here we need to use Lemma 4.4 below,
rather than a concrete construction in [Z, Lemma 3.7].
4.2. Recall that an A-module T is an r-cotilting module ([HR], [AR], [H], [Mi]) if the following
conditions are satisfied:
(i) inj.dimT ≤r;
(ii) Exti (T,T)=0 for i≥1;
A
(iii) there is an exact sequence 0→T →···→T →D(A )→0 with each T ∈add(T).
m 0 A j
For short,by m we denote the functor P(i)⊗ −: A-mod→Mon(Q,A), andby mwe denote
i k
thefunctorkQ⊗ −:A-mod→Mon(Q,A). Thenm(T)= m (T)=kQ⊗ T, ∀T ∈A-mod.
k i k
i∈LQ0
Lemma 4.2. ([LZ,Lemma 2.3]) We have adjoint pair (m ,F ) for each i∈Q , where functor F
i i 0 i
is defined in 2.4.
We also need the following fact.
Lemma 4.3. Let X = (X ,X ) ∈ Λ-mod and T ∈ A-mod. Then we have an isomorphism of
j α
abelian groups for each i∈Q , which is natural in both positions
0
ExtsΛ(mi(T),X)∼=ExtsA(T,Xi), ∀ s≥0.
Proof. The proofis sameasin [Z,Lemma 3.4]forQ=•→···→•. Forcompletenesswe include
a justification. Taking the i-th branch of an injective resolution 0 →X →I0 → I1 →··· of X,
Λ
by Lemma 2.3 0→X →I0 →I1 →··· is an injective resolution of X . On the other hand by
i i i A i
Lemma 4.2 we have the following commutative diagram
0 −−−−→ Hom (m (T),X) −−−−→ Hom (m (T),I0) −−−−→ Hom (m (T),I1) −−−−→ ···
Λ i Λ i Λ i
≀ ≀ ≀
  
  
0 −−−−→ Hom y(T,X ) −−−−→ Hom y(T,I0) −−−−→ Hom y(T,I1) −−−−→ ··· ,
A i A i A i
from this we see the assertion. (cid:4)
4.3. Let X be a full subcategory of A-mod. Following [AR] let X denote the full subcategory
of A-mod consisting of those A-modules X such that there is anbexact sequence 0 → X →
m
Xm−1 →···→X0 →X →0 with eachXi ∈X. Recallthat X is self-orthogonal if ExtsA(M,N)=
0, ∀ M, N ∈ X, ∀ s ≥ 1. In this case X ⊆ X⊥, where X⊥ = {X ∈ A-mod | Exti (M,X) =
A
0, ∀ M ∈X, ∀ i≥1}. b
MONOMORPHISM OPERATOR AND PERPENDICULAR OPERATOR 9
The following fact is of independent interest. It is a key step in the proof of Theorem 4.1.
Lemma 4.4. Let X be a self-orthogonal full subcategory of A-mod. Then
(i) X is closed under taking cokernels of monomorphisms.
b
(ii) X is closed under taking extensions.
b
(iii) If X is closed under taking kernels of epimorphisms, then so is X.
b
f
Proof. (i) Let 0→X −→Y →Z →0 be an exact sequence with X,Y ∈X. By definition there
exist exact sequences 0 → Xn → Xn−1 → ··· → X0 −c→0 X → 0, and 0 →bYn → Yn−1 → ··· →
Y −d→0 Y → 0 with X , Y ∈ X ∪{0}, 0 ≤ i ≤ n. Since X is self-orthogonal, f : X → Y induces
0 i i
a chain map f• : X• → Y•, where X• is the complex 0 → Xn → Xn−1 → ··· → X0 → 0, and
similarly for Y•. Consider the following commutative diagram in the bounded derived category
Db(A), with two rows being distinguished triangles
X• f• // Y• // Con(f•) // X•[1]
c0 d0
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
// // //
X Y Z X[1]
f
(note that the lower row is also a distinguished triangle since 0 → X −→ Y → Z → 0 is exact),
where Con(f•) is the mapping cone 0 → Xn → Xn−1⊕Yn → ··· → X0⊕Y1 −→∂ Y0 → 0. Since
c and d are isomorphisms in Db(A), we have Z ∼= Con(f•) in Db(A). It follows that the i-th
0 0
cohomology group of Con(f•) is isomorphic to the i-th cohomology group of the stalk complex Z
for eachi∈Z. InparticularCon(f•) is exactexceptatthe 0-thposition, andY /Im∂ ∼=Z. Thus
0
∂
0→Xn →Xn−1⊕Yn →···→X0⊕Y1 −→Y0 →Z →0
is exact. This proves Z ∈X.
b
(iii)canbesimilarlyproved,and(ii)canbeprovedbyaversionofHorse-shoeLemma. Weomit
the details. (Only (i) will be needed in the proof of Theorem 4.1.) (cid:4)
Lemma 4.5. Let T be an r-cotilting A-module. Then kQ⊗ T is an (r+1)-cotilting Λ-module
k
with EndΛ(kQ⊗kT)∼=(kQ⊗kEndA(T))op.
Proof. Let 0 → T → I → ··· → I → 0 be a minimal injective resolution of T. Then we
0 r
have the exact sequence 0 → kQ⊗ T → kQ⊗ I → ··· → kQ⊗ I → 0. By Lemma 2.3
k k 0 k r
inj.dim(kQ⊗ I )≤1, 0≤j ≤r, it follows that inj.dim(kQ⊗ T)≤r+1.
k j k
Since the branch (kQ⊗ T) is a direct sum of copies of T, by Lemma 4.3 we have
k i
Exts(kQ⊗ T,kQ⊗ T)= Exts(m (T),kQ⊗ T)
Λ k k M Λ i k
i∈Q0
∼= M ExtsA(T,(kQ⊗kT)i)=0, ∀ s≥1.
i∈Q0
Now, put X =add(kQ⊗ T). To see that kQ⊗ T is a cotilting Λ-module, it remains to claim
k k
D(Λ ) ∈ X, i.e., D(kQ )⊗ D(A ) ∈ X. In fact, since proj.dim D(kQ ) = 1, we have an
Λ kQ k A kQ
b b
10 KEYANSONG,PUZHANG
exact sequence 0 → P → P → D(kQ ) → 0 with P ,P being projective kQ-modules. So we
1 0 kQ 0 1
have the exact sequence 0→P ⊗ D(A )→P ⊗ D(A )→D(kQ )⊗ D(A )→0. Since T
1 k A 0 k A kQ k A
is a cotilting A-module, we have an exact sequence 0→T →···→T →D(A )→0 with each
m 0 A
T ∈add(T). Sowehavetheexactsequence0→P ⊗ T →···→P ⊗ T →P ⊗ D(A )→0,
j i k m i k 0 i k A
wherei=0,1,with eachP ⊗T ∈add(kQ⊗ T). Thus P ⊗ D(A )∈X andP ⊗ D(A )∈X.
i j k 0 k A 1 k A
By Lemma 4.4(i) we have D(kQ )⊗ D(A )∈X. b b
kQ k A
Finally, by Lemma 4.2 we have b
HomΛ(mi(T),mj(T))∼=HomA(T,(mj(T))i)=(EndA(T))mji,
wherem isthe numberofpathsofQfromj toi. Thusonecaneasilyseethatthereisanalgebra
ji
isomorphism
EndΛ(kQ⊗kT)∼= M HomΛ(mi(T),mj(T))∼=(kQ⊗kEndA(T))op.
i,j∈Q0
(In fact, if we label the vertices of Q as 1,··· ,n, such that if there is an arrow from j to i then
j >i. Then
kkm21 km31 ··· kmn1
0 k km32 ··· kmn2
kQ∼= 0 0 k ··· kmn3 ,
. . . .
.. .. .. .. 
 
0 0 0 ··· k n×n
and hence
EndA(T)EndA(T)m21 EndA(T)m31 ··· EndA(T)mn1
 0 EndA(T) EndA(T)m32 ··· EndA(T)mn2
kQ⊗kEndA(T)∼= 0.. 0.. EndA..(T) ··· EndA(..T)mn3 .)
 . . . . 
 0 0 0 ··· EndA(T) n×n
This completes the proof. (cid:4)
4.4. Proof of Theorem 4.1. By Lemma 4.5 kQ⊗ T is a cotilting Λ-module, and by Theorem
k
3.1 Mon(Q, ⊥T)= ⊥(kQ⊗ T).
k
If L is another cotilting Λ-module such that ⊥L=Mon(Q, ⊥T)= ⊥(kQ⊗ T), then
k
Exts((kQ⊗ T)⊕L,(kQ⊗ T)⊕L)=0, ∀ s≥1,
Λ k k
so(kQ⊗ T)⊕LisalsoacotiltingΛ-module. By[H]thenumberofpairwisenon-isomorphicdirect
k
summands of (kQ⊗ T)⊕L is equalto the one of kQ⊗ T,from whichthe proofis completed. (cid:4)
k k
5. Contravariantly finiteness of monomorphism categories
5.1. Let X be a full subcategory of A-mod and M ∈ A-mod. Recall from [AR] that a right
X-approximation of M is an A-map f : X −→ M with X ∈ X, such that the induced homomor-
phism Hom (X′,X) −→ Hom (X′,M) is surjective for X′ ∈ X. If every A-module M admits
A A
a right X-approximation, then X is contravariantly finite in A-mod. Dually one has the concept
of a covariantly finite subcategory. If X is both contravariantly and covariantly finite, then X is
functorially finite in A-mod. Due to H. Krause and Ø. Solberg [KS, Corollary 0.3], a resolving