Auslander-Reiten Translations in Monomorphism Categories

Auslander-Reiten Translations in Monomorphism 

Categories 

Bao-Lin Xiong 

(Joint work with P. Zhang and Y. H. Zhang) 

Department of Mathematics, Shanghai Jiao Tong University 

Shanghai, 2011.10.4 

Bao-Lin Xiong (SJTU) Auslander-Reiten Translations in Monomorphism Categories ISPN ’80 1 / 24

Motivation 

C. M. Ringel and M. Schmidmeier, 2008: 

1 The submodule category S(A) of an Artin algebra A has 

AR-sequences. 

2 τSX ∼ = Mimo τ CokX for X ∈ S(A), where τS (resp. τ) is the 

AR-translation in S(A) (resp. A-mod). 

3 If A is commutative uniserial then τ 6 S X ∼ = X for each 

indecomposable nonprojective object X ∈ S(A). 


Motivation 

C. M. Ringel and M. Schmidmeier, 2008: 

1 The submodule category S(A) of an Artin algebra A has 

AR-sequences. 

2 τSX ∼ = Mimo τ CokX for X ∈ S(A), where τS (resp. τ) is the 

AR-translation in S(A) (resp. A-mod). 

3 If A is commutative uniserial then τ 6 S X ∼ = X for each 

indecomposable nonprojective object X ∈ S(A). 

Question: Can we generalize the above theory? 


The notions 

A: an Artin algebra, A-mod: the category of all fin. gen. left A-modules 


The notions 


Morn(A): the morphism category 

� 

of A-mod, n ≥ 2 

Objects: X (φi ) = 

Xn 

� X1 

. 

Xn 

(φ i ) 

φn−1 �� 

Xn−1 

, φ i : X i+1 → X i are A-maps, i.e. 

φn−2 �� 

· · · 

φ2 �� 

X2 

φ1 �� 

X1 


The notions 


Morn(A): the morphism category 

� 

of A-mod, n ≥ 2 


Xn 

� X1 

. 

Xn 

(φ i ) 

φn−1 �� 

Xn−1 

, φ i : X i+1 → X i are A-maps, i.e. 

φn−2 �� 

· · · 

� f1 

φ2 �� 

X2 

φ1 �� 

X1 

Morphisms: f : X (φi ) → Y (θi ) is f = 

. 

, where fi : Xi → Yi are 

fn 

A-maps for 1 ≤ i ≤ n, such that the following diagram commutes 

fn 

Xn 

�� 

Yn 

φn−1 

θn−1 � 

�� 

Xn−1 

fn−1 

�� 

� Yn−1 

φn−2 

�� 

· · · 

θn−2 �� 

· · · 

� 

φ2 

θ2 � 

�� 

X2 

f2 

�� 

� Y2 

φ1 

θ1 � 

�� 

f1 

X1 

�� 

� Y1. 


The notions 

The monomorphism category Sn(A) is the full subcategory of 

Morn(A) consisting of all the objects X (φi ) where φ i : X i+1 −→ X i 

are monomorphisms for 1 ≤ i ≤ n − 1. 


The notions 

The monomorphism category Sn(A) is the full subcategory of 


are monomorphisms for 1 ≤ i ≤ n − 1. 

The epimorphism category Fn(A) is the full subcategory of 


are epimorphisms for 1 ≤ i ≤ n − 1. 


The kernel and cokernel functors 

Ker : Morn(A) −→ Sn(A), 

⎛ 

⎜ 

⎝ 

X1 

X2 

. 

Xn−1 

Xn 

⎞ 

⎟ 

⎠ 

(φ i ) 

⎛ 

⎜ 

↦→ ⎜ 

⎝ 

Xn 

Ker(φ1···φn−1) 

. 

Ker(φn−2φn−1) 

Ker φn−1 

where φ ′ i : Ker(φ i · · · φn−1) ↩→ Ker(φ i−1 · · · φn−1), 2 ≤ i ≤ n − 1, and 

φ ′ 1 : Ker(φ1 · · · φn−1) ↩→ Xn are the canonical monomorphisms. 

Bao-Lin Xiong (SJTU) Auslander-Reiten Translations in Monomorphism Categories ISPN ’80 5 / 24 

⎞ 

⎟ 

⎠ 

(φ ′ i ) 

,

The kernel and cokernel functors 

Cok : Morn(A) −→ Fn(A), 

⎛ 

⎜ 

⎝ 

X1 

X2 

. 

Xn−1 

Xn 

⎞ 

⎟ 

⎠ 

(φ i ) 

⎛ 

⎜ 

↦→ ⎜ 

⎝ 

Coker φ1 

Coker(φ1φ2) 

. 

Coker(φ1···φn−1) 

X1 

⎞ 

⎟ 

⎠ 

(φ ′′ 

i ) 

where φ ′′ 

i : Coker(φ1 · · · φ i+1) ↠ Coker(φ1 · · · φ i), 1 ≤ i ≤ n − 2, and 

φ ′′ 

n−1 : X1 ↠ Coker(φ1 · · · φn−1) are the canonical epimorphisms. 


,

The functor: Mono 

Mono : Morn(A) −→ Sn(A), 

⎛ 

⎜ 

⎝ 

X1 

X2 

. 

Xn−1 

Xn 

⎞ 

⎟ 

⎠ 

(φ i ) 

⎛ 

⎜ 

↦→ ⎜ 

⎝ 

X1 

Im φ1 

. 

Im(φ1···φn−2) 

Im(φ1···φn−1) 

where φ ′ i : Im(φ1 · · · φ i) ↩→ Im(φ1 · · · φ i−1), 2 ≤ i ≤ n − 1, and 

φ ′ 1 : Im φ1 ↩→ X1 are the canonical monomorphisms. 


⎞ 

⎟ 

⎠ 

(φ ′ i ) 

,

The object MimoX (φi) 

Let X (φi ) ∈ Morn(A). 

The object MimoX (φi ) ∈ Sn(A) is defined as follows. 

For each 1 ≤ i ≤ n − 1, fix an injective envelope 

Then we have an extension 

e ′ i : Ker φ i ↩→ IKer φ i. 

e i : X i+1 −→ IKer φ i. 


The object MimoX (φi) 

Define MimoX (φi ) to be the object 

⎛ 

⎜ 

⎝ 

That is 

Xn 

X1⊕IKer φ1⊕···⊕IKer φn−1 

X2⊕IKer φ2⊕···⊕IKer φn−1 

. 

Xn−1⊕IKer φn−1 

Xn 

⎞ 

⎟ 

⎠ 

θn−1 �� 

Xn−1 ⊕ IKer φn−1 

(θ i ) 

⎛ 

⎜ 

where θi = ⎜ 

⎝ 

θn−2 �� 

· · · 

φi 0 0 ··· 0 

ei 0 0 ··· 0 

0 1 0 ··· 0 

0 0 1 ··· 0 

. . . ··· . 

0 0 0 ··· 1 

⎞ 

⎟ 

⎠ 

(n−i+1)×(n−i) 

θ1 �� 

X1 ⊕ IKer φ1 ⊕ · · · ⊕ IKer φn−1 . 


.

The Auslander-Reiten translation in Sn(A) 

Theorem 2.1 

(i) The subcategories Sn(A) and Fn(A) are functorially finite in 

Morn(A) and hence have AR-sequences. 

(ii) For an object X (φi ) ∈ Sn(A), the Auslander-Reiten translate is 

given by 

τSX (φi ) ∼ = Mimo τ CokX (φi ) 

(1). 


Remark 2.2 

The process means: 

Give an object X (φi ) in Sn(A) 

τSX (φi ) ∼ = Mimo τ CokX (φi ). (1) 

Take the cokernel object X ′ (φ ′ i ) = CokX (φ i ). 

Apply τ to these maps φ ′ i (1 ≤ i ≤ n − 1). 

Represent τCokX (φi ) by an object X ′′ 

(φ ′′ 

i 

) = 

� X ′′ 

1 

. 

X ′′ 

n 

� 

(φ ′′ 

i ) 

where X ′′ 

1 , X ′′ 

2 , · · · , X ′′ 

n−1 have no nonzero injective direct 

summands. 

Apply Mimo, there is a well-defined object in Sn(A) up to 

isomorphism. 

in Morn(A) 


An example 

k: a field; A = k[X]/〈X 2 〉, S = k[X]/〈X〉, i : S −→ A, π : A −→ S. 

� � 

AS0 

� � 

SAA 

= Mimoτ 

� � 

S00 

= Mimo 

� � 

S00 

= 

τS 

τS 

τS 

τS 

� S00 

� SSS 

� AAS 

� 

� 

� 

(0,i) 

(0,0) 

(1,1) 

(i,1) 

� 

SSS 

= Mimoτ 

� 

00S 

= Mimoτ 

� 

0SA 

= Mimoτ 

� 

� 

� 

(1,π) 

(1,1) 

(0,0) 

(π,0) 

� 

SSS 

= Mimo 

� 

00S 

= Mimo 

� 

0S0 

= Mimo 

� 

� 

� 

(0,0) 

(1,1) 

(0,0) 

(0,0) 

= 

= 

= 

� SSS 

� AAS 

� AS0 

� 

� 

� 

(0,0) 

(1,1) 

—————————————————————————————– 

� � 

SS0 

� � 

0SS 

= Mimoτ 

� � 

0SS 

= Mimo 

� � 

ASS 

= 

τS 

τS 

� ASS 

� 

(0,1) 

(1,i) 

� 

SSA 

= Mimoτ 

� 

(1,0) 

(π,1) 

� 

SS0 

= Mimo 

� 

(1,0) 

(0,1) 

= 

� SS0 

� 

(i,1) 

(1,i) 

(1,i) 

(0,1) 


An example 

The Auslander-Reiten quiver of S3(A) looks like 

A A 

A 

00 � 

A 

�� 

0 � 

A 

�� 

�� 

� 

� 

� 

A �� 

�� 

�� 

�� 

� 

� 

� 

� �� 

� 

� 

�� 

� 

S � 

�� 

�� 

A � 

A �� 

�� 

S 

00 �� 

� 

S �� 

�� 

0 � 

A �� 

� 

� 

� 

S � 

S 

�� 

�� 

� 

� 

� 

� 

S 

�� 

�� 

�� 

�� 

� � 

� 

�� 

� 

S � 

�� 

�� 

A � �� 

S �� 

S �� 

0 � 

S �� 

� �� 

� 

S � 

S 

� 

� �� 

� �� 

0 

� 

� 

�� 

�� 

� �� 

S � �� 

S �� 

S �� 

00 

S 

Remark: This AR-quiver has been described by A.Moore. 


Applications to selfinjective algebras 

A: a selfinjective Artin algebra, 

A-mod: the stable category of A-mod 

Morn(A-mod): the morphism category of A-mod 

� 


� X1 

. 

Xn 

(φi ) 

, φi : Xi+1 → Xi in A-mod, 


Applications to selfinjective algebras 

A: a selfinjective Artin algebra, 

A-mod: the stable category of A-mod 

Morn(A-mod): the morphism category of A-mod 

� 


⎛ 

f1 

� X1 

⎞ 

. 

Xn 

(φi ) 

, φi : Xi+1 → Xi in A-mod, 

Morphisms: ⎝ ⎠ : X (φi . 

) → Y (θi ), fi : Xi → Yi such that the following 

fn 

diagram commutes in A-mod 

fn 

Xn 

�� 

Yn 

φn−1 

θn−1 

�� 

Xn−1 

fn−1 

�� 

�� 

Yn−1 

φn−2 

θn−2 

�� 

· · · 

�� 

· · · 

φ2 

θ2 

�� 

X2 

f2 

�� 

�� 

Y2 

φ1 

θ1 

�� 

f1 

X1 

�� 

�� 

Y1. 


The rotation of X (φi) 

For X (φi ) ∈ Morn(A-mod), we have the following commutative diagram 

with exact rows in A-mod, 

Xn 

φn−1 

�� 

Xn−1 

�� 

. 

φn−2 

φ3 

�� 

X3 

φ2 

�� 

X2 

φ1 

�� 

�� 

�� 

X1 ψn−1 

X1 

. 

X1 

�� 

X1 

�� 

Y 1 

n 

ψn−2 

�� 

�� 

Y 1 n−1 

�� 

. 

�� 

�� 

Y 1 

3 

�� 

�� 

Y 1 

2 

ψn−3 

ψ2 

ψ1 

�� 

Ω −1 Xn 

�� 

�� 

Ω−1Xn−1 �� 

. 

�� 

�� 

Ω−1X3 �� 

�� 

Ω−1X2. Bao-Lin Xiong (SJTU) Auslander-Reiten Translations in Monomorphism Categories ISPN ’80 15 / 24

The rotation of X (φi) 

The rotation RotX (φi ) of X (φi ) is defined to be 

(X1 

ψn−1 

��Y 

1 

n 

�� 

· · · ψ1 

��Y 

1 

2 ) ∈ Morn(A-mod) 

(here,a for convenience we write the rotation in a row). We remark that 

RotX (φi ) is well-defined. 

Lemma 3.1 

Let X (φi ) ∈ Morn(A). Then RotX (φi ) ∼ = Cok MimoX (φi ) in Morn(A-mod). 


For X (φi ) ∈ Morn(A-mod), define Ω−1X (φi ) to be 

� 

Ω−1 � 

X1 

. 

Ω −1 Xn 

(Ω −1 φ i ) 

∈ Morn(A-mod). 


For X (φi ) ∈ Morn(A-mod), define Ω−1X (φi ) to be 

� 

Ω−1 � 

X1 

. 

Ω −1 Xn 

(Ω −1 φ i ) 

Proposition 3.2 

∈ Morn(A-mod). 

Let A be a selfinjective algebra, X (φi ) ∈ Sn(A). Then there are the 

following isomorphisms in Morn(A-mod) 

(i) τ j 

S X (φ i ) ∼ = τ j Rot j X (φi ) for j ≥ 1. In particular, τSX (φi ) ∼ = τ CokX (φi ). 

(ii) τ s(n+1) 

S X (φi ) ∼ = τ s(n+1) Ω−s(n−1) X (φi ), ∀ s ≥ 1. 


Theorem 3.3 

Let A be a selfinjective algebra, and X (φi ) ∈ Sn(A). Then we have 

τ s(n+1) 

S X (φi ) ∼ = Mimo τ s(n+1) Ω −s(n−1) X (φi ), s ≥ 1. (2) 


Theorem 3.3 

Let A be a selfinjective algebra, and X (φi ) ∈ Sn(A). Then we have 

τ s(n+1) 

S X (φi ) ∼ = Mimo τ s(n+1) Ω −s(n−1) X (φi ), s ≥ 1. (2) 

Applying the above theorem to the selfinjective Nakayama algebras 

A(m, t), we get 

Corollary 3.4 

For an indecomposable nonprojective object X (φi ) ∈ Sn(A(m, t)), 

m ≥ 1, t ≥ 2, there are the following isomorphisms: 

(i) If n is odd, then τ m(n+1) 

S X (φi ) ∼ = X (φi ); 

(ii) If n is even, then τ 2m(n+1) 

S X (φi ) ∼ = X (φi ). 


An example 

Let A = kQ/〈δα, βγ, αδ − γβ〉, where Q is the quiver 2• 

�� 

α �� 

1• 

Then A is selfinjective with τ ∼ = Ω −1 and Ω 6 ∼ = id on the object of 

A-mod. The Auslander-Reiten quiver of A is 

3 

1 

1 

2 3 

�� 

2 �� 

1 

�� 

� 

�� 

� 

� �� 

�� 

�� 

3 � 

�� 

1 �� 

3 �� 

�� 

� 

� 

� 

� 

2 � 

� �� 

� 1 � 

� 

�� 

� 

� 

2 � 

� 

� 

� � 

� 

�� 

�� 

�� 

� 

�� 

� 

�� 

�� 

�� 

�� 

�� 

�� 

�� 

�� 

1 �� 

2 3 

2 3 1 �� 

�� 

1 

�� 

� �� 

� 

� � 

1 

�� 

� 

� 

2 3 

�� 

�� 

� 

�� 

� 

� �� 

�� 

� 

� �� 

�� 

� 

�� 

�� 

� 

1 �� 

�� 

2 �� 

�� 

2 �� 

� 

3 � 1 

3 

� �� 

�� 

�� 

�� 

2 �� 

1 

3 


δ 

�� 

β 

�� 

γ 

3•

An example 

Let X (φi ) be an indecomposable nonprojective object in Sn(A). 

By (2), for s ≥ 1 we have 

τ s(n+1) 

S X (φi ) ∼ = Mimo τ s(n+1) Ω −s(n−1) X (φi ) ∼ = Mimo Ω −2sn X (φi ) 

in Sn(A). 

Then we get 

(i) if n ≡ 0, or 3 (mod6), then τ n+1 

S X (φi ) ∼ = X (φi ); and 

(ii) if n ≡ ±1, or ± 2 (mod6), then τ 3(n+1) 

S X (φi ) ∼ = X (φi ). 


Serre functors of stable monomorphism categories 

A: a finite-dimensional selfinjective algebra over a field 

Sn(A) is a Frobenius category. 

Sn(A): the stable category of Sn(A) 

Sn(A) is a Hom-finite Krull-Schmidt triangulated category with 

suspension functor Ω −1 

S 

= Ω−1 

Sn(A) . Since Sn(A) has Auslander-Reiten 

sequences, it follows that Sn(A) has Auslander-Reiten triangles, and 

hence, by a theorem of Reiten and Van den Bergh, it has a Serre 

functor FS = F Sn(A), which coincides with Ω −1 

S �τS on the objects of 

Sn(A). 


Theorem 4.1 

Let A be a selfinjective algebra, and FS be the Serre functor of Sn(A). 

Then we have an isomorphism in Sn(A) for X (φi ) ∈ Sn(A) and for s ≥ 1 

F s(n+1) 

S X (φi ) ∼ = Mimo τ s(n+1) Ω −2sn X (φi ). (4.4) 

Moreover, if d1 and d2 are positive integers such that τ d1M ∼ = M and 

Ω d2M ∼ = M for each indecomposable nonprojective A-module M, then 

F N(n+1) 

S 

X (φi ) ∼ d1 

= X (φi ), where N = [ (n+1,d1) , 

d2 

(2n,d2) ]. 


Applying the above theorem to the selfinjective Nakayama algebras 

A(m, t), we get 

Corollary 4.2 

Let FS be the Serre functor of Sn(A(m, t)) with m ≥ 1, t ≥ 2, and X be 

an arbitrary object in Sn(A(m, t)). Then 

(i) If t = 2, then F N(n+1) 

S X ∼ = X, where N = m 

(m,n−1) . 

(ii) If t ≥ 3, then F N(n+1) 

S X ∼ = X, where N = m 

(m,t,n+1) . 


Thank you! 

E-mail: xiongbaolin@gmail.com

Auslander-Reiten Translations in Monomorphism Categories

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