auslander-reiten translations in monomorphism categories bao-lin.pdf · auslander-reiten...

Auslander-Reiten Translations in MonomorphismCategories

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

3 If A is commutative uniserial then τ6SX ∼= 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

Morn(A): the morphism category of A-mod, n ≥ 2

Objects: X(φi ) =

(X1...

Xn

)(φi )

, φi : Xi+1 → Xi are A-maps, i.e.

Xnφn−1 // Xn−1

φn−2 // · · · φ2 // X2φ1 // X1

Morphisms: f : X(φi ) → Y(θi ) is f =

(f1...fn

), where fi : Xi → Yi are

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

Xn

fn

φn−1

// Xn−1

fn−1

φn−2

// · · ·φ2

// X2 φ1

//

f2

X1

f1

Ynθn−1 // Yn−1

θn−2 // · · · θ2 // Y2θ1 // Y1.


The notions

The monomorphism category Sn(A) is the full subcategory ofMorn(A) consisting of all the objects X(φi ) where φi : Xi+1 −→ Xiare monomorphisms for 1 ≤ i ≤ n − 1.

The epimorphism category Fn(A) is the full subcategory ofMorn(A) consisting of all the objects X(φi ) where φi : Xi+1 −→ Xiare epimorphisms for 1 ≤ i ≤ n − 1.


The kernel and cokernel functors

Ker : Morn(A) −→ Sn(A),X1X2...

Xn−1Xn

(φi )

7→

Xn

Ker(φ1···φn−1)

...Ker(φn−2φn−1)

Kerφn−1

(φ′i )

,

where φ′i : Ker(φi · · ·φn−1) → Ker(φi−1 · · ·φn−1), 2 ≤ i ≤ n − 1, andφ′1 : Ker(φ1 · · ·φn−1) → Xn are the canonical monomorphisms.


The kernel and cokernel functors

Cok : Morn(A) −→ Fn(A),X1X2...

Xn−1Xn

(φi )

7→

Cokerφ1

Coker(φ1φ2)

...Coker(φ1···φn−1)

X1

(φ′′i )

,

where φ′′i : Coker(φ1 · · ·φi+1) Coker(φ1 · · ·φi), 1 ≤ i ≤ n − 2, andφ′′n−1 : X1 Coker(φ1 · · ·φn−1) are the canonical epimorphisms.


The functor: Mono

Mono : Morn(A) −→ Sn(A),X1X2...

Xn−1Xn

(φi )

7→

X1

Imφ1...

Im(φ1···φn−2)Im(φ1···φn−1)

(φ′i )

,

where φ′i : Im(φ1 · · ·φi) → Im(φ1 · · ·φi−1), 2 ≤ i ≤ n − 1, andφ′1 : Im φ1 → X1 are the canonical monomorphisms.


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

e′i : Ker φi → IKer φi .

Then we have an extension

ei : Xi+1 −→ IKer φi .


The object MimoX(φi)

Define MimoX(φi ) to be the objectX1⊕IKerφ1⊕···⊕IKerφn−1X2⊕IKerφ2⊕···⊕IKerφn−1

...Xn−1⊕IKerφn−1

Xn

(θi )

where θi =

φi 0 0 ··· 0ei 0 0 ··· 00 1 0 ··· 00 0 1 ··· 0...

...... ···

...0 0 0 ··· 1

(n−i+1)×(n−i)

.

That is

Xnθn−1 // Xn−1 ⊕ IKer φn−1

θn−2 // · · · 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 isgiven by

τSX(φi )∼= Mimo τ CokX(φi ) (1).


Remark 2.2

τSX(φi )∼= Mimo τ CokX(φi ). (1)

The process means:Give an object X(φi ) in Sn(A)

Take the cokernel object X ′(φ′i )

= CokX(φi ).

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 )

in Morn(A)

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 toisomorphism.


An example

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

τS

( AS0

)(0,i)

= Mimoτ( S

AA

)(1,π)

= Mimo( S

00

)(0,0)

=( S

00

)(0,0)

τS

( S00

)(0,0)

= Mimoτ( S

SS

)(1,1)

= Mimo( S

SS

)(1,1)

=( S

SS

)(1,1)

τS

( SSS

)(1,1)

= Mimoτ( 0

0S

)(0,0)

= Mimo( 0

0S

)(0,0)

=( A

AS

)(i,1)

τS

( AAS

)(i,1)

= Mimoτ( 0

SA

)(π,0)

= Mimo( 0

S0

)(0,0)

=( A

S0

)(1,i)

—————————————————————————————–

τS

( SS0

)(0,1)

= Mimoτ( 0

SS

)(1,0)

= Mimo( 0

SS

)(1,0)

=( A

SS

)(1,i)

τS

( ASS

)(1,i)

= Mimoτ( S

SA

)(π,1)

= Mimo( S

S0

)(0,1)

=( S

S0

)(0,1)


An example

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

A00

!!CCC

CCAA0

!!CCC

CCAAA

!!CCC

CC

S00

==

!!CCC

CCAS0

==

!!CCC

CCoo A

AS

==

!!CCC

CCoo S

SS

oo

SS0

==

!!CCC

CCASS

==

!!CCC

CCoo S

S0

==oo

SSS

== S00

==oo

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

Objects: X(φi ) =

(X1

...Xn

)(φi )

, φi : Xi+1 → Xi in A-mod,

Morphisms:

f1...fn

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

diagram commutes in A-mod

Xn

fn

φn−1// Xn−1

fn−1

φn−2// · · ·

φ2 // X2φ1 //

f2

X1

f1

Yn

θn−1// Yn−1

θn−2// · · ·

θ2 // Y2θ1 // Y1.


The rotation of X(φi)

For X(φi ) ∈ Morn(A-mod), we have the following commutative diagramwith exact rows in A-mod,

Xn //

φn−1

X1 ψn−1

// Y 1n

//

ψn−2

Ω−1Xn

Xn−1 //

φn−2

X1 // Y 1n−1

//

ψn−3

Ω−1Xn−1

...φ3

......ψ2

...

X3 //

φ2

X1 // Y 13

//

ψ1

Ω−1X3

X2

φ1 // X1 // Y 12

// Ω−1X2.


The rotation of X(φi)

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

(X1ψn−1

//Y 1n

// · · ·ψ1 //Y 1

2 ) ∈ Morn(A-mod)

(here,a for convenience we write the rotation in a row). We remark thatRotX(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−1X1

...Ω−1Xn

)(Ω−1φi )

∈ Morn(A-mod).

Proposition 3.2

Let A be a selfinjective algebra, X(φi ) ∈ Sn(A). Then there are thefollowing isomorphisms in Morn(A-mod)

(i) τ jSX(φi )

∼= τ j RotjX(φi ) for j ≥ 1. In particular, τSX(φi )∼= τ CokX(φi ).

(ii) τs(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

τs(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 algebrasA(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 τm(n+1)S X(φi )

∼= X(φi );

(ii) If n is even, then τ2m(n+1)S X(φi )

∼= X(φi ).


An example

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

1•β //

δoo 3•

γoo

Then A is selfinjective with τ ∼= Ω−1 and Ω6 ∼= id on the object ofA-mod. The Auslander-Reiten quiver of A is

312

<<<

<1

2 31

777

7777

7

3##F

FFF12

@@@

@

AAoo 3

1##GG

GGoo 2

%%KKKKKoo

12 3

##GGGG

;;wwww1

@@@

@

??~~~~oo 2 3

1

%%KKKKK

99sssss

CCoo 1

2 3oo

2

;;xxxx 13

<<<

<

??~~~~oo 2

1

;;wwwwoo 3

99sssssoo

213

AA


An example

Let X(φi ) be an indecomposable nonprojective object in Sn(A).By (2), for s ≥ 1 we have

τs(n+1)S X(φi )

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

in Sn(A).Then we get

(i) if n ≡ 0, or 3 (mod6), then τn+1S X(φi )

∼= X(φi ); and

(ii) if n ≡ ±1, or ± 2 (mod6), then τ3(n+1)S X(φi )

∼= X(φi ).


Serre functors of stable monomorphism categories

A: a finite-dimensional selfinjective algebra over a fieldSn(A) is a Frobenius category.

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

Sn(A) is a Hom-finite Krull-Schmidt triangulated category withsuspension functor Ω−1

S = Ω−1Sn(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 Serrefunctor FS = FSn(A), which coincides with Ω−1

S τS on the objects ofSn(A).


Theorem 4.1Let 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−2snX(φi ). (4.4)

Moreover, if d1 and d2 are positive integers such that τd1M ∼= M and

Ωd2M ∼= M for each indecomposable nonprojective A-module M, then

F N(n+1)S X(φi )

∼= X(φi ), where N = [ d1(n+1,d1)

, d2(2n,d2)

].


Applying the above theorem to the selfinjective Nakayama algebrasA(m, t), we get

Corollary 4.2Let 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: [email protected]


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