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?
Bao-Lin Xiong (SJTU) Auslander-Reiten Translations in Monomorphism Categories ISPN ’80 2 / 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?
Bao-Lin Xiong (SJTU) Auslander-Reiten Translations in Monomorphism Categories ISPN ’80 2 / 24
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.
Bao-Lin Xiong (SJTU) Auslander-Reiten Translations in Monomorphism Categories ISPN ’80 3 / 24
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.
Bao-Lin Xiong (SJTU) Auslander-Reiten Translations in Monomorphism Categories ISPN ’80 3 / 24
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.
Bao-Lin Xiong (SJTU) Auslander-Reiten Translations in Monomorphism Categories ISPN ’80 3 / 24
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.
Bao-Lin Xiong (SJTU) Auslander-Reiten Translations in Monomorphism Categories ISPN ’80 4 / 24
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.
Bao-Lin Xiong (SJTU) Auslander-Reiten Translations in Monomorphism Categories ISPN ’80 4 / 24
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.
Bao-Lin Xiong (SJTU) Auslander-Reiten Translations in Monomorphism Categories ISPN ’80 5 / 24
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.
Bao-Lin Xiong (SJTU) Auslander-Reiten Translations in Monomorphism Categories ISPN ’80 6 / 24
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.
Bao-Lin Xiong (SJTU) Auslander-Reiten Translations in Monomorphism Categories ISPN ’80 7 / 24
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 .
Bao-Lin Xiong (SJTU) Auslander-Reiten Translations in Monomorphism Categories ISPN ’80 8 / 24
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 .
Bao-Lin Xiong (SJTU) Auslander-Reiten Translations in Monomorphism Categories ISPN ’80 9 / 24
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).
Bao-Lin Xiong (SJTU) Auslander-Reiten Translations in Monomorphism Categories ISPN ’80 10 / 24
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.
Bao-Lin Xiong (SJTU) Auslander-Reiten Translations in Monomorphism Categories ISPN ’80 11 / 24
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)
Bao-Lin Xiong (SJTU) Auslander-Reiten Translations in Monomorphism Categories ISPN ’80 12 / 24
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.
Bao-Lin Xiong (SJTU) Auslander-Reiten Translations in Monomorphism Categories ISPN ’80 13 / 24
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.
Bao-Lin Xiong (SJTU) Auslander-Reiten Translations in Monomorphism Categories ISPN ’80 14 / 24
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.
Bao-Lin Xiong (SJTU) Auslander-Reiten Translations in Monomorphism Categories ISPN ’80 14 / 24
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.
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
(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).
Bao-Lin Xiong (SJTU) Auslander-Reiten Translations in Monomorphism Categories ISPN ’80 16 / 24
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.
Bao-Lin Xiong (SJTU) Auslander-Reiten Translations in Monomorphism Categories ISPN ’80 17 / 24
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.
Bao-Lin Xiong (SJTU) Auslander-Reiten Translations in Monomorphism Categories ISPN ’80 17 / 24
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 ).
Bao-Lin Xiong (SJTU) Auslander-Reiten Translations in Monomorphism Categories ISPN ’80 18 / 24
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 ).
Bao-Lin Xiong (SJTU) Auslander-Reiten Translations in Monomorphism Categories ISPN ’80 18 / 24
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
Bao-Lin Xiong (SJTU) Auslander-Reiten Translations in Monomorphism Categories ISPN ’80 19 / 24
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 ).
Bao-Lin Xiong (SJTU) Auslander-Reiten Translations in Monomorphism Categories ISPN ’80 20 / 24
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).
Bao-Lin Xiong (SJTU) Auslander-Reiten Translations in Monomorphism Categories ISPN ’80 21 / 24
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)
].
Bao-Lin Xiong (SJTU) Auslander-Reiten Translations in Monomorphism Categories ISPN ’80 22 / 24
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) .
Bao-Lin Xiong (SJTU) Auslander-Reiten Translations in Monomorphism Categories ISPN ’80 23 / 24
Thank you!
E-mail: [email protected]
Bao-Lin Xiong (SJTU) Auslander-Reiten Translations in Monomorphism Categories ISPN ’80 24 / 24