Auslander-Reiten Translations in Monomorphism Categories
Auslander-Reiten Translations in Monomorphism Categories
Auslander-Reiten Translations in Monomorphism Categories
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<strong>Auslander</strong>-<strong>Reiten</strong> <strong>Translations</strong> <strong>in</strong> <strong>Monomorphism</strong><br />
<strong>Categories</strong><br />
Bao-L<strong>in</strong> Xiong<br />
(Jo<strong>in</strong>t work with P. Zhang and Y. H. Zhang)<br />
Department of Mathematics, Shanghai Jiao Tong University<br />
Shanghai, 2011.10.4<br />
Bao-L<strong>in</strong> Xiong (SJTU) <strong>Auslander</strong>-<strong>Reiten</strong> <strong>Translations</strong> <strong>in</strong> <strong>Monomorphism</strong> <strong>Categories</strong> ISPN ’80 1 / 24
Motivation<br />
C. M. R<strong>in</strong>gel and M. Schmidmeier, 2008:<br />
1 The submodule category S(A) of an Art<strong>in</strong> algebra A has<br />
AR-sequences.<br />
2 τSX ∼ = Mimo τ CokX for X ∈ S(A), where τS (resp. τ) is the<br />
AR-translation <strong>in</strong> S(A) (resp. A-mod).<br />
3 If A is commutative uniserial then τ 6 S X ∼ = X for each<br />
<strong>in</strong>decomposable nonprojective object X ∈ S(A).<br />
Bao-L<strong>in</strong> Xiong (SJTU) <strong>Auslander</strong>-<strong>Reiten</strong> <strong>Translations</strong> <strong>in</strong> <strong>Monomorphism</strong> <strong>Categories</strong> ISPN ’80 2 / 24
Motivation<br />
C. M. R<strong>in</strong>gel and M. Schmidmeier, 2008:<br />
1 The submodule category S(A) of an Art<strong>in</strong> algebra A has<br />
AR-sequences.<br />
2 τSX ∼ = Mimo τ CokX for X ∈ S(A), where τS (resp. τ) is the<br />
AR-translation <strong>in</strong> S(A) (resp. A-mod).<br />
3 If A is commutative uniserial then τ 6 S X ∼ = X for each<br />
<strong>in</strong>decomposable nonprojective object X ∈ S(A).<br />
Question: Can we generalize the above theory?<br />
Bao-L<strong>in</strong> Xiong (SJTU) <strong>Auslander</strong>-<strong>Reiten</strong> <strong>Translations</strong> <strong>in</strong> <strong>Monomorphism</strong> <strong>Categories</strong> ISPN ’80 2 / 24
The notions<br />
A: an Art<strong>in</strong> algebra, A-mod: the category of all f<strong>in</strong>. gen. left A-modules<br />
Bao-L<strong>in</strong> Xiong (SJTU) <strong>Auslander</strong>-<strong>Reiten</strong> <strong>Translations</strong> <strong>in</strong> <strong>Monomorphism</strong> <strong>Categories</strong> ISPN ’80 3 / 24
The notions<br />
A: an Art<strong>in</strong> algebra, A-mod: the category of all f<strong>in</strong>. gen. left A-modules<br />
Morn(A): the morphism category<br />
�<br />
of A-mod, n ≥ 2<br />
Objects: X (φi ) =<br />
Xn<br />
� X1<br />
.<br />
Xn<br />
(φ i )<br />
φn−1 ��<br />
Xn−1<br />
, φ i : X i+1 → X i are A-maps, i.e.<br />
φn−2 ��<br />
· · ·<br />
φ2 ��<br />
X2<br />
φ1 ��<br />
X1<br />
Bao-L<strong>in</strong> Xiong (SJTU) <strong>Auslander</strong>-<strong>Reiten</strong> <strong>Translations</strong> <strong>in</strong> <strong>Monomorphism</strong> <strong>Categories</strong> ISPN ’80 3 / 24
The notions<br />
A: an Art<strong>in</strong> algebra, A-mod: the category of all f<strong>in</strong>. gen. left A-modules<br />
Morn(A): the morphism category<br />
�<br />
of A-mod, n ≥ 2<br />
Objects: X (φi ) =<br />
Xn<br />
� X1<br />
.<br />
Xn<br />
(φ i )<br />
φn−1 ��<br />
Xn−1<br />
, φ i : X i+1 → X i are A-maps, i.e.<br />
φn−2 ��<br />
· · ·<br />
� f1<br />
φ2 ��<br />
X2<br />
φ1 ��<br />
X1<br />
Morphisms: f : X (φi ) → Y (θi ) is f =<br />
.<br />
, where fi : Xi → Yi are<br />
fn<br />
A-maps for 1 ≤ i ≤ n, such that the follow<strong>in</strong>g diagram commutes<br />
fn<br />
Xn<br />
��<br />
Yn<br />
φn−1<br />
θn−1 �<br />
��<br />
Xn−1<br />
fn−1<br />
��<br />
� Yn−1<br />
φn−2<br />
��<br />
· · ·<br />
θn−2 ��<br />
· · ·<br />
�<br />
φ2<br />
θ2 �<br />
��<br />
X2<br />
f2<br />
��<br />
� Y2<br />
φ1<br />
θ1 �<br />
��<br />
f1<br />
X1<br />
��<br />
� Y1.<br />
Bao-L<strong>in</strong> Xiong (SJTU) <strong>Auslander</strong>-<strong>Reiten</strong> <strong>Translations</strong> <strong>in</strong> <strong>Monomorphism</strong> <strong>Categories</strong> ISPN ’80 3 / 24
The notions<br />
The monomorphism category Sn(A) is the full subcategory of<br />
Morn(A) consist<strong>in</strong>g of all the objects X (φi ) where φ i : X i+1 −→ X i<br />
are monomorphisms for 1 ≤ i ≤ n − 1.<br />
Bao-L<strong>in</strong> Xiong (SJTU) <strong>Auslander</strong>-<strong>Reiten</strong> <strong>Translations</strong> <strong>in</strong> <strong>Monomorphism</strong> <strong>Categories</strong> ISPN ’80 4 / 24
The notions<br />
The monomorphism category Sn(A) is the full subcategory of<br />
Morn(A) consist<strong>in</strong>g of all the objects X (φi ) where φ i : X i+1 −→ X i<br />
are monomorphisms for 1 ≤ i ≤ n − 1.<br />
The epimorphism category Fn(A) is the full subcategory of<br />
Morn(A) consist<strong>in</strong>g of all the objects X (φi ) where φ i : X i+1 −→ X i<br />
are epimorphisms for 1 ≤ i ≤ n − 1.<br />
Bao-L<strong>in</strong> Xiong (SJTU) <strong>Auslander</strong>-<strong>Reiten</strong> <strong>Translations</strong> <strong>in</strong> <strong>Monomorphism</strong> <strong>Categories</strong> ISPN ’80 4 / 24
The kernel and cokernel functors<br />
Ker : Morn(A) −→ Sn(A),<br />
⎛<br />
⎜<br />
⎝<br />
X1<br />
X2<br />
.<br />
Xn−1<br />
Xn<br />
⎞<br />
⎟<br />
⎠<br />
(φ i )<br />
⎛<br />
⎜<br />
↦→ ⎜<br />
⎝<br />
Xn<br />
Ker(φ1···φn−1)<br />
.<br />
Ker(φn−2φn−1)<br />
Ker φn−1<br />
where φ ′ i : Ker(φ i · · · φn−1) ↩→ Ker(φ i−1 · · · φn−1), 2 ≤ i ≤ n − 1, and<br />
φ ′ 1 : Ker(φ1 · · · φn−1) ↩→ Xn are the canonical monomorphisms.<br />
Bao-L<strong>in</strong> Xiong (SJTU) <strong>Auslander</strong>-<strong>Reiten</strong> <strong>Translations</strong> <strong>in</strong> <strong>Monomorphism</strong> <strong>Categories</strong> ISPN ’80 5 / 24<br />
⎞<br />
⎟<br />
⎠<br />
(φ ′ i )<br />
,
The kernel and cokernel functors<br />
Cok : Morn(A) −→ Fn(A),<br />
⎛<br />
⎜<br />
⎝<br />
X1<br />
X2<br />
.<br />
Xn−1<br />
Xn<br />
⎞<br />
⎟<br />
⎠<br />
(φ i )<br />
⎛<br />
⎜<br />
↦→ ⎜<br />
⎝<br />
Coker φ1<br />
Coker(φ1φ2)<br />
.<br />
Coker(φ1···φn−1)<br />
X1<br />
⎞<br />
⎟<br />
⎠<br />
(φ ′′<br />
i )<br />
where φ ′′<br />
i : Coker(φ1 · · · φ i+1) ↠ Coker(φ1 · · · φ i), 1 ≤ i ≤ n − 2, and<br />
φ ′′<br />
n−1 : X1 ↠ Coker(φ1 · · · φn−1) are the canonical epimorphisms.<br />
Bao-L<strong>in</strong> Xiong (SJTU) <strong>Auslander</strong>-<strong>Reiten</strong> <strong>Translations</strong> <strong>in</strong> <strong>Monomorphism</strong> <strong>Categories</strong> ISPN ’80 6 / 24<br />
,
The functor: Mono<br />
Mono : Morn(A) −→ Sn(A),<br />
⎛<br />
⎜<br />
⎝<br />
X1<br />
X2<br />
.<br />
Xn−1<br />
Xn<br />
⎞<br />
⎟<br />
⎠<br />
(φ i )<br />
⎛<br />
⎜<br />
↦→ ⎜<br />
⎝<br />
X1<br />
Im φ1<br />
.<br />
Im(φ1···φn−2)<br />
Im(φ1···φn−1)<br />
where φ ′ i : Im(φ1 · · · φ i) ↩→ Im(φ1 · · · φ i−1), 2 ≤ i ≤ n − 1, and<br />
φ ′ 1 : Im φ1 ↩→ X1 are the canonical monomorphisms.<br />
Bao-L<strong>in</strong> Xiong (SJTU) <strong>Auslander</strong>-<strong>Reiten</strong> <strong>Translations</strong> <strong>in</strong> <strong>Monomorphism</strong> <strong>Categories</strong> ISPN ’80 7 / 24<br />
⎞<br />
⎟<br />
⎠<br />
(φ ′ i )<br />
,
The object MimoX (φi)<br />
Let X (φi ) ∈ Morn(A).<br />
The object MimoX (φi ) ∈ Sn(A) is def<strong>in</strong>ed as follows.<br />
For each 1 ≤ i ≤ n − 1, fix an <strong>in</strong>jective envelope<br />
Then we have an extension<br />
e ′ i : Ker φ i ↩→ IKer φ i.<br />
e i : X i+1 −→ IKer φ i.<br />
Bao-L<strong>in</strong> Xiong (SJTU) <strong>Auslander</strong>-<strong>Reiten</strong> <strong>Translations</strong> <strong>in</strong> <strong>Monomorphism</strong> <strong>Categories</strong> ISPN ’80 8 / 24
The object MimoX (φi)<br />
Def<strong>in</strong>e MimoX (φi ) to be the object<br />
⎛<br />
⎜<br />
⎝<br />
That is<br />
Xn<br />
X1⊕IKer φ1⊕···⊕IKer φn−1<br />
X2⊕IKer φ2⊕···⊕IKer φn−1<br />
.<br />
Xn−1⊕IKer φn−1<br />
Xn<br />
⎞<br />
⎟<br />
⎠<br />
θn−1 ��<br />
Xn−1 ⊕ IKer φn−1<br />
(θ i )<br />
⎛<br />
⎜<br />
where θi = ⎜<br />
⎝<br />
θn−2 ��<br />
· · ·<br />
φi 0 0 ··· 0<br />
ei 0 0 ··· 0<br />
0 1 0 ··· 0<br />
0 0 1 ··· 0<br />
. . . ··· .<br />
0 0 0 ··· 1<br />
⎞<br />
⎟<br />
⎠<br />
(n−i+1)×(n−i)<br />
θ1 ��<br />
X1 ⊕ IKer φ1 ⊕ · · · ⊕ IKer φn−1 .<br />
Bao-L<strong>in</strong> Xiong (SJTU) <strong>Auslander</strong>-<strong>Reiten</strong> <strong>Translations</strong> <strong>in</strong> <strong>Monomorphism</strong> <strong>Categories</strong> ISPN ’80 9 / 24<br />
.
The <strong>Auslander</strong>-<strong>Reiten</strong> translation <strong>in</strong> Sn(A)<br />
Theorem 2.1<br />
(i) The subcategories Sn(A) and Fn(A) are functorially f<strong>in</strong>ite <strong>in</strong><br />
Morn(A) and hence have AR-sequences.<br />
(ii) For an object X (φi ) ∈ Sn(A), the <strong>Auslander</strong>-<strong>Reiten</strong> translate is<br />
given by<br />
τSX (φi ) ∼ = Mimo τ CokX (φi )<br />
(1).<br />
Bao-L<strong>in</strong> Xiong (SJTU) <strong>Auslander</strong>-<strong>Reiten</strong> <strong>Translations</strong> <strong>in</strong> <strong>Monomorphism</strong> <strong>Categories</strong> ISPN ’80 10 / 24
Remark 2.2<br />
The process means:<br />
Give an object X (φi ) <strong>in</strong> Sn(A)<br />
τSX (φi ) ∼ = Mimo τ CokX (φi ). (1)<br />
Take the cokernel object X ′ (φ ′ i ) = CokX (φ i ).<br />
Apply τ to these maps φ ′ i (1 ≤ i ≤ n − 1).<br />
Represent τCokX (φi ) by an object X ′′<br />
(φ ′′<br />
i<br />
) =<br />
� X ′′<br />
1<br />
.<br />
X ′′<br />
n<br />
�<br />
(φ ′′<br />
i )<br />
where X ′′<br />
1 , X ′′<br />
2 , · · · , X ′′<br />
n−1 have no nonzero <strong>in</strong>jective direct<br />
summands.<br />
Apply Mimo, there is a well-def<strong>in</strong>ed object <strong>in</strong> Sn(A) up to<br />
isomorphism.<br />
<strong>in</strong> Morn(A)<br />
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An example<br />
k: a field; A = k[X]/〈X 2 〉, S = k[X]/〈X〉, i : S −→ A, π : A −→ S.<br />
� �<br />
AS0<br />
� �<br />
SAA<br />
= Mimoτ<br />
� �<br />
S00<br />
= Mimo<br />
� �<br />
S00<br />
=<br />
τS<br />
τS<br />
τS<br />
τS<br />
� S00<br />
� SSS<br />
� AAS<br />
�<br />
�<br />
�<br />
(0,i)<br />
(0,0)<br />
(1,1)<br />
(i,1)<br />
�<br />
SSS<br />
= Mimoτ<br />
�<br />
00S<br />
= Mimoτ<br />
�<br />
0SA<br />
= Mimoτ<br />
�<br />
�<br />
�<br />
(1,π)<br />
(1,1)<br />
(0,0)<br />
(π,0)<br />
�<br />
SSS<br />
= Mimo<br />
�<br />
00S<br />
= Mimo<br />
�<br />
0S0<br />
= Mimo<br />
�<br />
�<br />
�<br />
(0,0)<br />
(1,1)<br />
(0,0)<br />
(0,0)<br />
=<br />
=<br />
=<br />
� SSS<br />
� AAS<br />
� AS0<br />
�<br />
�<br />
�<br />
(0,0)<br />
(1,1)<br />
—————————————————————————————–<br />
� �<br />
SS0<br />
� �<br />
0SS<br />
= Mimoτ<br />
� �<br />
0SS<br />
= Mimo<br />
� �<br />
ASS<br />
=<br />
τS<br />
τS<br />
� ASS<br />
�<br />
(0,1)<br />
(1,i)<br />
�<br />
SSA<br />
= Mimoτ<br />
�<br />
(1,0)<br />
(π,1)<br />
�<br />
SS0<br />
= Mimo<br />
�<br />
(1,0)<br />
(0,1)<br />
=<br />
� SS0<br />
�<br />
(i,1)<br />
(1,i)<br />
(1,i)<br />
(0,1)<br />
Bao-L<strong>in</strong> Xiong (SJTU) <strong>Auslander</strong>-<strong>Reiten</strong> <strong>Translations</strong> <strong>in</strong> <strong>Monomorphism</strong> <strong>Categories</strong> ISPN ’80 12 / 24
An example<br />
The <strong>Auslander</strong>-<strong>Reiten</strong> quiver of S3(A) looks like<br />
A A<br />
A<br />
00 �<br />
A<br />
��<br />
0 �<br />
A<br />
���<br />
��<br />
�<br />
�<br />
�<br />
A ��<br />
��<br />
��<br />
���<br />
�<br />
�<br />
�<br />
� ��<br />
�<br />
�<br />
��<br />
�<br />
S �<br />
��<br />
��<br />
A �<br />
A ��<br />
��<br />
S<br />
00 ��<br />
�<br />
S ��<br />
��<br />
0 �<br />
A ��<br />
�<br />
�<br />
�<br />
S �<br />
S<br />
���<br />
��<br />
�<br />
�<br />
�<br />
�<br />
S<br />
��<br />
��<br />
���<br />
��<br />
� �<br />
�<br />
��<br />
�<br />
S �<br />
��<br />
��<br />
A � ��<br />
S ��<br />
S ��<br />
0 �<br />
S ��<br />
� ��<br />
�<br />
S �<br />
S<br />
�<br />
� ��<br />
� ���<br />
0<br />
�<br />
�<br />
��<br />
��<br />
� ��<br />
S � ��<br />
S ��<br />
S ��<br />
00<br />
S<br />
Remark: This AR-quiver has been described by A.Moore.<br />
Bao-L<strong>in</strong> Xiong (SJTU) <strong>Auslander</strong>-<strong>Reiten</strong> <strong>Translations</strong> <strong>in</strong> <strong>Monomorphism</strong> <strong>Categories</strong> ISPN ’80 13 / 24
Applications to self<strong>in</strong>jective algebras<br />
A: a self<strong>in</strong>jective Art<strong>in</strong> algebra,<br />
A-mod: the stable category of A-mod<br />
Morn(A-mod): the morphism category of A-mod<br />
�<br />
Objects: X (φi ) =<br />
� X1<br />
.<br />
Xn<br />
(φi )<br />
, φi : Xi+1 → Xi <strong>in</strong> A-mod,<br />
Bao-L<strong>in</strong> Xiong (SJTU) <strong>Auslander</strong>-<strong>Reiten</strong> <strong>Translations</strong> <strong>in</strong> <strong>Monomorphism</strong> <strong>Categories</strong> ISPN ’80 14 / 24
Applications to self<strong>in</strong>jective algebras<br />
A: a self<strong>in</strong>jective Art<strong>in</strong> algebra,<br />
A-mod: the stable category of A-mod<br />
Morn(A-mod): the morphism category of A-mod<br />
�<br />
Objects: X (φi ) =<br />
⎛<br />
f1<br />
� X1<br />
⎞<br />
.<br />
Xn<br />
(φi )<br />
, φi : Xi+1 → Xi <strong>in</strong> A-mod,<br />
Morphisms: ⎝ ⎠ : X (φi .<br />
) → Y (θi ), fi : Xi → Yi such that the follow<strong>in</strong>g<br />
fn<br />
diagram commutes <strong>in</strong> A-mod<br />
fn<br />
Xn<br />
��<br />
Yn<br />
φn−1<br />
θn−1<br />
��<br />
Xn−1<br />
fn−1<br />
��<br />
��<br />
Yn−1<br />
φn−2<br />
θn−2<br />
��<br />
· · ·<br />
��<br />
· · ·<br />
φ2<br />
θ2<br />
��<br />
X2<br />
f2<br />
��<br />
��<br />
Y2<br />
φ1<br />
θ1<br />
��<br />
f1<br />
X1<br />
��<br />
��<br />
Y1.<br />
Bao-L<strong>in</strong> Xiong (SJTU) <strong>Auslander</strong>-<strong>Reiten</strong> <strong>Translations</strong> <strong>in</strong> <strong>Monomorphism</strong> <strong>Categories</strong> ISPN ’80 14 / 24
The rotation of X (φi)<br />
For X (φi ) ∈ Morn(A-mod), we have the follow<strong>in</strong>g commutative diagram<br />
with exact rows <strong>in</strong> A-mod,<br />
Xn<br />
φn−1<br />
��<br />
Xn−1<br />
��<br />
.<br />
φn−2<br />
φ3<br />
��<br />
X3<br />
φ2<br />
��<br />
X2<br />
φ1<br />
��<br />
��<br />
��<br />
X1 ψn−1<br />
X1<br />
.<br />
X1<br />
��<br />
X1<br />
��<br />
Y 1<br />
n<br />
ψn−2<br />
��<br />
��<br />
Y 1 n−1<br />
��<br />
.<br />
��<br />
��<br />
Y 1<br />
3<br />
��<br />
��<br />
Y 1<br />
2<br />
ψn−3<br />
ψ2<br />
ψ1<br />
��<br />
Ω −1 Xn<br />
��<br />
��<br />
Ω−1Xn−1 ��<br />
.<br />
��<br />
��<br />
Ω−1X3 ��<br />
��<br />
Ω−1X2. Bao-L<strong>in</strong> Xiong (SJTU) <strong>Auslander</strong>-<strong>Reiten</strong> <strong>Translations</strong> <strong>in</strong> <strong>Monomorphism</strong> <strong>Categories</strong> ISPN ’80 15 / 24
The rotation of X (φi)<br />
The rotation RotX (φi ) of X (φi ) is def<strong>in</strong>ed to be<br />
(X1<br />
ψn−1<br />
��Y<br />
1<br />
n<br />
��<br />
· · · ψ1<br />
��Y<br />
1<br />
2 ) ∈ Morn(A-mod)<br />
(here,a for convenience we write the rotation <strong>in</strong> a row). We remark that<br />
RotX (φi ) is well-def<strong>in</strong>ed.<br />
Lemma 3.1<br />
Let X (φi ) ∈ Morn(A). Then RotX (φi ) ∼ = Cok MimoX (φi ) <strong>in</strong> Morn(A-mod).<br />
Bao-L<strong>in</strong> Xiong (SJTU) <strong>Auslander</strong>-<strong>Reiten</strong> <strong>Translations</strong> <strong>in</strong> <strong>Monomorphism</strong> <strong>Categories</strong> ISPN ’80 16 / 24
For X (φi ) ∈ Morn(A-mod), def<strong>in</strong>e Ω−1X (φi ) to be<br />
�<br />
Ω−1 �<br />
X1<br />
.<br />
Ω −1 Xn<br />
(Ω −1 φ i )<br />
∈ Morn(A-mod).<br />
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For X (φi ) ∈ Morn(A-mod), def<strong>in</strong>e Ω−1X (φi ) to be<br />
�<br />
Ω−1 �<br />
X1<br />
.<br />
Ω −1 Xn<br />
(Ω −1 φ i )<br />
Proposition 3.2<br />
∈ Morn(A-mod).<br />
Let A be a self<strong>in</strong>jective algebra, X (φi ) ∈ Sn(A). Then there are the<br />
follow<strong>in</strong>g isomorphisms <strong>in</strong> Morn(A-mod)<br />
(i) τ j<br />
S X (φ i ) ∼ = τ j Rot j X (φi ) for j ≥ 1. In particular, τSX (φi ) ∼ = τ CokX (φi ).<br />
(ii) τ s(n+1)<br />
S X (φi ) ∼ = τ s(n+1) Ω−s(n−1) X (φi ), ∀ s ≥ 1.<br />
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Theorem 3.3<br />
Let A be a self<strong>in</strong>jective algebra, and X (φi ) ∈ Sn(A). Then we have<br />
τ s(n+1)<br />
S X (φi ) ∼ = Mimo τ s(n+1) Ω −s(n−1) X (φi ), s ≥ 1. (2)<br />
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Theorem 3.3<br />
Let A be a self<strong>in</strong>jective algebra, and X (φi ) ∈ Sn(A). Then we have<br />
τ s(n+1)<br />
S X (φi ) ∼ = Mimo τ s(n+1) Ω −s(n−1) X (φi ), s ≥ 1. (2)<br />
Apply<strong>in</strong>g the above theorem to the self<strong>in</strong>jective Nakayama algebras<br />
A(m, t), we get<br />
Corollary 3.4<br />
For an <strong>in</strong>decomposable nonprojective object X (φi ) ∈ Sn(A(m, t)),<br />
m ≥ 1, t ≥ 2, there are the follow<strong>in</strong>g isomorphisms:<br />
(i) If n is odd, then τ m(n+1)<br />
S X (φi ) ∼ = X (φi );<br />
(ii) If n is even, then τ 2m(n+1)<br />
S X (φi ) ∼ = X (φi ).<br />
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An example<br />
Let A = kQ/〈δα, βγ, αδ − γβ〉, where Q is the quiver 2•<br />
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1•<br />
Then A is self<strong>in</strong>jective with τ ∼ = Ω −1 and Ω 6 ∼ = id on the object of<br />
A-mod. The <strong>Auslander</strong>-<strong>Reiten</strong> quiver of A is<br />
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δ<br />
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3•
An example<br />
Let X (φi ) be an <strong>in</strong>decomposable nonprojective object <strong>in</strong> Sn(A).<br />
By (2), for s ≥ 1 we have<br />
τ s(n+1)<br />
S X (φi ) ∼ = Mimo τ s(n+1) Ω −s(n−1) X (φi ) ∼ = Mimo Ω −2sn X (φi )<br />
<strong>in</strong> Sn(A).<br />
Then we get<br />
(i) if n ≡ 0, or 3 (mod6), then τ n+1<br />
S X (φi ) ∼ = X (φi ); and<br />
(ii) if n ≡ ±1, or ± 2 (mod6), then τ 3(n+1)<br />
S X (φi ) ∼ = X (φi ).<br />
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Serre functors of stable monomorphism categories<br />
A: a f<strong>in</strong>ite-dimensional self<strong>in</strong>jective algebra over a field<br />
Sn(A) is a Frobenius category.<br />
Sn(A): the stable category of Sn(A)<br />
Sn(A) is a Hom-f<strong>in</strong>ite Krull-Schmidt triangulated category with<br />
suspension functor Ω −1<br />
S<br />
= Ω−1<br />
Sn(A) . S<strong>in</strong>ce Sn(A) has <strong>Auslander</strong>-<strong>Reiten</strong><br />
sequences, it follows that Sn(A) has <strong>Auslander</strong>-<strong>Reiten</strong> triangles, and<br />
hence, by a theorem of <strong>Reiten</strong> and Van den Bergh, it has a Serre<br />
functor FS = F Sn(A), which co<strong>in</strong>cides with Ω −1<br />
S �τS on the objects of<br />
Sn(A).<br />
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Theorem 4.1<br />
Let A be a self<strong>in</strong>jective algebra, and FS be the Serre functor of Sn(A).<br />
Then we have an isomorphism <strong>in</strong> Sn(A) for X (φi ) ∈ Sn(A) and for s ≥ 1<br />
F s(n+1)<br />
S X (φi ) ∼ = Mimo τ s(n+1) Ω −2sn X (φi ). (4.4)<br />
Moreover, if d1 and d2 are positive <strong>in</strong>tegers such that τ d1M ∼ = M and<br />
Ω d2M ∼ = M for each <strong>in</strong>decomposable nonprojective A-module M, then<br />
F N(n+1)<br />
S<br />
X (φi ) ∼ d1<br />
= X (φi ), where N = [ (n+1,d1) ,<br />
d2<br />
(2n,d2) ].<br />
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Apply<strong>in</strong>g the above theorem to the self<strong>in</strong>jective Nakayama algebras<br />
A(m, t), we get<br />
Corollary 4.2<br />
Let FS be the Serre functor of Sn(A(m, t)) with m ≥ 1, t ≥ 2, and X be<br />
an arbitrary object <strong>in</strong> Sn(A(m, t)). Then<br />
(i) If t = 2, then F N(n+1)<br />
S X ∼ = X, where N = m<br />
(m,n−1) .<br />
(ii) If t ≥ 3, then F N(n+1)<br />
S X ∼ = X, where N = m<br />
(m,t,n+1) .<br />
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Thank you!<br />
E-mail: xiongbaol<strong>in</strong>@gmail.com<br />
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