Cent. Eur. J. Math. • 12(6) • 2014 • 813-823
DOI: 10.2478/s11533-013-0385-7
Central European Journal of Mathematics
Orbit algebras that are invariant under
stable equivalences of Morita type
Research Article
Zygmunt Pogorzały1∗
1 Faculty of Mathematics and Computer Science, Nicholaus Copernicus University, ul. Chopina 12/18, 87-100 Toruń, Poland
Received 18 April 2013; accepted 10 September 2013
Abstract: In this note we show that there are a lot of orbit algebras that are invariant under stable equivalences of Morita
type between self-injective algebras. There are also indicated some links between Auslander–Reiten periodicity
of bimodules and noetherianity of their orbit algebras.
MSC:
16E40, 16B50, 16D50, 16B10
Keywords: Stable equivalence of Morita type • Self-injective algebra • Orbit algebra of a bimodule
© Versita Sp. z o.o.
1.
Introduction
Let K be a fixed field. Consider an additive K -category A. For every K -linear functor F : A → A and any fixed object X
in A we define two algebras A(F ; X ) and A(X ; F ) of the functor F in X as follows. For the algebra A(F ; X ), its K -linear
structure is that of
∞
M
HomA (F n (X ), X ).
n=0
Multiplication of a homogeneous element u : F i (X ) → X of degree i by a homogeneous element v : F j (X ) → X of degree j
is given by
F i (v)
u
u · v = F j+i (X ) −−→ F i (X ) −
→X .
∗
E-mail: zypo@mat.umk.pl
813
Unauthenticated
Download Date | 10/2/16 8:15 PM
Orbit algebras that are invariant under stable equivalences of Morita type
However, for the algebra A(F ; X ), its K -linear structure is that of
∞
M
HomA (X , F n (X )).
n=0
Multiplication of a homogeneous element a : X → F i (X ) of degree i by a homogeneous element b : X → F j (X ) of degree
j is given by
F j (a)
b
a?b = X →
− F j (X ) −−→ F j+i (X ) .
Such algebras have nice properties in case F is a self-equivalence. They were studied by several authors (see [3, 7, 8,
10, 11, 19, 20]). A general reason to study these algebras is the local analysis of the endofunctor F . In many cases we
are not able to describe all indecomposable objects of A and morphisms between them. In these cases the considered
orbit algebras describe the local structure of A at a given object.
A particular instance of an algebra A(F ; X ) is a factor HH(A) of the Hochschild cohomology algebra HH(A), where A is
a finite dimensional self-injective associative K -algebra with a unit. Actually HH(A) ∼
= A(ΩAe ; A), where Ae = A ⊗K Aop
e
e
is the enveloping algebra of A and ΩAe : mod(A ) → mod(A ) is the Heller’s loop space functor on the stable category of
the left finite dimensional Ae -modules (compare [9]).
Another example of an algebra A(F ; X ) is the so-called Auslander–Reiten orbit algebra A(τAe ; A), where τAe : mod(Ae ) →
mod(Ae ) is the Auslander–Reiten translate (see [1, 2]). It was proved in [14, 15] that if two self-injective finite dimensional
K -algebras A, B are stably equivalent of Morita type then the algebras HH(A) and HH(B) are isomorphic as well as the
algebras A(τAe ; A) and A(τBe ; B). We have to point that computing of these algebras is a little bit technical task. In the
paper we shall show only a trivial example of an application of our main result. In a subsequent paper we shall apply
the results below to show that some self-injective algebras of polynomial growth are not stably equivalent of Morita
type.
In the paper we shall show that there are many invariants of stable equivalences of Morita type between self-injective
algebras. Our main result is the following
Theorem 1.1.
Let A, B be finite dimensional self-injective K -algebras that are stably equivalent of Morita type. Let B MA , A NB be
bimodules that determine a stable equivalence of Morita type between A and B. For every indecomposable non-projective
A-A-bimodule X that is projective as a left A-module and as a right A-module denote by Y the indecomposable nonprojective direct summand in the B-B-bimodule B M ⊗A X ⊗A NB . Then the following conditions are satisfied:
(a) There are K -algebra isomorphisms A τ e ; τ ne (X ) ∼
= A τ e ; τ me (Y ) and A τ ne (X ); τ e ∼
= A τ me (Y ); τ e for any
A
A
B
B
A
A
B
B
integers n, m.
∼
n
m
(b) There are K -algebra isomorphisms A ΩAe ; ΩnAe (X ) ∼
= A ΩBe ; Ωm
Be (Y ) and A ΩAe (X ); ΩAe = A ΩBe (Y ); ΩBe for
any integers n, m.
For studying the above algebras of a self-injective algebra A, it is important to know [3, 4, 6, 17, 19] whether a
bimodule X is an ΩAe -periodic Ae -modules as well as a τAe -periodic Ae -module. The following result characterises the
orbit algebras A(τAe ; M) and A(M; τAe ) for a τAe -periodic Ae -module M. The result shows that the necessary condition
for the τAe -periodicity of an Ae -module M is the noetherianity of its orbit algebras.
Theorem 1.2.
Let A be an indecomposable finite dimensional self-injective K -algebra that is non-simple. Let M be a left Ae -module
that is τAe -periodic. Then the algebra A(τAe ; M) (resp., A(M; τAe )) is right (resp., left) noetherian.
The paper is composed in the following way. In Section 2 we collected basic notions and notations. Section 3 contains
a proof of Theorem 1.1. In Section 4 we consider some bimodules that can be used in studying of τAe -periodicity and
ΩAe -periodicity of Ae -modules. Section 5 contains a proof of Theorem 1.2. In the paper, we shall freely use some basic
notions from representations theory of artin algebras. Their descriptions and properties can be found in [1].
814
Unauthenticated
Download Date | 10/2/16 8:15 PM
Z. Pogorzały
2.
Preliminaries
For an associative finite dimensional K -algebra A with a unity, we shall denote by mod(A) the category of the left finite
dimensional A-modules and the homomorphisms between them. The stable category mod(A) of mod(A) (or shortly of A)
modulo projective objects is the quotient category mod(A)/P, where P is the two-sided ideal in mod(A) consisting of
the morphisms that factorize through projective A-modules. For any objects X , Y in mod(A), we shall denote the space
of morphisms Hommod(A) (X , Y ) by Hom A (X , Y ). Any element in Hom A (X , Y ) is a coset f of a morphism f ∈ HomA (X , Y )
modulo P(X , Y ). In particular, if A is a self-injective algebra then there is a selfequivalence τA : mod(A) → mod(A) that
is called Auslander–Reiten translate [1, 2].
Recall that the enveloping algebra Ae of an algebra A is the algebra A ⊗K Aop , where Aop stands for the opposite algebra
of A. It is well known that for two finite dimensional self-injective K -algebras A and B, the algebra A ⊗K B is self-injective
too. In particular, the enveloping algebra Ae of a self-injective algebra A is also self-injective.
Every left finite dimensional A ⊗K Bop -module can be interpreted as a finite dimensional A-B-bimodule with a central
action of K , and conversely, every such bimodule can be considered as a left A ⊗K Bop -module. Then we shall not
distinguish between left A ⊗K Bop -modules and A-B-bimodules with a central action of K .
A finite dimensional A-B-bimodule X is said to be left-right projective provided that it is projective as a left A-module
and as a right B-module. Further we shall denote the full subcategory of mod(A ⊗K Bop ) spanned by the left-right
projective A-B-bimodules by lrp(A ⊗K Bop ).
Two finite dimensional K -algebras A and B are stably equivalent if there is an equivalence Φ : mod(A) → mod(B).
These algebras are said to be stably equivalent of Morita type [5] provided that there exist an A-B-bimodule N and a
B-A-bimodule M such that the following conditions are satisfied:
• M, N are left-right projective bimodules,
• M ⊗A N ∼
= B ⊕ Π as B-B-bimodules for some projective B-B-bimodule Π,
• N⊗ M ∼
= A ⊕ Σ as A-A-bimodules for some projective A-A-bimodule Σ.
B
It is clear that if A, B are stably equivalent of Morita type then the functor B M ⊗A · : mod(A) → mod(B) induces an
∼
=
→ mod(B). Moreover, we shall say equivalently that the above bimodules M and N determine a
equivalence mod(A) −
stable equivalence of Morita type between A and B. Nice examples of stably equivalent algebras of Morita type are
derived equivalent self-injective algebras [18].
3.
Stable equivalences of algebras
For a given pair of finite dimensional K -algebras A, B and a fixed indecomposable object X in lrp(A ⊗K Bop ), we shall
denote by
τA⊗ Bop
K
mod X
ΩA⊗ Bop
K
(A ⊗K Bop ),
resp.,
mod X
(A ⊗K Bop ),
the full subcategory of mod(A ⊗K Bop ) formed by the finite direct sums of objects that are isomorphic to those of the form
n
n
τA⊗
op (X ) (resp., ΩA⊗ Bop (X )), where n ranges the integers.
KB
K
Lemma 3.1.
Let A, B be finite dimensional self-injective K -algebras. Let the bimodules B MA and A NB determine a stable equivalence
of Morita type between A and B. For a given indecomposable object X of lrp(Ae ) denote by Y an indecomposable object
of lrp(Be ) that is isomorphic to B M ⊗A X ⊗A NB . Then the following conditions are satisfied:
τ e
τ e
(a) There exists an equivalence F : mod A (Ae ) → mod B (Be ) such that for every integer n we have F τ ne (X ) ∼
= τ ne (Y )
τ e
X
Y
A
B
in mod YB (Be ).
Ω e
Ω e
(b) There exists an equivalence G : mod XA (Ae ) → mod YB (Be ) such that for every integer n we have G ΩnAe (X ) ∼
= ΩnBe (Y )
ΩBe
e
in mod Y (B ).
815
Unauthenticated
Download Date | 10/2/16 8:15 PM
Orbit algebras that are invariant under stable equivalences of Morita type
Proof.
Consider an indecomposable object X in lrp(Ae ). Choose an indecomposable Y in lrp(Be ) such that
τ e
τ e
∼
Y = B M ⊗A X ⊗A NB in mod(Be ). Now we define a functor F : mod XA (Ae ) → mod YB (Be ) as follows. For every object Z
τAe
τ e
in mod X (Ae ) we put F (Z ) = M ⊗A Z ⊗A N. For every morphism f : Z1 → Z2 in mod XA (Ae ) we put F (f) = idM ⊗ f ⊗ idN .
It is easily seen that F is a well-defined functor.
τ e
τ e
Now we shall define a functor H : mod YB (Be ) → mod XA (Ae ) similarly. We put H(U) = N ⊗B U ⊗B M for every object U
τ e
τ e
in mod YB (Be ). Furthermore, we put H(g) = idN ⊗ g ⊗ idM for any morphism g : U1 → U2 in mod YB (Be ).
τ e
τ e
We deduce from [15, Lemmas 2.3 and 2.4] that for every object Z in mod A (Ae ) we have HF (Z ) ∼
= Z in mod A (Ae ). Now
X
τ e
X
consider a morphism f : Z1 → Z2 in mod XA (Ae ). Then
HF (f) = H idM ⊗ f ⊗ idN = idN ⊗ idM ⊗ f ⊗ idN ⊗ idM = idA⊕Σ ⊗ f ⊗ idA⊕Σ = idA ⊗ f ⊗ idA = f.
Similarly one shows that the composed functor F H is isomorphic to the identity functor. Therefore F is an equivalence
of categories, and (a) is shown.
In order to show (b), we have to define a functor G : mod XA (Ae ) → mod YB (Be ). For every object Z in modX A (Ae ), we put
Ω e
G(Z ) = M ⊗A X ⊗A N. For every morphism f : Z1 → Z2 in mod XA (Ae ), we put G(f) = idM ⊗ f ⊗ idN . Again it is easily seen
Ω e
Ω e
that G is a well-defined functor. We define a functor L : mod YB (Be ) → mod XA (Ae ) similarly, putting L(U) = N ⊗B U ⊗B M
ΩBe
Ω e
e
for every object U in mod Y (B ), and L(g) = idN ⊗ g ⊗ idM for every morphism g : U1 → U2 in mod YB (Be ).
Ω e
Ω e
Ω e
Ω e
Ω e
We infer by [12, Proposition 2.9] that for every object Z in mod XA (Ae ) we have LG(Z ) ∼
= Z in mod XA (Ae ). Further for
ΩAe
any morphism f : Z1 → Z2 in mod X (Ae ) we obtain
LG(f) = L(idM ⊗ f ⊗ idN ) = idN ⊗ idM ⊗ f ⊗ idN ⊗ idM = idA⊕Σ ⊗ f ⊗ idA⊕Σ = idA ⊗ f ⊗ idA = f.
Similarly one shows that the composed functor GL is isomorphic to the identity functor. Consequently, G is an equivalence.
Lemma 3.2.
Let A, B be finite dimensional self-injective K -algebras. Let X ∈ lrp(Ae ), Y ∈ lrp(Be ) be indecomposable objects. Then
the following conditions are satisfied:
(a) If A(τAe ; X ) ∼
= A(τBe ; Y ) then we have a K -algebra isomorphism A τAe ; τAne (X ) ∼
= A τBe ; τBme (Y ) for any integers
m, n.
(b) If A(X ; τ e ) ∼
= A(Y ; τ e ) then we have a K -algebra isomorphism A τ ne (X ); τ e ∼
= A τ me (Y ); τ e for any inteA
B
A
A
B
B
gers m, n.
Proof.
We deduce from [15, Corollary 3.3] that for any integers m, n and indecomposable objects X ∈ lrp(Ae ),
Y ∈ lrp(Be ) we have isomorphisms τAne (X ) ∼
= τAne (A) ⊗A X in lrp(Ae ) and τBme (Y ) ∼
= τBme (B) ⊗B Y in lrp(Be ). Since
n
τAe (A) ⊗A · : mod(A) → mod(B) is a stable selfequivalence of Morita type, we infer by Lemma 3.1 (a) that A(τAe ; X ) ∼
=
A τAe ; τAne (X ) and A(X ; τAe ) ∼
= A τAne (X ); τAe as K -linear spaces.
Now we shall show that for any morphism f : τAs e (X ) → τAt e (X ), the following statement holds:
idτAne (A) ⊗ τAe (f) ⊗ idτA−ne (A) = τAe idτAne (A) ⊗ f ⊗ idτA−ne (A) ,
where s, t are any integers and τAe (f) is a representative of the coset τAe (f). Consider the following commutative diagram:
0
/ τAe τAs e (X )
0
τAe (f)
/ τAe τAt e (X )
/ E
/ F
/ τAs e (X )
f1
/ 0
f
/ τAt e (X )
/ 0
816
Unauthenticated
Download Date | 10/2/16 8:15 PM
Z. Pogorzały
whose rows are Auslander–Reiten sequences. Moreover, if f is an isomorphism then f1 and τAe (f) are also isomorphisms.
Then we obtain the following commutative diagram:
0
/ τAe τAs e (X )
0
/ τAe τAt e (X )
/ [E]
e
h
ef
1
/ [F ]
/ τAs e (X )
/ 0
/ τAt e (X )
/ 0
f
e stands for idτAne (A) ⊗ g ⊗ idτ −ne (A)
whose rows are exact, where [Z ] stands for τAne (A) ⊗A Z ⊗A τA−n
e (A) for every object Z and g
A
for every morphism g. Then we infer by [17, Proposition 2.4] that
τAe idτAne (A) ⊗ f ⊗ idτA−ne (A) = idτAne (A) ⊗ τAe (f) ⊗ idτA−ne (A) .
Similarly one shows that
−1
n (A) ⊗ f ⊗ id −n
idτAne (A) ⊗ τA−1
id
.
=
τ
e (f) ⊗ idτ −n
e
τ
(A)
τ
(A)
A
Ae
Ae
Ae
Applying the above equalities and Lemma 3.1 (a), we obtain that F h ◦ τAt e (g) = F (h) ◦ τAt e (F (g)) for any morphisms
τ
τ
e
e
g : τAs e (X ) → X and h : τAt e (X ) → X , where F : mod XA (Ae ) → mod τAne (X ) (Ae ) is an equivalence that is induced by the
A
functor τAne (A) ⊗A · : mod(Ae ) → mod(Ae ). Thus the above K -linear isomorphism from A(τAe ; X ) to A τAe ; τAne (X ) is an
isomorphism of K -algebras.
Similarly one proves that A(τBe ; Y ) ∼
m. Consequently, the isomorphism A(τAe ; X ) ∼
= A τBe ; τBme (Y ) for any integer
=
∼
n
m
A(τ e ; Y ) induces an isomorphism A τ e ; τ e (X ) = A τ e ; τ e (Y ) for any integers m, n. Similar arguments lead to
B
A
B
A
B
condition (b). We omit the details.
Lemma 3.3.
Let A, B be finite dimensional self-injective K -algebras. Let X ∈ lrp(Ae ), Y ∈ lrp(Be ) be indecomposable objects. Then
the following conditions are satisfied:
(a) If A(Ω e ; X ) ∼
= A(Ω e ; Y ) then we have a K -algebra isomorphism A Ω e ; Ωne (X ) ∼
= A Ω e ; Ωme (Y ) for any integers
A
B
A
B
A
B
m, n.
(b) If A(X ; ΩAe ) ∼
= A(Y ; ΩBe ) then we have a K -algebra isomorphism A ΩnAe (X ); ΩAe ∼
= A Ωm
Be (Y ); ΩBe for any integers
m, n.
Proof. We deduce from [12, Proposition 2.9] that for any integers m, n and indecomposable objects X ∈ lrp(Ae ),
∼ m
e
Y ∈ lrp(Be ) we have ΩnAe (X ) ∼
Ωn (A) ⊗A · : mod(A) →
= ΩnAe (A) ⊗A X in lrp(Ae ) and Ωm
Be (Y ) = ΩBe (B) ⊗B Y in lrp(B ). Since
Ae
mod(A) is a stable selfequivalence of Morita type, so we infer by Lemma 3.1 (b) that A ΩAe ; X ∼
= A ΩAe ; ΩnAe (X ) and
∼
A(X ; ΩAe ) = A ΩnAe (X ); ΩAe as K -linear spaces.
Now we shall show that for any morphism f : ΩsAe (X ) → ΩtAe (X ), it holds
e idΩn (A) ⊗ f ⊗ idΩ−n (A) ,
idΩnAe (A) ⊗ ΩAe (f) ⊗ idΩ−n
=
Ω
A
(A)
e
A
Ae
Ae
where s, t are any integers, and ΩAe (f) is a representative of the coset ΩAe (f). Let the following diagram:
0
/ ΩAe ΩsAe (X )
0
ΩAe (f)
/ ΩAe ΩtAe (X )
/ P
f1
/ Q
/ ΩsAe (X )
/ 0
f
/ ΩtAe (X )
/ 0
817
Unauthenticated
Download Date | 10/2/16 8:15 PM
Orbit algebras that are invariant under stable equivalences of Morita type
be commutative, whose rows are exact, where P, Q are projective Ae -modules. Then we obtain the following commutative
diagram:
/ ΩAe (ΩsAe (X ))
/ [P]
/ ΩsAe (X )
/ 0
0
^
Ω
Ae (f)
/ ΩAe (ΩtAe (X ))
0
ef
1
/ [Q]
ef
/ ΩtAe (X )
/ 0
e
n
whose rows are exact, where [Z ] stands for ΩnAe (A) ⊗A Z ⊗A Ω−n
Ae (A) for every term Z and h stands for idΩAe (A) ⊗ h ⊗ idΩ−n
Ae (A)
for every morphism h. Then it is clear that
= idΩnAe (A) ⊗ ΩAe (f) ⊗ idΩ−n
ΩAe idΩnAe (A) ⊗ f ⊗ idΩ−n
.
(A)
e
Ae
A
Dual arguments show that
−1
.
idΩnAe (A) ⊗ Ω−1
=
Ω
e idΩne (A) ⊗ f ⊗ idΩ−n
(A)
(A)
A
Ae (f) ⊗ idΩ−n
e
e
A
A
A
Applying the above equalities and Lemma 3.1 (b), we obtain that G h ◦ ΩtAe (g) = G(h) ◦ ΩtAe (G(g)) for any morphisms
g : ΩsAe (X ) → X and h : ΩtAe (X ) → X , where G : mod XA (Ae ) → mod ΩAne (X ) (Ae ) is an equivalence that is induced by the
A
functor Ωne (A) ⊗ · : mod(AE ) → mod(Ae ). Therefore the above K -linear isomorphism A(Ω e ; X ) ∼
= A Ω e ; Ωne (X ) is a
Ω e
Ω e
A
A
A
A
A
K -algebra isomorphism.
∼
Similarly one proves that A(ΩBe ; Y ) ∼
= A ΩBe ; Ωm
Be (Y ) for any integer m. Thus the isomorphism A(ΩAe ; X ) = A(ΩBe ; Y )
∼
n
m
induces an isomorphism A ΩAe ; ΩAe (X ) = A ΩBe ; ΩBe (Y ) for any integers m, n, which finishes the proof of (a). Similar
arguments lead to (b). We omit the details.
Lemma 3.4.
Let A, B be finite dimensional self-injective K -algebras. Let X ∈ lrp(Ae ), Y ∈ lrp(Be ) be indecomposable objects. Then
the following conditions are satisfied:
τ e
τ e
(a) If there exists an equivalence F : mod XA (Ae ) → mod YB (Be ) that commutes with τAe and τBe and such that F (X ) ∼
=Y
τ e
in mod B (Be ) then we have K -algebra isomorphisms A(τ e ; X ) ∼
= A(τ e ; Y ) and A(X ; τ e ) ∼
= A(Y ; τ e ).
Y
A
B
A
B
Ω e
Ω e
(b) If there exists an equivalence G : mod XA (Ae ) → mod YB (Be ) that commutes with ΩAe and ΩBe and such that G(X ) ∼
=Y
ΩBe
∼
e
e
e
in mod Y (Be ) then we have K -algebra isomorphisms A(ΩAe ; X ) ∼
A(Ω
;
Y
)
and
A(X
;
Ω
)
A(Y
;
Ω
).
=
B
A =
B
Proof.
τ e
τ e
Suppose that there exists an equivalence F : mod XA (Ae ) → mod YB (Be ) that commutes with τAe and τBe and
τ e
τ e
such that F (X ) ∼
= τBne (Y ) in mod YB (Be ) for every integer n. Since F is an
= Y in mod YB (Be ). Then we have F (τAne (X )) ∼
equivalence, we have
F Hom Ae (τAne (X ), X ) ∼
= Hom Be F (τAne (X )), F (X ) ∼
= Hom Be τBne (Y ), Y
for any n ≥ 0. Moreover, for any morphisms f : τAs e (X ) → X , g : τAt e (X ) → X , s, t ≥ 0, we have f · g = f ◦ τAs e (g). Since
F commutes with τAe and τBe , so we have F (f · g) = F (f ◦ τAs e (g)) = F (g) ◦ F (τAs e (g)) = F (f) ◦ τBs e (F (g)) = F (f) · F (g).
Consequently, we obtain that A(τAe ; X ) ∼
= A(τBe ; Y ). But
F HomAe (X , τAne (X )) ∼
= Hom Be F (X ), F (τAne (X )) ∼
= Hom Be Y , τBne (Y ) ,
because F is an equivalence. Furthermore, for any morphisms f : X → τAs e (X ), g : X → τAt e (X ), s, t ≥ 0, we have
f ? g = τAt e (f) ◦ g. Since F commutes with τAe and τBe , so we have
F (f ? g) = F τAt e (f) ◦ g = F τAt e (f) ◦ F (g) = τBt e (F (f)) ◦ F (g) = F (f) ? F (g).
Finally, we obtain that A(X ; τAe ) ∼
= A(Y ; τBe ) as well, and condition (a) is proved. Condition (b) can be proved similarly.
We omit the details.
818
Unauthenticated
Download Date | 10/2/16 8:15 PM
Z. Pogorzały
Proof of Theorem 1.1.
Let A and B be finite dimensional self-injective K -algebras that are stably equivalent of
Morita type. Let bimodules B MA , A NB determine such an equivalence. Then we deduce from Lemma 3.1 (a) that there
τ e
τ e
τ e
is an equivalence F : mod XA (Ae ) → mod YB (Be ) such that F (X ) ∼
= Y in mod YB (Be ). Then condition (a) is a direct
consequence of Lemmas 3.2 and 3.4 (a). In order to prove condition (b), we use Lemma 3.1 (b) and further Lemmas 3.3
and 3.4 (a).
Example.
α
α
α
0
1
2
Consider the cyclic quiver Q whose vertices are 0, 1, 2 and whose arrows are 0 −
→
1, 1 −
→
2, 2 −
→
0. Suppose that K
is an algebraically closed field. Consider two ideals I1 , I2 in the path algebra KQ. Let I1 be generated by the paths of
length 3, and I2 be generated by the paths of length 4. Then A = KQ/I1 and B = KQ/I2 are self-injective Nakayama
algebras. Furthermore, we know from [16] that these algebras are τ-periodic over their enveloping algebras. Their period
is 3. Applying the techniques given in [16] one can show that A(A; τAe ) ∼
6= A(B; τBe ). Consequently, A and B are not
stably equivalent of Morita type.
4.
Invariance of some bimodules
For a finite dimensional self-injective K -algebra A, consider two non-zero objects X , Y from mod(Ae ). Then we can
consider the following K -linear space:
A(X ; τAe ; Y ) =
∞
M
j
Hom Ae τAi e (X ), τAe (Y ) .
i,j=0
Further we can define a multiplication of elements of the space by elements of A(Y ; τAe ) on the left, and by elements of
j
A(τAe ; X ) on the right. The multiplication for homogeneous elements is as follows: for h : τAi e (X ) → τAe (Y ), f : Y → τAt e (Y )
j
s
i
and g : τAe (X ) → X we put f ? h · g = τAe (f) ◦ h ◦ τAe (g). Extending this multiplication bilinearly for any morphisms, we
obtain a structure of an A(Y ; τAe )-A(τAe ; X )-bimodule on A(X ; τAe ; Y ). This bimodule reflects the local structure of the
category mod(Ae ) between two objects X , Y . This approach seems to be reasonable, because Ae usually is wild.
Similarly one can define a structure of an A(Y ; ΩAe )-A(ΩAe ; X )-bimodule on the K -linear space
A(X ; ΩAe ; Y ) =
∞
M
j
Hom Ae ΩiAe (X ), ΩAe (Y ) .
i,j=0
Proposition 4.1.
Let A, B be finite dimensional self-injective K -algebras that are stably equivalent of Morita type. Let the bimodules
e
e
A NB and B MA determine this stable equivalence. If X , Y ∈ lrp(A ) and U, V ∈ lrp(B ) are indecomposable objects such
∼
e
that U ∼
M
⊗
X
⊗
N,
V
M
⊗
Y
⊗
N
in
lrp(B
)
then
the
following
conditions
are
satisfied:
=
=
A
A
A
A
(a) There is a structure of an A(Y ; τAe )-A(τAe ; X )-bimodule on A(U; τBe ; V ) such that A(X ; τAe ; Y ) and A(U; τBe ; V ) are
isomorphic A(Y ; τAe )-A(τAe ; X )-bimodules.
(b) There is a structure of an A(Y ; ΩAe )-A(ΩAe ; X )-bimodule on A(U; ΩBe ; V ) such that A(X ; ΩAe ; Y ) and A(U; ΩBe ; V )
are isomorphic A(Y ; ΩAe )-A(ΩAe ; X )-bimodules.
We know from Theorem 1.1 that A(Y ; τAe ) ∼
= A(V ; τBe ) and A(τAe ; X ) ∼
= A(τBe ; U). Thus there is a structure
of an A(Y ; τAe )-A(τAe ; X )-bimodule on A(U; τBe ; V ), because it is transmitted by the above isomorphisms. Furthermore,
we infer by [13, Theorem 1] that the functor B M ⊗A · ⊗A NB : lrp(Ae ) → lrp(Be ) induces an equivalence F : lrp(Ae ) →
lrp(Be ) with F (A) ∼
= B. A straightforward verification shows that this equivalence gives an isomorphism A(X ; τAe ; Y ) ∼
=
A(U; τBe ; V ) of A(Y ; τAe )-A(τAe ; X )-bimodules, which proves (a). Condition (b) can be proved similarly. We omit the
details.
Proof.
Now we shall study some properties of the bimodules A(X ; τAe ; Y ), because they can be applied for studying the
τAe -periodicity of Ae -modules.
819
Unauthenticated
Download Date | 10/2/16 8:15 PM
Orbit algebras that are invariant under stable equivalences of Morita type
Lemma 4.2.
Let A be a finite dimensional self-injective K -algebra. Let X , Y ∈ lrp(Ae ) be non-zero indecomposable objects. Then,
for every non-negative integers i, j, the following conditions are satisfied:
j
(a) A τAi e (X ); τAe ; τAe (Y ) is an A(Y ; τAe )-A(τAe ; X )-subbimodule of A(X ; τAe ; Y ). Moreover,
A τAi e (X ); τAe ; τAi e (Y ) ∼
= A(X ; τAe ; Y )
as A(Y ; τAe )-A(τAe ; X )-bimodules.
j
(b) A ΩiAe (X ); ΩAe ; ΩAe (Y ) is an A(Y ; ΩAe )-A(ΩAe ; X )-subbimodule of A(X ; ΩAe ; Y ). Moreover,
A ΩiAe (X ); ΩAe ; ΩiAe (Y ) ∼
= A(X ; ΩAe ; Y )
as A(Y ; ΩAe )-A(ΩAe ; X )-bimodules.
j
We start with verifying that A τAi e (X ); τAe ; τAe (Y ) is an A(Y ; τAe )-A(τAe ; X )-subbimodule of A(X ; τAe ; Y ).
L
t+j
j
∞
s+i
It is obvious that A τAi e (X ); τAe ; τAe (Y ) =
s,t=0 Hom Ae τAe (X ), τAe (Y ) is a K -linear subspace of A(X ; τAe ; Y ) =
L∞
t+j
s+i
s
t
r
s,t=0 Hom Ae τAe (X ), τAe (Y ) . Consider any homogeneous element h : τAe (X ) → τAe (Y ). Choose f : Y → τAe (Y )
t+j
t+j+r
j
q
q+s+i
s+i
i
and g : τAe (X ) → X . Then f ? h · g = τAe (f) ◦ h ◦ τAe (g) ∈ Hom Ae τAe (X ), τAe (Y ) ⊂ A τAe (X ); τAe ; τAe (Y ) .
j
Hence A τAi e (X ); τAe ; τAe (Y ) is indeed an A(Y ; τAe )-A(τAe ; X )-subbimodule of A(X ; τAe ; Y ). Similarly one verifies that
j
A ΩiAe (X ); ΩAe ; ΩAe (Y ) is an A(Y ; ΩAe )-A(ΩAe ; X )-subbimodule of A(X ; ΩAe ; Y ).
Now consider a K -linear map φi : A τAi e (X ); τAe ; τAi e (Y ) → A(X ; τAe ; Y ) given by the following formula φi (h) = τA−ie (h)
t+i
−i
e
for any homogeneous h : τAs+i
e (X ) → τAe (Y ). Since τAe is a selfequivalence of the category mod(A ) whose quasi-inverse
i
is τAe , so φi is an isomorphism of K -linear spaces. Moreover, observe that φi is an isomorphism of A(Y ; τAe )-A(τAe ; X )
q
t+i
r
-bimodules. Indeed, consider homogeneous morphisms h : τAs+i
e (X ) → τAe (Y ), f : Y → τAe (Y ), g : τAe (X ) → X . Then
Proof.
s+i
−i
t+i
s+i
t
−i
s
φi (f ? h · g) = φi τAt+i
e (f) ◦ h ◦ τAe (g) = τAe τAe (f) ◦ h ◦ τAe (g) = τAe (f) ◦ τAe (h) ◦ τAe (g) = f ? φi (h) · g
which shows (a). In order to show condition (b), we similarly consider the map κi : A ΩiAe (X ); ΩAe ; ΩiAe (Y ) → A(X ; ΩAe ; Y )
given by the formula κi (h) = Ω−i
Ae (h). We omit the details.
Lemma 4.3.
Let A be a finite dimensional self-injective K -algebra. Let X ∈ lrp(Ae ) be non-zero indecomposable object. Then the
following conditions are satisfied for every non-negative integer i:
∼
i+1
(a) A τAi e (X ); τAe ; τAi e (X ) /A τAi+1
= A(X ; τAe ; X )/A τAe (X ); τAe ; τAe (X ) as A(X ; τAe )-A(τAe ; X )-bimoe (X ); τAe ; τAe (X )
dules.
(b) A(X ; τAe ; X )/A τAe (X ); τAe ; τAe (X ) is a cyclic A(X ; τAe )-A(τAe ; X )-bimodule.
(c) A Ωi (X ); Ω e ; Ωi (X ) /A Ωi+1 (X ); Ω e ; Ωi+1 (X ) ∼
= A(X ; Ω e ; X )/A Ω e (X ); Ω e ; Ω e (X ) as A(X ; Ω e )-A(Ω e ; X )A
Ae
Ae
A
Ae
A
Ae
A
A
A
A
A
bimodules.
(d) A(X ; ΩAe ; X )/A ΩAe (X ); ΩAe ; ΩAe (X ) is a cyclic A(X ; ΩAe )-A(ΩAe ; X )-bimodule.
Proof.
We know from Lemma 4.2 (a) that there is the following commutative diagram of A(X ; τAe )-A(τAe ; X )-bimodules:
i+1
/ A τAi+1
e (X ); τAe ; τAe (X )
0
0
/ A τAi e (X ); τAe ; τAi e (X )
C
φi
/ A τAe (X ); τAe ; τAe (X )
A
D
/ Wi
/ 0
/ W0
/ 0
φi
/ A(X ; τAe ; X )
B
820
Unauthenticated
Download Date | 10/2/16 8:15 PM
Z. Pogorzały
whose rows are exact, where
i+1
Wi = A τAi e (X ); τAe ; τAi e (X ) /A τAi+1
e (X ); τAe ; τAe (X ) ,
W0 = A(X ; τAe ; X )/A τAe (X ); τAe ; τAe (X ) ,
A, C are inclusions and φi is the isomorphism that is the restriction of φi . Then there exists an A(X ; τAe )-A(τAe ; X )
-bimodule isomorphism δi : Wi → W0 that completes the above diagram to a commutative one. In the way condition (a)
is shown. In order to prove condition (c), we proceed similarly.
Now consider the factor A(X ; τAe )-A(τAe ; X )-bimodule A(X ; τAe ; X )/A τAe (X ); τAe ; τAe (X ) . We shall denote by [h] the
coset of any element h ∈ A(X ; τAe ; X ) modulo A τAe (X ); τAe ; τAe (X ) . Observe that one can indicate the unique generator
[idX ] in this factor bimodule. Indeed, for any homogeneous element h ∈ Hom Ae (X , τAi e (X )) we have [h] = [h ? idX ] =
h ? [idX ]. Moreover, for any homogeneous element g ∈ Hom Ae (τAi e (X ), X ) we have [g] = [idX · g] = [idX ] · g. Therefore
condition (b) is satisfied. One can prove condition (d) in a similar way.
5.
Noetherianity of orbit algebras
We start this section with proving Theorem 1.2.
Proof of Theorem 1.2.
We know from [17, Proposition 17] that A(τAe ; M) is right noetherian. Now we shall show
that A(M; τAe ) is left noetherian. This part of the proof is similar to the proof of [17, Proposition 17]. If M is an
Ae -module that is τAe -periodic then there is a minimal positive integer n such that τAne (M) ∼
= M. Then the left Ae -module
∼
n−1
2
e
M = M ⊕ τAe (M) ⊕ τAe (M) ⊕ · · · ⊕ τAe (M) is also a τAe -periodic A -module and τAe (M) = M.
We shall show that A(M; τAe ) is left noetherian. Consider an isomorphism g : M → τAe (M). Then gEnd Ae (M) =
End Ae (M)g. Furthermore, for every finite dimensional Ae -module N, it holds Hom Ae N, τAi e (M) = gi Hom Ae (N, M),
where i ranges all non-negative integers. Notice that End Ae (M) is a finite dimensional K -algebra, hence it is left
noetherian. Then
M
M i
A(M; τAe ) =
g Hom Ae (M, M)
Hom Ae M, τAi e (M) =
i≥0
i≥0
by the above remarks, where the multiplication of homogeneous elements is of the form gi h · gj f = gi+j h1 f for h1
satisfying hgj = gj h1 .
Now let C be a left ideal in the algebra A(M, τAe ), Ci = C ∩ gi Hom Ae (M, M). Assume that m1 is the minimal
non-negative integer such that Cm1 6= 0. Since Cm1 is a finite dimensional K -linear space, so denote by f1 , . . . , fs1
its K -basis. Now assume to the contrary that C is not finitely generated. Then C is not generated by f1 , . . . , fs1 .
Thus there is an integer m2 > m1 such that we can find an element f ∈ Cm2 that does not belong to the subspace
(gm2 −m1 f1 , . . . , gm2 −m1 fs1 ) spanned by gm2 −m1 f1 , . . . , gm2 −m1 fs1 . Since f1 , . . . , fs1 are linearly independent in Cm1 and g is
an isomorphism, so the elements gm2 −m1 f1 , . . . , gm2 −m1 fs1 are linearly independent in Cm2 . Therefore there is a K -basis
gm2 −m1 f1 , . . . , gm2 −m1 fs1 , fs1 +1 , . . . , fs2 of the space Cm2 and s2 > s1 .
Assume that for some j ≥ 2 there is a K -basis f10 , . . . , fs0 1 , fs0 1 +1 , . . . , fs0 2 , . . . , fs0 j−1 +1 , . . . , fs0 j of the space Cmj that is
constructed in the above way, where sj > sj−1 > . . . > s1 > 1. Since C is not finitely generated, so there are mj+1 > mj
and f ∈ Cmj+1 such that f does not belong to the subspace that is spanned by
gmj+1 −mj f10 , . . . , gmj+1 −mj fs0 1 , . . . , gmj+1 −mj fs0 j−1 +1 , . . . , gmj+1 −mj fs0 j .
Similarly, we know that the last system of elements from Cmj+1 is linearly independent and we can extend it to a K -basis
of Cmj+1 . In this way we obtain a system of linearly independent elements
gmj+1 −mj f10 , . . . , gmj+1 −mj fs0 1 , . . . , gmj+1 −mj fs0 j−1 +1 , . . . , gmj+1 −mj fs0 j , fsj +1 , . . . , fsj+1
satisfying sj+1 > sj . Consequently, for every l > dimK End Ae (M) we have dimK Cml ≥ l > dimK End Ae (M) which is
impossible. Thus C is finitely generated and A(M; τAe ) is left noetherian. Finally, since M is a direct summand in M,
so A(M; τAe ) is left noetherian as well.
821
Unauthenticated
Download Date | 10/2/16 8:15 PM
Orbit algebras that are invariant under stable equivalences of Morita type
Theorem 5.1.
Let A be an indecomposable self-injective finite dimensional K -algebra that is not simple. If X is a left Ae -module that
is not projective then the following conditions are satisfied:
(a) If X is τAe -periodic then the A(X ; τAe )-A(τAe ; X )-bimodule A(X ; τAe ; X ) is finitely generated.
(b) If X is ΩAe -periodic then the A(X ; ΩAe )-A(ΩAe ; X )-bimodule A(X ; ΩAe ; X ) is finitely generated.
Assume that X is τAe -periodic and let n be the minimal positive integer such that τAne (X ) ∼
=
X . We have to show that the A(X ; τAe )-A(τAe ; X )-bimodule A(X ; τAe ; X ) is finitely generated. In particular, we shall
show that idX , idτAe (X ) , . . . , idτ n−1
is a system of generators of this bimodule.
e (X )
Proof of Theorem 1.2.
A
Notice that for any homogeneous element f : τAme (X ) → τAi e (X ) with m ≥ i, 0 ≤ i < n,
f = idτ i e (X ) · τA−ie (f).
A
Similarly, for any homogeneous element g : τAi e (X ) → τAme (X ) with m ≥ i, 0 ≤ i < n,
g = τA−ie (g) ? idτ i e (X ) .
A
Further notice idτ kn+j (X ) can be obtained as
Ae
−j
−j
idτ kn+j (X ) = τAe (f −1 ) ? idτ j e (X ) · τAe (f)
A
Ae
kn+j
j
for any k ≥ 1, 0 ≤ j < n, where f : τAe (X ) → τAe (X ) is an isomorphism. Consequently, we obtain that the elements
idX , idτAe (X ) , . . . , idτ n−1
generate the A(X ; τAe )-A(τAe ; X )-bimodule A(X ; τAe ; X ). Similar arguments show condition (b).
e (X )
A
From the proof of Theorem 5.1 we have
Corollary 5.2.
Let A be a finite dimensional indecomposable self-injective K -algebra that is not simple. If X is a nonprojective
Ae -module then the following conditions are satisfied:
(a) If X is τAe -periodic whose period is n then the A(X ; τAe )-A(τAe ; X )-bimodule A(X ; τAe ; X ) has an n-element set of
homogeneous generators.
(b) If X is ΩAe -periodic whose period is n then the A(X ; ΩAe )-A(ΩAe ; X )-bimodule A(X ; ΩAe ; X ) has an n-element set
of homogeneous generators.
Acknowledgements
The author is deeply indebt the referees for their comments and remarks.
The research has been supported by the research grant No. DEC-2011/02/A/ST1/00216 of the Polish National Center.
References
[1] Assem I., Simson D., Skowroński A., Elements of the Representation Theory of Associative Algebras, I, London Math.
Soc. Stud. Texts, 65, Cambridge University Press, Cambridge, 2006
822
Unauthenticated
Download Date | 10/2/16 8:15 PM
Z. Pogorzały
[2] Auslander M., Reiten I., Representation theory of Artin algebras III. Almost split sequences, Comm. Algebra, 1975,
3(3), 239–294
[3] Bergh P.A., Orbit algebras and periodicity, Colloq. Math., 2009, 114(2), 245–252
[4] Brenner S., Butler M.C.R., King A.D., Periodic algebras which are almost Koszul, Algebr. Represent. Theory, 2002,
5(4), 331–367
[5] Broué M., Equivalences of blocks of group algebras, In: Finite-Dimensional Algebras and Related Topics, Ottawa,
August 10–18, 1992, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 424, Kluwer, Dordrecht, 1994, 1–26
[6] Dugas A., Periodic resolutions and self-injective algebras of finite type, J. Pure Appl. Algebra, 2010, 214(6), 990–1000
[7] Green E.L., Snashall N., Solberg Ø., The Hochschild cohomology ring of a selfinjective algebra of finite representation
type, Proc. Amer. Math. Soc., 2003, 131(11), 3387–3393
[8] Happel D., Hochschild cohomology of finite-dimensional algebras, In: Séminaire d’Algèbre Paul Dubreil et MariePaul Malliavin, 39ème Année, Paris, 1987/1988, Lecture Notes in Math., 1404, Springer, Berlin, 1989, 108–126
[9] Heller A., The loop-space functor in homological algebra, Trans. Amer. Math. Soc., 1960, 96, 382–394
[10] Kerner O., Minimal approximations, orbital elementary modules, and orbit algebras of regular modules, J. Algebra,
1999, 217(2), 528–554
[11] Lenzing H., Wild canonical algebras and rings of automorphic forms, In: Finite-Dimensional Algebras and Related
Topics, Ottawa, August 10–18, 1992, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 424, Kluwer, Dordrecht, 1994,
191–212
[12] Linckelmann M., Stable equivalences of Morita type for self-injective algebras and p-groups, Math. Z., 1996, 223(1),
87–100
[13] Pogorzały Z., Left-right projective bimodules and stable equivalences of Morita type, Colloq. Math., 2001, 88(2),
243–255
[14] Pogorzały Z., Invariance of Hochschild cohomology algebras under stable equivalences of Morita type, J. Math. Soc.
Japan, 2001, 53(4), 913–918
[15] Pogorzały Z., A new invariant of stable equivalences of Morita type, Proc. Amer. Math. Soc., 2002, 131(2), 343–349
[16] Pogorzały Z., On Galois coverings of the enveloping algebras of self-injective Nakayama algebras, Comm. Algebra,
2003, 31(6), 2985–2999
[17] Pogorzały Z., On the Auslander–Reiten periodicity of self-injective algebras, Bull. London Math. Soc., 2004, 36(2),
156–168
[18] Rickard J., Derived equivalences as derived functors, J. London Math. Soc., 1991, 43(1), 37–48
[19] Schulz R., Boundedness and periodicity of modules over QF rings, J. Algebra, 1986, 101(2), 450–469
[20] Snashall N., Solberg Ø., Support varieties and Hochschild cohomology rings, Proc. London Math. Soc., 2004, 88(3),
705–732
823
Unauthenticated
Download Date | 10/2/16 8:15 PM