On the tameness of trivial extension algebras by

149 (1996)

On the tameness of trivial extension algebras

by

Ibrahim A s s e m (Sherbrooke, Qu´e.) and Jos´ e Antonio d e l a P e ˜ n a (M´exico)

Abstract. For a finite dimensional algebra A over an algebraically closed field, let T (A) denote the trivial extension of A by its minimal injective cogenerator bimodule. We prove that, if T _A is a tilting module and B = End T _A , then T (A) is tame if and only if T (B) is tame.

Introduction. Let k be an algebraically closed field. In this paper, an algebra A is always assumed to be associative, with an identity and finite dimensional over k. We denote by mod A the category of finitely gener- ated right A-modules, and by mod A the stable module category whose objects are the A-modules, and the set of morphisms from M _A to N _A is Hom _A (M, N ) = Hom A (M, N )/P(M, N ), where P(M, N ) is the subspace of all morphisms factoring through projective modules. Two algebras R and S are called stably equivalent if the categories mod R and mod S are equiva- lent. There are several important problems of the representation theory of algebras which are formulated in terms of the stable equivalence of two self- injective algebras (see, for instance, [9, 19]). But few things are known. For instance, it is not yet known whether for two stably equivalent self-injective algebras R and S, the tameness of R implies that of S. We consider this problem in the following context.

Let A be an algebra, and D = Hom _k (−, k) denote the usual duality on mod A. The trivial extension T (A) of A (by the minimal injective con- generator bimodule DA) is defined to be the k-algebra whose vector space structure is that of A ⊕ DA, and whose multiplication is defined by

(a, q)(a ⁰ , q ⁰ ) = (aa ⁰ , aq ⁰ + qa ⁰ )

for a, a ⁰ ∈ A and q, q ⁰ ∈ _A (DA) _A . Trivial extensions are a special class of self-injective (actually, of symmetric) algebras. They have been extensively

1991 Mathematics Subject Classification: Primary 16G60.

[171]

studied in the representation theory of algebras (see, for instance, [2, 3, 8, 10, 12, 17]), in particular, in the context of tilting theory. Following [7], an A-module T _A is called a tilting module if Ext ² _A (T, −) = 0, Ext ¹ _A (T, T ) = 0 and there is an exact sequence

0 → A A → T _A ⁰ → T _A ⁰⁰ → 0,

with T ⁰ , T ⁰⁰ in the additive category add T consisting of the direct sums of direct summands of T . A tilting triple (B, _B T _A , A) consists of an alge- bra A, a tilting module T _A and B = End T . For a tilting triple (B, T, A), Tachikawa and Wakamatsu have constructed in [17] a stable equivalence functor S : mod T (A) → mod T (B). Our main result is the following:

Theorem. Let (B, _B T _A , A) be a tilting triple. Then T (A) is tame if and only if T (B) is tame.

Recall from [5] that an algebra C is called tame if, for each d ∈ N, there is a finite number of k[t] − C-bimodules M 1 , . . . , M _s(d) which are free and finitely generated left k[t]-modules and such that, for all but at most finitely many non-isomorphic indecomposable C-modules X with dim _k X = d, there is an isomorphism X → (k[t]/ht − λi) ⊗ ^∼ _k[t] M i , for some 1 ≤ i ≤ s(d) and λ ∈ k. In this case, we let µ _C (d) denote the least possible s(d). We say that C is of polynomial growth (or domestic) if µ _C (d) ≤ d ^m for some m ∈ N (or µ C (d) ≤ K for some K ∈ N, respectively).

The representation theory of the trivial extension of polynomial growth is well known. Namely, the representation-finite trivial extensions are trivial extensions of tilted algebras of Dynkin type (by [8]), the representation- infinite domestic trivial extensions are either trivial extensions of tilted al- gebras of Euclidean type or quotients of a trivial extension T (A) of some representation-infinite algebra A of Euclidean type e A n by the group Z 2 (by [3, 10, 11]), and the non-domestic trivial extensions of polynomial growth are trivial extensions of tubular algebras (by [10, 12]). But the representation theory of the tame trivial extensions which are not of polynomial growth is still completely unknown. Our theorem ensures that the tameness of these algebras is preserved under tilting of the original algebra.

The main technique used for the proof of the theorem is the geometric setting developed in [14]. The proof essentially reduces to showing that the construction of S yields a constructible function on objects S : mod T (A) → mod T (B). Then one applies [14], (4.3).

1. The Tachikawa–Wakamatsu stable equivalence functor

1.1. Let A be a finite dimensional k-algebra assumed moreover to be ba-

sic and connected. We shall use freely, and without further reference, facts

about the module category mod A and the Auslander–Reiten translations

τ = D Tr and τ ⁻¹ = Tr D, as in [4, 15]. For tilting theory, we refer the reader to [1]. Recall in particular that if (B, _B T _A , A) is a tilting triple, then T _A induces a torsion theory (T , F) in mod A, where T (or F) denotes the full subcategory of mod A generated by T A (or cogenerated by τ T A , respec- tively).

Given an A-module M , the evaluation morphism ε _M : Hom _A (T, M )⊗ _B T

→ M defined by f ⊗ t 7→ f (t) (where f ∈ Hom _A (T, M ), t ∈ T ) is functorial, and is an isomorphism if and only if M ∈ T . Similarly, to a B-module X corresponds a functorial morphism η _X : X _B → Hom _A (T, X ⊗ _B T ) defined by x 7→ (t 7→ x ⊗ t) (where x ∈ X, t ∈ T ).

Finally, we have canonical isomorphisms DT ⊗ _B T → DA and T ⊗ ^∼ _B DT → DB, which we shall consider as identifications throughout this ^∼ paper.

1.2. Torsion resolutions. Let (B, T, A) be a tilting triple, and (T , F) be the corresponding torsion theory in mod A. For an A-module M _A , an exact sequence of the form

0 −→ M −→ V ^α

0 β

−→ T 0 −→ 0

with V 0 ∈ T and T 0 ∈ add T is called a torsion resolution for M . By [17], each module M _A admits a torsion resolution

0 −→ M −→ V (M ) ^α

−→ T (M ) −→ 0 ^β

such that T (M ) = P ⊗ _B T , where P _B is a projective cover of Ext ¹ _A (T, M ), and which is minimal in the following sense: for any other torsion resolution for M ,

0 −→ M −→ V ^α

^{0 β} −→ T

⁰ −→ 0,

there exists T ⁰⁰ ∈ add T such that we have a commutative diagram with exact rows:

0 M V ⁰ T ⁰ 0

0 M V (M ) ⊕ T ⁰⁰ T (M ) ⊕ T ⁰⁰ 0 //     

α

//      

β

//       //

//

 _α

0



//

 _β

M

0 0 1



// //

In fact, the module V (M ) is constructed as follows:

(a) If M ∈ T , then V (M ) = M .

(b) If M ∈ F, then V (M ) → K ⊗ ^∼ B T , where K B is the kernel of the projective cover morphism P _B → Ext ¹ _A (T, M ).

(c) In general, if 0 → tM → M → M/tM → 0 is the canonical sequence

of M in the torsion theory (T , F), the torsion resolution of M is obtained as

the middle column in the following commutative diagram with exact rows

and columns:

0 0

0 tM M M/tM 0

0 tM V (M ) V (M/tM ) 0

T (M ) T (M/tM )

0 0

²² ²²

//       // ²² // ²² //

// // ²² // ²² //

²²

_____

_____

²²

Lemma. Let t = dim _k T , s _A = dim _k A, s _B = dim _k B and M _A be an A-module with d = dim k M . Then dim k V (M ) ≤ d + dt ² s ³ _A s B .

P r o o f. Using the middle row in the above diagram, we see that dim k V (M ) = dim k tM +dim k V (M/tM ). Since tM ⊆ M , we have dim k tM

≤ d. Put N = M/tM . We have a short exact sequence of B-modules 0 −→ K −→ P _B −→ Ext ^{p 1} _A (T, N ) −→ 0,

where p is a projective cover. Hence V (N ) → K ⊗ ^∼ _B T yields dim _k V (N ) ≤ t dim _k K. On the other hand,

dim _k K ≤ dim _k P _B ≤ s _B dim _k Ext ¹ _A (T, N ).

In order to bound dim _k Ext ¹ _A (T, N ), we shall use the Auslander–Reiten for- mula Ext ¹ _A (T, N ) → D Hom ^∼ _A (τ ⁻¹ N, T ) and a minimal projective presenta- tion in mod A ^op :

A P ₁ −→ ^f

_A P ₀ −→ ^f

_A DN −→ 0.

Indeed, the latter yields dim _k P ₁ ≤ s _A dim _k Ker f ₀ and dim _k Ker f ₀ ≤ s A dim k DN = s A dim k N . Applying the functor Hom A (−, A) yields an ex- act sequence

Hom _A (P ₀ , A) −→ Hom _A (P ₁ , A) −→ τ ⁻¹ N −→ 0.

Hence, dim _k τ ⁻¹ N ≤ dim _k Hom _A (P ₁ , A) ≤ s _A dim _k P ₁ ≤ s ³ _A dim _k N . On the other hand, since N = M/tM , we have dim _k N ≤ dim _k M = d. The result then follows from the inequalities

dim _k V (M/tM ) ≤ t dim _k K ≤ ts _B dim _k Ext ¹ _A (T, N )

≤ ts B dim k Hom A (τ ⁻¹ N, T ) ≤ ts B dim k T dim k τ ⁻¹ N

≤ t ² s _B s ³ _A d.

1.3. The stable equivalence. For an algebra A, let T (A) denote the trivial extension of A by DA. We shall use the following equivalent description of mod T (A) (see [6]). Let C be the category whose objects are the pairs (M, ϕ) where M is an A-module and ϕ : M ⊗ A DA → M is an A-linear map such that ϕ(ϕ ⊗ DA) = 0, and where a morphism f : (M, ϕ) → (M ⁰ , ϕ ⁰ ) is an A-linear map f : M → M ⁰ such that ϕ ⁰ (f ⊗DA) = f ϕ. Then C → mod T (A). ^∼ Throughout this paper, we shall identify these two categories. In particular, any A-module M induces a T (A)-module M ⊕ (M ⊗ DA),  _{0 0}

1 0


that will be denoted by

It is well known that the image of the canonical embedding mod A → mod T (A) actually lies inside the stable category (see, for instance, [17]).

Let (M, ϕ) be an arbitrary T (A)-module. As an A-module, M admits a minimal torsion resolution

0 −→ M −→ V (M ) ^α

−→ T (M ) −→ 0. ^β

Now, define a B-linear map

φ M : M ⊗ A DT B −→ Hom A (T, V (M )) ⊕ [Hom A (T, V (M )) ⊗ B DB]

by the formula φ

=

"

Hom _A (T, α _M ) Hom _A (T, −ϕ)η _{M ⊗DT} (ε ⁻¹ _{V (M )} ⊗ DT )(α M ⊗ DT )

# .

The construction of φ M may be visualised as follows:

Hom _A (T, M ⊗ _A DT ⊗ _B T ) Hom _A (T, M ⊗ _A DA) Hom _A (T, M )

M ⊗ _A DT Hom _A (T, V (M ))⊕

[Hom _A (T, V (M )) ⊕ _B DB]

V (M ) ⊗ _A DT Hom _A (T, V (M )) ⊗ _B T ⊗ _A DT

∼ // ^Hom

^(T,−ϕ) //

Hom

(T,α

)

²² η

OO

α

⊗DT

²² _∼

ε

⊗DT

//

∼

OO

The source and the target of φ _M are each endowed with a natural T (B)- module structure. Namely, M ⊗ _A DT has the T (B)-module structure in- duced by the morphism

M ⊗ A DT ⊗ B DB −→ M ⊗ ^∼ A DT ⊗ B T ⊗ A DT

−→ M ⊗ ∼ _A DA ⊗ _A DT −−−−−→ M ⊗ ^−ϕ⊗DT _A DT,

while Hom _A (T, V (M ))⊕[Hom _A (T, V (M ))⊗ _B DB] induces the T (B)-module

One shows that φ M is T (B)-linear, so that S(M, ϕ) = Coker φ M is a T (B)-module. We have thus defined a map on objects S : mod T (A) → mod T (B). One can show that for an A-module M ∈ T , we have S(M, 0) = Hom _A (T, M ), that the image of a projective T (A)-module is a projective T (B)-module, and, finally, that S is a functor from mod T (A) to mod T (B).

We have the following theorem:

Theorem [17]. Let (B, T, A) be a tilting triple. Then the functor S : mod T (A) → mod T (B) is an equivalence.

2. Geometrisation of the problem

2.1. The proof of our theorem relies on the methods of algebraic geom- etry and the criteria for tameness developed in [13, 14]. For more details on the constructions used, we refer the reader to [14].

First, we identify our A- and T (A)-modules with points of constructible sets in appropriate affine spaces.

Let A be an algebra, and 1 = a ₁ , a ₂ , . . . , a _s be a k-basis of A such that a i a j = P _s

m=1 λ ^(m) _ij a m for all 1 ≤ i, j ≤ m (the scalars λ ^(m) _ij are the so-called structure constants of A). Recall that an A-module M of k-dimension d may be identified with a representation M : A → End _k (k ^d ), thus, each of the basis elements a i corresponds to a d × d matrix M (a i ). We define, for each d ∈ N, mod _A (d) to be the (closed) subset of the affine space Q _s

i=1 k ^d×d consisting of the s-tuples of d × d matrices M = (M (a ₁ ), . . . , M (a _s )) such that M (a 1 ) is the identity matrix and M (a i )M (a j ) = P _s

m=1 λ ^(m) _ij M (a m ) for all 1 ≤ i, j ≤ s.

Recall that a vector space category is a pair (K, |−|) consisting of a k-linear category and a faithful k-linear functor |−| : K → mod k.

Following [14], we say that the pair (K, |−|) is geometrisable if there exists an increasing function σ : N → N inducing, for each d ∈ N, a function {X ∈ K : dim _k |X| = d} → k ^σ(d) (denoted by X 7→ e X) satisfying

(G1) the image K(d) is constructible as a subset of the affine space k ^σ(d) ; (G2) there is a function µ : K(d) → K such that dim k |µ( e X)| = d and

µ( e X) ^∼ = e X; moreover, µ( e X) −→ X. ^∼

Examples. (a) The module category mod A is a vector space category

taking |−| : mod A → mod k to be the forgetful functor. Let σ(d) = sd ² ,

for d ∈ N. As above, let 1 = a 1 , a 2 , . . . , a s be a k-basis of A. The map

M 7→ f M = (M (a ₁ ), . . . , M (a _s )) ∈ k ^σ(d) defines a geometrisation of mod A.

(b) Let K A be the category whose objects are triples (M, f, N ), where M, N are A-modules, and f : M → N is A-linear, and whose morphisms (h, g) : (M, f, N ) → (M ⁰ , f ⁰ , N ⁰ ) are pairs of A-linear maps such that f ⁰ h = gf . Then K A is a vector space category with |(M, f, N )| = |M |⊕|N | (where, on the right, |−| : mod A → mod k denotes the forgetful functor, as in (a)).

Let σ(d) = (s+1)d ² . For a triple (M, f, N ) with dim _k |M | = m, dim _k |N | = n and d = m+n, we define (M, f, N ) ^∼ to be the (s+1)-tuple of d×d matrices

 M (a 1 ) 0 0 N (a ₁ )

 , . . . ,

 M (a s ) 0 0 N (a _s )

 ,

 0 f 0 0



.

The images of this map form a closed subset K A (m, n) of k ^σ(d) . Thus K _A (d) = S

d=m+n K _A (m, n) is closed in k ^σ(d) . Therefore K _A is geometris- able.

2.2. Let us now consider mod T (A). For (M, ϕ) ∈ mod T (A), we set

|(M, ϕ)| to equal the underlying k-space |M | of M . Thus mod T (A) is a vector space category. The following lemma is shown in [14], we only indicate the main steps of the proof.

Lemma. The vector space category mod T (A) is geometrisable.

S k e t c h o f p r o o f. Let σ(d) = 2sd ² , where a 1 = 1, . . . , a s is a basis of A. Let m ≤ sd. Consider the functor F = − ⊗ _A DA : mod A → mod A.

By [14], (2.2), the set F ^{−1 d} _m

= {M ∈ mod _A (d) : dim _k |F M | = m} is constructible. The set of pairs (M, ϕ) in mod A (d) × k ^d×m such that ϕ : F M → f g M is A-linear and ϕ(ϕ ⊗ DA) = 0 is a constructible subset C(m) of k ^sd

× k ^d×m ⊆ k ^σ(d) . Since, for any N ∈ mod A , dim k |F N | ≤ s dim k |N |, it follows that mod _{T (A)} (d) = S

m≤sd C(m) is a constructible subset of k ^σ(d) . For (M, ϕ) ∈ mod T (A) with dim _k |M | = d, we thus set (M, ϕ) ^∼ = ( f M , ϕ) ∈ mod _{T (A)} (d).

2.3. In 1.3, we defined the function on objects S : mod T (A) → mod T (B) which induces a stable equivalence S : mod T (A) → mod T (B).

Let U, V be two vector space categories. A function on objects f : U → V is called an object-function if X → Y in U implies f (X) ^∼ → f (Y ) in V. An ^∼ object-function f : U → V, where U, V are geometrisable categories, is called constructible [14] if, for each d, m ∈ N, the following are satisfied:

(C1) the set f ^{−1 d} _m

= {M ∈ U(d) : dim k |f (M )| = m} is constructible, and empty for m large enough;

(C2) there exist c ∈ N and a constructible subset C ⊆ f ^{−1 d} _m

×k ^c ×V(m)

such that the following diagram (where π ₁ , π ₃ denote the respective

projection morphisms) commutes:

C ⊆ f ^{−1 d} _m

× k ^c × V(m) V(m)

f ^{−1 d} _m

π

²²

π

//

mmmm mmmm f mmmm m66

in the sense that f π 1 (w) ^∼ −→ π ^∼ 3 (w) ^∼ , for w ∈ C.

(C3) π ₁ (C) = f ^{−1 d} _m
.

The following examples are treated in detail in [14].

Examples. (a) The composition of constructible object-functions is con- structible.

(b) Let _B X _A be a bimodule. Then − ⊗ _B X _A : mod B → mod A and

− ⊗ B X A : K B → K A yield constructible object-functions.

(c) Let f : _B X _A → _B Y _A be a morphism of B-A-bimodules. Then −⊗ _B f : mod B → K _A yields a constructible object-function.

(d) The functors Ker, Coker : K _A → mod A yield constructible object- functions.

(e) Let B T A be a bimodule and η : 1 mod B → Hom B (T, − ⊗ B T ) be the functorial morphism defined in 1.1. Then η induces a constructible object- function η : mod B → K _B .

(f) Let _B T _A be a bimodule and ε : Hom _A (T, −) ⊗ _B T → 1 _{mod A} be the (evaluation) functorial morphism defined in 1.1. Then ε induces a con- structible object-function ε : mod A → K A . If T A is a tilting module and B = End T _A , then ε _M is invertible for each M ∈ T and ε ⁻¹ : T → K _A is also constructible (here, T denotes, as usual, the torsion class induced by T A in mod A).

2.4. Proposition. The object function S : mod T (A) → mod T (B) is constructible.

P r o o f. By the examples in 2.3 above, and the construction of S given in 1.3, it suffices to show that, if M ∈ mod A and

0 −→ M −→ V (M ) ^α

−→ T (M ) −→ 0 ^β

is a minimal torsion resolution of M , then the object-function s : mod A → K _A given by M 7→ (V (M ), β _M , T (M )) is constructible.

Let (T , F) be the torsion theory in mod A induced by the tilting triple

(B, T, A). We claim that, for any d ∈ N, the sets F(d) = {M ∈ mod _A (d) :

M ∈ F} and T (d) = {M ∈ mod A (d) : M ∈ T } are constructible. Indeed, by

[14], (2.5), the functors F = Hom _A (T, −) and F ⁰ = Ext ¹ _A (T, −) from mod A

to mod B induce constructible object-functions. Hence F ^{−1 d} ₀

= {M ∈ mod _A (d) : F (M ) = 0} = F(d) and F ^{0 −1 d} ₀

= {M ∈ mod _A (d) : F ⁰ (M ) = 0}

= T (d) are constructible.

Consider the object-function c : mod A → K A × K A defined by M 7→

((tM, i, M ), (M, p, M/tM )), where

0 −→ tM −→ M ⁱ −→ M/tM −→ 0 ^p

is the canonical sequence of M in the torsion theory (T , F). We now claim that c is constructible. Indeed, for e ≤ d, let

C(d, e)

= {(M ⁰ , i, M, p, M ⁰⁰ ) ∈ T (e) × k ^e×d × mod _A (d) × k ^d×(d−e) × F(d − e) : the sequence 0 −→ M ⁰ −→ M ⁱ −→ M ^{p 00} −→ 0 is exact}.

Clearly, C(d, e) is constructible. Therefore c is constructible.

The object-function f : F → K A defined by f (M ) = (K ⊗ B T, j ⊗ B T , P 0 ⊗ B T ), where

0 −→ K −→ P ^j 0 p

−→ Ext ¹ _A (T, M ) −→ 0

is an exact sequence, and P ₀ is a projective cover of Ext ¹ _A (T, M ), is also a constructible object-function. Indeed, the coordinates of f are obtained by composition of the following object-functions:

(a) Ext ¹ _A (T, −) : mod A → mod B,

(b) P ₀ : mod B → mod B, such that P ₀ (X) is a projective cover of X, (c) Ker : K _B → mod B,

(d) − ⊗ B T : mod B → mod A,

each of which is constructible (see 2.3 above, or [14]). Hence f is con- structible.

We proceed to show that s is constructible. Let d ∈ N and m ≤ d + dt ² s ³ _A s _B (as in 1.2). Choose also e ≤ d. Consider the set C(d, e, m) of 16- tuples

(M, M ⁰ , i, p, M ⁰⁰ , j, V ⁰⁰ , σ, P, q, L, α, V, β, α ⁰ , β ⁰ ) such that

(a) M ∈ mod A (d), c(M ) = ((M ⁰ , i, M ), (M, p, M ⁰⁰ )) ∈ C(d, e),

(b) f (M ⁰⁰ ) = (V ⁰⁰ , σ, P ) and 0 −→ M ⁰⁰ −→ V ^{j 00} −→ L −→ 0 is exact in ^q mod A,

(c) V ∈ mod A (m) and the following diagram is commutative, with exact

rows and columns:

0 0

0 M ⁰ M M ⁰⁰ 0

0 M ⁰ V V ⁰⁰ 0

L L

0 0

²² ²²

//      

i //

α

²²

p //

j

²² //

// ^α

//

β

²²

β

//

q

²² //

²²

_______

_______

²²

The set C(d, e, m) is constructible and s ⁻¹ _(m,m−d) ^d

= [

e≤d

π ₁ C(d, e, m)

is constructible. Moreover, the diagram

C(d, e, m) K A ((m, m − d))

s ⁻¹ _(m,m−d) ^d

⊆ mod _A (d)

π

²²

π //

iiiii iiiii s iiiii ii44

(where π, π ₁ denote the respective projections) commutes. Hence s is con- structible.

2.5. P r o o f o f t h e t h e o r e m. As pointed out in the introduction, our theorem immediately follows from the above proposition and the criterion for tameness in [14], (4.3). Indeed, let X be a non-projective T (B)-module.

Then there exists a T (A)-module M and a projective T (B)-module P such that S(M ) −→ X ⊕ P . In the terminology of [14], (4.1), we say that S ^∼ constructively almost covers mod T (B). A direct application of [14], (4.3) completes the proof.

R e m a r k. Let (B, T, A) be a tilting triple. Then T (A) is domestic repres- entation-infinite (or non-domestic of polynomial growth) if and only if T (B) is domestic representation-infinite (or non-domestic of polynomial growth, respectively). This follows from [3, 10, 11, 12].

Acknowledgements. This paper was written while the second author

was visiting the first. The first author gratefully acknowledges partial sup-

port from the Natural Sciences and Engineering Council of Canada and the

Universit´e de Sherbrooke, and the second author gratefully acknowledges

partial support from the Consejo Nacional de Ciencia y Tecnolog´ıa and the Universit´e de Sherbrooke.

References

[1] I. A s s e m, Tilting theory—an introduction, in: Topics in Algebra, Banach Center Publ. 26, Part 1, PWN, Warszawa, 1990, 127–180.

[2] I. A s s e m, D. H a p p e l and O. R o l d ´a n, Representation-finite trivial extension al- gebras, J. Pure Appl. Algebra 33 (1984), 235–242.

[3] I. A s s e m, J. N e h r i n g and A. S k o w r o ´ n s k i, Domestic trivial extension of simply connected algebras, Tsukuba J. Math. 13 (1989), 31–72.

[4] M. A u s l a n d e r and I. R e i t e n, Representation theory of artin algebras III , IV , Comm. Algebra 3 (1975), 239–294 and 5 (1977), 443–518.

[5] Yu. A. D r o z d, Tame and wild matrix problems, in: Representation Theory II, Lec- ture Notes in Math. 832, Springer, Berlin, 1980, 240–258.

[6] R. M. F o s s u m, P. A. G r i f f i t h and I. R e i t e n, Trivial Extensions of Abelian Cat- egories, Lecture Notes in Math. 456, Springer, Berlin, 1975.

[7] D. H a p p e l and C. M. R i n g e l, Tilted algebras, Trans. Amer. Math. Soc. 274 (2) (1982), 399–443.

[8] D. H u g h e s and J. W a s c h b ¨ u s c h, Trivial extensions of tilted algebras, Proc. Lon- don Math. Soc. (3) 46 (1983), 347–364.

[9] R. M a r t´ın e z - V i l l a, Properties that are left invariant under stable equivalence, Comm. Algebra 18 (12) (1990), 4141–4169.

[10] J. N e h r i n g, Trywialne rozszerzenia wielomianowego wzrostu [Trivial extensions of polynomial growth], Ph.D. thesis, Nicholas Copernicus Univ., 1989 (in Polish).

[11] —, Polynomial growth trivial extensions of non-simply connected algebras, Bull.

Polish Acad. Sci. Math. 36 (1988), 441–445.

[12] J. N e h r i n g and A. S k o w r o ´ n s k i, Polynomial growth trivial extensions of simply connected algebras, Fund. Math. 132 (1989), 117–134.

[13] J. A. d e l a P e ˜ n a, Functors preserving tameness, ibid. 137 (1991), 177–185.

[14] —, Constructible functors and the notion of tameness, Comm. Algebra, to appear.

[15] C. M. R i n g e l, Tame Algebras and Integral Quadratic Forms, Lecture Notes in Math. 1099, Springer, Berlin, 1984.

[16] H. T a c h i k a w a, Selfinjective algebras and tilting theory, in: Representation Theory I. Finite Dimensional Algebras, Lectures Notes in Math. 1177, Springer, Berlin, 1986, 272–307.

[17] H. T a c h i k a w a and T. W a k a m a t s u, Tilting functors and stable equivalences for selfinjective algebras, J. Algebra 109 (1987), 138–165.

D´ EPARTEMENT DE MATH´ EMATIQUES INSTITUTO DE MATEM ´ ATICAS

ET INFORMATIQUE UNIVERSIDAD NACIONAL

UNIVERSIT´ E DE SHERBROOKE AUT ´ ONOMA DE M´ EXICO

SHERBROOKE, QU´ EBEC M´ EXICO 04510, D.F.

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E-mail: IBRAHIM.ASSEM@DMI.USHERB.CA E-mail: JAP@PENELOPE.MATEM.UNAM.MX

Received 27 February 1995;

in revised form 21 September 1995