ON THE REPRESENTATION TYPE OF CERTAIN TRIVIAL EXTENSIONS

Let A ∼= kQ/I be a basic and connected finite dimension algebra over closed field k. In this note show that in case B = A[M ] is a tame one-point extension of a tame concealed algebra A by an indecomposable module M , then the trivial extension T (B) = B ∝ DB is tame if and only if the module M is regular. 324 C. Novoa and J. A. de la Peña


Introduction.
Throughout this paper, k denotes an algebraically closed field.By algebra A, we mean always a basic, connected and finite dimensional algebra over k ( associative with 1).We denote by modA the category of finitely generated right A-modules, and D b (A) the derived category of bounded complexs over the abelian category modA (see [H]).
The concept of repetitive algebra was introduced by Hughes -Waschbush ( [HW]) in 1983, where their main interest was to obtain the classification of the finite representation self-injective algebras.In section 2 we recall some known facts about repetitive algebra.In this note, we will use the properties of repetitive categories to study the representation type of the trivial extension T (B) = B ∝ DB, where B is a one-point extension of a tame concealed algebra by an indecomposable module.
In section 3 we establish our main theorem on the representation type of the trivial extension T (B).For that purpose, we prove that there exist a strong relation between the trivial extension T (B) and the class of clannish algebras introduced by Crawley-Boevey in [C-B].As a consequence of our main theorem we show that all tree algebra with non-negative Euler form χ A of corank χ A ≤ 2, have trivial extension of tame representation type.
This paper was done during a postdoctoral stay of the first named author at UNAM.Both authors thankfully acknowledge the finantial support of UNAM and CONACyT, México.

Preliminaries.
We recall that a quiver Q = (Q 0 , Q 1 ) is an oriented graph, where Q 0 is the set of vertices and Q 1 is the set of arrows.The ordinary quiver associated to an algebra A will be denote by Q A .The k-algebra A will be called triangular when Q A has no oriented cycles.For each vertex i of Q A , we shall denote by e i the corresponding primitive idempotent of A, and by S i the corresponding simple A-module.We denote P i (respectively I i ) the projective cover (respectively, the injective envelope) of S i .A bound quiver algebra A ∼ = kQ/I will sometimes be considered as a k-category.
Let H be a Krull-Schmidt category.By definition, the quiver Γ(H) of H has as vertices the isomorphism classes [M ] of indecomposable objects M ∈ H, and there are many arrows [M ] −→ [N ] as the dimension of the space of irreducible maps from M to N in H (see V II.1 [ARS]).If H =modA or D b (A), then Γ(H) is a translation quiver (see 2.1 in [R]).The quiver Γ(modA), or Γ A , is called the Auslander-Reiten quiver of A. A translation quiver Γ is called a tube (see V III.4 in [ARS]), if it contains cyclic paths and its topological realization is |Γ| = S 1 × R + 0 (where S 1 is the unit circle and R + 0 is the set of nonnegative real numbers).A k-category A is called A-free whenever it contains no full sub category B ∼ = kQ where the underlying graph of Q is A n , for some n.
For the basic definitions and results of tilting theory, we refer the reader to [A1].Two finite-dimensional k-algebras A and B are called tilting-cotilting equivalent, if there exist a sequence of algebras A = A 0 , A 1 , ..., A m+1 = B and a sequence of modules A i and T i A i is either a tilting or cotilting module (see [A1]).

The one-point extension (respectively, coextension) of an algebra A by an A-module M will be denoted by A[M ] (respectively, [M ]A).
Let A be a triangular algebra and i a sink in Q A .The reflection S + i A (see [HW]), of A is defined as the quotient of the one-point extension A[I i ] by the bilateral ideal generated by e i .Dually, starting with a source j, we define the reflection S − j A. By a polynomial-growth critical algebra, shortly pg-critical algebra (see 3 in [NS]) we mean an alebra A satisfying the following conditions: 1) A or A op is of one of the following form: where C is a representation infinite tilted algebra of type D n with (4 ≤ n), with a complete slice in the preinjective component, and M (respectively, N ) is an indecomposable regular C-module of regular length 2 (respectively, regular length 1) lying in a tube T in Γ C having n − 2 rays, and t + 1 (2 ≤ t) is the number of objects in C [N, t] which are not in C.
2) Every proper convex sub category of A is of polynomial growth.
In particular, we say that the algebra having n − 2 rays.
Proposition 2.1 (1.4 in [P1]).A pg-critical algebra A is derivedequivalent to an algebra given by the following quiver: Figure 1.
With the commutative relations, indicated by dotted edges.
Let A be a finite-dimensional k-algebra, and D = Hom k (−, k) denote the standard duality on modA.The repetitive algebra A (see [HW]) of A is the self-injective, locally finite-dimensional algebra without identity, defined by: where matrices have only finitely many non-zero entries, addition is the usual addition of matrices, and multiplication is induced from the canonical bimodule structure of DA = Hom k (A, k) and the zero map DA ⊗ DA → 0. It was proved in [W] that if T A is a tilting module and B = EndT A , then mod A ∼ = mod B, where mod denote a stable category in the sense of chapter X in [ARS].
The repetitive algebra A was introduced as the Galois covering (see [G]) of the trivial extension T (A) = A ∝ DA of A by its minimal injective cogenerator DA.Let ν the Nakayama automorphism of A and G =< ν >.We consider A as k-category, then we have the Galois cover functor: F : A −→ ( A/G), where each element of A corresponds to an orbit.This functor induces the push-down functor F λ : mod A −→ mod( A/G) and pull-up functor F .: mod( A/G) −→ mod A, and by 2.2 in [HW] we know that T (A) ∼ = A/G.
A k-algebra A is called (ν A )-exhaustive, when the push-down functor F λ : mod A −→ modT (A) associated to the Galois cover functor We say that the k-algebra A is of locally finite support, if for each indecomposable projective module P , the isomorphism class of the indecomposable projective module P is such that the number of indecomposable module M , with Hom A (P, M ) = 0 and Hom A (P , M ) = 0 is finite.
In particular, in [LDS] it is show that: If A is of locally finite support if and only if the gldim A (global dimension) is strong and finite, that is, the complexes of the derived category has bounded length.
In [LDS] it was proved that if a k-algebra A is locally support finite, then A is ν A -exhaustive.Now, the following theorem given by Assem and Skowroński in [AS2], establishes a classification of the repetitive algebra A which are locally support finite.

Representation of T (A).
Let Q be a quiver, Sp be a subset of the loops of Q, and R be a set of relations for Q.We call the element of Sp special loops, the remaining arrows are called ordinary.Let R Sp := {x 2 − x : x ∈ Sp}, and write (R) for the ideal in kQ/(R Sp ) generated b imagen of the element, of R and denote J the ideal of kQ/(R Sp ) generated by the ordinary arrows.
A triple (Q, Sp, R) as above is called clannish (see 2.5 in [C-B]) if the following conditions hold: 1) (R) ⊂ J 2 2) for any vertex of Q at most 2 arrows start, and at most 2 arrows stop; 3) for every ordinary arrow β there is at most one arrow α with αβ / ∈ R, and at most one arrow γ with βγ / ∈ R. We consider now the following lemma.
Lemma 3.1.Let A be a 2-tubular k-algebra.Then the trivial extension T (A) is tame, and the category modT (A) is equivalent to a category modC, where C is clannish.
Proof.Let A be a 2-tubular k-algebra.By lemma 2.1, we have that , where D is given by the quiver in figure 1.
Hence, the ordinary quiver of the trivial extension T (D) is given by: Figure 2. with commutativity relations given by: uβ We considered now, the following clannish algebra C, given by the following quiver: We defined the functor G in the following form: Hence, from the relations of Clannish algebra C, is easy verify that this definition of the functor G, defined a module in modT (D).By the construction these functors F and G we have that F • G = 1 modT (D) and G • F = 1 mod C .But, is known that an algebra Clannish is tame (see [C-B]), then we have that the trivial extension T (D) is tame.Since, a derived equivalence induces a stable equivalence between the trivials extensions (see [HW]), then modT (A) ∼ = modT (D) then by Krause (see [Kr]) we have that the trivial extension T (A) is tame.
Proposition 3.1 ( prop. 2 in [DS]).Let R be a locally bounded k-category, and G be the group of the k-linear automorphism of R acting freely on the objects of R. If R/G is tame, then R also is tame.
Theorem 3.1.Let A be a tame concealed algebra, M be an indecomposable module in modA, and assume B = A[M ] is of tame representation.Then the trivial extension T (B) is tame if and only if the module M is regular.
Proof.Assume T (B) is tame.How, T (B) ∼ = B/(ν), then by Proposition 3.1, we have that B = A[M ] is tame.Moreover, by lemma 3 in Ringel [R2] if A[M ] is tame, then the module M is regular or preinjective.
We consider the C algebra, given by the quiver in the following figure: We consider now the algebra D defined by the following quiver: Here the module M is the same as before.Then, by Barot-Lenzing (see [BL]), we have that: ).Using the similar construction as the lemma 3.1 with clannish algebras is not dificult to see that modD is equivalent to a clannish algebra E given by the following quiver: Before to state the next result, we consider A ∼ = kQ/I be an algebra, where Q is a quiver without oriented cycles.Let χ A : Z Q 0 −→ Z and q A : Z Q 0 −→ Z be the quadratic forms defined by: where v = (v 1 , ..., v n ) , r(i, j) = dim k e j (I/(IJ + JI))e i , and J is the ideal generated by arrows of the quiver Q.The quadratic form χ A is called the Euler form of the algebra A, and q A is called the Tits form.Its know that if gldimA ≤ 2, then χ A = q A .
As the consequence of our theorem 3.1, we have the following corollary.
Corollary 3.1.Let A ∼ = kQ/I, where Q is a tree, such that Euler form χ A is non-negative and corankχ A ≤ 2. Then the tivial extension T (A) is tame.
Proof.By Barot-De la Peña (see [BP]) we have that the algebra A is domestic tubular, tubular or 2-tubular.Therefore, by the above theorem we have that the trivial extension T (A) is tame.

Figure 4 .
Figure 4.where the dotted lines is zero relation.We have that C := A p−1q[M ]   where the module M is defined by: 1 0 ... M := 0 0 0 ... 0 that is lies in the mouth the tube of rank Figure 5.