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 generated right

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... 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 equivalent. There are several important problems of the representation theory of algebras which are formulated in terms of the stable equivalence of two selfinjective 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 congenerator 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 , q ) = (aa , aq + qa ) for a, a ∈ A and q, q ∈ A (DA) A . Trivial extensions are a special class of self-injective (actually, of symmetric) algebras. They have been extensively