On the tameness of trivial extension algebras

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 TA is a tilting module and B = EndTA, 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 modA the category of finitely generated right A-modules, and by modA the stable module category whose objects are the A-modules, and the set of morphisms from MA to NA is HomA(M,N) = HomA(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 modR and modS 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 = Homk(−, k) denote the usual duality on modA. 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 1991 Mathematics Subject Classification: Primary 16G60.