Representations of non-splittable extension algebras

Abstract

Let A be an algebra over a field K and D a self-duality D: mod A 2 mod AoP, where AoP is the opposite algebra of A and mod A the category of finitely generated left A-modules. By an extension for short we understand an extension algebra T over A with kernel DA in the sense of Cartan-Eilenberg [3]: 0 -+ DA + T -+p A --+ 0, where p is an algebra epimorphism. In the case where A is hereditary, by means of the Heller function QAXDA, the Auslander-Reiten quiver rAwDA of the trivial extension A K DA is completely determined by rA [lo]. Moreover, for any extension T, Tr is isomorphic to rAwDA. It seems that this fact suggests the existence of a closer connection between some categories of modules over T and over A 1x DA, though mod T is not in general equivalent to mod A K DA. In this paper we are concerned with categorical relations between non-splittable extensions and the trivial extensions, and we shall establish some relation between them for some class of algebras to which hereditary algebras belong. We prove the following, where mod, A denotes the category of finitely generated left A-modules without projective summands.