Let A be a finite-dimensional algebra over an algebraically closed field k. In order to study the category mod A of finitely generated left A-modules, we may assume that A is basic and connected. By [8], there is a finite quiver (i.e. oriented graph) Q A and a surjective homomorphism of algebras ν : kQ A → A such that I ν = ker ν ⊂ (kQ + A ) 2 , where kQ A denotes the path algebra associated to Q A and kQ + A denotes the ideal of kQ A generated by all arrows in Q A . For each pair (Q A , I ν ), called a presentation of A, we can define the fundamental group π 1 (Q A , I ν ) (see [9,13] or Section 1.2 below). Then there is surjective homomorphism of algebras F : A ν → A defined by the action of π 1 (Q A , I ν ) on A ν , called the universal Galois covering of A with respect to ν. As in [8], we shall consider algebras as locally finite k-categories.
Galois coverings have proved to be a powerful tool for the study of the module category mod A. Indeed, A is representation-finite if and only if A ν is locally support-finite and locally representation-finite; in this case, if char k = 2, then A ν = k Q/ Ĩ where Q is a quiver without oriented cycles [4,9,14]. If A is tame, then A is tame [6] but the converse does not hold [11]. Certain A-modules may be described via the push-down functor F λ : mod A ν → mod A, see [5].
In this paper we shall consider only triangular algebras, that is, algebras A = kQ A /I ν such that Q A has no oriented cycles. In this case, a vertex x in Q A is said to be separating if for the indecomposable decomposition rad
with t s, of the induced full subquiver Q (x) A of Q A with vertices those y which are not predecessors