Let A be a finite-dimensional algebra over an algebraically closed field k. In order to study the category modA 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) QA and a surjective homomorphism of algebras ν : kQA → A such that Iν = kerν ⊂ (kQA)2, where kQA denotes the path algebra associated to QA and kQ + A denotes the ideal of kQA generated by all arrows in QA. For each pair (QA, Iν), called a presentation of A, we can define the fundamental group π1(QA, Iν) (see [9,13] or Section 1.2 below). Then there is surjective homomorphism of algebras F : Aν →A defined by the action of π1(QA, Iν) on Aν , 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 modA. Indeed, A is representation-finite if and only if Aν is locally support-finite and locally representation-finite; in this case, if chark = 2, then Aν = kQ/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 Fλ : mod Aν → modA, see [5]. In this paper we shall consider only triangular algebras, that is, algebras A= kQA/Iν such that QA has no oriented cycles. In this case, a vertex x in QA is said to be separating if for the indecomposable decomposition radPx = M1 ⊕ · · · ⊕ Ms , there is a decomposition into connected components Q A = Q1 ∐ . . . ∐ Qt , with t s, of the induced full subquiver Q A of QA with vertices those y which are not predecessors