Incidence algebras and algebraic fundamental group

One of the main tools for the study of the category of finite dimensional modules over a basic algebra, over an algebraically closed field k is its presentation as quiver and relations. This theory is mainly due to P. Gabriel (see for example [GRo]). More precisely, it has been proved that for all finite dimensional and basic algebras over an algebraically closed field k, there exists a unique quiver Q and an admissible ideal I of the algebra kQ, the path quiver algebra of Q, such that A is isomorphic to kQ/I. Such a couple (Q, I) is called a presentation of A by quiver and relations. For each presentation (Q, I), we can construct an algebraic fundamental group Π1(Q, I). I will present here three results. First, the fundamental group of an incidence algebra has a geometric representation (see [Rey1] or [Bus]). Indeed, the algebraic fundamental group is isomorphic to a topological fundamental group of a simplicial complex. Second, to give a geometric vision of all algebraic fundamental groups, we construct for each presentation (Q, I) an incidence algebra A and show that there is an exact sequence of groups of the following form :