TITLE: Minimal semi-injective models in categories of quiver representations.

ABSTRACT: It is a classic result that every module has an minimal injective resolution. More generally, every complex of modules, M, has a minimal semi-injective (or DG-injective) model; that is, there exists a quasi-isomorphism M -> I where I is a minimal semi-injective complex. The existence of such models can be found in works of Avramov-Foxby-Halperin (unpublished) and Garcia Rozas (1999). A (chain) complex can be viewed as a representation of the quiver ... -> 1 -> 0 -> -1 -> ... with the relations that any two consecutive arrows compose to zero. It is possible to extend the notions of "quasi-isomorphism", "minimality", and "semi-injectivity" to representations of (certain) other quivers with relations; for example, to differential modules, which are representations of the Jordan quiver with the relation that the square of the loop is zero. We will explain these notions and prove the existence of minimal semi-injective models in this more general context. The talk is based on joint work (in progress) with Peter Jørgensen from Aarhus University.