| Résume||(report on joint work with Henrik Holm) Let R be a k-algebra. Given a cotorsion pair (A,B) in Mod(R), Gillespie's Theorem shows how to construct a model category structure on C(Mod R), the category of chain complexes over Mod(R). There is an associated homotopy category H.
If (A,B) is the trivial cotorsion pair (projective modules, everything), then H is the derived category D(Mod R). Several other important triangulated categories can also be obtained from the construction.
Chain complexes over R are the Mod(R)-valued representations of a certain quiver with relations: Linearly oriented A double infinity modulo the composition of any two consecutive arrows. We show that Gillespie's Theorem generalises to arbitrary self-injective quivers with relations, providing us with many new model category structures.|