Gorenstein flat representations of left rooted quivers

We study Gorenstein flat objects in the category Rep ( Q , R ) of representations of a left rooted quiver Q with values in Mod ( R ) , the category of all left R-modules, where R is an arbitrary associative ring. We show that a representation X in Rep ( Q , R ) is Gorenstein flat if and only if for each vertex i the canonical homomorphism φ i X : ⊕ a : j → i X ( j ) → X ( i ) is injective, and the left R-modules X ( i ) and Coker φ i X are Gorenstein flat. As an application, we obtain a Gorenstein flat model structure on Rep ( Q , R ) in which we give explicit descriptions of the subcategories of trivial, cofibrant and fibrant objects.