Gorenstein flat representations of left rooted quivers

Abstract

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 φiX:⊕a:j→iX(j)→X(i) is injective, and the left R-modules X(i) and CokerφiX 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.