Homological theory of representations having pure acyclic injective resolutions

Let $Q$ be a quiver and $R$ an associative ring. A representation by $R$-modules of $Q$ is called strongly fp-injective if it admits a pure acyclic injective resolution in the category of representations. It is shown that such representations possess many nice properties. We characterize strongly fp-injective representations under some mild assumptions, which is closely related to strongly fp-injective $R$-modules. Subsequently, we use such representations to define relative Gorenstein injective representations, called Gorenstein strongly fp-injective representations, and give an explicit characterization of the Gorenstein strongly fp-injective representations of right rooted quivers. As an application, a model structure in the category of representations is given.