Separated monic representations I: Gorenstein-projective modules

For a finite acyclic quiver Q, an ideal I of a path algebra kQ generated by monomial relations, and a finite-dimensional k-algebra A, we introduce the separated monic representations of a bound quiver (Q,I) over A. They differ from the (usual) monic representations. The category smon(Q,I,A) of the separated monic representations of (Q,I) over A coincides with the category mon(Q,I,A) of the (usual) monic representations if and only if I=0 and each vertex of Q is the ending vertex of at most one arrow. We give properties of the structural maps of separated monic representations, and prove that smon(Q,I,A) is a resolving subcategory of rep(Q,I,A). We introduce the condition (G). Let Λ:=A⊗kQ/I. By the equivalence rep(Q,I,A)≅Λ-mod of categories, the main result claims that a Λ-module is Gorenstein-projective if and only if it is in smon(Q,I,A) and has a local A-Gorenstein-projective property (G). As consequences, the separated monic Λ-modules are exactly the projective Λ-modules if and only if A is semi-simple; and they are exactly the Gorenstein-projective Λ-modules if and only if A is self-injective, and if and only if smon(Q,I,A) is a Frobenius category.