Let Λ be the path algebra of a finite quiver Q over a finite-dimensional algebra A. Then Λ-modules are identified with representations of Q over A. This yields the notion of monic representations of Q over A. If Q is acyclic, then the Gorenstein-projective Λ-modules can be explicitly determined via the monic representations. As an application, A is self-injective if and only if the Gorenstein-projective Λ-modules are exactly the monic representations of Q over A.Key words and phrases. representations of a quiver over an algebra, monic representations, Gorenstein-projective modules * The corresponding author.

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We generalize the monomorphism category from quiver (with monomial relations) to arbitrary finite dimensional algebras by a homological definition. Given two finite dimension algebras A and B, we use the special monomorphism category Mon(B, A-Gproj) to describe some Gorenstein projective bimodules over the tensor product of A and B. If one of the two algebras is Gorenstein, we give a sufficient and necessary condition for Mon(B, A-Gproj) being the category of all Gorenstein projective bimodules. In addition, if both A and B are Gorenstein, we can describe the category of all Gorenstein projective bimodules via filtration categories. Similarly, in this case, we get the same result for infinitely generated Gorenstein projective bimodules.

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Given a field k, a finite-dimensional k-algebra A, and a finite acyclic quiver Q, let AQ be the path algebra of Q over A. Then the category of representations of Q over A is equivalent to the category of AQ-modules. The main result of this paper explicitly describes the strongly Gorenstein-projective AQ-modules via the separated monic representations with a local strongly Gorenstein-property. As an application, a necessary and sufficient condition is given on when each Gorenstein-projective AQmodule is strongly Gorenstein-projective. As a direct result, for an integer t ≥ 2, let A = k[x]/ x t , each Gorenstein-projective AQ-module is strongly Gorensteinprojective if and only if A = k[x]/ x 2 .