Gorenstein Projective Modules And Cluster-tilted Algebras

In1995, Enochs and Jenda [EJ1] introduced the concept of Gorenstein projectivemodules. It was extensively studied by many outstanding experts. Now, Gorensteinprojective modules have become a main ingredient in the relative homological algebra[EJ2] and in the representation theory of algebras (e.g.[AR2, GZ1, IKM, B3]); andplay a central role in the Tate cohomology of algebras (e.g.[AM, Buch]). Explicitlyconstructing all the Gorenstein projective modules is a fundamental problem, but ithas not been fully resolved. Recently, our discussion group work very hard on thisquestion and got some good results. This paper is a part of these results, containingmainly two parts:Firstly, the relationship between the monic A-representations of a quiver Q overan algebra A and Gorenstein projective AQ-modules are studied. As we all know,the path algebra kQ is hereditary. Thus, it is a Gorenstein algebra. But in general,A kkQ is not a Gorenstein algebra. Let Λ be the path algebra of Q over a finitedimensional algebra A. So we have Λ～=A kkQ, and Λ-mod=～Rep(Q, A) the categoryof all the finite dimensional A-representations of Q. Hence, we can study Gorensteinprojective A kkQ-modules via A-representations of Q. This yields the notion ofmonic representations of Q over A. The full subcategory of Rep(Q, A) consisting ofthe monic representations of Q over A is denoted by Mon(Q, A). It is proved thatMon(Q, A) is a resolving and functorially finite subcategory in Rep(Q, A), and hencehas Auslander-Reiten sequences.If Q is an acyclic quiver, we can explicitly describe all the Gorenstein projectiveΛ-modules via the monic representations plus an extra condition. As an application,A is self-injective if and only if the Gorenstein projective Λ-modules are exactly themonic representations of Q over A; if and only if Mon(Q, A) is a Frobenius category.Secondly, we study the constructions and some properties of Gorenstein projectivemodules over cluster tilting algebras of type afne An. By [KR], we know that clustertilting algebras are1-Gorenstein algebras. Thus, Gorenstein projective modules overcluster tilting algebras are exactly the torsionless modules. For cluster tilting algebrasof type afne An, we prove that all indecomposable Gorenstein projective modules arelocal, and nonprojective indecomposable Gorenstein projective modules are uniserialmodules; the number of nonisomorphism indecomposable Gorenstein projective mod- ules only dependents on the number of arrows and3-cycles in the ordinary quiver of acluster tilting algebra, and the relation expression is given.