Quiver Grassmannians and Auslander varieties for wild algebras

Abstract

Let k be an algebraically closed field and Λ a finite-dimensional k-algebra. Given a Λ-module M, the set Ge(M) of all submodules of M with dimension vector e is called a quiver Grassmannian. If D,Y are Λ-modules, then we consider Hom(D,Y) as a Γ(D)-module, where Γ(D)=End(D)op, and the Auslander varieties for Λ are the quiver Grassmannians of the form GeHom(D,Y). Quiver Grassmannians, thus also Auslander varieties are projective varieties and it is known that every projective variety occurs in this way. There is a tendency to relate this fact to the wildness of quiver representations and the aim of this note is to clarify these thoughts: We show that for an algebra Λ which is (controlled) wild, any projective variety can be realized as an Auslander variety, but not necessarily as a quiver Grassmannian.