2013/05/17 by Claus Michael Ringel, Ringel, Claus Michael
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Commutative Algebra and Its Applications #FOS: Mathematics #Representation Theory (math.RT) #math.RT
paper · pdf · doi:10.48550/arxiv.1305.4003
On the basis of vivid feedback, the references to the literature were adjusted and corrected
openalex publication_date 2013/05/17 · arxiv created 2013/05/20 · arxiv updated 2013/05/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let k be an algebraically closed field and A a finite-dimensional k-algebra. Given an A-module M, the set Ge(M) of all submodules of M with dimension vector e is called a quiver Grassmannian. If D,Y are A-modules, then we consider Hom(D,Y) as a B-module, where B is the opposite of the endomorphism ring of D, and the Auslander varieties for A are the quiver Grassmannians of the form Ge Hom(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 A which is (controlled) wild, any projective variety can be realized as an Auslander variety, but not necessarily as a quiver Grassmannian.