Quiver Grassmannians and Auslander varieties for wild algebras

TL;DR

This paper explores the relationship between quiver Grassmannians, Auslander varieties, and the wildness of algebras, showing that wild algebras can realize any projective variety as an Auslander variety, but not always as a quiver Grassmannian.

Contribution

It demonstrates that for controlled wild algebras, any projective variety can be represented as an Auslander variety, clarifying the connection to algebra wildness.

Findings

Any projective variety can be realized as an Auslander variety for wild algebras.

Not all projective varieties can be realized as quiver Grassmannians for wild algebras.

The distinction between Auslander varieties and quiver Grassmannians is significant in the context of wild algebras.

Abstract

Let k be an algebraically closed field and A a finite-dimensional k-algebra. Given an A-module M, the set G_e(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 G_e 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.