On quiver Grassmannians and orbit closures for representation-finite algebras

TL;DR

This paper introduces a new algebraic construction called the projective quotient algebra and demonstrates its use in desingularizing quiver Grassmannians and orbit closures for representation-finite algebras, generalizing previous results.

Contribution

It constructs the projective quotient algebra for representation-finite algebras and applies it to desingularize geometric objects like quiver Grassmannians and orbit closures.

Findings

Existence of a unique tilting and cotilting module for Auslander algebras.

Construction of desingularizations for quiver Grassmannians and orbit closures.

Generalization of previous desingularization results.

Abstract

We show that Auslander algebras have a unique tilting and cotilting module which is generated and cogenerated by a projective-injective; its endomorphism ring is called the projective quotient algebra. For any representation-finite algebra, we use the projective quotient algebra to construct desingularizations of quiver Grassmannians, orbit closures in representation varieties, and their desingularizations. This generalizes results of Cerulli Irelli, Feigin and Reineke.