Finitistic and Representation Dimensions

TL;DR

This paper proves several important conjectures in the theory of artin algebras for those that can be realized as endomorphism algebras of modules over representation-finite algebras, linking algebraic properties to their representations.

Contribution

It establishes the validity of key conjectures for a class of artin algebras related to representation-finite algebras and explores conditions involving quasi-hereditary algebras and their extensions.

Findings

Finitistic dimension conjecture holds for certain artin algebras.

Gorenstein Symmetry and Nakayama conjectures are confirmed in this context.

Conditions on quasi-hereditary algebras imply the finitistic dimension conjecture.

Abstract

We prove that the finitistic dimension conjecture, the Gorenstein Symmetry Conjecture, the Wakamatsu-tilting conjecture and the generalized Nakayama conjecture hold for artin algebras which can be realized as endomorphism algebras of modules over representation-finite algebras. Note it is a question whether or not all artin algebras have such realizations. It was also shown that if every quasi-hereditary algebras has a left idealized extension which is a monomial algebra or an algebra whose representation dimension is not more than 3, then the finitistic dimension conjecture holds.