Teresa Conde

The Gabriel-Roiter measure and the finiteness of the representation dimension

Abstract:

The induction scheme used in Roiter's proof of the first Brauer-Thrall conjecture prompted Gabriel to introduce an invariant, known as the Gabriel-Roiter measure. The usefulness of the Gabriel-Roiter measure is not limited to the first Brauer-Thrall conjecture: Ringel has used it to give new proofs of results established by himself, Auslander and Tachikawa in the 70's. It turns out that the Gabriel-Roiter measure can also be used to provide an alternative proof of the finiteness of the representation dimension for Artin algebras, a result originally shown by Iyama in 2002. The concept of Gabriel-Roiter measure can be extended to abelian length categories and every such category has multiple Gabriel-Roiter measures. The aim of this talk is to clarify the following refined version of Iyama's theorem: given any object X and any Gabriel-Roiter measure m in an abelian length category, there exists an object X' which depends on X and m, such that the endomorphism ring of the direct sum of X with X' is quasihereditary, and hence has finite global dimension.