Odysseas Giatagantzidis (Thessaloniki)

Radical-preserving homomorphisms and the Finitistic Dimension Conjecture

Abstract:

The Finitistic Dimension Conjecture (FDC), which states that the (little) finitistic dimension of any artin algebra is finite, is one of the most important homological conjectures in the representation theory of artin algebras as it implies all other major homological conjectures. Although it has been confirmed for many particular classes of artin algebras, it remains open for over six decades.

One of the most general and efficient reduction techniques for the finiteness of the finitistic dimension are due to Small and Small, Kirkman and Kuzmanovich. Their work from 1968 and 1992 respectively implies that given any epimorphism of artin algebras f: B -> A with kernel inside the (Jacobson) radical of B, the global, little and big finitistic dimension of B is bounded above by the respective dimension of A plus the projective dimension of A as a right B-module. We generalize this result for any homomorphism f: B -> A of artin algebras as long as the kernel of f remains inside the radical of B and the image of the radical of B under f lies inside the radical of A. As a consequence, we recover and extend results from the literature (e.g. arrow removal operation) and open the way for new applications.