Some Techniques of the Finitistic Dimension Conjecture and Their Applications

Tony Guo (Waltham, MA)

Date: January 10, 2025

Abstract: The finitistic dimension conjecture states that for every finite-dimensional algebra, there is a uniform upper bound (called the finitistic dimension) on the finite projective dimension of its finitely generated modules. The conjecture is a sufficient condition for numerous other homological conjectures including the Nakayama conjecture and Gorenstein symmetry conjecture. There have been studies on algebras with special structures and properties, such as algebras with monomial relations, vanishing radical powers, and small representation dimension. Under these constraints, we are able to understand the projective resolutions well or make general statements on their behaviors. In a similar vein, there are lower and upper bounds of the finitistic dimension such as the depth, phi-dimension, delooping level, and derived delooping level.

In this talk, we introduce the derived delooping level as an improvement of the delooping level and discuss its relationship with the phi-dimension. We also apply the new invariant to a special construction of triangular matrix algebras to obtain another sufficient condition for the finitistic dimension conjecture. Throughout the talk, we mention a number of open questions related to the new technique.