Grant: Discovery Grants Program - Individual

Given an artin algebra A, one considers the class of finitely generated A-modules, which we denote by mod A. We can then compute the projective dimension of each such A-module M and we denote this invariant by pd M. The latter is either finite or infinite. We then define the finitistic dimension of A to be: fin.dim. A = sup{pd M : M in mod A and pd M is finite}. In the 60's, Bass conjectured that fin.dim. A is always finite. Since then, this has been shown for very few classes. We recently showed that algebras with at most three radical layers of infinite projective dimension satisfy the conjecture, generalizing a result of Green and Huisgen-Zimmermann dating from 1991. In this research program, we want to study the finitistic dimension from different points of view. We will consider relative homology, stratifying systems and algebras whose simple modules all have infinite projective dimension. While our goal is not to show the conjecture in general, we hope that the results and tools we are developing will help the mathematical community to better master this invariant.