Summer School 2001:
Homological conjectures for finite dimensional algebras

I. Ext-algebras, Koszul algebras, and the finitistic dimension conjecture.

I will present connections between understanding the structure of Ext-algebra and the finitistic dimension conjecture. With this in mind I will look at special classes of Ext-algebras and their modules, including Koszul algebras and show that certain subcategories of modules have finite finitistic dimension.

I will present methods of constructing projective resolutions. These will include both some old techniques and some new ones. Underlying the new techniques is the theory of Groebner bases, which will be assumed. I will show how the newer techniques lead to proving a generalization of the fact that the category of Koszul modules over a Koszul algebra has finite finitistic dimension.