SOME HOPF-ALGEBRA INVARIANTS AND NEW RESULTS ON THEIR RELATIONS WITH EACH OTHER

In 1987, two very interesting invariants, called silp (supremum over the injective dimension of projectives) and spli (supremum over the projective dimension of injectives) were introduced for general rings by Gedrich and Gruenberg, and a simple-looking relationship between them was proved for the case where the ring was a K-projective Hopf K-algebra with K a commutative noetherian ring of finite self-injective dimension. Ever since the original Gedrich and Gruenberg paper, the question of to what extent this relationship between silp and spli can be generalized has remained interesting, mainly for the Noetherian condition on K. It is still an open question as to whether the result holds in general without requiring K to be Noetherian. In my talk, I will first start with the general usefulness of these invariants and then highlight my contributions to removing the Noetherian hypothesis for group algebras and strongly graded group algebras. Finally, I will show how the simple relation between silp and spli is connected to stratification questions for stable categories associated with the finite-dimensional modules over several classes of Hopf algebras.