On the Relation Between Finitistic and Good Filtration Dimensions

Abstract

In this paper, we discuss generalizations of the concepts of good filtration dimension and Weyl filtration dimension, introduced by Friedlander and Parshall for algebraic groups, to properly stratified algebras. We introduce the notion of the finitistic Δ-filtration dimension for such algebras and show that the finitistic dimension for such an algebra is bounded by the sum of the finitistic Δ-filtration dimension and the -filtration dimension. In particular, the finitistic dimension must be finite. We also conjecture that this bound is exact when the algebra has a simple preserving duality. We give several examples of well-known algebras where this is the case, including many of the Schur algebras, and blocks of category 𝒪. We also give an explicit combinatorial formula for the global dimension in this case.

Key Words:

• Properly stratified algebra
• Finistic dimension
• Good filtration dimension
• Tilting modules

Acknowledgments

The research was done under a project supported by the Royal Swedish Academy of Sciences. The first author was also partially supported by the Swedish Research council. The second author thanks the Maths Department at Uppsala for their hospitality during her visit. We thank V. Dlab for information and discussions concerning the reference (Ágoston et al., 2002). We also thank the referee and A. Frisk for many helpful remarks and suggestions, which led to improvements in the paper.

Notes

^#Communicated by Dieter Happel.

†Dedicated to the memory of Sheila Brenner.