Necessity of finite (weak) global dimension assumptions in Theorem 8.6

Determine whether the finiteness assumptions in Theorem 8.6 are necessary; namely, whether, assuming H is left and right faithfully flat over a right coideal subalgebra A with C=H/HA+, one must require either that A has finite weak global dimension or that C has finite global dimension in order for properties (0.1)–(0.4) on projectivity/injectivity of Hopf modules to hold.

Background

Theorem 8.6 shows that under two-sided faithful flatness of H over A, if either A has finite weak global dimension or C has finite global dimension, then all four homological properties (0.1)–(0.4) for the associated Hopf module categories are satisfied.

The authors explicitly state uncertainty about whether these finiteness conditions are required, and provide in Section 9 partial results on vanishing of Ext and Tor as approximations to the desired full conclusion without finiteness hypotheses.