Some homological properties of category O. III

We prove that thick category O associated to a semi-simple complex finite dimensional Lie algebra is extension full in the category of all modules. We also prove the weak Alexandru conjecture both for regular blocks of thick category O and the associated categories of Harish–Chandra bimodules, but disprove it for singular blocks.We prove that thick category O associated to a semi-simple complex finite dimensional Lie algebra is extension full in the category of all modules. We also prove the weak Alexandru conjecture both for regular blocks of thick category O and the associated categories of Harish–Chandra bimodules, but disprove it for singular blocks.A