Some Homological Properties of Category

Abstract We compute projective dimension of translated simple modules in the regular block Bernstein–Gelfand–Gelfand category $\mathcal{O}$ terms Kazhdan–Lusztig combinatorics. This allows us to determine which projectives can appear at last step a minimal resolution for module, confirming conjecture by Johan Kåhrström. also derive some inequalities, Lusztig’s $\textbf{a}$-function, possible degrees top (or socle) module live. Finally, we prove that Kostant’s problem is equivalent homological decomposing $\mathcal O$. gives conjectural answer basis and addresses yet another