Some homological properties of the category

In the first part of this paper the projective dimension of the structural modules in the BGG category O is studied. This dimension is computed for simple, standard and costandard modules. For tilting and injective modules an explicit conjecture relating the result to Lusztig’s a-function is formu-lated (and proved for type A). The second part deals with the extension algebra of Verma modules. It is shown that this algebra is in a natural way Z2-graded and that it has two Z-graded Koszul subalgebras. The dimension of the space Ext1 into the projective Verma module is determined. In the last part several new classes of Koszul modules and modules, represented by linear complexes of tilting modules, are constructed