Some homological properties of the category $\mathcal{O}$

Details Some homological properties of the category $\mathcal{O}$ Some homological properties of the category $\mathcal{O}$

2006-07-24

arxiv.org/abs/math/0607589v1

Pacific J. Math. 232 (2007), no. 2, 313--341.

Mathematics

Representation Theory

30 pages

Scientific paper

In the first part of this paper the projective dimension of the structural modules in the BGG category $\mathcal{O}$ 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 $\mathbf{a}$-function is formulated (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 $\mathbb{Z}^2$-graded and that it has two $\mathbb{Z}$-graded Koszul subalgebras. The dimension of the space $\mathrm{Ext}^1$ 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.

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