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- W2022207421 abstract "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." @default.
- W2022207421 created "2016-06-24" @default.
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- W2022207421 date "2006-07-24" @default.
- W2022207421 modified "2023-09-27" @default.
- W2022207421 title "Some homological properties of the category $mathcal{O}$" @default.
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