Abstract
We show that there exists a natural non-degenerate pairing of the homomorphism space between two neighbor standard modules over a quasi-hereditary algebra with the first extension space between the corresponding costandard modules. This pairing happens to be a special representative in a general family of pairings involving standard, costandard and tilting modules. In the graded case, under some “Koszul-like” assumptions (which we prove are satisfied for example for the blocks of the category $\mathcal{O}$), we obtain a non-degenerate pairing between certain graded homomorphism and graded extension spaces. This motivates the study of the category of linear complexes of tilting modules for graded quasi-hereditary algebras. We show that this category realizes the module category for the quadratic dual of the Ringel dual of the original algebra. As a corollary we obtain that in many cases Ringel and Koszul dualities commute.
Citation
Volodymyr Mazorchuk. Serge Ovsienko. "A pairing in homology and the category of linear complexes of tilting modules for a quasi-hereditary algebra." J. Math. Kyoto Univ. 45 (4) 711 - 741, 2005. https://doi.org/10.1215/kjm/1250281654
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