A pairing in homology and the category of linear complexes of tilting modules for a quasi-hereditary algebra

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.