## Journal of Mathematics of Kyoto University

- J. Math. Kyoto Univ.
- Volume 45, Number 4 (2005), 711-741.

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

Volodymyr Mazorchuk and Serge Ovsienko

#### 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.

#### Article information

**Source**

J. Math. Kyoto Univ., Volume 45, Number 4 (2005), 711-741.

**Dates**

First available in Project Euclid: 14 August 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.kjm/1250281654

**Digital Object Identifier**

doi:10.1215/kjm/1250281654

**Mathematical Reviews number (MathSciNet)**

MR2226627

**Zentralblatt MATH identifier**

1147.16010

**Subjects**

Primary: 16E30: Homological functors on modules (Tor, Ext, etc.)

Secondary: 16G20: Representations of quivers and partially ordered sets

#### Citation

Mazorchuk, Volodymyr; Ovsienko, Serge. A pairing in homology and the category of linear complexes of tilting modules for a quasi-hereditary algebra. J. Math. Kyoto Univ. 45 (2005), no. 4, 711--741. doi:10.1215/kjm/1250281654. https://projecteuclid.org/euclid.kjm/1250281654