On split quasi-hereditary covers and Ringel duality

TL;DR

This paper introduces new homological invariants to connect Ringel duality, quasi-hereditary covers, and double centraliser properties, providing tools for constructing and analyzing algebra covers and dualities.

Contribution

It develops relative dominant and codominant dimensions as invariants, linking them to quasi-hereditary covers and Ringel duality, with applications to Iwahori-Hecke algebras and category O.

Findings

Relative codominant dimension constructs quasi-hereditary covers.

New proof of Ringel self-duality for category O blocks.

Homological invariants behave well under change of rings.

Abstract

In this paper, we develop two new homological invariants called relative dominant dimension with respect to a module and relative codominant dimension with respect to a module. These are used to establish precise connections between Ringel duality, split quasi-hereditary covers and double centraliser properties. These homological invariants are studied over Noetherian algebras which are finitely generated and projective as a module over the ground ring and they are shown to behave nicely under change of rings techniques. It turns out that relative codominant dimension with respect to a summand of a characteristic tilting module is a useful tool to construct quasi-hereditary covers of Noetherian algebras and measure their quality. In particular, this homological invariant is used to construct split quasi-hereditary covers of quotients of Iwahori-Hecke algebras using Ringel duality of $q$-Schur algebras. Combining techniques of cover theory with relative dominant dimension theory we obtain a new proof for Ringel self-duality of the blocks of the Bernstein-Gelfand-Gelfand category $\mathcal{O}$.