Characteristic tilting modules and Ringel duality in the Noetherian world

TL;DR

This paper strengthens the theoretical foundations of Ringel duality for split quasi-hereditary algebras over Noetherian rings, providing new descriptions and properties of certain subcategories and their equivalences.

Contribution

It introduces new descriptions of resolving subcategories and proves that equivalences between these subcategories lift to Morita equivalences preserving quasi-hereditary structures.

Findings

Resolved subcategory descriptions for split quasi-hereditary algebras

Proved lifting of equivalences to Morita equivalences

Established preservation of quasi-hereditary structure under equivalences

Abstract

The foundations of Ringel duality for split quasi-hereditary algebras over commutative Noetherian rings are strengthened. Several descriptions and properties of the smallest resolving subcategory containing all standard modules over split quasi-hereditary algebras over commutative Noetherian rings are provided. In particular, given two split quasi-hereditary algebras $A$ and $B$, we prove that any exact equivalence between the smallest resolving subcategory containing all standard modules over $A$ and the smallest resolving subcategory containing all standard modules over $B$ lifts to a Morita equivalence between $A$ and $B$ which preserves the quasi-hereditary structure.