Tilting and cotilting modules over concealed canonical algebras

TL;DR

This paper investigates infinite dimensional tilting and cotilting modules over concealed canonical algebras, revealing their construction, classification, and relation to noncommutative geometry, with distinctions based on algebra type and slope.

Contribution

It introduces new classification results for tilting modules over concealed canonical algebras, including the distinction between rational and irrational slopes, and links to noncommutative geometric frameworks.

Findings

Tilting modules are constructed via universal localization in the domestic case.

For tubular algebras, tilting modules of rational slope are similar to the domestic case.

Unique tilting modules exist for each irrational slope, up to equivalence.

Abstract

We study infinite dimensional tilting modules over a concealed canonical algebra of domestic or tubular type. In the domestic case, such tilting modules are constructed by using the technique of universal localization, and they can be interpreted in terms of Gabriel localizations of the corresponding category of quasi-coherent sheaves over a noncommutative curve of genus zero. In the tubular case, we have to distinguish between tilting modules of rational and irrational slope. For rational slope the situation is analogous to the domestic case. In contrast, for any irrational slope, there is just one tilting module of that slope up to equivalence. We also provide a dual description of infinite dimensional cotilting modules and a classification result for the indecomposable pure-injective modules.