The telescope conjecture for hereditary rings via Ext-orthogonal pairs

TL;DR

This paper investigates Ext-orthogonal pairs in module categories over hereditary rings, establishing their properties, and proving the telescope conjecture for derived categories in this context, while providing counterexamples for higher global dimension rings.

Contribution

It characterizes Ext-orthogonal pairs generated by a single module and proves the telescope conjecture for hereditary rings, linking homological epimorphisms and universal localizations.

Findings

Homological epimorphisms and universal localizations coincide for hereditary rings.

The telescope conjecture holds for the derived category of hereditary rings.

Counterexamples show the conjecture fails for rings with global dimension 2.

Abstract

For the module category of a hereditary ring, the Ext-orthogonal pairs of subcategories are studied. For each Ext-orthogonal pair that is generated by a single module, a 5-term exact sequence is constructed. The pairs of finite type are characterized and two consequences for the class of hereditary rings are established: homological epimorphisms and universal localizations coincide, and the telescope conjecture for the derived category holds true. However, we present examples showing that neither of these two statements is true in general for rings of global dimension 2.