Telescope conjecture, idempotent ideals, and the transfinite radical

TL;DR

This paper explores the telescope conjecture in module categories of artin algebras, linking it to idempotent ideals and proving it for specific algebra classes using properties of Krull-Schmidt categories.

Contribution

It establishes the equivalence between the telescope conjecture and idempotent ideal generation, and proves the conjecture for certain classes of algebras.

Findings

Telescope conjecture is equivalent to idempotent ideals being generated by identities.

Proved the conjecture for domestic standard selfinjective algebras.

Proved the conjecture for domestic special biserial algebras.

Abstract

We show that for an artin algebra $\Lambda$, the telescope conjecture for module categories is equivalent to certain idempotent ideals of mod-$\Lambda$ being generated by identity morphisms. As a consequence, we prove the conjecture for domestic standard selfinjective algebras and domestic special biserial algebras. We achieve this by showing that in any Krull-Schmidt category with local d.c.c. on ideals, any idempotent ideal is generated by identity maps and maps from the transfinite radical.