Telescope conjecture, idempotent ideals, and the transfinite radical

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.

14 pages, a comment on the Krull-Gabriel dimension added to section 4

References in corpus (2)

• The countable Telescope Conjecture for module categories
• Derived categories, resolutions, and Brown representability

Cited by in corpus (5)

• Triangular Matrix Categories I: Dualizing Varieties and generalized one-point extension
• Variations of the telescope conjecture and Bousfield lattices for localized categories of spectra
• Biserial algebras via subalgebras and the path algebra of D_4
• Telescope conjecture via homological residue fields with applications to schemes
• Homological theory of $k$-idempotent ideals in dualizing varieties