# Telescope conjecture for homotopically smashing t-structures over commutative noetherian rings

## TL;DR

This paper proves that homotopically smashing t-structures over commutative noetherian rings are always compactly generated, extending Neeman's telescope conjecture, and explores implications for pure-injective cosilting objects and derivators.

## Contribution

It establishes the compact generation of homotopically smashing t-structures over commutative noetherian rings, generalizing the telescope conjecture in this context.

## Key findings

- Homotopically smashing t-structures are compactly generated.
- Extension of the telescope conjecture to commutative noetherian rings.
- Cofinite type results for pure-injective cosilting objects.

## Abstract

We show that any homotopically smashing t-structure in the derived category of a commutative noetherian ring is compactly generated. This generalizes the validity of the telescope conjecture for commutative noetherian rings due to Neeman. As another consequence, we obtain a cofinite type result for pure-injective cosilting objects. We also give a formulation of telescope conjecture for homotopically smashing t-structures in underlying triangulated categories of certain Grothendieck derivators.