Classifying smashing ideals in derived categories of valuation domains

TL;DR

This paper classifies smashing ideals in derived categories of valuation domains, constructs rings where the telescope conjecture fails, and shows the Krull dimensions of different spectra can differ arbitrarily.

Contribution

It provides a complete classification of smashing ideals for derived categories of valuation domains and constructs new examples where the telescope conjecture fails.

Findings

Complete classification of smashing ideals for valuation domains

Construction of rings where the telescope conjecture fails

Demonstration that Krull dimensions of spectra can differ arbitrarily

Abstract

Building on results of Bazzoni-\v{S}\v{t}ov\'{\i}\v{c}ek, we give a complete classification of the frame of smashing ideals for the derived category of a finite dimensional valuation domain. In particular, we give an explicit construction of an infinite family of commutative rings such that the telescope conjecture fails and which generalise an example of Keller. As a consequence, we deduce that the Krull dimension of the Balmer spectrum and the Krull dimension of the smashing spectrum can differ arbitrarily for rigidly-compactly generated tensor-triangulated categories.