Big categories, big spectra

TL;DR

This paper introduces a new topological invariant called the big spectrum for tensor-triangulated categories, along with novel support notions, providing insights into the structure of localizing ideals and the telescope conjecture.

Contribution

It develops the concept of the big spectrum and new support theories based on smashing subcategories and localizing ideals, advancing the understanding of tensor-triangulated categories.

Findings

The big spectrum exists with at least one big prime over each Balmer prime.

A morphism from the smashing subcategory frame to the Balmer spectrum detects telescope conjecture failures.

Examples illustrate the application of the new invariants and support theories.

Abstract

We introduce a new topological invariant of a rigidly-compactly generated tensor-triangulated category and two new notions of support. The first is based on smashing subcategories: it is unknown whether the frame of smashing subcategories is spatial in general, but supposing it is as in the examples we understand, we show it is equipped with a morphism to the Balmer spectrum which detects the failure of the telescope conjecture and we develop the corresponding support theory. The new invariant, the big spectrum, results from taking the entire collection of localizing ideals seriously and considering prime localizing ideals. Although there are, in principle, a proper class of localizing ideals, we are able to prove the existence of at least one big prime lying over every Balmer prime. We conclude with a pair of examples illustrating our constructions.