Strong generators in tensor triangulated categories

TL;DR

This paper proves that in certain tensor triangulated categories with connected spectra, no proper non-zero thick tensor ideals can be strongly generated, impacting the understanding of their subcategory structures.

Contribution

It establishes the non-existence of proper non-zero strongly generated thick tensor ideals in connected rigid tensor triangulated categories, including categories of perfect complexes over rings.

Findings

No proper non-zero thick tensor ideals are strongly generated in such categories.

Specifically, perfect complexes over rings without non-trivial idempotents lack strongly generated subcategories.

The result applies to categories with connected Balmer spectrum.

Abstract

We show that in an essentially small rigid tensor triangulated category with connected Balmer spectrum there are no proper non-zero thick tensor ideals admitting strong generators. This proves, for instance, that the category of perfect complexes over a commutative ring without non-trivial idempotents has no proper non-zero thick subcategories that are strongly generated.