The spectrum of a well-generated tensor triangulated category

TL;DR

This paper investigates the structure of localizing tensor ideals in well-generated tensor triangulated categories, revealing how their associated spaces can be refined and extended for larger regular cardinals, with applications to known and new examples.

Contribution

It introduces conditions under which the space of localizing tensor ideals can be obtained by refining the topology of the compact objects' space, extending the understanding to larger regular cardinals.

Findings

The space for a well-generated category can be obtained by refining the topology of the compact subcategory's space.

New spaces are identified for regular cardinals larger than countable infinity.

The framework applies to several known examples, providing a unified perspective.

Abstract

For a tensor triangulated category and any regular cardinal $\alpha$ we study the frame of $\alpha$-localizing tensor ideals and its associated space of points. For a well-generated category and its frame of localizing tensor ideals we provide conditions such that the associated space is obtained by refining the topology of the corresponding space for the triangulated subcategory of $\alpha$-compact objects. This is illustrated by several known examples for $\alpha=\aleph_0$, and new spaces arise for $\alpha>\aleph_0$.