On some spectral spaces associated to tensor triangulated categories

TL;DR

This paper studies a specific closure operator on the space of thick submodules in a tensor triangulated category, proving that its fixed points form a spectral space with a monoid structure.

Contribution

It introduces a finite type closure operator on thick submodules and demonstrates that its fixed points form a spectral space with a compatible topological monoid structure.

Findings

The fixed points of the closure operator form a spectral space.

The space of fixed points has a natural topological monoid structure.

The closure operator is of finite type, ensuring certain topological properties.

Abstract

We consider a closure operator $c$ of finite type on the space $SMod(\mathcal M)$ of thick $\mathcal K$-submodules of a triangulated category $\mathcal M$ that is a module over a tensor triangulated category $(\mathcal K,\otimes,1)$. Our purpose is to show that the space $SMod^c(\mathcal M)$ of fixed points of the operator $c$ is a spectral space that also carries the structure of a topological monoid.