Triangulated categories with a compact silting object, Brown-Comenetz duality and Brown representability theorems

TL;DR

This paper develops a dual framework for triangulated categories with compact silting objects using Brown--Comenetz duality, leading to new representability theorems, localization results, and insights into recollements.

Contribution

It introduces an intrinsic non-compact subcategory and characterizes it, extending Neeman's results with a dual perspective and new applications.

Findings

Established a dual framework for triangulated categories with silting objects.

Proved representability theorems for bounded and low-bounded subcategories.

Derived localization results and analyzed restrictions of recollements.

Abstract

In this paper, we establish a dual framework for Neeman's results concerning triangulated categories with compact silting objects by employing Brown--Comenetz duality. This framework introduces an intrinsic non-compact subcategory, provides its characterization, and demonstrates representability theorems for both the low-bounded and bounded subcategories. Additionally, it elucidates how recollements are restricted to short exact sequences on the dual (non-compact) side. Several localization results and specific applications are also derived.