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

Abstract

The paper develops a Brown--Comenetz dual framework for Neeman's representability theorems for triangulated categories with a single compact generator (Invent. math., 244:531-616, 2026). Starting from a locally Hom-finite approximable triangulated category, we use the Brown--Comenetz duals of compact objects to construct a triangulated subcategory $\E$, which plays the role of an injective-side analogue of the compact subcategory $\T^c$. We introduce the intrinsic subcategory $\T_c^+$, dual to Neeman's subcategory $\T_c^-$, and characterize its objects by strong $\E$-coapproximating systems and homotopy inverse limits. Under the compact silting hypothesis, we prove Brown representability theorems identifying $(\T_c^+)^{\op}$ with locally finite $\E$-homological functors and $(\T_c^b)^{\op}$ with finite $\E$-homological functors. We also establish localization results for recollements on the Brown--Comenetz side and derive applications to derived categories of finite-dimensional algebras.