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

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 $\text{E}$, which plays the role of an injective-side analogue of the compact subcategory ${\text{T}}^c$. We introduce the intrinsic subcategory ${\text{T}}_c^{+}$, dual to Neeman’s subcategory ${\text{T}}_c^{-}$, and characterize its objects by strong $\text{E}$-coapproximating systems and homotopy inverse limits. Under the compact silting hypothesis, we prove Brown representability theorems identifying $({\text{T}}_c^{+}{)}^{\text{op}}$ with locally finite $\text{E}$-homological functors and $({\text{T}}_c^b{)}^{\text{op}}$ with finite $\text{E}$-homological functors. We also establish localization results for recollements on the Brown–Comenetz side and derive applications to derived categories of finite-dimensional algebras.