The Popescu-Gabriel theorem for triangulated categories

The Popescu-Gabriel theorem states that each Grothendieck abelian category is a localization of a module category. In this paper, we prove an analogue where Grothendieck abelian categories are replaced by triangulated categories which are well generated (in the sense of Neeman) and algebraic (in the sense of Keller). The role of module categories is played by derived categories of small differential graded categories. An analogous result for topological triangulated categories has recently been obtained by A. Heider.

💡 Research Summary

The paper establishes a triangulated‑category analogue of the classical Popescu‑Gabriel theorem, which asserts that every Grothendieck abelian category can be realized as a localization of a module category. In the triangulated setting the authors replace Grothendieck abelian categories with well‑generated triangulated categories in the sense of Neeman and require that these categories be algebraic in the sense of Keller—that is, they are equivalent to the derived category of a small differential graded (dg) category. The main result shows that any such well‑generated algebraic triangulated category 𝒞 can be obtained as a smashing localization of the derived category D(A) of a suitable small dg‑category A.

The proof proceeds in two principal stages. First, using Neeman’s theory of well‑generated categories, the authors exhibit a set of compact generators G for 𝒞. The compact objects generate a thick subcategory which can be identified with the perfect derived category of a small dg‑category A built from the endomorphism dg‑algebra of G (more precisely, A = End(G)^{op}). Because 𝒞 is algebraic, it is already equivalent to the derived category of some dg‑category; the construction of A shows that one can choose A to be small and pre‑triangulated.

Second, the authors consider the derived category D(A) of this small dg‑category. D(A) is a compactly generated triangulated category whose compact objects are precisely the perfect A‑modules. By forming a smashing localization with respect to a set S of morphisms that become isomorphisms in 𝒞, they obtain a new triangulated category D(A)