Scalar extensions of triangulated categories

Given a triangulated category over a field $K$ and a field extension $L/K$, we investigate how one can construct a triangulated category over $L$. Our approach produces the derived category of the base change scheme $X_L$ if the category one starts with is the bounded derived category of a smooth projective variety $X$ over $K$ and the field extension is finite and Galois. We also investigate how the dimension of a triangulated category behaves under scalar extensions.

💡 Research Summary

The paper “Scalar extensions of triangulated categories” addresses the problem of extending a triangulated category defined over a base field (K) to a triangulated category over a larger field (L) when a field extension (L/K) is given. The author’s motivation comes from algebraic geometry: for a smooth projective variety (X) over (K) the bounded derived category of coherent sheaves (D^{b}(X)) is a (K)-linear triangulated category, and one would like a canonical construction that produces an (L)-linear triangulated category which, in the geometric situation, coincides with the derived category of the base‑changed variety (X_{L}=X\times_{\operatorname{Spec}K}\operatorname{Spec}L).

The paper proceeds in several stages. First, in Section 2 the author recalls the classical scalar‑extension construction for additive and abelian categories. An object of the base‑changed category (\mathcal C_{L}) is a pair ((C,f)) where (C) is an object of (\mathcal C) and (f\colon L\to\operatorname{End}{\mathcal C}(C)) is a (K)-algebra homomorphism, i.e. an (L)-module structure on (C). Morphisms are required to be compatible with these structures. Lemmas 2.2 and 2.3 show that (\mathcal C{L}) is again additive (resp. abelian) and carries a natural (L)-linear structure. The construction behaves well with respect to functors, equivalences, and Galois descent: if (L/K) is finite Galois with group (G), then (\mathcal C) is equivalent to the category of (G)-equivariant objects in (\mathcal C_{L}).

The main difficulty for triangulated categories is the non‑functoriality of cones, which prevents a naïve transfer of the additive construction. To overcome this, Section 3 introduces differential graded (DG) categories and the notion of a pre‑triangulated DG‑category. In a DG‑category the Hom‑spaces are complexes, composition respects the differential, and cones are functorial. The homotopy category (H^{0}(\mathcal A)) of a pre‑triangulated DG‑category (\mathcal A) is a triangulated category, and any “enhanced” triangulated category can be realized in this way.

Section 4 contains the core construction. Given a (K)-linear pre‑triangulated DG‑category (\mathcal A) whose homotopy category is the triangulated category (T), the author defines a DG‑category (\mathcal A_{L}:=\mathcal A\otimes_{K}L) by tensoring each Hom‑complex with (L). The homotopy category (H^{0}(\mathcal A_{L})) is taken as the scalar extension (T_{L}). However, a direct definition does not automatically give the expected geometric result, so the author refines the construction (Propositions 4.9 and 4.11) by imposing a compatibility condition on the (L)-module structures of objects and morphisms, ensuring that the resulting triangulated category is independent of the chosen DG‑enhancement up to a natural equivalence.

The first main theorem (Main Result 1) states:

1. For any triangulated category (T) that admits a DG‑enhancement, there is a canonical way to produce an (L)-linear triangulated category (T_{L}).
2. If (T\simeq D^{b}(X)) for a smooth projective variety (X) over (K) and (L/K) is finite Galois, then (T_{L}) is equivalent to (D^{b}(X_{L})). Moreover, when (L/K) is merely finite (not necessarily Galois), the same holds for any Noetherian scheme (X).

The paper also discusses an alternative approach using algebraic triangulated categories and derived categories of DG‑categories, which, while more elegant, involves more sophisticated DG‑objects.

Section 5 studies the behavior of Rouquier’s dimension under scalar extension. The second main theorem (Main Result 2) shows that if (\mathcal C) is an abelian category with enough injectives and a set of generators, and (L/K) is a finite Galois extension, then \