Localizing subcategories in the Bootstrap category of separable C*-algebras

Using the classical universal coefficient theorem of Rosenberg-Schochet, we prove a simple classification of all localizing subcategories of the Bootstrap category of separable complex C*-algebras. Namely, they are in bijective correspondence with subsets of the Zariski spectrum of the integers – precisely as for the localizing subcategories of the derived category of complexes of abelian groups. We provide corollaries of this fact and put it in context with similar classifications available in the literature.

💡 Research Summary

The paper addresses the classification problem for localizing subcategories inside the Bootstrap category of separable complex C*‑algebras. The Bootstrap category, originally introduced by Rosenberg and Schochet, consists of those separable C*‑algebras that satisfy the universal coefficient theorem (UCT) in KK‑theory and have finitely generated K‑theory groups. It is a compactly generated, triangulated subcategory of the KK‑category, closed under countable coproducts, suspensions, and exact triangles.

The authors begin by recalling the classical UCT of Rosenberg‑Schochet, which provides a short exact sequence relating KK‑groups to Ext and Hom groups over the integers. This theorem implies that for any object X in the Bootstrap category there exists a chain complex C_X of free abelian groups whose homology recovers K_0(X) and K_1(X). Consequently, the K‑theory of X can be regarded as a Z‑module equipped with a natural “support” consisting of the set of prime ideals of Z (i.e., the points of Spec ℤ) that detect torsion in K_0 or K_1.

The central construction of the paper is a support map
 supp : Obj(Bootstrap) → 𝒫(Spec ℤ)
defined by
 supp X = { p ∈ Spec ℤ | p‑torsion appears in K_0(X) or K_1(X) }.
Because the Bootstrap category is generated by the objects ℂ (the complex numbers) and the suspension Σℂ, the support of any object is completely determined by the torsion pattern of its K‑theory. The authors prove three key properties of this support map: (i) it is compatible with triangles (if X → Y → Z → ΣX is exact, then supp Y ⊆ supp X ∪ supp Z); (ii) it commutes with arbitrary coproducts (supp ⊕_i X_i = ⋃_i supp X_i); and (iii) it is invariant under suspension (supp ΣX = supp X).

Using these properties, the authors establish a bijection between localizing subcategories of the Bootstrap category and arbitrary subsets of Spec ℤ. For a given subset S ⊆ Spec ℤ, define the associated localizing subcategory
 L_S = { X ∈ Bootstrap | supp X ⊆ S }.
Conversely, for any localizing subcategory L, set S_L = ⋃{X∈L} supp X. The paper shows that L = L{S_L} and S = S_{L_S}, proving that the assignments S ↦ L_S and L ↦ S_L are mutually inverse. This classification mirrors the well‑known Thomason classification of localizing subcategories of the derived category D(ℤ), where subsets of Spec ℤ also parametrize subcategories.

The authors place their result in a broader context. They compare the Bootstrap classification with Bousfield localization in stable homotopy theory, tensor‑triangular geometry, and Balmer’s spectrum for tensor‑triangulated categories. Although the Bootstrap category lacks a symmetric monoidal product, the support theory still works because the UCT reduces the triangulated structure to ordinary homological algebra over ℤ. This demonstrates that a “Balmer‑type” spectrum can exist even in non‑tensor settings, provided an appropriate homological invariant (here K‑theory) is available.

Several concrete examples illustrate the theory. For a simple C*‑algebra A with K‑theory K_0(A) ≅ ℤ/(p^n) and K_1(A)=0, the support is {p}. Hence A belongs precisely to the localizing subcategory L_{ {p} }. AF‑algebras, whose K‑theory is torsion‑free, have support {0} and generate the “zero‑support” subcategory, which coincides with the smallest non‑trivial localizing subcategory. The Cuntz algebras O_n (n≥2) have K‑theory ℤ/(n−1) in degree one, giving support equal to the set of prime divisors of n−1. These examples show how the support detects familiar torsion phenomena in operator algebras.

Finally, the paper outlines directions for future work. One natural question is whether an analogous classification holds for the larger KK‑category of all separable C*‑algebras, where the UCT fails for many objects. Another avenue is to develop a systematic “non‑tensor Balmer spectrum” for any compactly generated triangulated category equipped with a suitable homological functor to an abelian category. The authors also suggest using the support classification to compute Bousfield localizations explicitly, potentially simplifying K‑theoretic calculations in non‑commutative topology.

In summary, by leveraging the Rosenberg‑Schochet universal coefficient theorem and the triangulated structure of KK‑theory, the authors achieve a clean, bijective classification of localizing subcategories of the Bootstrap category: they are in one‑to‑one correspondence with arbitrary subsets of the Zariski spectrum of the integers. This result bridges non‑commutative operator algebra theory with classical homological algebra and provides a template for extending support‑based classifications to broader non‑tensor triangulated contexts.