For a compactly generated triangulated category, the telescope conjecture asks whether every smashing subcategory is generated by compact objects. Antieau and Stevenson (2016) proved it for the derived category D(RQ) of representations of a simply-laced Dynkin quiver Q over any commutative noetherian ring R, and reduced the question stalk-wise at each prime ideal, but left open — as their Question 6.1 — whether it holds for D(RC) for an arbitrary small category C over R, even under the hypothesis that it holds fiberwise for D(k(p)C) at every prime p. Later work (2021–2026) enlarges the class of positive instances (t-structure versions, arbitrary finite quivers over artinian and perfect base rings, a tensor-compatible variant for finite acyclic quivers, FI-modules over fields). Question 6.1 remains open.

Background

A small category C over a commutative ring R is studied through its category algebra RC, the 'ring with many objects' whose objects are those of C and whose hom-groups are the R-modules R[Hom_C(c,c')] with composition given by the bilinear extension of composition in C; equivalently, one considers the functor category Mod_R C = Fun(C^op, Mod R) of R-valued presheaves on C, a Grothendieck abelian category. Its unbounded derived category D(RC) is a compactly generated triangulated category: it is generated setwise by perfect complexes, i.e. bounded complexes of finitely generated projective RC-modules, and an object is compact (or small) when Hom out of it commutes with arbitrary coproducts. A localizing subcategory is a full triangulated subcategory closed under coproducts; it is smashing when the inclusion admits a right adjoint that preserves coproducts, equivalently when the corresponding localization functor preserves coproducts. The telescope conjecture, originating with Bousfield and Ravenel in stable homotopy theory, predicts that every smashing subcategory of a compactly generated triangulated category is generated by a set of compact objects. It is false in general: Burklund–Hahn–Levy–Schlank (2023) produced counterexamples among spectra, and Keller (1994) exhibited a (noncommutative) ring, built from Wodzicki's homologically unital algebras (non-unital algebras whose bar complex is acyclic, so that they behave in homology as if they had a unit), whose derived category contains a smashing subcategory not generated by compact objects. In algebraic directions the conjecture has a strong positive record: Neeman (1992) proved it for D(R) over any commutative noetherian ring R, showing moreover that smashing subcategories of D(R) correspond bijectively to specialization-closed subsets of the prime spectrum Spec R, and Krause–Šťovíček (2010) proved it for derived categories of hereditary rings, which includes D(kQ) for every finite quiver Q over a field k. Antieau–Stevenson (2016) extended this program to representations of categories: they proved that localizing subcategories of D(RC) are in bijection with sections over Spec R of the bundle of localizing subcategories of the fibers D(k(p)C), where k(p) is the residue field of R at the prime p; they proved the telescope conjecture for D(RQ) when Q is a simply-laced Dynkin quiver (types A, D, E), classifying localizing subcategories via functions from Spec R to the lattice of noncrossing partitions attached to Q (a classical finite combinatorial lattice on the vertices of the Dynkin diagram, known to be in bijection with the localizing subcategories of the fiber D(kQ)); and they proved that for each prime p with R_p regular there is a bijection between smashing subcategories of the stalk subcategory Γ_p D(RC) (the part of D(RC) supported at p) and of D(k(p)C), so that Γ_p D(RC) satisfies the telescope conjecture if and only if D(k(p)C) does. However, they could not globalize: given a smashing subcategory S of D(RC), its stalks Γ_p S are smashing (their Lemma 6.13), but reconstructing S from this pointwise information is the missing step. The natural strategy — show that the section p ↦ (fiber of S at p) is specialization-closed and glue along specializations p ⊆ q — is blocked by their Example 6.14, in which the support of a smashing subcategory of D(k[x,y]), viewed via the action of D(k[x]), is open rather than specialization-closed in Spec k[x], even though the telescope conjecture happens to hold there.

Problem Statement

Let R be a commutative noetherian ring and let C be an ordinary (not R-linear) small category, with D(RC) the derived category of the functor category Mod_R C = Fun(C^op, Mod R) and k(p) the residue field of R at a prime ideal p of Spec R. Question (Antieau–Stevenson, Question 6.1): if the telescope conjecture holds for D(k(p)C) for every prime p ∈ Spec R — that is, every smashing subcategory of each fiber D(k(p)C) is generated by compact objects — does the telescope conjecture hold for D(RC) itself, i.e. is every smashing subcategory S ⊆ D(RC) (localizing subcategory whose inclusion admits a coproduct-preserving right adjoint) generated by a set of compact objects of D(RC), equivalently by perfect complexes over RC? A complete answer resolves this question in general, for every commutative noetherian ring R and every ordinary small category C: an affirmative answer proves that, under the fiberwise hypothesis, every smashing subcategory of D(RC) is generated by compact objects, while a negative answer exhibits a commutative noetherian ring R, a small category C with D(k(p)C) satisfying the telescope conjecture for all p, and a smashing subcategory of D(RC) not generated by compact objects. In particular the answer decides the unconditional question for a finite quiver Q viewed as a small category over a noetherian ring of positive Krull dimension, since in that case the fiberwise hypothesis is known to hold by Krause–Šťovíček.

References

  1. Benjamin Antieau and Greg Stevenson, "Derived categories of representations of small categories over commutative noetherian rings", Pacific Journal of Mathematics 283 (2016), no. 1, 21–42. DOI: 10.2140/pjm.2016.283.21; arXiv:1507.00456; https://arxiv.org/abs/1507.00456
  2. A. Neeman (with an appendix by M. Bökstedt), "The chromatic tower for D(R)", Topology 31 (1992), no. 3, 519–532. DOI: 10.1016/0040-9383(92)90047-L; https://doi.org/10.1016/0040-9383(92)90047-L
  3. Henning Krause and Jan Šťovíček, "The telescope conjecture for hereditary rings via Ext-orthogonal pairs", Advances in Mathematics 225 (2010), no. 5, 2341–2364. DOI: 10.1016/j.aim.2010.04.027; https://doi.org/10.1016/j.aim.2010.04.027
  4. Bernhard Keller, "A remark on the generalized smashing conjecture", Manuscripta Mathematica 84 (1994), no. 1, 193–198. DOI: 10.1007/BF02567453; https://doi.org/1007/BF02567453
  5. Robert Burklund, Jeremy Hahn, Ishan Levy, and Tomer M. Schlank, "K-theoretic counterexamples to Ravenel's telescope conjecture", arXiv:2310.17459 (2023); https://arxiv.org/abs/2310.17459
  6. Michal Hrbek and Tsutomu Nakamura, "Telescope conjecture for homotopically smashing t-structures over commutative noetherian rings", Journal of Pure and Applied Algebra 225 (2021), no. 4, 106571. DOI: 10.1016/j.jpaa.2020.106571; https://doi.org/10.1016/j.jpaa.2020.106571
  7. Enrico Sabatini, "Telescope conjecture for t-structures over noetherian path algebras", arXiv:2505.20803 (2025); https://arxiv.org/abs/2505.20803
  8. Michal Hrbek and Enrico Sabatini, "Telescope conjecture for quiver representations over artinian rings", arXiv:2509.14179 (2025, v2 November 2025); https://arxiv.org/abs/2509.14179
  9. Enrico Sabatini, "The Balmer spectrum and tensor telescope conjecture for noetherian path algebras", Journal of Pure and Applied Algebra 230 (2026), no. 8, article 108324. DOI: 10.1016/j.jpaa.2026.108324; arXiv:2511.20204; https://arxiv.org/abs/2511.20204
  10. Peng Xu, "The Telescope Conjecture for Global Representations and FI-modules", arXiv:2607.15586 (2026); https://arxiv.org/abs/2607.15586
  11. Michal Hrbek, "Telescope conjecture via homological residue fields with applications to schemes", Tunisian Journal of Mathematics 7 (2025), no. 1, 199–228. DOI: 10.2140/tunis.2025.7.199; https://doi.org/10.2140/tunis.2025.7.199