Spatiality of the frame of smashing tensor ideals of a rigidly-compactly generated tensor-triangulated category

For a rigidly-compactly generated tensor-triangulated category $T$ , the smashing tensor ideals of $T$ form a frame $\mathscr{S}^\otimes(T)$ (Balmer–Krause–Stevenson, 2020). Balchin and Stevenson's Big categories, big spectra' builds a support theory and a smashing spectrum on the hypothesis that this frame is spatial, i.e. isomorphic to the lattice of open subsets of a topological space; however, the originally claimed proof of spatiality in general was retracted in version 4 of that paper, whose appendix exhibits counterexamples to the available methods of proof. The open problem is to decide, for every rigidly-compactly generated tensor-triangulated category $T$, whether the frame $\mathscr{S}^\otimes(T)$ is spatial — equivalently, whether the meet-prime smashing ideals separate any two smashing ideals $S_2 \subsetneq S_1$ — or to exhibit a rigidly-compactly generated counterexample exhibiting genuine pointless' topology. Recent work (2024–2026) confirms the question remains open: non-spatial smashing frames exist outside the rigidly-compactly generated world, the frame is now known to be $\omega_1$ -coherent, and a conditional local-to-global criterion for spatiality is available, but neither a general positive answer nor a rigid counterexample is known; even the case of the derived category $D(R)$ of an arbitrary commutative ring is open.

Background

A tensor-triangulated (tt-)category is a triangulated category carrying a symmetric monoidal tensor product compatible with the triangulated structure (the tensor is exact in each variable). The central examples are 'big' categories such as the derived category $D(R)$ of a commutative ring, the stable homotopy category of spectra, and the stable module category of a finite group; these are typically not small, and their subcategories of compact (i.e. small, finitely presentable) objects play a decisive role. A tt-category $T$ is called rigidly-compactly generated if it is compactly generated and its compact objects $T^c$ are rigid, meaning every compact object admits a tensor dual; this is the standard standing hypothesis for big tt-geometry and holds in all the examples above.

For the essentially small compact part $T^c$ , Balmer's spectrum $\mathrm{Spc}(T^c)$ — the set of prime thick tensor ideals of $T^c$ with a spectral topology — classifies the thick tensor ideals of $T^c$ , and this 'small' theory is well developed. Passing to the big category $T$ , the analogous classification problem concerns localizing tensor ideals (full triangulated subcategories closed under coproducts, tensor, and retracts). A localizing tensor ideal $S \subseteq T$ is called smashing if the corresponding localization functor is given by tensoring with an idempotent object; equivalently, $S$ is the kernel of a smashing localization. The telescope conjecture (TC) for $T$ predicts that every localizing tensor ideal is generated by compact objects, or equivalently that the lattice of localizing tensor ideals is completely determined by the Balmer spectrum of $T^c$ ; it is known to hold in some examples and to fail in others.

Balmer, Krause, and Stevenson (Math. Proc. Camb. Phil. Soc. 168 (2020) 323–343) proved that the poset $\mathscr{S}^\otimes(T)$ of smashing tensor ideals of a rigidly-compactly generated tt-category $T$ , ordered by inclusion, is always a set and forms a frame: a complete lattice in which finite meets distribute over arbitrary joins, the lattice-theoretic abstraction of the lattice of open subsets of a topological space (pointless topology). A frame is called spatial if it is isomorphic to the lattice of open subsets of an actual topological space; equivalently, it has enough points, where a point is a frame morphism onto the two-element frame. For $\mathscr{S}^\otimes(T)$ the points correspond bijectively to the meet-prime smashing ideals (those smashing ideals $P \neq T$ such that $S_1 \wedge S_2 \subseteq P$ implies $S_1 \subseteq P$ or $S_2 \subseteq P$ , where $\wedge$ is the frame-theoretic meet, which need not be intersection). If the frame is spatial, the resulting space $\mathrm{Spc}^s(T)$ is the smashing spectrum, and Balchin–Stevenson show (conditionally on spatiality) that it carries a support theory refining the Balmer spectrum: it comes with a canonical continuous map to the Hochster dual of $\mathrm{Spc}(T^c)$ that is a homeomorphism exactly when TC holds, so the smashing spectrum detects the failure of the telescope conjecture.

An early version of Balchin–Stevenson claimed that $\mathscr{S}^\otimes(T)$ is always spatial; the argument was based on an incorrect identification of the meet of two smashing ideals with their intersection, and version 4 of the paper retracts the claim and devotes an appendix to counterexamples showing that this identification fails (already for $T = D(\mathrm{QCoh}\,\mathbb{P}^1_k)$ ) and that related model-theoretic approaches via the Ziegler spectrum (the set of isomorphism classes of indecomposable pure-injective objects, where pure-injective means injective with respect to pure monomorphisms) stumble because an intersection of exact ideals need not be exact. A simple positive fact is that if TC holds for $T$ then $\mathscr{S}^\otimes(T)$ is spatial; also, every coherent frame (one generated by finitely presented elements under finite meets and arbitrary joins) is spatial. The question is therefore exactly the interface between the telescope conjecture and the lattice theory of smashing ideals: is spatiality, a strictly weaker property than TC, nevertheless valid for every rigidly-compactly generated tt-category?

Problem Statement

Let $T$ be a rigidly-compactly generated tensor-triangulated category and let $\mathscr{S}^\otimes(T)$ be its frame of smashing tensor ideals (Balmer–Krause–Stevenson). Determine whether $\mathscr{S}^\otimes(T)$ is spatial for every such $T$ : equivalently, is it true that for any two smashing ideals $S_1, S_2$ of $T$ with $S_2 \subsetneq S_1$ there exists a meet-prime smashing ideal $P$ of $T$ with $S_2 \subseteq P$ and $S_1 \not\subseteq P$ ? Equivalently, does the smashing spectrum $\mathrm{Spc}^s(T)$ exist as an honest topological space whose lattice of open subsets is $\mathscr{S}^\otimes(T)$ for every rigidly-compactly generated $T$ , or does some rigidly-compactly generated tensor-triangulated category exhibit genuine pointless topology, i.e. a smashing frame with too few (or no separating) points? The special case — is the smashing frame of $D(R)$ spatial for every commutative ring $R$ ? — is open, but the problem as posed concerns the full rigidly-compactly generated class; in particular it is known that spatiality follows from the telescope conjecture and fails in the larger class of presentably symmetric monoidal stable $\infty$ -categories without rigidity.

References

Scott Balchin, Greg Stevenson, `Big categories, big spectra', arXiv:2109.11934 (2021, v4 2023), DOI 10.48550/arXiv.2109.11934, https://arxiv.org/abs/2109.11934
Paul Balmer, Henning Krause, Greg Stevenson, `The frame of smashing tensor-ideals', Math. Proc. Cambridge Philos. Soc. 168 (2020), no. 2, 323–343, DOI 10.1017/s0305004118000725, arXiv:1701.05937, https://arxiv.org/abs/1701.05937
Henning Krause, Janina C. Letz, `The spectrum of a well-generated tensor triangulated category', Bull. Lond. Math. Soc. 55 (2023) 680–705, DOI 10.1112/blms.12749, arXiv:2203.03249, https://arxiv.org/abs/2203.03249
Charalampos Verasdanis, `Stratification and the smashing spectrum', Math. Z. 305 (2023), no. 4, Article 54, DOI 10.1007/s00209-023-03378-4, arXiv:2204.01082, https://arxiv.org/abs/2204.01082
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