Classifying the frame of smashing ideals of D(R) for commutative rings beyond valuation domains

For a commutative ring $R$, the smashing ideals of the unbounded derived category $D(R)$ — the tensor-closed localizing subcategories whose quotient functor admits an exact right adjoint — form a frame $\mathrm{Sm}(D(R))$, and the telescope conjecture for $D(R)$ asks whether the natural inflation map from the frame of thick tensor-ideals of the compact objects $D(R)^c$ (equivalently, by Thomason's classification, the lattice of Thomason subsets of the Zariski spectrum $\mathrm{Spec}\,R$, i.e. the frame of open sets of its Hochster dual) to $\mathrm{Sm}(D(R))$ is an isomorphism. This classification is complete — at the level of a bijection between smashing subcategories and equivalence classes of homological epimorphisms — for commutative rings of weak global dimension at most one (Bazzoni–Šťovíček), and known as an explicit frame only for commutative noetherian rings (Neeman, where the telescope conjecture holds) and for finite dimensional valuation domains (Balchin–Tecklenburg, who also built an infinite family of rings generalising Keller's counterexample where the telescope conjecture fails and where the smashing spectrum stays one-dimensional while the Balmer spectrum has arbitrary Krull dimension). The open problem is to extend this classification to commutative rings beyond these classes — e.g. infinite dimensional valuation domains at the explicit-frame level and rings of higher weak or Krull dimension — and to characterise in general the possible ways $\mathrm{Sm}(D(R))$ can differ from the lattice of Thomason subsets of the Balmer spectrum, i.e. to delimit exactly where the telescope conjecture for derived categories of commutative rings holds and fails.

Background

In tensor-triangular geometry one studies a triangulated category $\mathcal{T}$ equipped with a compatible symmetric monoidal structure. For a commutative ring $R$, the central example is the unbounded derived category $D(R)$ of $R$-modules with its derived tensor product; it is compactly generated, its compact objects $D(R)^c$ being the perfect complexes (bounded complexes of finitely generated projective modules), and the Balmer spectrum $\mathrm{Spc}(D(R)^c)$ — the topological space of prime thick tensor-ideals of the compact category — is canonically homeomorphic to the Zariski spectrum $\mathrm{Spec}\,R$ (Thomason, Neeman). A localizing subcategory $\mathcal{S} \subseteq \mathcal{T}$ is a full triangulated subcategory closed under arbitrary coproducts and, here, under tensoring with $\mathcal{T}$; it is a smashing subcategory (its associated tensor-ideal is a smashing ideal) if the quotient functor $\mathcal{T} \to \mathcal{T}/\mathcal{S}$ admits an exact right adjoint, i.e. if it arises from a smashing localization. Balmer–Krause–Stevenson proved that the collection $\mathrm{Sm}(\mathcal{T})$ of smashing ideals of a rigidly-compactly generated tensor-triangulated category (one where every compact object has a $\otimes$-dual, so the tensor unit is compact) is a set and forms a frame — a lattice with finite meets and arbitrary joins where finite meets distribute over arbitrary joins — and that inflating a thick ideal of compacts to the localizing ideal it generates gives a frame homomorphism $\mathrm{infl}\colon \mathrm{Th}(\mathcal{T}^c) \to \mathrm{Sm}(\mathcal{T})$. The (generalized) telescope conjecture for $\mathcal{T}$, originating in Ravenel's conjecture for the stable homotopy category and studied algebraically as the generalized smashing conjecture, is precisely the statement that $\mathrm{infl}$ is an isomorphism; for $\mathcal{T}=D(R)$ this asks whether every smashing ideal is generated by perfect complexes, so that $\mathrm{Sm}(D(R))$ would be identified with the lattice of Thomason subsets of $\mathrm{Spec}\,R$. Here a Thomason subset is a union of closed subsets of $\mathrm{Spec}\,R$ each of whose complements is quasi-compact (for noetherian $R$ these are exactly the specialization-closed subsets), and by Thomason's classification this lattice of thick tensor-ideals of $D(R)^c$ is equivalently the frame of open sets of the Hochster dual of $\mathrm{Spec}\,R$ — the space with the same points but with the specialization order (and hence the topology) reversed. When the frame $\mathrm{Sm}(\mathcal{T})$ is spatial — meaning it is the frame of open sets of some topological space, which is itself an open question in general (Balchin–Stevenson) — that space is the smashing spectrum $\mathrm{Spcs}(\mathcal{T})$, and the telescope conjecture becomes the statement that the natural comparison map from the smashing spectrum to the (Hochster-dual) Balmer spectrum is a homeomorphism. The known state of the classification for $D(R)$: Neeman proved the telescope conjecture for commutative noetherian rings, so there $\mathrm{Sm}(D(R))$ is the lattice of specialization-closed (equivalently Thomason) subsets of $\mathrm{Spec}\,R$. Keller gave the first counterexample, a non-noetherian valuation domain (an integral domain whose ideals are totally ordered by inclusion), where the telescope conjecture fails. Bazzoni–Šťovíček then handled all rings of weak global dimension at most one (the weak global dimension is the supremum of flat dimensions of all modules; rings of weak global dimension at most one are the weak-dimension analogue of hereditary rings — rings of global dimension at most one — without a noetherian hypothesis, and the class contains all valuation domains of every rank, since over a valuation domain every torsion-free module is flat), showing smashing subcategories correspond to equivalence classes of homological epimorphisms — ring maps $R \to S$ that are epimorphisms with $\mathrm{Tor}^R_i(S,S)=0$ for $i \geq 1$ — classifying smashing localizations for any valuation domain in these terms, and giving a criterion for the telescope conjecture that holds, for instance, for von Neumann regular rings (rings in which every element $a$ satisfies $a \in a^2R$, so that every prime quotient is a field) and, among Prüfer domains (integral domains in which every finitely generated ideal is invertible, equivalently whose localization at every prime ideal is a valuation domain), exactly for the strongly discrete ones (those in which no nonzero prime ideal is idempotent). Balchin–Tecklenburg completed the explicit frame-level picture for finite dimensional valuation domains: for their valuation domains $A^{\star}_n$ of Krull dimension $n$ the number of smashing ideals is a Fibonacci number $\mathrm{Fib}_{2n+3}$ while there are only $n+2$ thick ideals, the smashing spectrum has $2n+1$ points and is always one-dimensional, and for an arbitrary $n$-dimensional valuation domain $R$ the frame $\mathrm{Sm}(D(R))$ is an explicit subframe of $\mathrm{Sm}(D(A^{\star}_n))$. In particular the Balmer spectrum can have arbitrarily large Krull dimension while the smashing spectrum stays one-dimensional, and telescope-conjecture failures are abundant. What remains completely open is everything beyond these classes: an explicit frame-level description of $\mathrm{Sm}(D(R))$ — its order, meet and join structure, cardinality invariants, and smashing spectrum — for rings of weak or Krull dimension $\geq 2$ and for infinite dimensional valuation domains, where the Bazzoni–Šťovíček bijection with homological epimorphisms identifies the smashing localizations but no explicit frame is known, and any general characterization of the possible frames and comparison maps that arise from commutative rings.

Problem Statement

Determine the frame $\mathrm{Sm}(D(R))$ of smashing ideals of the unbounded derived category $D(R)$ of a commutative ring $R$ beyond the classes where it is already known, and characterize in general how this frame can differ from the frame of thick ideals of the perfect complexes. Concretely, the task comprises two independent parts, a classification part (1) and a telescope-conjecture part (2); the target combines a classification satisfying (1) with a delimitation satisfying (2). (1) Give an explicit classification of $\mathrm{Sm}(D(R))$ in terms of ring-theoretic data, namely (1-general) for arbitrary commutative rings of higher weak or Krull dimension, and, as the two named sub-families, (1a) for infinite dimensional valuation domains at the level of explicit frames, i.e. beyond the bijection between smashing subcategories and equivalence classes of homological epimorphisms established by Bazzoni–Šťovíček for arbitrary valuation domains and beyond the explicit frame classification achieved by Balchin–Tecklenburg only for finite dimensional valuation domains, and (1b) for commutative rings of weak global dimension at least two. (2) Delimit the telescope conjecture for derived categories of commutative rings, namely (2a) determine for which commutative rings the inflation frame homomorphism $\mathrm{infl}\colon \mathrm{Th}(D(R)^c) \to \mathrm{Sm}(D(R))$ — from the lattice of Thomason subsets of the Zariski (= Balmer) spectrum of $R$, equivalently the frame of open sets of its Hochster dual — is an isomorphism, (2b) classify the ways it can fail, and (2c) describe the possible smashing spectra $\mathrm{Spcs}(D(R))$ that arise (subject to spatiality, which may itself need to be settled along the way). A complete answer would place the known cases — noetherian rings (Neeman), weak global dimension at most one (Bazzoni–Šťovíček), and finite dimensional valuation domains (Balchin–Tecklenburg) — into one coherent classification covering all commutative rings.

References

1. Scott Balchin and Florian Tecklenburg, "Classifying smashing ideals in derived categories of valuation domains", 2024 (arXiv:2407.11791); published as Journal of Pure and Applied Algebra 229 (2025), no. 3, article 107917, DOI 10.1016/j.jpaa.2025.107917, https://arxiv.org/abs/2407.11791
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