A strong finiteness condition for smashing localisations

Isabel Longbottom

REVIEW 2 major objections 3 minor 17 references

A strong finiteness condition for smashing localisations

T0 review · 2 major / 3 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read Every finite localisation of p-local spectra is compactly central, and over all spectra the compactly central localisations are exactly the finite ones touching only finitely many primes.

desk verdict Genuinely new framework and a solid proof that L^f_n is compactly central; the Sp classification rests on an unjustified and false modified fracture square. read the letter →

arxiv 2509.07344 v1 pith:27RY6XPP submitted 2025-09-09 math.AT

Isabel Longbottom This is my paper · ORCID

classification math.AT MSC 55P4255P6055N20

keywords smashinglocalisationcompactlycentralfiniteL^f_nmapMoravaK-theoryp-localspectrachromatic

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper tries to prove that a strong finiteness condition — compact centrality — captures the natural finite chromatic localisations of spectra. Its main theorem is that each L^f_n, the finite localisation of p-local spectra whose kernel is generated by the type ≥ n+1 finite spectra, is compactly central: it arises from a single central map S_(p) → A with A finite. Since the thick subcategory theorem identifies the finite localisations of p-local spectra with the L^f_n, every finite localisation of Sp_(p) is compactly central. Passing to the full category of spectra, a nonzero finite localisation is compactly central exactly when it is the identity on p-local spectra for all but finitely many primes. The point of the condition is that a compactly central localisation packs the whole smashing localisation into one finite datum — a map out of the sphere whose left- and right-tensorings agree — making the infinite chromatic information explicit and checkable.

What carries the argument

The central object is the central map: a map α: 1 → A whose two composites A → A⊗A — tensoring on the left or on the right — are homotopic. From a central map the paper forms J_α, the free E_1 algebra on the E_0-algebra α; centrality makes J_α an idempotent algebra (Corollary 2.27), hence a smashing localisation L_α ≃ J_α ⊗ (−), and J_α is the sequential colimit of the tensor powers A^{⊗r} (Proposition 2.36). For maps between finite p-local spectra, centrality is detected by Morava K-theory: each K(n)_*α is zero or an isomorphism. Maps with exactly those K(n)-conditions are called algebraically central, and Theorem 3.12 bridges the gap: a large p-power tensor power of an algebraically centra

What would settle it

Compute, for the finite localisation F of Sp with parameters n_p = 1 at every prime p, the homotopy pullback of ∏_p L_(p)FS and HQ over L_Q(∏_p L_(p)FS), and compare it with FS: if the natural map FS → that pullback is not an equivalence, the fracture square of Lemma 3.17 fails, Corollary 3.20's uniqueness collapses, and Theorem B is refuted. For Theorem A, a direct check is available: compute K(n)-homology of the generalised Smith–Toda map α_m from Section 3.4 and verify the centrality homotopy for α_m^{⊗p^N} at the explicit exponent N forced by Corollary 3.30.

Watch

Extended reading notes

Core claim

The central discovery: the smashing localisation L^f_n of p-local spectra — kernel generated by the finite spectra of type at least n+1 — is compactly central: it is induced by a central map β: S_(p) → A with A finite, via the free E_1 algebra J_β, which is idempotent and gives the localisation by tensoring. The route runs through algebraically central maps between finite spectra — K(n)-homology an isomorphism below height n and zero from n up — and the proof that a large p-power tensor power of an algebraically central map is central (Theorem 3.12), using K(n)-nilpotence detection and a binomial-coefficient valuation bound (Corollary 3.30). Explicit algebraically central maps for every heig

Load-bearing premise

Lemma 3.17 assumes, without proof or reference, that the idempotent algebra of any finite localisation of spectra is recovered as the homotopy pullback of its p-localisations and its rationalisation — a 'slightly modified' arithmetic fracture square with p-localisations in place of p-completions. The square is not valid for arbitrary idempotent algebras, and the uniqueness half of the global classification (Corollary 3.20), and hence Theorem B, rests on it.

Editorial extensions

If this is right

Every finite localisation of p-local spectra — L^f_n for −1 ≤ n ≤ ∞ — is compactly central, so each is presented by a single central map S_(p) → A with A finite (Theorem A).
On the whole stable category, a nonzero finite localisation is compactly central if and only if it is nontrivial at only finitely many primes; equivalently, its parameters n_p are infinite except on a finite set (Theorem B).
The localisation produced by a compact central map is read off from the type of its cofibre: type m gives L^f_{m−1}, type 0 gives the zero localisation, and type ∞ the identity (Theorem 3.7); finite composites of compactly central localisations remain compactly central (Lemma 2.66).
Finiteness and compact centrality coincide on Sp_(p) but not on Sp: p-localisation and rationalisation are finite yet not compactly central, and the disproof of the telescope conjecture shows smashing does not imply finiteness (Corollary 2.52).
Compactly central localisations of Sp with prescribed p-local behaviour can be assembled independently at each prime and combined by tensor product, giving an explicit construction for every finite-localisation parameter family supported on finitely many primes (Lemma 3.22 and Theorem 3.23).

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The global half of the classification rises or falls with the modified arithmetic fracture square of Lemma 3.17; if that square fails for some finite localisation, Theorem B's uniqueness statement would need repair while the p-local Theorem A would survive intact.
Compact centrality offers a practical obstruction test for smashing localisations: a candidate localisation that admits no map from the sphere into a finite spectrum with the required K(n)-homology profile cannot be compactly central, which gives a concrete way to probe the gap between L_n and L^f_n.
The same free-algebra machine — central maps into compact objects plus a nilpotence-detecting family of homology theories — should transfer to any presentably symmetric monoidal stable ∞-category whose compact objects are dualisable, yielding compactly-central versions of finite localisation classifications beyond spectra.
The proof of Theorem 3.12 encodes a quantitative bound: the exponent N with α^{⊗p^N} central is forced by the nilpotence index and torsion order of ε = x − y via the valuation bound of Corollary 3.30, so the minimal such N is in principle computable from data about the map α itself.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

You should know two things. First, the central notion is worth taking seriously: compactly central localisations are a real finiteness condition, and Theorem A — L^f_n is compactly central — is a new result with a credible proof. The technique of taking a large tensor power of an algebraically central map to force centrality is clever and seems to work; the combinatorial Proposition 3.31 is the sort of thing you can check. Second, the global classification in Theorem B has a load-bearing hole. Lemma 3.17 asserts a 'slightly modified' arithmetic fracture square with p-localisations in place of p-completions. That square is false as stated. For the identity finite localisation, the proposed pullback would make S the fibre product of HQ and ∏ S_(p); the homotopy fibres give (∏ Z_(p))/Z on one side and ((∏ Z_(p))⊗Q)/Q on the other, and these are not isomorphic. So the square is not a pullback. This is the only mechanism offered for uniqueness in Corollary 3.20, which feeds directly into Theorem B. The classification may still be true and recoverable via a different route, e.g. from finite acyclics and the thick subcategory theorem, but as written the Sp-side is unproved. Credit where due: Theorem A is the real content, and the paper is careful about the background. The connection between centrality and Miller finiteness is useful. I did not find circularity; definitions are restrictive and constructions are explicit. The references look right, including the recent telescope-conjecture counterexample. Who is this for? People working on smashing localisations and the telescope conjecture, especially on the p-local side. A serious referee should be sent this, not a desk reject, because Theorem A is likely correct and the framework is likely useful. But the referee must demand a fix for Lemma 3.17 — either a proof or a replacement argument — before the classification of compactly central localisations of Sp can be trusted. I'd cite it once it's fixed; I wouldn't cite Theorem B in its present form.

Referee Report

2 major / 3 minor

Summary. The paper introduces a class of smashing localisations called compactly central, defined via a central map from the unit to a compact object, and studies them in the categories Sp and Sp_(p). The main positive result is Theorem A/3.14: the finite localisation L_n^f of p-local spectra is compactly central, exhibited by constructing algebraically central maps from Smith–Toda complexes and showing that large tensor powers of such maps are central. The paper further claims Theorem B/3.23, a classification of compactly central localisations of Sp as precisely those finite localisations that are the identity on Sp_(p) for all but finitely many primes. The p-local part is developed through K(n)-homology criteria and a computation of the localisation associated to a compact central map. The global passage from p-local data to Sp relies on Lemma 3.17, which asserts a modified arithmetic fracture square with p-localisations.

Significance. If correct, the paper would establish a new and fairly explicit finiteness mechanism for smashing localisations: every finite localisation of Sp_(p) is generated by a central map from the unit to a compact spectrum, and the L_n^f localisation admits a compact central presentation. This is a genuinely strong structural result, and the construction via algebraically central maps and tensor powers is concrete and interesting. The paper also gives an explicit classification of compactly central localisations of Sp, which would be a valuable counterpart to Miller's finite localisations. The author is careful to separate background material and includes useful appendices. However, the global classification depends on an asserted fracture square that is false as stated, so the significance of Theorem B is currently not supported.

major comments (2)

[§3.3, Lemma 3.17] The proof of Lemma 3.17 asserts, without proof or reference, a 'slightly modified' arithmetic fracture square with p-localisations in place of p-completions: FS is claimed to be the homotopy pullback of ∏_p L_(p)FS and HQ over L_Q(∏_p L_(p)FS). This is not a valid pullback in general. For the identity finite localisation F=id, the asserted square becomes S → HQ ← ∏_p S_(p). The homotopy fibre of S → ∏_p S_(p) has π_0 of its cofibre equal to (∏_p Z_(p))/Z, whereas the homotopy fibre of HQ → L_Q(∏_p S_(p)) has π_0 of its cofibre equal to ((∏_p Z_(p))⊗Q)/Q, a Q-vector space. These spectra are not equivalent, so the square is not a homotopy pullback. Since Corollary 3.20 (uniqueness of the parameter family {n_p}) and hence Theorem 3.23 rest directly on this Lemma, the global classification is not proved. The p-local Theorem 3.14 appears unaffected, but the passage from Sp_(p) to Sp needs a c
[§3.3, Corollary 3.20 and Theorem 3.23] The classification of finite localisations of Sp by arbitrary parameter families {n_p} and the subsequent classification of compactly central localisations both depend on Lemma 3.17's uniqueness assertion. Because that Lemma's proof uses the invalid fracture square, the statements 'there is a unique finite localisation F satisfying F|_{Sp_(p)} ≃ L^f_{n_p}' and 'a nonzero finite localisation F is compactly central iff n_p = ∞ for all but finitely many p' are not established by the arguments provided. The author should either prove the needed uniqueness by a correct method (for example, using the thick subcategory theorem directly on finite spectra) or clearly restrict the claims to the p-local setting.

minor comments (3)

[§2.4, Lemma 2.48] The sentence 'If every mapping space to DZ⊗LX is contractible then by Yoneda DZ⊗LX itself is contractible' is imprecise in a stable ∞-category: an object with contractible mapping spaces from all objects is zero, not merely contractible. This is a wording issue and does not affect the argument.
[§3.4, Construction 3.25] The notation '(⋆) Y_m → S_(p) → X_m' is called an exact sequence, but in the stable ∞-category setting it is a fibre/cofibre sequence. The intended meaning is clear, but the terminology could mislead.
[§3.3, Lemma 3.19] The phrase 'after gluing all the F_ps together we still have L_Q F is rationalisation' is vague. Since the preceding uniqueness argument is invalid, this step needs to be made precise independently of Lemma 3.17.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: explicit Smith-Toda constructions drive Theorems A and B; Lemma 3.17's unproved fracture square is a correctness gap, not a circular step.

full rationale

Circularity score 0. The paper's central object, 'compactly central localisation', is defined by existence of a central map with compact domain/cofibre; it is not defined as 'finite localisation' or as 'L^f_n'. Theorems A and B are proved by explicit constructions: Section 3.4 constructs algebraically central maps from generalised Smith-Toda complexes (Proposition 3.24), Theorem 3.12 shows a high tensor power of such a map is genuinely central using K(n)-homology and p-adic binomial-coefficient estimates (Proposition 3.31), and Theorem 3.7 then identifies the resulting smashing localisation as L^f_{m-1} using the external Hopkins-Smith thick subcategory theorem. No parameter is fitted and then renamed a prediction; no central claim is assumed in the definition of the objects it concerns. The paper contains no self-citations by the author; all cited external results (Lurie, Miller, Hopkins-Smith, Ravenel, Bousfield) are used as standard mathematical inputs and are not doing the work of the new constructions. The only serious issue is correctness, not circularity: Lemma 3.17 asserts a 'slightly modified' arithmetic fracture square with p-localisations (quote: 'we will use a slightly modified version of the usual arithmetic fracture square, featuring p-localisations instead of p-completions') without proof, and the reviewer's counterexample (identity localisation) suggests it is false as stated. That gap would undermine the uniqueness part of Corollary 3.20 and hence the proof of Theorem B, but an unsupported or false lemma is not a circular derivation: the conclusion is not contained in the lemma's inputs by construction. Therefore no circular step is identified.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The central claims rest on standard domain theorems of chromatic homotopy theory and Lurie's Higher Algebra. The sole ad hoc assumption is the modified fracture square in Lemma 3.17, which is introduced without proof and appears invalid in general. No free parameters or invented entities appear; the parameters n_p in the classification are the output, not fitted inputs.

assumptions (8)

domain assumption Hopkins-Smith thick subcategory theorem: thick subcategories of p-local finite spectra are exactly type ≥ n for n ≥ 0.
Used in Lemma 3.5, Theorem 3.7 and Section 3.3 to identify localisations L_β with L^f_n and to classify finite localisations.
domain assumption Hopkins-Ravenel: E(n)-localisation of Sp_(p) is smashing.
Background in Section 3.1; used to state L_n ≃ L_{K(≤n)} and to contextualise smashing localisations.
domain assumption Nilpotence detection: the Morava K-theories {K(n)} jointly detect smash nilpotence of maps of finite spectra.
Used in Theorem 3.7 and Corollary 3.10 to conclude nilpotence from vanishing K(n)-homology.
domain assumption Miller's theorem: every finite localisation of Sp is smashing.
Used in Corollary 2.49 and throughout Section 3.3.
standard math Lurie, Higher Algebra Props 4.8.2.7/4.8.2.9: idempotent maps and idempotent algebras are in bijection with smashing localisations.
Foundation of Section 2; used in Lemma 2.3 and Corollary 2.27.
standard math Ohkawa's theorem: there is a set of Bousfield classes.
Used in Theorem 2.65 to show smashification exists.
domain assumption Existence of generalised Smith-Toda complexes of every finite type.
Used in Proposition 3.24 to construct algebraically central maps α_m.
ad hoc to paper The modified arithmetic fracture square with p-localisations (Lemma 3.17) is a homotopy pullback.
Assumed in the proof of Lemma 3.17 without proof or reference; not a standard theorem and likely false in general for idempotent algebras.

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Cite this review

Pith. "Pith review of A strong finiteness condition for smashing localisations." pith.science (2026). https://pith.science/paper/27RY6XPP

@misc{pith2026250907344,
  author       = {Pith},
  title        = {Pith review of: A strong finiteness condition for smashing localisations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/27RY6XPP}},
  note         = {Machine review of arXiv:2509.07344}
}

abstract

We define a class of smashing localisations which we call compactly central, and classify compactly central localisations of $Sp_{(p)}$ and of $Sp$. Our main result is that $L_n^f$ is a compactly central localisation. A map $\alpha: 1 \to A$ in a presentably symmetric monoidal $\infty$-category $\mathscr{C}$ is central if there exists a homotopy $\alpha \otimes id_A \simeq id_A \otimes \alpha: A \to A \otimes A$. A central map $\alpha$ can be used to produce a smashing localisation $L_\alpha$ of $\mathscr{C}$, because the free $\mathbb{E}_1$ algebra on the $\mathbb{E}_0$ algebra $\alpha$ is an idempotent commutative algebra. When both the monoidal unit and $A$ are compact, we call $L_\alpha$ compactly central. We show that when $\mathscr{C}$ is (compactly generated) rigid, all compactly central localisations are finite in the sense of Miller. Not all finite localisations of $Sp$ are compactly central. To exhibit $L_n^f$ as compactly central, we determine properties of the $K(n)$-homology of a map between $p$-local finite spectra which ensure that some tensor power of the map is central.

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Reference graph

Works this paper leans on

17 extracted references · 16 canonical work pages

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Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.

[1] [1]

A short introduction to the telescope and chromatic splitting con- jectures

Tobias Barthel. “A short introduction to the telescope and chromatic splitting con- jectures”. In:Bousfield Classes and Ohkawa’s Theorem: Nagoya, Japan, August 28- 30, 2015. Springer. 2020, pp. 261–273. 31

work page 2015

[2] [2]

The localization of spectra with respect to homology

Aldridge K Bousfield. “The localization of spectra with respect to homology”. In: Topology18.4 (1979), pp. 257–281. 46, 48

work page 1979

[3] [3]

Robert Burklund et al.K-theoretic counterexamples to Ravenel’s telescope conjec- ture. 2023. arXiv:2310.17459 [math.AT].url:https://arxiv.org/abs/2310. 17459. 25, 48

arXiv 2023

[4] [4]

Ohkawa’s theorem: there is a set of Bous- field classes

William Dwyer and John Palmieri. “Ohkawa’s theorem: there is a set of Bous- field classes”. In:Proceedings of the American Mathematical Society129.3 (2001), pp. 881–886. 28

work page 2001

[5] [5]

Nilpotence and Stable Homotopy Theory II

Michael J. Hopkins and Jeffrey H. Smith. “Nilpotence and Stable Homotopy Theory II”. In:Annals of Mathematics148.1 (1998), pp. 1–49.issn: 0003486X.url:http: //www.jstor.org/stable/120991. 2, 32–34

work page 1998

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Cohomological Bousfield classes

Mark Hovey. “Cohomological Bousfield classes”. In:Journal of Pure and Applied Algebra103.1 (1995), pp. 45–59. 47

work page 1995

[7] [7]

The structure of the Bousfield lattice

Mark Hovey and John Palmieri. “The structure of the Bousfield lattice”. In:arXiv preprint(1998).url:https://arxiv.org/abs/math/9801103v1. 48

work page Pith review arXiv 1998

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Mark Hovey and Neil Strickland.MoravaK-theories and localisation. Vol. 666. American Mathematical Society, 1999. 47

work page 1999