{"id":"64e33b4d-2ee4-4f83-94cb-356c20e5bd02","arxiv_id":"2608.10681","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For any connective E2-ring R with R_Q ≠ 0, the ∞-category of R-linear localizing motives is not compactly generated.","lead":"This paper proves that the category of localizing motives is not compactly generated for any connective ring spectrum whose rationalization is nonzero. The result settles a structural question in algebraic K-theory and extends Efimov's relative theorem to the absolute case.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 2.1 generalizes Mathew's theorem to all dualizable motives on the strength of a single unverified sentence; if smooth-properness is used in any other step, the main contradiction collapses.","rationale":"The central argument is short and each step after Theorem 2.1 is straightforward: the square-zero extension provides the required A[S^1] summand, base change preserves non-compact-generation, and Corollary 2.3's colimit argument is sound assuming triviality of S^1-actions on compact objects. The real soft spot is Theorem 2.1 itself. The paper explicitly acknowledges that Mathew's NMot^ω_A only covers smooth and proper categories and that dualizable motives are not expected to be representable by such categories, yet it asserts without proof that the only use of smooth-properness is a lifting step supplied by Corollary 1.3. Corollary 1.3 only gives filtered-colimit descent for dualizable motives; it does not, by itself, produce a smooth and proper lift. Thus the proof has a gap in the reduction. Since the gap is a missing justification of a cited result's adaptation rather than an identified false step, the appropriate verdict remains CONDITIONAL: the main theorem is plausible and likely true, but the author should expand the proof of Theorem 2.1 or provide a direct proof of the triviality of the S^1-action for all dualizable motives. I agree with the reader's weakest-assumption identification.","tokens_in":3499,"tokens_out":14525,"duration_ms":140965,"concrete_test":"Extract from [Mat20, §3] and the proof of Corollary 4.8 the exact statement and proof of the lifting/spreading-out lemma. For each occurrence of smoothness or properness, determine whether it is used only to construct a lift to a compact connective E∞-algebra R, or also used elsewhere (e.g., to prove HH is dualizable, to construct the trace, or to apply Kaledin's theorem). Then check whether Corollary 1.3 supplies a lift with exactly those properties. A decisive minimal test: take R = S, A = Q and a dualizable Q-linear motive arising via Corollary 1.3 from a dualizable S-linear motive; attempt to represent it by a smooth proper S-linear category. If the representation is impossible or the proof uses smooth-properness at any other point, Theorem 2.1 needs a separate proof.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 2, Theorem 2.1, is the load-bearing step. It asserts that for a connective rational E∞-ring A, every dualizable A-linear motive M has trivial S^1-action on HH(M/A), citing [Mat20, Cor 4.8] plus a claim that Mathew's proof for smooth proper categories extends to all dualizable motives. The extension rests on a single unproved sentence: 'the only place in which Mathew uses that he is dealing with smooth and proper A-linear categories ... is to show that they lift to a smooth and proper R-linear category ... By Corollary 1.3, this is also the case for dualizable motives.' This is not demonstrated. Corollary 1.3 states that (Mot^loc_−)^dbl preserves filtered colimits; it implies that a dualizable motive over a filtered colimit base descends to a dualizable motive over a compact connective base. It does not by itself produce a smooth and proper category: over a compact connective E∞-ring, dualizable motives are not known (and not expected) to all be represented by smooth proper categories, as the paper itself notes. The proof must additionally show that the dualizable R-linear motive obtained from Corollary 1.3 lies in the image of the motive functor on smooth proper R-linear categories, or that Mathew's argument never uses smoothness/properness except in the descent step. Neither is supplied. If Mathew's proof uses smooth-properness to guarantee finiteness of HH, the existence of a duality trace, or to run Kaledin's degeneration argument, then Theorem 2.1 is unsupported and Corollary 2.3 cannot be applied to the square-zero motive.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims that for every connective E2-ring spectrum R with nonzero rationalization R_Q, the infinity-category Mot^loc_R of localizing motives is not compactly generated; in particular, this holds for R = S. The proof combines Efimov's rigidity theorem (compact = dualizable for Mot^loc), a theorem of Mathew on triviality of S^1-actions on Hochschild homology over connective rational E-infinity rings, and a square-zero extension construction producing a motive whose Hochschild homology contains A[S^1] as a summand. A colimit-of-compacts argument then yields a contradiction.","tokens_in":3810,"tokens_out":11424,"duration_ms":121400,"significance":"If the proof is completed, the result is significant: it resolves a natural open question about the absolute localizing motive category and extends Efimov's non-compact-generation result from a relative case to a broad class of bases including the sphere spectrum. The proof is concise and builds on very recent heavy machinery (Efimov's rigidity, RSW25, Stefanich), and it explicitly notes its limitations (nonconnective and p-local bases are not covered). The square-zero-extension trick is elegant, and the paper gives a clear structural reason why compact generation fails. The main weakness is that the load-bearing extension of Mathew's theorem to all dualizable motives is asserted rather than demonstrated.","major_comments":[{"comment":"The theorem is load-bearing, and its proof is not self-contained. The author asserts in a single sentence that Mathew's argument for smooth and proper A-linear categories extends to all dualizable A-linear motives because the only use of smooth/properness is to lift to a smooth and proper R-linear category, and that Corollary 1.3 supplies such a lift. But Corollary 1.3 only produces a dualizable motive over a compact connective base; it does not imply that this dualizable motive lies in the image of the motive functor from smooth and proper R-linear categories. The paper itself concedes that not all dualizable motives are expected to be of this form. The proof must therefore either (i) show that the dualizable R-linear motive obtained from Corollary 1.3 is represented by a smooth and proper R-linear category, or (ii) give a step-by-step check that Mathew's proof never uses smoothness or properness except in the lifting step, e.g., for finiteness of HH, for the existence of a duality trace, or for Kaledin's degeneration argument. Without one of these, the contradiction in Corollary 2.3 has no foundation.","section":"Section 2, Theorem 2.1"},{"comment":"The statement that the internal left adjoint Mot^loc_R -> Mot^loc_A preserves Ind((Mot^loc_-)^omega) requires that the functor preserves compact objects. The paper states rigidity, and hence compact = dualizable, for commutative ring spectra, while R is only an E2-ring. Please clarify how this preservation step is justified for E2 R; if it follows from Efimov's rigidity for arbitrary rigid bases, cite the precise statement, and otherwise prove the preservation of compactness directly.","section":"Section 2, proof of the Theorem (last paragraph)"}],"minor_comments":[{"comment":"The phrase 'This is clear' in the proof of Proposition 1.2 hides a nontrivial point: every C-motivic equivalence should be exhibited as a filtered colimit of base changes of C_i-motivic equivalences. Please give a few more details here, since this proposition feeds directly into Corollary 1.3.","section":"Section 1, Proposition 1.2"},{"comment":"The notation A[S^1] is used without definition. Please specify that it denotes the free A-module on S^1_+ with the free S^1-action, and note explicitly that a retract of a trivial S^1-module is trivial, which is the contradiction.","section":"Section 2, Corollary 2.3"},{"comment":"The sentence 'This \"is\" [Mat20, Corollary 4.8] and the second paragraph of Section 3 in loc. cit..' is grammatically awkward and should be rephrased.","section":"Section 2, Theorem 2.1"},{"comment":"The standalone 'Warning 2.2' could be integrated into the proof of Corollary 2.3, where the distinction between 'trivial' and 'unipotent' actions is actually used.","section":"Section 2, Warning 2.2"},{"comment":"Reference [Ram24b] is dated 2026 while [Ram24a] is dated 2024; please check that the years are consistent with the arXiv postings.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The main theorem is plausible and the overall strategy is attractive, but the manuscript currently depends on a very terse, unverified extension of Mathew's theorem. The editor may want to ensure that the author supplies a full justification before acceptance; the reliance on several very recent and partly unpublished sources (Efimov's papers, RSW25, and the author's own work) makes this particularly important."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a real result that extends Efimov's non-compact generation from Mot^loc_{Q[x]} to Mot^loc_R for any connective E2 ring R with R_Q≠0, in particular R=S. The argument is a clean observation: combine Mathew's theorem (trivial S^1-action on HH of smooth-proper categories) with Efimov's rigidity/duality and a square-zero extension trick. If the key reduction holds, the theorem is proved in four pages.\n\nWhat's new and good: The absolute case was open, and the proof is short because it reuses heavy machinery. The square-zero extension construction is neat, and the extension from E∞ to E2 via π_0(R)_Q is handled correctly. I believe the result is true.\n\nThe main soft spot is exactly the one the stress-test flags. Theorem 2.1 asserts Mathew's theorem holds for all dualizable A-linear motives, and the proof is one sentence claiming Mathew only uses smooth-properness to lift to a compact connective base. Corollary 1.3 does give descent of dualizable motives to a compact base, but the author never shows that Mathew's subsequent argument (Kaledin's degeneration, finiteness of HH, duality trace) runs on dualizable motives as opposed to smooth-proper categories. Since dualizable motives are not expected to be representable by smooth-proper categories, this is a real gap. It may well be fixable—Mathew's methods might only need dualizability—but the paper as written doesn't supply the needed analysis. A serious referee should ask for a fuller proof of Theorem 2.1.\n\nA minor note: the last step 'internal left adjoint preserves Ind(compacts)' is also terse; it's probably fine because base change on rigid categories preserves dualizables (which equal compacts), but it would help to say so.\n\nOtherwise, the citation pattern is responsible and the reliance on Efimov and Mathew is explicit. The paper is honest about what it does and doesn't do (nonconnective and p-primary cases left open).\n\nBottom line: worth serious refereeing. It's a significant within-field result, likely correct, but the current form is conditional on a missing proof detail. I'd suggest asking for that before acceptance.","headline":"Proves Mot^loc is not compactly generated for any connective E2 ring with R_Q≠0, by a short argument from Mathew; the key generalization to dualizable motives needs a filled-in proof.","tokens_in":4333,"tokens_out":4293,"would_cite":true,"duration_ms":45158,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["19D55","18N60"],"pacs":[],"model":"deepseek-v4-flash","headline":"Localizing motives are not compactly generated over any connective E2-ring with nonzero rationalization, including the sphere.","keywords":["localizing motives","compactly generated","∞-categories","algebraic K-theory","Hochschild homology","topological Hochschild homology","rigid categories","E2-ring spectra"],"falsifier":"Exhibit a connective rational $E_\\infty$-ring $A$ and a dualizable $A$-linear motive $M$ whose Hochschild homology has a non-trivial $S^1$-action, for example containing $A[S^1]$ as a retract. Such an example would directly contradict the key input from [Mat20] and, with it, the proof that $\\mathrm{Mot}^{\\mathrm{loc}}_A$ is not compactly generated.","tokens_in":3290,"feed_emoji":"♾️","tokens_out":16150,"duration_ms":123754,"temperature":0.7,"pith_summary":"The paper proves that the $\\infty$-category $\\mathrm{Mot}^{\\mathrm{loc}}_R$ of $R$-linear localizing motives — the universal target for localizing invariants such as algebraic K-theory — is not compactly generated whenever $R$ is a connective $E_2$-ring spectrum whose rationalization is nonzero, in particular for the sphere spectrum. This settles a question left open by the recent rigidity theorem of [Efi25b], which showed $\\mathrm{Mot}^{\\mathrm{loc}}_R$ is dualizable and self-dual but did not decide compact generation. The proof is a short contradiction: compact objects in $\\mathrm{Mot}^{\\mathrm{loc}}_R$ are dualizable by that rigidity theorem, and a theorem of [Mat20] says that dualizable motives over connective rational rings have a trivial $S^1$-action on their Hochschild homology. The paper then constructs a specific motive, coming from the square-zero extension $R \\oplus \\Sigma R$, whose Hochschild homology contains a nontrivial $S^1$-summand $R[S^1]$; such a summand cannot be a retract of a trivial action, so the motive cannot be a filtered colimit of compact objects. The result rules out the convenient possibility that localizing motives over these bases form a compactly generated category, so any structural description must work without compact generators.","feed_headline":"Localizing motives are never compactly generated over rational bases","feed_subtitle":"Even over the sphere spectrum, the infinity-category of localizing motives has no compact generators.","key_machinery":"This proof rests on three load-bearing pieces: (1) the rigidity theorem of [Efi25b], which identifies compact objects in $\\mathrm{Mot}^{\\mathrm{loc}}_R$ with dualizable objects and gives a self-duality; (2) the triviality theorem of [Mat20], which says that over a connective rational $E_\\infty$-ring every dualizable motive has a trivial $S^1$-action on its Hochschild homology, with the extension to all dualizable motives supplied by the filtered-colimit preservation of Corollary 1.3; and (3) the computation in [Hes94] that the square-zero extension $S = R \\oplus \\Sigma R$ has Hochschild homology containing $R[S^1]$ as a summand. The mechanism is to combine (1) and (2) to show every compact object has trivial $S^1$-Hochschild homology, then use (3) to create a single motive whose Hochschild homology has a nontrivial $S^1$-summand, forcing the contradiction that no compact generation can exist.","core_discovery":"The central claim is that for any connective $E_2$-ring spectrum $R$ with $R_{\\mathbb{Q}} \\neq 0$, the $\\infty$-category $\\mathrm{Mot}^{\\mathrm{loc}}_R$ of $R$-linear localizing motives is not compactly generated. The argument assumes, for contradiction, that $\\mathrm{Mot}^{\\mathrm{loc}}_R$ is compactly generated, meaning every object is a filtered colimit of compact objects. By the rigidity theorem of [Efi25b], compact objects in $\\mathrm{Mot}^{\\mathrm{loc}}_R$ are exactly the dualizable objects. By a theorem of [Mat20] — extended from smooth and proper categories to all dualizable motives via Corollary 1.3 — every dualizable motive over a connective rational $E_\\infty$-ring has a trivial $S^1$-action on its Hochschild homology. The paper then constructs a specific motive, the $R$-linear motive of the square-zero extension $S = R \\oplus \\Sigma R$, whose Hochschild homology contains $R[S^1]$ as a retract (a fact supplied by [Hes94]). Because $R[S^1]$ is compact, that retract would have to factor through one of the compact pieces in any colimit decomposition, yielding a retraction of a trivial $S^1$-action onto a nontrivial one. That contradiction establishes the theorem.","pith_inferences":["The paper's closing remark suggests a stronger statement: $\\mathrm{Mot}^{\\mathrm{loc}}_R$ may be non-compactly generated for every nonzero $E_2$-ring spectrum, not just those with nonzero rationalization. A natural next step is to test whether the method can be adapted to nonconnective rings or to rings supported at a single prime $p$, where the rational triviality theorem no longer applies.","If non-compact generation is as universal as suggested, then attempts to develop a 'derived Morita theory' or a presentation of localizing motives via generators and relations will need a different notion of finiteness, perhaps one based on dualizability rather than compactness.","A concrete experimental check of the paper's mechanism: compute the object $A \\otimes_R S$ in $\\mathrm{Mot}^{\\mathrm{loc}}_A$ for a concrete base such as $R = \\mathbb{Q}$. The theorem predicts this object is not compact; verifying this in an explicit model would test the delicate extension of the triviality theorem to all dualizable motives."],"forward_implications":["As a direct consequence, $\\mathrm{Mot}^{\\mathrm{loc}}_S$ over the sphere spectrum is not compactly generated, so it cannot be described as the ind-category of a small subcategory of compact objects.","For every connective $E_2$-ring with nonzero rationalization, relative localizing motive categories fail to be compactly generated, making this the generic behavior rather than a pathology.","The proof gives a concrete criterion: any motive whose Hochschild homology contains a nontrivial $S^1$-summand such as $R[S^1]$ is not in $\\mathrm{Ind}((\\mathrm{Mot}^{\\mathrm{loc}}_R)^\\omega)$, hence cannot be a filtered colimit of compact objects.","The argument shows that compact objects in $\\mathrm{Mot}^{\\mathrm{loc}}_R$ are extremely special (they all have trivial $S^1$-action on Hochschild homology over rational bases), which is a structural constraint on any future description of the category."],"supporting_citations":[{"why":"Introduces the category of localizing motives $\\mathrm{Mot}^{\\mathrm{loc}}$ and its universal property, the object of study.","marker":"[BGT13]"},{"why":"Supplies Theorem 2.1: $S^1$-actions on Hochschild homology are trivial for dualizable motives over connective rational $E_\\infty$-rings.","marker":"[Mat20]"},{"why":"Provides Theorem 1.1, the rigidity theorem identifying compact objects with dualizable objects in $\\mathrm{Mot}^{\\mathrm{loc}}_R$ and giving self-duality.","marker":"[Efi25b]"},{"why":"Gives the computation that the square-zero extension $S = R \\oplus \\Sigma R$ has $\\mathrm{HH}(S/R)$ containing $R[S^1]$ as a summand.","marker":"[Hes94]"},{"why":"Supplies background on dualizable categories and the preservation of filtered colimits used in Corollary 1.3.","marker":"[Ram24a]"}],"fun_headline_variants":["Localizing motives resist compact generation","No compact generators for localizing motives","Compact generation fails for localizing motives","Localizing motives are never compactly generated","Infinity-category of localizing motives has no compact generators"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the triviality theorem of [Mat20], originally proved for smooth and proper categories, really does extend to all dualizable $A$-linear motives, with Corollary 1.3 supplying the only step that needed smooth-and-properness; if that extension fails, the contradiction argument collapses.","fun_headline_variants_meta":{"raw":{"variants":["Localizing motives resist compact generation","No compact generators for localizing motives","Compact generation fails for localizing motives","Localizing motives are never compactly generated","Infinity-category of localizing motives has no compact generators"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000256,"raw_usage":{"total_tokens":1524,"prompt_tokens":845,"completion_tokens":679,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":461,"completion_tokens_details":{"reasoning_tokens":613}},"tokens_in":461,"tokens_out":679,"duration_ms":5813,"temperature":1.0,"reasoning_tokens":613,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T19:30:44.939794+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Exhibit a connective rational $E_\\infty$-ring $A$ and a dualizable $A$-linear motive $M$ whose Hochschild homology has a non-trivial $S^1$-action, for example containing $A[S^1]$ as a retract. Such an example would directly contradict the key input from [Mat20] and, with it, the proof that $\\mathrm{Mot}^{\\mathrm{loc}}_A$ is not compactly generated.","supporting_citations":[],"review_version":1}