{"id":"fa43e77d-ff52-4679-b686-34bbc0cf9059","arxiv_id":"2607.07137","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A general descendability criterion for topological six-functor formalisms yields h-descent for étale motivic spectra and rational motivic cohomology, plus arc-descent in weights ≤1.","lead":"Motivic cohomology theories are shown to satisfy strong 'descent' gluing conditions, meaning computations on a space can be assembled from computations on covers. The paper proves a general criterion and applies it to rational and étale motivic spectra, including new arc-descent results and an isomorphism between two pushforward functors for stacks.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified; residual risk is the external p-completed rigidity input in §4.4.","rationale":"I followed the main proof chain in good faith. Theorem C is internally coherent: the localization argument, the stratification by finite étale/radicial covers, and the algebraic-space dévissage all appear to work. The applications in Sections 3–5 depend on Theorem C plus known properties of SHQ,+ and SH_et; I found no circularity or missing step in the paper's own arguments. The previously acknowledged issue in [Kha3] is addressed by Theorem 5.1. The only truly load-bearing external premise is Bachmann's p-completed rigidity equivalence, which the reader also identified as the weakest assumption. I agree with that identification: if the equivalence is not available in the required symmetric monoidal/six-functor form for algebraic spaces, then Theorem B and its consequences do not follow. However, there is no evidence that the cited theorem fails in this setting, and the paper's internal reasoning provides no concrete error. The verdict therefore remains ACCEPT, with the same moderate confidence as the reader.","tokens_in":24679,"tokens_out":37123,"duration_ms":355535,"concrete_test":"Verify [Bac2, Thm. 3.1] in the exact setting used here: for qcqs algebraic spaces X with p invertible, check that the equivalence SH_et(X)^∧_p ≃ Shv(X_et)^∧_p is symmetric monoidal and intertwines f_* (equivalently f_!) for finite étale f. A minimal computational test is to take X = Spec(Z[1/p]) and f a finite G-torsor of order p, then compute the descendability index of the unit 1_X → f_*1_Y in both categories. If the indices disagree, Proposition 4.6 is false; if they agree, the reduction in §4.4 is sound.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I could not identify an internal inconsistency in the paper's argument. The central criterion Theorem 1.21 is supported by its own proof chain: the EGA IV_4 stratification step, Rydh's dévissage, Lemma 1.23, and Corollary 1.10 fit together as written, and the acknowledged gap in [Kha3, Thm. A.7] is explicitly repaired by Theorem 5.1. The single most load-bearing unverified input is the quoted Bachmann rigidity equivalence SH_et(X)^∧_p ≃ Shv(X_et)^∧_p used in §4.4. As cited, [Bac2, Thm. 3.1] is a deep statement about motivic spectra over schemes with p inverted, and the paper applies it to qcqs algebraic spaces and to the subcategory SH_et rather than to the full SH. For the reduction to (4.6.2.a) to be valid, this equivalence must be symmetric monoidal and compatible with the relevant f_* and f_! functors for finite étale morphisms. The paper does not spell out that compatibility. If it fails, Proposition 4.6, and hence Theorem B / Corollary 4.2, lose their main input. This is a residual risk attached to an external theorem, not an observed defect in the paper's own derivations.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper works in the author's 'topological weave' formalism for six-functor formalisms on algebraic spaces and Artin stacks. The central result is Theorem 1.21 (Theorem C): in a weave satisfying the localization property, descendability of every finite étale surjection and every finite radicial surjection of qcqs algebraic spaces is equivalent to descendability of every finitely presented surjection. The proof uses the EGA IV_4 stratification theorem, Rydh's dévissage for algebraic spaces, and a stacky dévissage variant (Theorem 5.1). Applications include: h-descent for rational motivic sheaves DM_Q on all algebraic spaces (Theorem A / Corollary 3.3); h-descent for étale motivic spectra SH_et on finite-dimensional noetherian algebraic spaces whose residue fields have uniformly bounded étale cohomological dimension (Theorem B / Corollary 4.2); v- and arc-hyperdescent for rational motivic cohomology C_mot(-;Q(r)) in weights r≤1, conditionally on a Beilinson–Soulé-type vanishing (Theorem 3.6, Corollary 3.7); and the forgetting-supports isomorphism f_!≃f_* for proper DM-type and tame Artin morphisms (Theorems 5.2 and 5.4). The paper also explicitly repairs a gap in [Kha3, Thm. A.7].","tokens_in":24934,"tokens_out":9883,"duration_ms":95957,"significance":"If the main results hold, they are substantial and useful. The paper gives a clean, flexible criterion for descendability and uses it to prove several results that were folklore or only known in restricted settings: h-descent for DM_Q on arbitrary algebraic spaces, h-descent for SH_et under bounded cohomological dimension assumptions, and v-/arc-hyperdescent for low-weight rational motivic cohomology. A notable strength is the paper's honesty about the gap in the previously claimed 'forgetting supports' isomorphism and its repair via the descendability framework. The proofs are detailed and internally coherent, and the reliance on external deep results is clearly signposted. The main residual risk is the unverified compatibility of Bachmann's p-completed rigidity equivalence with the six-functor structures used in §4.4; this is a fixable point of presentation rather than an observed mathematical error.","major_comments":[{"comment":"The proof of Proposition 4.6 in the p-complete case invokes Bachmann's rigidity equivalence SH_et(X)^∧_p ≃ Shv(X_et)^∧_p ([Bac2, Thm. 3.1]) and then works entirely in the category C = Shv(X_et)^∧_p. To make this reduction valid, the equivalence must be compatible with the structures used in the subsequent argument: the symmetric monoidal structure, the functors f_*, f_! and f^* for finite étale f, the t-structure, and the identification of homotopy sheaves with étale sheaves. The manuscript states only an equivalence of categories and does not spell out these compatibilities. Lemma 4.8, for instance, computes cohomological dimension in C by comparing H^s(Σ∞_+U;G) with étale cohomology of the underlying algebraic space; this requires that the rigidification identifies the relevant mapping spectra and the p-completion of étale cohomology. Since Proposition 4.6 — and hence Theorem B / Corol","section":"§4.4, around (4.6.2.a)"},{"comment":"Theorem B in the introduction asserts h-descent for DM_et(-;Λ) for every commutative ring Λ. The body of the paper (§4) only proves h-descent for SH_et (and SH_et[1/2]) on the relevant class of algebraic spaces; I could not find a proof of the DM_et assertion. The footnote says it 'can similarly be derived' from work of Ayoub and Cisinski–Déglise, but a formal theorem statement should be supported either by a proof in the text or by an explicit reference to a result that contains exactly this statement. As written, this is a missing proof of an advertised theorem, even if it does not affect the paper's other main conclusions.","section":"Introduction, Theorem B"}],"minor_comments":[{"comment":"There are several typesetting glitches: e.g., in Remark 1.2 and the proof of Lemma 1.7 the symbol 'lim←/leftr⫯g⊸tl⫯nen' appears instead of a proper lim←/Tot notation. The abstract also has 'satisfyh-descent' with a missing space. Please proofread the final version.","section":"Throughout"},{"comment":"The phrase 'By radditivity' would be clearer as 'By radditivity of the weave' or 'Since the weave is radditive'. Also, the term 'radditivity' is used without definition; it appears in [Kha2], but a one-line reminder in §2.1 would help.","section":"Corollary 1.17 proof"},{"comment":"The proof of Lemma 4.3 cites [MT, Thm. A] for the vanishing of the η-periodization SH_et[1/2][η^{-1}]. Since this is a very recent external result, a brief statement of the hypotheses (fields? arbitrary base?) would help the reader verify that it applies in the required generality.","section":"§4.2, Lemma 4.3"},{"comment":"The identification C_mot(-;Q(1)) ≃ Γ_cdh(-;G_m⊗Q)[-1] is stated as following from [BEM, Thm. 1.1(6)]. The subsequent vanishing H^s_cdh(U;G_m)_Q=0 for s≤0 may be obvious to experts, but a short explanation of the cdh sheafification step would improve readability.","section":"Corollary 3.7(c)"}],"recommendation":"major_revision","confidential_remarks":"The paper is a solid contribution to motivic homotopy theory, and the main criterion (Theorem 1.21) is plausible and well-supported by the internal proof chain. The two major comments above are both addressable: one asks for a more precise statement of an external rigidity input, the other asks for a proof or explicit reference for the DM_et part of Theorem B. I would be comfortable with acceptance after these points are clarified. One editorial note: the paper relies heavily on the author's own 'weaves' framework ([Kha], [Kha2]) and on recent preprints ([MT], [BEM]); the introduction could do more to orient a non-expert reader on the axioms of a weave."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This is a strong paper. The central contribution is Theorem C (1.21): in any topological weave satisfying localization, descendability of every finite étale surjection and every finite radicial surjection of qcqs algebraic spaces is equivalent to descendability of every finitely presented surjection. That reduction is genuinely useful, and the proof, running through the EGA IV_4 stratification and Rydh's dévissage, is carefully written and internally coherent. The applications are real: h-descent for SH_et under uniformly bounded residue-field cohomological dimension, h-descent for rational motivic sheaves on all algebraic spaces, v- and arc-hyperdescent for rational motivic cohomology in weights ≤1, and the patched forgetting-supports isomorphism f_! ≃ f_* for proper DM-type and tame Artin morphisms. The paper also explicitly flags and repairs a gap in the author's earlier [Kha3, Thm A.7] using descendability; that's the right way to handle it.\n\nNow the soft spots, in proportion. The most load-bearing external input is Bachmann's rigidity equivalence SH_et(X)^∧_p ≃ Shv(X_et)^∧_p in §4.4. As quoted it is for schemes with p invertible; the paper applies it to qcqs algebraic spaces and to SH_et rather than full SH, and does not spell out the compatibility with the f_* and f_! functors needed for Proposition 4.6. If that compatibility fails, Theorem B loses its main input. I don't see evidence that it fails, but a referee should ask for it. The rest of the external baggage — EGA, [EK], [AOV]/[AHR], the 'weaves' framework — is normal for this area; the paper leans on it heavily, but the results it cites are mostly established in print.\n\nOne caveat on the arc/v-hyperdescent: the general statement is conditional on a Beilinson-Soulé type vanishing assumption (BS_r), and the unconditional corollary covers only weights r ≤1. That is stated honestly, but it's worth remembering the title overpromises a little if you only read the abstract.\n\nCitation pattern: fine. Self-citations to [Kha] and [Kha2] are to the formalism the paper is built on, not a shell game. The distinction between folklore (rational h-descent) and genuinely new (the criterion, the étale h-descent under cd-bounds, the low-weight arc-descent, the patched forgetting-supports theorem) is drawn carefully.\n\nBottom line: for people working on motivic homotopy theory, descent, or six-functor formalisms over stacks, this is a useful paper and it deserves a serious referee. I would send it out; minor revisions likely, mainly around spelling out the rigidity compatibility in §4.4.","headline":"A genuinely useful criterion for h-descent in topological weaves, with a clean proof and real applications; the main residual risk is the Bachmann rigidity input in §4.4.","tokens_in":25479,"tokens_out":3004,"would_cite":true,"duration_ms":26445,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14F42","14F20","14A20"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves a criterion reducing descendability of all finitely presented surjections of algebraic spaces to the finite étale and finite radicial cases, and derives h-descent for rational motivic sheaves and étale motivic spectra, arc","keywords":["descendability","h-descent","topological weaves","rational motivic sheaves","étale motivic spectra","arc-descent","algebraic spaces","forgetting supports"],"falsifier":"Construct a qcqs algebraic space X and a prime p invertible on X for which the canonical map from the p-completion of étale motivic spectra to the p-completion of sheaves of spectra on the étale site is not an equivalence; that would defeat the input on which Proposition 4.6 and hence Theorem B rest. Alternatively, exhibit a weave satisfying the localization property where all finite étale and finite radicial surjections are descendable but some finitely presented surjection is not, which would disprove Theorem 1.21.","tokens_in":24505,"feed_emoji":"🧩","tokens_out":6582,"duration_ms":58884,"temperature":0.7,"pith_summary":"The paper establishes a criterion: in a topological weave—an abstract six-functor formalism for algebraic geometry—descendability of all finitely presented surjections of algebraic spaces is equivalent to descendability of just the finite étale and finite radicial surjections, provided the weave satisfies the localization property. Descendability is a strong form of descent: the unit sheaf can be built from the direct image of the unit along the cover using finite limits and tensor products. The criterion is then applied to prove h-descent for rational motivic sheaves on all algebraic spaces, h-descent for étale motivic spectra on bases with uniformly bounded étale cohomological dimension, v- and arc-hyperdescent for rational motivic cohomology in weights at most 1, and the forgetting-supports isomorphism for proper DM-type and tame Artin morphisms. The paper matters because it reduces a large class of descent statements to a simple check on two kinds of covers.","feed_headline":"One criterion makes every finitely presented surjection descendable","feed_subtitle":"Checking only finite étale and finite radicial covers now suffices for h-descent across algebraic spaces.","key_machinery":"The central object is descendability: a morphism of E∞-algebras is descendable when the base unit lies in the smallest thick subcategory generated by the target algebra and closed under tensoring with arbitrary modules, equivalently when the pro-object of partial totalizations of the Čech nerve is essentially constant. The engine is the bootstrapping theorem (Theorem 1.21), which, under the localization property of the weave, lifts descendability from finite étale and finite radicial surjections to all finitely presented surjections. The proof uses two lemmas: one glues descendability along closed immersions with open complements, and one reduces an arbitrary finitely presented surjection, v","core_discovery":"The central claim is Theorem C (Theorem 1.21): for any weave satisfying the localization property, descendability of every finite étale surjection and every finite radicial surjection of qcqs algebraic spaces is equivalent to descendability of every finitely presented surjection. The proof bootstraps from these two special cases by stratifying a given surjection via the classical stratification theorem and dévissage for algebraic spaces, then gluing descendability along closed/open decompositions. As direct consequences, the paper proves h-descent for the plus part of rational motivic spectra on all algebraic spaces, and for étale motivic spectra on noetherian finite-dimensional bases whose","pith_inferences":["The criterion suggests that h-descent for any six-functor formalism of motivic type is essentially controlled by finite covers: once finite étale and finite radicial maps descend, descent for all finitely presented surjections follows automatically under localization, so future descent proofs in new settings can focus on checking just those two cases.","The arc-descent result for weights at most 1 is unconditional, while the general-weight version is reduced to a vanishing condition on valuation rings; this indicates that arc-descent for rational motivic cohomology is governed entirely by negative-weight cohomology vanishing, a concrete algebro-geometric property one could verify or refute by computations on Henselian valuation rings.","The descendability strengthening means totalization commutes with any exact functor, not merely with the six-functor operations; this opens a route to constructing ∗-direct images and dualities on Artin stacks where exact functors must pass through totalizations, which is likely to be useful beyond the forgetting-supports applications given here.","A natural testable extension would be to ask whether the stratification and dévissage inputs admit analogues for derived or spectral algebraic spaces, which could extend the criterion to non-classical bases without substantially new ideas."],"forward_implications":["Every finitely presented surjection of qcqs algebraic spaces is descendable in the rational motivic sheaf weave, so rational motivic sheaves satisfy h-descent on all algebraic spaces, strengthening earlier folklore results restricted to quasi-excellent noetherian schemes.","Étale motivic spectra satisfy h-descent on noetherian finite-dimensional algebraic spaces with residue fields of uniformly bounded étale cohomological dimension, and after inverting 2 on the larger class with uniformly bounded virtual étale cohomological dimension.","Rational motivic cohomology in weights at most 1 satisfies v- and arc-hyperdescent on algebraic spaces; in general weights the same conclusion follows from a conjectural Beilinson–Soulé type vanishing condition on valuation rings.","The forgetting-supports map f_! → f_* is invertible for proper Deligne–Mumford morphisms and for proper tame Artin morphisms with equicharacteristic base, in rational motivic sheaves, yielding ∗-direct image along such morphisms.","Because descendability is preserved by symmetric monoidal exact functors, the h-descent results transfer automatically to any Q-linear oriented topological weave, and to any topological weave satisfying étale descent over the stated bases."],"fun_headline_variants":["Checking finite étale and radicial covers now yields full descent","Descent follows from just two types of finite covers","A criterion that cuts descent checking to two cover types","New criterion: descent from finite étale and radicial surjections","For h-descent, only finite étale and radicial matter now"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"For the étale-motivic application, the proof relies on an external rigidity equivalence identifying p-completed étale motivic spectra with p-completed sheaves of spectra on the small étale site for primes p invertible on the base; if that equivalence fails, the h-descent theorem for étale motivic spectra does not follow.","fun_headline_variants_meta":{"raw":{"variants":["Checking finite étale and radicial covers now yields full descent","Descent follows from just two types of finite covers","A criterion that cuts descent checking to two cover types","New criterion: descent from finite étale and radicial surjections","For h-descent, only finite étale and radicial matter now"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000182,"raw_usage":{"total_tokens":1081,"prompt_tokens":612,"completion_tokens":469,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":356,"completion_tokens_details":{"reasoning_tokens":383}},"tokens_in":356,"tokens_out":469,"duration_ms":4639,"temperature":1.0,"reasoning_tokens":383,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T08:08:17.086406+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct a qcqs algebraic space X and a prime p invertible on X for which the canonical map from the p-completion of étale motivic spectra to the p-completion of sheaves of spectra on the étale site is not an equivalence; that would defeat the input on which Proposition 4.6 and hence Theorem B rest. Alternatively, exhibit a weave satisfying the localization property where all finite étale and finite radicial surjections are descendable but some finitely presented surjection is not, which would disprove Theorem 1.21.","supporting_citations":[],"review_version":2}