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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

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2026-08-02 08:08 UTC pith:BMTF7NOA

load-bearing objection 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. the 2 major comments →

arxiv 2607.07137 v2 pith:BMTF7NOA submitted 2026-07-08 math.AG

Descendability and descent in topological weaves

Adeel A. Khan This is my paper
classification math.AG MSC 14F4214F2014A20
keywords descendabilityh-descenttopological weavesrational motivic sheavesétale motivic spectraarc-descentalgebraic spacesforgetting supports
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

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.

Core claim

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

What carries the argument

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

Load-bearing premise

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.

What would settle it

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.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • 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.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • 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.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

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].

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 (2)
  1. [§4.4, around (4.6.2.a)] 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
  2. [Introduction, Theorem B] 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.
minor comments (4)
  1. [Throughout] 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.
  2. [Corollary 1.17 proof] 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.
  3. [§4.2, Lemma 4.3] 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.
  4. [Corollary 3.7(c)] 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.

Circularity Check

0 steps flagged

No significant circularity; residual risk is an external rigidity input, not a circular derivation.

full rationale

The paper's central claims are derived, not assumed. Theorem 1.21 is proved by a genuine dévissage using EGA IV_4 17.16.4, Rydh's results, Lemma 1.23, and Corollary 1.10; the equivalence is not an identity because the nontrivial direction (i)⇒(ii) uses external stratification and dévissage theorems, while (ii)⇒(i) is immediate from the fact that finite étale and finite radicial surjections are finitely presented surjections. The applications to rational motivic sheaves and étale motivic spectra are obtained by combining this criterion with external results of Cisinski–Déglise, Bachmann, Bhatt–Mathew, and Rydh, rather than by restating those inputs. The self-citations to the author's own framework [Kha], [Kha2], [EK], [DFJK], and [Kha3] are foundational or already published peer-reviewed theorems; none of them is used as a substitute for the target conclusion. In particular, the paper explicitly flags the gap in [Kha3, Thm. A.7] in the introduction and repairs it with Theorem 5.1, which is an honest correction rather than a circular appeal. The only fragile load-bearing input identified is Bachmann's rigidity equivalence SH_et(X)^∧_p ≃ Shv(X_et)^∧_p cited in §4.4; reliance on an external theorem, even if deep or imperfectly matched to the present generality, is a correctness risk, not circularity. No fitted parameters are renamed as predictions, and no conclusion is equivalent to its hypothesis by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 8 axioms · 0 invented entities

No numbers are fitted and no new entities are postulated. The central claims are conditional on the topological-weave framework and on cited deep theorems, all declared. The main explicit hypotheses are (BS_r) for the conditional arc-descent statement and the bounded-cohomological-dimension hypotheses for SH_et.

axioms (8)
  • domain assumption The category of topological weaves (Khan, 'Weaves') and its properties: localization, nil-invariance, continuity, geometric generation, conservativity of stalks, radditivity, lisse extension.
    The paper's entire formalism; invoked throughout §1, §2, and §5. These are prior results by the author and collaborators, not proved here.
  • standard math EGA IV_4 Prop. 17.16.4: a finitely presented surjection of qcqs schemes admits a stratification with finite radicial and finite étale sections.
    Used in the proof of Theorem 1.21 (§1.5) to reduce arbitrary surjections to the covered cases.
  • standard math Rydh's dévissage and noetherian approximation for algebraic spaces and stacks ([HR, Ex. 3.2, Prop. 3.3], [Ry2, Thm. D, Props. B.2/B.3]).
    Used to pass from schemes to algebraic spaces in Theorem 1.21 and Corollary 1.16.
  • domain assumption Topological invariance of SH and SH_et for finite radicial surjections under the stated invertibility hypotheses ([EK, Thm. 2.1.1], Theorem 1.20).
    Needed for descendability of radicial covers in Theorems A and B.
  • domain assumption Bachmann's rigidity equivalence SH_et(X)^∧_p ≃ Shv(X_et)^∧_p for p invertible on X ([Bac2, Thm. 3.1]).
    Used in §4.4 to prove descendability of finite G-torsors; a central input for Theorem B.
  • domain assumption Generalized Beilinson–Soulé vanishing (BS_r): for each r there exists N_r with H^m_mot(V;Q(r))=0 for m<−N_r on finite rank henselian valuation rings.
    Hypothesis of Theorem 3.6; verified for r≤1 in Corollary 3.7 to get the unconditional arc-descent claim.
  • domain assumption Keel–Mori existence, [AOV, Thm. 3.2] local structure of tame stacks, and [AHR, Thm. 19.9] exact sequence of stabilizer groups.
    Used in the proof of Theorem 5.4 for the tame Artin forgetting-supports case.
  • domain assumption Conservativity of stalks for SH_et on S0 ([Bac3, Cor. 5.12]).
    Used in Lemma 4.4 and Proposition 4.6 to reduce to field-level checks.

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read the original abstract

We prove a criterion for a finitely presented surjection of algebraic spaces to be descendable in a topological weave. We apply this to show that \'etale motivic spectra satisfy $h$-descent on noetherian finite-dimensional schemes with residue fields of uniformly bounded \'etale cohomological dimension. We also show that rational motivic cohomology satisfies arc-descent in weights $\le 1$, and we construct the ``forgetting supports'' isomorphism $f_! \simeq f_*$ for a proper DM-type morphism of Artin stacks, in rational motivic sheaves.

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