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Once again on an analogue of the certain Voevodsky theorem

T0 review · 2 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Over a perfect field, every A1-invariant quasi-stable ZF*-presheaf of abelian groups has equal Zariski and Nisnevich sheafifications, and all positive-degree cohomology groups agree—the full framed analogue of Voevodsky's theorem.

desk verdict Real result, one honest field-scope gap: the cohomology half only works for infinite perfect fields. read the letter →

arxiv 2506.06795 v1 pith:QBZIAJ6O submitted 2025-06-07 math.KT

Ivan Panin , Dimitrii Tyurin This is my paper
classification math.KT MSC 14F4214F2019E15
keywords A1-invarianceframedcorrespondencesZF*-presheavesquasi-stablepresheavesNisnevichsheafificationZariskicohomologyGerstenresolutionVoevodskytheorem
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 establishes the full framed analogue of Voevodsky's theorem for presheaves with transfers. Its main theorem (2.6) says that for any $\mathbb{A}^1$-invariant quasi-stable $\mathbb{Z}F_*$-presheaf $F$ of abelian groups over a perfect field, the Zariski sheafification $F_{Zar}$ equals the Nisnevich sheafification $F_{Nis}$, and for every smooth $X$ and every $n>0$ the cohomology groups $H^n_{Zar}(X,F_{Zar})$ and $H^n_{Nis}(X,F_{Nis})$ are equal. This completes the earlier framed-presheaf analogue, which had shown that $F_{Nis}$ and its Nisnevich cohomology presheaves remain $\mathbb{A}^1$-invariant and quasi-stable but had not brought back the Zariski topology. The equality matters because Zariski cohomology is the more manageable side in practice: framed-presheaf cohomology can now be studied with Zariski-local arguments, exactly as cohomology of presheaves with transfers could after Voevodsky's result.

What carries the argument

A $\mathbb{Z}F_*$-presheaf is an additive contravariant functor from the category of framed correspondences to abelian groups; such a presheaf is $(\ast)$ when it is $\mathbb{A}^1$-invariant and quasi-stable, meaning pullback along the zero-section frame $\sigma_X$ is an isomorphism for every $X$. Two tools carry the argument. First, Theorem 2.3: for a $(\ast)$-presheaf $F$, the restriction $F(\mathrm{Spec}\,\mathcal{O}_{X,x})\to F(\mathrm{Spec}\,k(X))$ is injective, so a section is determined by its generic stalk. Second, Lemma 2.5 (a geometric moving lemma): given a closed subset $D\subset X$, a framed correspondence from a local scheme $U=\mathrm{Spec}\,\mathcal{O}_{X,x}$ to $X$ can be factored through $X\setminus D$ after composing with the $N$-fold power of the frame $\sigma_X$, at the cost of a harmless power of $\sigma_X$. This factorization lets the authors compare a generic section of $F_{Nis}$ with a Zariski section of $F$ and conclude the stalk map is an isomorphism. For cohomology, the resolution (4.1) of $F_{Nis}$ by skyscraper sheaves $(F_{Nis})^{-d}$ over points of codimension $d$ is the Gersten-type mechanism that identifies the two cohomology theories.

What would settle it

Compute the stalk $(H^1_{Nis})_x(X,F_{Nis})$ for a $(\ast)$-presheaf over a finite perfect field at a codimension-one point $x$; if it is nonzero, the Gersten-type resolution (4.1) fails there and the cohomology equality of Theorem 2.6 does not follow from the present proof. A broader test is to check the analogous stalk in degree $n$ at codimension $d\ne n$; any nonzero value would break the flabbiness that identifies Zariski and Nisnevich cohomology.

Watch

Extended reading notes

Core claim

The central claim, Theorem 2.6, is that over a perfect field the class of $(\ast)$-presheaves (those that are $\mathbb{A}^1$-invariant and quasi-stable) is closed under taking the Zariski sheafification and under taking Zariski cohomology. The proof shows that the natural map $F_{Zar}\to F_{Nis}$ is an isomorphism on stalks by combining an injectivity statement for local rings with a geometric moving lemma: any framed correspondence defined on the generic stalk can be shifted away from a closed subset without changing its value. For cohomology, the paper upgrades a Gersten-type flabby resolution of $F_{Nis}$, whose $d$-th term at a codimension-$d$ point is $(F_{Nis})^{-d}$, into a flabby resolution of $F_{Zar}$; because both sheaves coincide, the two cohomologies must agree. As a by-product, every $H^n_{Zar}(-,F_{Zar})$ inherits the structure of an $\mathbb{A}^1$-invariant quasi-stable $\mathbb{Z}F_*$-presheaf.

Load-bearing premise

The cohomology equality rests on a statement the authors borrow from the predecessor paper: at any point of codimension $d$, the Nisnevich cohomology sheaf of a $(\ast)$-presheaf is zero in degrees other than $d$, and in degree $d$ it is the expected shifted presheaf. Remark 4.2 notes that the recorded proof of this statement covers only infinite perfect fields, while the main theorem is stated over every perfect field.

Editorial extensions

If this is right

  • Framed-presheaf cohomology can be computed Zariski-locally: for every smooth $X$, $H^n_{Zar}(X,F_{Zar})\cong H^n_{Nis}(X,F_{Nis})$, so Nisnevich-local arguments can be replaced by Zariski-local ones.
  • The Zariski sheafification $F_{Zar}$ is itself an $\mathbb{A}^1$-invariant quasi-stable $\mathbb{Z}F_*$-sheaf, since it coincides with $F_{Nis}$ and the latter carries that structure by the main theorem of [GP1].
  • All Zariski cohomology presheaves $X\mapsto H^n_{Zar}(X,F_{Zar})$ are $\mathbb{A}^1$-invariant quasi-stable $\mathbb{Z}F_*$-presheaves, so the class of $(\ast)$-presheaves is closed under taking Zariski cohomology.
  • The result completes the framed analogue of Voevodsky's theorem: sheaf coincidence, homotopy invariance of cohomology, and Zariski–Nisnevich cohomology equality now hold in parallel to the presheaves-with-transfers case.

Reading between the lines

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

  • If the missing finite-field case of the borrowed stalk computation is supplied, the theorem would make framed-motivic cohomology computations over finite perfect fields as straightforward as over infinite ones, since the Zariski–Nisnevich gap disappears.
  • The Gersten-type resolution (4.1) points to a local-to-global recipe: cohomology in degree $n$ is assembled from values $(F_{Nis})^{-d}$ at codimension-$d$ points, so in principle $H^n$ can be computed by residue-type data along a filtration by subvarieties.
  • Because quasi-stability (invertibility of $\sigma_X^*$) is used to absorb the powers of $\sigma_X$ in the moving lemma, the method leaves open whether plain $\mathbb{A}^1$-invariance without quasi-stability would still force $F_{Zar}=F_{Nis}$; testing that would map the boundary of the theorem.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper claims a full analogue of Voevodsky's theorem for A^1-invariant quasi-stable ZF_*-presheaves of abelian groups on Sm/k over a perfect field k: the Zariski sheafification F_Zar equals the Nisnevich sheafification F_Nis, and for every smooth X and every n>0 the Zariski cohomology H^n_Zar(X,F_Zar) equals H^n_Nis(X,F_Nis). Section 3 proves the sheaf equality by comparing stalks at local rings, using injectivity of the map to the generic point and a moving lemma (Lemmas 2.5 and 3.3). Section 4 derives the cohomology equality from a flabby Gersten-type resolution of F_Nis, whose stalks are identified with the help of Theorem 4.1, quoted from [D, Theorem 6.4].

Significance. If the theorem holds in the stated generality, it is a meaningful extension of Voevodsky's classical theorem to framed correspondences and completes the Zariski-topology half of the Garkusha–Panin main theorem. The paper is short, clearly organized, and honest about its reliance on published inputs; the Section 3 argument is explicit and the proof of Lemma 3.3 is detailed. The main weakness is that a load-bearing quoted theorem is only available for infinite perfect fields, so the stated finite-field case is not proved here.

major comments (2)
  1. [§4, Theorem 4.1 and Remark 4.2] The cohomology equality H^n_Zar(X,F_Zar)=H^n_Nis(X,F_Nis) in Theorem 2.6 is stated for an arbitrary perfect field k, but the proof uses Theorem 4.1 to identify (H^n_Nis)_x(X,F_Nis) with (F_Nis)^{-codim(x)}(Spec k(x)) for n equal to the codimension and with 0 otherwise. As Remark 4.2 explicitly states, the proof of [D, Theorem 6.4] has been repaired in [DP, Theorem 3.8] only for infinite perfect fields; no argument for finite perfect fields is given. Since this stalk identification is the bridge between the flabby resolution and the equality of cohomology groups, the cohomology half of the main theorem is unsupported for finite k. The statement should be restricted to infinite perfect fields, or a proof valid for finite fields must be supplied.
  2. [§2, Lemma 2.5 and Theorem 2.3] The quotations of [GP1, Theorem 3.15(3)] and [GP1, Assertion 9.9] do not record the field hypotheses under which those results are proved. The Section 3 proof of F_Zar=F_Nis uses Lemma 2.5 as the moving input and Theorem 2.3 for injectivity at local rings; if those results also require k to be infinite, as the related [GP1, Theorem 3.15(5)] restriction mentioned in Remark 4.2 suggests, then the sheaf-level equality would inherit the same limitation. The manuscript should state the precise hypotheses of the quoted results and reconcile them with the perfect-field formulation of Theorem 2.6.
minor comments (4)
  1. [§2] In the definition of ZF_n(Y,X), the text contains the placeholder reference '[ссылка на статью]' and the typo 'абалевой'; both should be fixed before publication.
  2. [§4] The sentence introducing G^{-1} states that for any (∗)-presheaf and n>0 one has (G_Nis)^{-n} = (G^{-n})_Nis and refers to 'our article, Theorem 4.1'; this citation is confusing because Theorem 4.1 in the present paper is the quoted [D, Theorem 6.4], not this equality. Please clarify the intended reference.
  3. [§4] In the definition of G^{-1}, the map τ^* is written as G(X) ≅ F(X × A^1) → G(X × G_m); the symbol F should be G, or the notation should be explained.
  4. [Throughout] The spacing in formulas such as 'FN is' and 'FZar' is irregular in several places, including the abstract and Theorem 2.6; a careful copyedit is needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the proof is a deduction from earlier published theorems whose hypotheses do not include the target statement.

full rationale

The paper's main derivation is self-contained as a logical deduction once the cited prior results are granted. Section 3 establishes F_Zar = F_Nis by checking stalks: it uses the injectivity theorem [GP1, Thm. 3.15(3)], the transfer structure on F_Nis and its cohomology presheaves [GP1, Thm. 1.1, Cor. 2.17], and the moving lemma [GP1, Assertion 9.9]. None of these inputs asserts that Zariski and Nisnevich sheafifications coincide; that equality is the output of the argument, not an input. Section 4 derives the cohomology equality from a general flabby resolution supplied by [P1, Cor. 9.2] together with the Gersten-type stalk computation quoted as Theorem 4.1 from [D, Thm. 6.4] (with the characteristic-2 repair from [DP]). The resolution plus the already-proved sheaf equality gives H^n_Zar(X,F_Zar)=H^n_Nis(X,F_Nis) by standard flabby-resolution bookkeeping, not by assuming that equality. The cited works include earlier papers by the same authors, but they are parameter-free theorems with fixed stated hypotheses and were not derived from the present theorem; this is legitimate self-citation, not load-bearing circularity. Remark 4.2 does disclose a genuine limitation: the quoted stalk identification is currently justified only for infinite perfect fields, whereas Theorem 2.6 is stated for all perfect fields. That is a correctness/completeness gap for finite perfect fields, but it is not an instance of circularity, since the missing support is an unproved extension of a prior theorem rather than a reduction of the conclusion to its own assumptions.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The paper introduces no new entities or fitted parameters. It depends on a chain of established theorems about framed correspondences, Nisnevich sheafification, and Gersten resolutions, mostly from the same research group but published earlier.

assumptions (7)
  • domain assumption The category ZF* of framed correspondences and its quotient relation are as defined in [GP1] and [V3].
    Used throughout Section 2 to define ZF*-presheaves and their A1-invariance and quasi-stability.
  • domain assumption Theorem 2.2: F_Nis and H^n_Nis(-,F_Nis) admit unique ZF*-structures and are (∗)-presheaves.
    Imported from [GP1, Theorem 1.1 and Corollary 2.17], extended to characteristic 2 by [DP]. It supplies the transfer structure on Nisnevich sheafification used in the proof.
  • domain assumption Theorem 2.3: for U=Spec(O_X,x) and a (∗)-presheaf F, F(U) to F(Spec k(X)) is injective.
    Imported from [GP1, Theorem 3.15(3)]; used in Corollary 2.4 and in the proof of surjectivity of a_U.
  • domain assumption Lemma 2.5: the moving lemma for framed correspondences relating [j][φ] to [σ^N_X][can].
    Imported from [GP1, Proposition 9.9]; key geometric input in Proposition 3.1.
  • domain assumption Theorem 4.1: the identification (H^n_Nis)_x(X,F_Nis) = (F_Nis)^{-d}(Spec k(x)) for n=d, and zero otherwise.
    Imported from [D, Theorem 6.4]; the paper notes in Remark 4.2 that the proof is only set up for infinite perfect fields, though the statement is used for perfect fields.
  • domain assumption Panin's Corollary 9.2 in [P1] provides the flabby Gersten resolution for F_Nis on X_Zar from the cohomology theory structure.
    Used in Section 4 to produce the resolution whose terms are then rewritten via Theorem 4.1.
  • domain assumption k is a perfect field.
    Stated in Theorem 2.6 and Section 2; the proof's cited resolution theorem is noted only for infinite perfect fields.

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Pith. "Pith review of Once again on an analogue of the certain Voevodsky theorem." pith.science (2026). https://pith.science/paper/QBZIAJ6O

@misc{pith2026250606795,
  author       = {Pith},
  title        = {Pith review of: Once again on an analogue of the certain Voevodsky theorem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QBZIAJ6O}},
  note         = {Machine review of arXiv:2506.06795}
}
abstract

Suppose that $F$ is an $\mathbb{A}^{1}$-invariant quasi-stable $\mathbb{Z}F_{\ast}$-presheaf. Then its Zariski sheafification $F_{Zar}$ coincides with its Nisnevich sheafification $F_{Nis}$. Moreover, if $X\in Sm/k$ is $k$-smooth, then for any $n$ there is equality $H^{n}_{Zar}(X, F_{Zar})=H^{n}_{Nis}(X,F_{Nis})$.

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Reference graph

Works this paper leans on

10 extracted references · 10 canonical work pages

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