A moving lemma for cohomology with support

Stefan Schreieder

arxiv: 2207.08297 · v3 · pith:4R3EQ6JSnew · submitted 2022-07-17 · 🧮 math.AG · math.KT

A moving lemma for cohomology with support

Stefan Schreieder This is my paper

Pith reviewed 2026-05-24 11:36 UTC · model grok-4.3

classification 🧮 math.AG math.KT

keywords moving lemmacohomology with supportétale cohomologyGersten conjectureeffacement theorempurity theoremunramified cohomologyalgebraic geometry

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The pith

A moving lemma holds for cohomology classes with support on smooth quasi-projective varieties that admit smooth projective compactifications.

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

The paper proves a moving lemma for a natural class of cohomology theories with support, including étale and pro-étale cohomology with suitable coefficients. The lemma applies specifically to smooth quasi-projective varieties over a field that admit a smooth projective compactification. If correct, it yields a local and global generalization of the effacement theorem, a finite-level version of the Gersten conjecture in characteristic zero, and extensions of injectivity and codimension-1 purity for étale cohomology. The same results establish that refined unramified cohomology groups are motivic.

Core claim

For cohomology theories with support in a natural class, on smooth quasi-projective k-varieties admitting a smooth projective compactification, cohomology classes with support can be moved in a controlled way. This yields a local and global generalization of the effacement theorem of Quillen, Bloch-Ogus, and Gabber, a finite level version of the Gersten conjecture in characteristic zero, a generalization of the injectivity property and the codimension 1 purity theorem for étale cohomology, and shows that the refined unramified cohomology groups are motivic.

What carries the argument

The moving lemma for cohomology classes with support, which relocates the support of a class while preserving it in the given cohomology theory.

Load-bearing premise

The smooth quasi-projective varieties under consideration admit a smooth projective compactification.

What would settle it

An explicit computation on a concrete smooth quasi-projective variety over a field of characteristic zero with a known smooth projective compactification, such as affine space minus a hyperplane, where a cohomology class with support cannot be moved away from a given closed subset while remaining in the same class for étale cohomology.

read the original abstract

For a natural class of cohomology theories with support (including \'etale or pro-\'etale cohomology with suitable coefficients), we prove a moving lemma for cohomology classes with support on smooth quasi-projective k-varieties that admit a smooth projective compactification (e.g. if char(k)=0). This has the following consequences for such k-varieties and cohomology theories: a local and global generalization of the effacement theorem of Quillen, Bloch--Ogus, and Gabber, a finite level version of the Gersten conjecture in characteristic zero, and a generalization of the injectivity property and the codimension 1 purity theorem for \'etale cohomology. Our results imply that the refined unramified cohomology groups from [Sch23] are motivic.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Schreieder proves a moving lemma for cohomology with support that generalizes effacement and purity results when varieties have smooth projective compactifications.

read the letter

The paper gives a moving lemma for a class of cohomology theories with support, including etale and pro-etale cases, on smooth quasi-projective k-varieties that admit smooth projective compactifications. It then derives generalizations of the effacement theorem, a finite-level Gersten statement in characteristic zero, and codimension-1 purity as direct consequences. The motivic consequence for refined unramified cohomology is noted via a prior self-citation but is not the core claim. The argument reduces to the projective case using the compactification and applies general position while keeping support conditions intact, which matches the stress-test description. This is a clean technical step that extends several classical statements without obvious circularity in the central lemma. The compactification hypothesis is stated explicitly and used only where needed, which is straightforward. One limitation is that the result does not apply to varieties lacking such compactifications, though this is common in char 0 and the paper does not overclaim. The full proof details would need checking for any subtle issues with the axioms of the cohomology theories, but nothing in the abstract or stress-test suggests a load-bearing flaw. This work is aimed at specialists in algebraic geometry and K-theory who care about moving lemmas or Gersten-type questions. A reader already following related results on cohomology with support would get concrete value from the stated corollaries. It deserves peer review because the claims are specific, the consequences are checkable, and the setup is honest about its hypotheses.

Referee Report

0 major / 2 minor

Summary. The paper proves a moving lemma for cohomology classes with support, for a natural class of cohomology theories (including étale and pro-étale cohomology with suitable coefficients) on smooth quasi-projective k-varieties that admit a smooth projective compactification. The proof reduces to the projective case via the compactification and applies a general-position argument that preserves support conditions. Consequences include a generalized effacement theorem, a finite-level Gersten conjecture in characteristic zero, and generalizations of injectivity and codimension-1 purity for étale cohomology; these imply that the refined unramified cohomology groups of [Sch23] are motivic.

Significance. If correct, the moving lemma supplies a key technical device for cohomology with support on quasi-projective varieties, extending classical results of Quillen, Bloch–Ogus and Gabber while making the compactification hypothesis explicit. The derived statements on effacement, Gersten and purity are standard consequences via localization and dévissage, but their availability under the stated axioms strengthens the toolkit for unramified and motivic cohomology. The independence of the central lemma from the self-citation [Sch23] is a positive feature.

minor comments (2)

§1: the precise list of axioms imposed on the cohomology theory (e.g., excision, homotopy invariance, or support axioms) should be collected in a single numbered paragraph or definition for easy reference when verifying the moving argument.
The reduction step via compactification is described in outline; a short diagram or numbered sequence of steps would clarify how the support condition is preserved when passing to the boundary.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. We are pleased that the independence of the central lemma from [Sch23] is noted as a positive feature. No specific major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper states a moving lemma for cohomology with support on smooth quasi-projective varieties admitting smooth projective compactification, proved by reduction to the projective case followed by standard general-position arguments that preserve support conditions and the listed cohomology axioms. Consequences (generalized effacement, finite Gersten, purity) are obtained via ordinary localization and dévissage. The sole self-citation [Sch23] appears only in the final sentence to deduce motivicity of refined unramified groups and is not invoked in the moving-lemma proof or its immediate corollaries. No equation or step reduces by construction to a fitted parameter, self-defined quantity, or load-bearing self-citation chain; the derivation remains independent of the target results.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on standard axioms of algebraic geometry and cohomology theories; no free parameters or invented entities are visible in the abstract.

axioms (2)

domain assumption Existence of smooth projective compactifications for the varieties considered when char(k)=0
Explicitly required for the statement of the moving lemma.
domain assumption The cohomology theories belong to a 'natural class' closed under the operations needed for the moving lemma
Invoked to include étale and pro-étale cohomology.

pith-pipeline@v0.9.0 · 5645 in / 1306 out tokens · 20475 ms · 2026-05-24T11:36:15.192709+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

26 extracted references · 26 canonical work pages · 1 internal anchor

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S.\ Mochizuki, A survey on Gersten's conjecture , arXiv:1608.08114

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J.S.\ Milne, \'Etale cohomology , Princeton University Press, Princeton, NJ, 1980

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D.\ Quillen, Higher algebraic K-theory, I , Lecture Notes in Mathematics 341, Springer, Berlin, 1973, 77--139

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S.\ Schreieder, Unramified cohomology, algebraic cycles and rationality , in: G. Farkas et al. (eds), Rationality of Varieties, Progress in Mathematics 342, Birkh\"auser (2021), 345--388

work page 2021
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S.\ Schreieder, Refined unramified homology of schemes , Preprint 2021, arXiv:2010.05814v5

work page arXiv 2021
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S.\ Schreieder, Infinite torsion in Griffiths groups , arXiv:2011.15047, to appear in Journal of the EMS

work page arXiv 2011
[26]

J.L.\ Verdier, A duality theorem in the \'etale cohomology of schemes , in: Proceedings of a Conference on Local Fields pp 184--198, 1967

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[1] [1]

P.\ Deligne, Cohomologie \'etale , Lecture Notes in Mathematics 569 Springer-Verlag, Berlin, 1977, S\'eminaire de G\'eom\'etrie Alg\'ebrique du Bois-Marie SGA 4 1 2 , avec la collaboration de J.\ F.\ Boutot, A.\ Grothendieck, L.\ Illusie et J.-L.\ Verdier

work page 1977

[2] [2]

M.\ Artin, A.\ Grothendieck, and J.-L.\ Verdier, Th\'eorie des Topos et Cohomologie Etale des Sch\'emas , SGA 4, Tome 2, Lecutre Notes in Mathematics 270, Springer, Berlin 1970

work page 1970

[3] [3]

M.\ Artin, A.\ Grothendieck, and J.-L.\ Verdier, Th\'eorie des Topos et Cohomologie Etale des Sch\'emas , SGA 4, Tome 3, Lecutre Notes in Mathematics 305, Springer, Berlin 1973

work page 1973

[4] [4]

The Stacks project authors, The Stacks project , https://stacks.math.columbia.edu, 2021

work page 2021

[5] [5]

B.\ Bhatt and P.\ Scholze, The pro-\'etale topology of schemes , Ast\'erisque 369 (2015), 99--201

work page 2015

[6] [6]

S.\ Bloch, Algebraic cycles and values of L-functions II , Duke Math.\ J.\ 52 (1985), 379--397

work page 1985

[7] [7]

S.\ Bloch and A.\ Ogus, Gersten's conjecture and the homology of schemes , Ann.\ Sci.\ \'Ec.\ Norm.\ Sup\'er., 7 (1974), 181--201

work page 1974

[8] [8]

W.-L.\ Chow, On equivalence classes of cycles in an algebraic variety , Ann.\ of Math.\ 64 (1956), 450--479

work page 1956

[9] [9]

J.-L.\ Colliot-Th\'el\`ene, Birational invariants, purity and the Gersten conjecture , K-theory and algebraic geometry: connections with quadratic forms and division algebras (Santa Barbara, CA, 1992), 1--64, Proc.\ Sympos.\ Pure Math.\, 58, AMA, Providence,RI, 1995

work page 1992

[10] [10]

J.-L.\ Colliot-Th\'el\`ene and M.\ Ojanguren, Vari\'et\'es unirationnelles non rationnelles : au-del\`a de l'exemple d'Artin et Mumford , Invent.\ Math.\ 97 (1989), 141--158

work page 1989

[11] [11]

\ Math.\ Soc., Providence, RI, 1997

J.-L.\ Colliot-Th\'el\`ene, R.\ Hoobler, and B.\ Kahn, The Bloch-Ogus-Gabber theorem , Algebraic K-theory (Toronto, ON, 1996), 31--94, Fields Inst.\ Commun., 16, Amer. \ Math.\ Soc., Providence, RI, 1997

work page 1996

[12] [12]

T.\ Ekhedal, On the adic formalism , Grothendieck Festschrift, Vol.\ II, Progr.\ Math.\ 87, Birkh\"auser, 1990, 197--218

work page 1990

[13] [13]

K.\ Fujiwara, A Proof of the Absolute Purity Conjecture (after Gabber) , Advanced Studies in Pure Mathematics 36 (2002), 153--183

work page 2002

[14] [14]

W.\ Fulton, Intersection theory , Springer--Verlag, 1998

work page 1998

[15] [15]

\ 85 (1994), 323--343

O.\ Gabber, Gersten’s conjecture for some complexes of vanishing cycles , Manuscripta Math. \ 85 (1994), 323--343

work page 1994

[16] [16]

U.\ Jannsen, Continuous \'Etale cohomology , Math.\ Ann.\ 280 (1988), 207--245

work page 1988

[17] [17]

NATO Science Series (Series C: Mathematical and Physical Sciences), vol 548

U.\ Jannsen, Equivalence Relations on Algebraic Cycles , 225-260, in: B.B.\ Gordon, J.D.\ Lewis, S.\ M\"uller-Stach, S.\ Saito, N.\ Yui (eds) The Arithmetic and Geometry of Algebraic Cycles. NATO Science Series (Series C: Mathematical and Physical Sciences), vol 548. Springer, Dordrecht, 2000

work page 2000

[18] [18]

M.\ Levine, Mixed Motives , Math.\ Surveys and Monographs 57 AMS, Providence, 1998

work page 1998

[19] [19]

Springer-Verlag, 2005

M.\ Levine, Mixed Motives , in: Handbook of K-theory, vol.\ 1, E.M.\ Friedlander, D.R.\ Grayson, eds., 429--522. Springer-Verlag, 2005

work page 2005

[20] [20]

S.\ Mochizuki, A survey on Gersten's conjecture , arXiv:1608.08114

work page internal anchor Pith review Pith/arXiv arXiv

[21] [21]

J.S.\ Milne, \'Etale cohomology , Princeton University Press, Princeton, NJ, 1980

work page 1980

[22] [22]

D.\ Quillen, Higher algebraic K-theory, I , Lecture Notes in Mathematics 341, Springer, Berlin, 1973, 77--139

work page 1973

[23] [23]

Farkas et al

S.\ Schreieder, Unramified cohomology, algebraic cycles and rationality , in: G. Farkas et al. (eds), Rationality of Varieties, Progress in Mathematics 342, Birkh\"auser (2021), 345--388

work page 2021

[24] [24]

S.\ Schreieder, Refined unramified homology of schemes , Preprint 2021, arXiv:2010.05814v5

work page arXiv 2021

[25] [25]

S.\ Schreieder, Infinite torsion in Griffiths groups , arXiv:2011.15047, to appear in Journal of the EMS

work page arXiv 2011

[26] [26]

J.L.\ Verdier, A duality theorem in the \'etale cohomology of schemes , in: Proceedings of a Conference on Local Fields pp 184--198, 1967

work page 1967