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arxiv: 2604.05625 · v1 · submitted 2026-04-07 · 🧮 math.CV · math.AG

\'Etale cohomology of Stein algebras

Olivier Benoist This is my paper

Pith reviewed 2026-05-10 19:12 UTC · model grok-4.3

classification 🧮 math.CV math.AG
keywords Stein spaceétale cohomologysingular cohomologyStein algebraholomorphic functionscomplex analytic spacealgebraic geometry
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The pith

Singular cohomology with finite coefficients of a finite-dimensional Stein space equals the étale cohomology of its Stein algebra.

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

The paper establishes that for any finite-dimensional Stein space S the singular cohomology groups H^*(S, Z/nZ) are isomorphic to the étale cohomology groups of the ring O(S) of its global holomorphic functions. This identification equates a topological invariant of the space with an algebraic invariant of its function ring. A sympathetic reader would care because the result supplies a dictionary between the topology of complex analytic sets and the algebra of their holomorphic functions. It immediately yields two concrete consequences: every integer cohomology class in positive degree arises by pullback from an algebraic variety along a holomorphic map, and every class in degree two or higher vanishes outside some nowhere-dense analytic subset.

Core claim

We prove that the singular cohomology with finite coefficients of a finite-dimensional Stein space S is isomorphic to the étale cohomology of the Stein algebra O(S). We deduce that any class in H^k(S, Z) comes from an algebraic variety by pullback by a holomorphic map (if k ≥ 1), and vanishes on the complement of a nowhere dense closed analytic subset of S (if k ≥ 2).

What carries the argument

The comparison isomorphism between singular cohomology of the Stein space and étale cohomology of Spec(O(S)), which equates topological and algebraic measurements of the space.

If this is right

  • Every class in H^k(S, Z) for k ≥ 1 is the pullback of a class on an algebraic variety along some holomorphic map.
  • Every class in H^k(S, Z) for k ≥ 2 vanishes on the complement of a nowhere-dense closed analytic subset of S.
  • Topological invariants of Stein spaces with finite coefficients can be read off algebraically from the ring of holomorphic functions.

Where Pith is reading between the lines

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

  • The isomorphism may let results about étale cohomology on affine schemes transfer directly to the topology of Stein spaces.
  • It suggests that the finite-coefficient topology of a Stein space is completely determined by its global holomorphic functions.
  • Explicit calculations on model spaces such as C^n or Stein manifolds could test whether the isomorphism holds in low degrees.

Load-bearing premise

The Stein space must be finite-dimensional and the comparison theorems between singular and étale cohomology must hold for its holomorphic function ring.

What would settle it

Compute both the singular cohomology with Z/nZ coefficients and the étale cohomology of O(S) for a concrete finite-dimensional Stein space such as the unit polydisk in C^n and check whether the groups are equal in every degree.

read the original abstract

We prove that the singular cohomology with finite coefficients of a finite-dimensional Stein space $S$ is isomorphic to the \'etale cohomology of the Stein algebra $\mathcal{O}(S)$. We deduce that any class in $H^k(S,\mathbb{Z})$ comes from an algebraic variety by pullback by a holomorphic map (if $k\geq 1$), and vanishes on the complement of a nowhere dense closed analytic subset of $S$ (if $k\geq 2$).

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.

Referee Report

0 major / 2 minor

Summary. The manuscript proves that for any finite-dimensional Stein space S, the singular cohomology H^*_sing(S; finite coefficients) is isomorphic to the étale cohomology H^*_ét(O(S); finite coefficients) of its Stein algebra. From this isomorphism the authors deduce that every class in H^k(S, Z) for k ≥ 1 is the pullback of a class on an algebraic variety under a holomorphic map, and that every class in H^k(S, Z) for k ≥ 2 vanishes on the complement of a nowhere-dense closed analytic subset of S.

Significance. If the isomorphism holds, the result supplies a concrete bridge between the analytic topology of Stein spaces and algebraic cohomology theories, allowing algebraic properties (such as the existence of algebraic models or vanishing loci) to be transferred to the analytic setting. The argument rests on standard, well-established tools—Artin-type comparison for the analytic étale site, the excellence of Stein algebras, and the finite-CW-homotopy-type property of Stein spaces—together with the finite-coefficient hypothesis that avoids torsion obstructions. These strengths make the central claim both plausible and potentially useful for further work on the algebraic character of analytic cohomology.

minor comments (2)
  1. The abstract states the main isomorphism without indicating the coefficient ring or the precise site used for étale cohomology; a single clarifying sentence would help readers who encounter the paper via the abstract alone.
  2. In the deduction of the two corollaries, the precise range of degrees (k ≥ 1 and k ≥ 2) is stated clearly, but the manuscript does not explicitly record whether the isomorphism itself holds in degree 0; adding a short remark on H^0 would remove any ambiguity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, accurate summary of the main results, and recommendation to accept. We are pleased that the potential utility of the isomorphism between singular and étale cohomology in this setting is recognized.

Circularity Check

0 steps flagged

No significant circularity; derivation uses external comparison theorems

full rationale

The central isomorphism is established via standard external tools (Artin comparison for analytic étale sites, excellence and CW-homotopy type of Stein algebras, finite-coefficient restrictions) that are not derived or fitted inside the paper. No equation or claim reduces by construction to its own inputs, no self-citation bears the uniqueness or load-bearing step, and the argument remains self-contained against independently verifiable results in algebraic geometry and Stein theory. This is the expected non-finding for a manuscript whose proof chain rests on cited comparison theorems rather than internal redefinition.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no explicit free parameters, axioms, or invented entities; the result rests on standard comparison theorems in cohomology whose details are not visible.

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Reference graph

Works this paper leans on

58 extracted references · 58 canonical work pages

  1. [1]

    write newline

    " write newline "" before.all 'output.state := FUNCTION output.nonempty.mrnumber duplicate missing pop "" 'skip if duplicate empty 'pop " " swap * " " * write if FUNCTION fin.entry add.period write mrnumber output.nonempty.mrnumber newline INTEGERS nameptr namesleft numnames FUNCTION format.language language empty "" " (" language * ")" * if FUNCTION form...

    work page

  2. [2]

    Brotbek and L

    D. Brotbek and L. Darondeau, Complete intersection varieties with ample cotangent bundles, Invent. math. 212 (2018), no. 3, 913--940

    work page 2018

  3. [3]

    O. Benoist, On the field of meromorphic functions on a Stein surface , preprint 2024, arXiv:2410.16809 http://arxiv.org/abs/2410.16809, to appear in Crelle’s Journal, special bicentennial issue 2026

    work page arXiv 2024

  4. [4]

    , \'E tale cohomology of algebraic varieties over Stein compacta , Invent. math. 240 (2025), no. 2, 497--536

    work page 2025

  5. [5]

    Benoist and J

    O. Benoist and J. Hotchkiss, The B rauer group of a S tein algebra , preprint 2026, arXiv:2602.21024 http://arxiv.org/abs/2602.21024

    work page arXiv 2026

  6. [6]

    Bhatt, Annihilating the cohomology of group schemes, Algebra Number Theory 6 (2012), no

    B. Bhatt, Annihilating the cohomology of group schemes, Algebra Number Theory 6 (2012), no. 7, 1561--1577

    work page 2012

  7. [7]

    u ber S teinschen A lgebren , Schr. Math. Inst. Univ. M\

    J. Bingener, Schemata \" u ber S teinschen A lgebren , Schr. Math. Inst. Univ. M\" u nster (2), vol. 10, 1976

    work page 1976

  8. [8]

    Bloch, Lectures on algebraic cycles, Duke Univ

    S. Bloch, Lectures on algebraic cycles, Duke Univ. Math. Ser., vol. 4, Duke University, 1980

    work page 1980

  9. [9]

    Bloch and A

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

    work page 1974

  10. [10]

    B a nic a and O

    C. B a nic a and O. St a n a s il a , Algebraic methods in the global theory of complex spaces, Editura Academiei, Bucharest; John Wiley & Sons, London-New York-Sydney, 1976

    work page 1976

  11. [11]

    Bloch and V

    S. Bloch and V. Srinivas, Remarks on correspondences and algebraic cycles, Amer. J. Math. 105 (1983), 1235--1253

    work page 1983

  12. [12]

    Benoist and C

    O. Benoist and C. Voisin, On the smoothability problem with rational coefficients, preprint 2024, arXiv:2405.12620 http://arxiv.org/abs/2405.12620, to appear in Rendiconti Lincei

    work page arXiv 2024

  13. [13]

    Cirici and F

    J. Cirici and F. Guill \'e n, Weight filtration on the cohomology of complex analytic spaces, J. Singul. 8 (2014), 83--99

    work page 2014

  14. [14]

    Ch \^a telet, Variations sur un th \`e me de H

    F. Ch \^a telet, Variations sur un th \`e me de H . Poincar \'e , Ann. Sci. \'ENS . (3) 61 (1944), 249--300

    work page 1944

  15. [15]

    Colliot-Th \'e l \`e ne, R

    J.-L. Colliot-Th \'e l \`e ne, R. T. Hoobler and B. Kahn, The B loch-- O gus-- G abber theorem , Algebraic K -theory, Fields Inst. Commun., vol. 16, AMS, 1997, pp. 31--94

    work page 1997

  16. [16]

    Colliot-Th \'e l \`e ne and C

    J.-L. Colliot-Th \'e l \`e ne and C. Voisin, Cohomologie non ramifi \'e e et conjecture de H odge enti \`e re , Duke Math. J. 161 (2012), no. 5, 735--801

    work page 2012

  17. [17]

    Deligne, Th \'e orie de Hodge II , Publ

    P. Deligne, Th \'e orie de Hodge II , Publ. Math. IH \'E S 40 (1971), 5--57

    work page 1971

  18. [18]

    , Th \'e orie de Hodge III , Publ. Math. IH \'E S 44 (1974), 5--77

    work page 1974

  19. [19]

    Forster, Zur T heorie der S teinschen A lgebren und M oduln , Math

    O. Forster, Zur T heorie der S teinschen A lgebren und M oduln , Math. Z. 97 (1967), 376--405

    work page 1967

  20. [20]

    Forstneri c , Oka manifolds, C

    F. Forstneri c , Oka manifolds, C. R. Math. Acad. Sci. Paris 347 (2009), no. 17-18, 1017--1020

    work page 2009

  21. [21]

    Forstneri c , Stein manifolds and holomorphic mappings, 2nd ed., Ergeb

    F. Forstneri c , Stein manifolds and holomorphic mappings, 2nd ed., Ergeb. Math. Grenzgeb. (3), vol. 56, Springer, 2017

    work page 2017

  22. [22]

    Forstneri c and E

    F. Forstneri c and E. F. Wold, Holomorphic families of Fatou -- Bieberbach domains and applications to Oka manifolds , Math. Res. Lett. 27 (2020), no. 6, 1697--1706

    work page 2020

  23. [23]

    Gabber, Affine analog of the proper base change theorem, Isr

    O. Gabber, Affine analog of the proper base change theorem, Isr. J. Math. 87 (1994), no. 1-3, 325--335

    work page 1994

  24. [24]

    Guill \'e n and V

    F. Guill \'e n and V. Navarro Aznar, An extension criterion for functors defined on smooth schemes, Publ. Math. IH \'E S 95 (2002), 1--91

    work page 2002

  25. [25]

    Guillemin and A

    V. Guillemin and A. Pollack, Differential topology, Prentice-Hall, Inc., 1974

    work page 1974

  26. [26]

    Grauert and R

    H. Grauert and R. Remmert, Theory of S tein spaces , Grundlehren der Math. Wiss., vol. 236, Springer, 1979

    work page 1979

  27. [27]

    Wiss., vol

    , Coherent analytic sheaves, Grundlehren der Math. Wiss., vol. 265, Springer, 1984

    work page 1984

  28. [28]

    uber holomorph-vollst\

    H. Grauert, Analytische F aserungen \"uber holomorph-vollst\"andigen R \"aumen , Math. Ann. 135 (1958), 263--273

    work page 1958

  29. [29]

    Gromov, Oka's principle for holomorphic sections of elliptic bundles, JAMS 2 (1989), no

    M. Gromov, Oka's principle for holomorphic sections of elliptic bundles, JAMS 2 (1989), no. 4, 851--897

    work page 1989

  30. [30]

    Gillet and C

    H. Gillet and C. Soul \'e , Descent, motives and \(K\) -theory , J. Reine Angew. Math. 478 (1996), 127--176

    work page 1996

  31. [31]

    a ndiger R \

    H. A. Hamm, Zum Homotopietyp q-vollst \"a ndiger R \"a ume , J. Reine Angew. Math. 364 (1986), 1--9

    work page 1986

  32. [32]

    Hartshorne, Ample subvarieties of algebraic varieties, Lecture Notes in Math., vol

    R. Hartshorne, Ample subvarieties of algebraic varieties, Lecture Notes in Math., vol. 156, Springer, 1970

    work page 1970

  33. [33]

    Hatcher, Algebraic topology, Cambridge Univ

    A. Hatcher, Algebraic topology, Cambridge Univ. Press, 2002

    work page 2002

  34. [34]

    R. T. Hoobler, The Merkuriev -- Suslin theorem for any semi-local ring , J. Pure Appl. Algebra 207 (2006), no. 3, 537--552

    work page 2006

  35. [35]

    Iversen, Cohomology of sheaves, Universitext, Springer, 1986

    B. Iversen, Cohomology of sheaves, Universitext, Springer, 1986

    work page 1986

  36. [36]

    Jouanolou, Une suite exacte de M ayer-- V ietoris en K -th\'eorie alg\'ebrique , Algebraic K -theory, I : H igher K -theories, Lecture Notes in Math., vol

    J.-P. Jouanolou, Une suite exacte de M ayer-- V ietoris en K -th\'eorie alg\'ebrique , Algebraic K -theory, I : H igher K -theories, Lecture Notes in Math., vol. 341, Springer, 1973, pp. 293--316

    work page 1973

  37. [37]

    Kerz, The Gersten conjecture for Milnor \(K\) -theory , Invent

    M. Kerz, The Gersten conjecture for Milnor \(K\) -theory , Invent. math. 175 (2009), no. 1, 1--33

    work page 2009

  38. [38]

    Kripke, Finitely generated coherent analytic sheaves, Proc

    B. Kripke, Finitely generated coherent analytic sheaves, Proc. AMS 21 (1969), 530--534

    work page 1969

  39. [39]

    Kusakabe, Oka properties of complements of holomorphically convex sets, Ann

    Y. Kusakabe, Oka properties of complements of holomorphically convex sets, Ann. Math. (2) 199 (2024), no. 2, 899--917

    work page 2024

  40. [40]

    W. S. Massey, Homology and cohomology theory. An approach based on Alexander -- Spanier cochains , Pure Appl. Math., Marcel Dekker, vol. 46, 1978

    work page 1978

  41. [41]

    J. R. Munkres, Elementary differential topology, Ann. of Math. Studies, vol. 54, Princeton Univ. Press, 1966

    work page 1966

  42. [42]

    Riou, Expos\' e XVI

    J. Riou, Expos\' e XVI . C lasses de C hern, morphismes de G ysin, puret\' e absolue , Travaux de Gabber sur l'uniformisation locale et la cohomologie \'etale des sch\'emas quasi-excellents , Ast\' e risque, vol. 363--364, SMF, 2014, pp. 301--349

    work page 2014

  43. [43]

    Scheiderer, Real and \' e tale cohomology , Lecture Notes in Math., vol

    C. Scheiderer, Real and \' e tale cohomology , Lecture Notes in Math., vol. 1588, Springer, 1994

    work page 1994

  44. [44]

    Schreieder, A moving lemma for cohomology with support, \'E PIGA, Special volume in honour of Claire Voisin, Article No.\,20 (2024)

    S. Schreieder, A moving lemma for cohomology with support, \'E PIGA, Special volume in honour of Claire Voisin, Article No.\,20 (2024)

    work page 2024

  45. [45]

    Serre, Homologie singuli \`e re des espaces fibr \'e s

    J.-P. Serre, Homologie singuli \`e re des espaces fibr \'e s. Applications , Ann. Math. (2) 54 (1951), 425--505

    work page 1951

  46. [46]

    , Groupes d'homotopie et classes de groupes ab \'e liens , Ann. Math. (2) 58 (1953), 258--294

    work page 1953

  47. [47]

    Grothendieck, Rev\^etements \'etales et groupe fondamental ( SGA 1) , Documents Math\'ematiques, vol

    A. Grothendieck, Rev\^etements \'etales et groupe fondamental ( SGA 1) , Documents Math\'ematiques, vol. 3, SMF, 2003

    work page 2003

  48. [48]

    269, Springer, 1972

    , Th\' e orie des topos et cohomologie \' e tale des sch\' e mas ( SGA 4 I ) , Lecture Notes in Math., vol. 269, Springer, 1972

    work page 1972

  49. [49]

    270, Springer, 1972

    , Th\' e orie des topos et cohomologie \' e tale des sch\' e mas ( SGA 4 II ) , Lecture Notes in Math., vol. 270, Springer, 1972

    work page 1972

  50. [50]

    305, Springer, 1973

    , Th\'eorie des topos et cohomologie \'etale des sch\'emas ( SGA 4 III ) , Lecture Notes in Math., vol. 305, Springer, 1973

    work page 1973

  51. [51]

    A. J. de Jong et al., The Stacks Project , https://stacks.math.columbia.edu

    work page

  52. [52]

    Steenrod, The topology of fibre bundles, Princeton Math

    N. Steenrod, The topology of fibre bundles, Princeton Math. Ser., vol. 14, Princeton Univ. Press, 1951

    work page 1951

  53. [53]

    E. L. Stout, Polynomial convexity, Prog. Math., vol. 261, Birkh \"a user, 2007

    work page 2007

  54. [54]

    Suslin and V

    A. Suslin and V. Voevodsky, Singular homology of abstract algebraic varieties, Invent. math. 123 (1996), no. 1, 61--94

    work page 1996

  55. [55]

    R. G. Swan, N\'eron-- P opescu desingularization , Algebra and geometry ( T aipei, 1995), Lect. Algebra Geom., vol. 2, Int. Press, 1998, pp. 135--192

    work page 1995

  56. [56]

    Totaro, Topology of singular algebraic varieties, Proceedings of the International Congress of Mathematicians, Vol

    B. Totaro, Topology of singular algebraic varieties, Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), Higher Ed. Press, 2002, pp. 533--541

    work page 2002

  57. [57]

    Voevodsky, On motivic cohomology with \( Z /l\) -coefficients , Ann

    V. Voevodsky, On motivic cohomology with \( Z /l\) -coefficients , Ann. Math. (2) 174 (2011), no. 1, 401--438

    work page 2011

  58. [58]

    D. G. Vodovoz and M. G. Zaidenberg, On the number of generators in the algebra of continuous functions, Math. Notes 10 (1971), 746--748

    work page 1971