pith. sign in

arxiv: 2312.15782 · v4 · submitted 2023-12-25 · 🧮 math.AG · math.AC

Cancellation and splitting of Symplectic modules in the critical range and Euler class group

Rakesh Pawar , Husney Parvez Sarwar This is my paper

Pith reviewed 2026-05-24 04:49 UTC · model grok-4.3

classification 🧮 math.AG math.AC
keywords symplectic modulescancellationsplittingEuler class groupChow groupA1-homotopy categoryPostnikov towerssmooth affine varieties
0
0 comments X

The pith

Symplectic modules over smooth affine varieties cancel in the critical range via A1-homotopy methods.

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

The paper proves a cancellation theorem for symplectic modules that parallels Fasel's result for projective modules. It achieves this by analyzing Postnikov towers in the A1-homotopy category and showing that certain top cohomology groups with coefficients in homotopy sheaves vanish. The same vanishing yields a splitting result akin to Murthy's theorem for these modules. It further uses the vanishing to establish that the (d-1)th Euler class group is isomorphic to the (d-1)th Chow group for smooth affine varieties of dimension d in some cases. A reader would care because the results connect the structure of modules over polynomial rings to classical geometric invariants.

Core claim

We prove that symplectic modules satisfy cancellation and splitting in the critical range by carefully analyzing Postnikov towers in the A1-homotopy category and establishing the vanishing of top cohomology with coefficients in some homotopy sheaves. This gives an analog of Fasel's projective module cancellation result and Murthy's splitting theorem. As an application of the vanishing, we partially answer a question about the isomorphism of the (d-1)th Euler class group and the (d-1)th Chow group for smooth affine varieties of dimension d.

What carries the argument

Postnikov towers in the A1-homotopy category, used to prove vanishing of top cohomology groups with coefficients in homotopy sheaves.

If this is right

  • Symplectic modules of rank at least the critical value cancel over the ring of the variety.
  • Such modules split off a hyperbolic summand under the stated rank and dimension conditions.
  • The (d-1)th Euler class group equals the (d-1)th Chow group for the smooth affine varieties where the vanishing applies.
  • Splitting and cancellation results extend the known theorems from projective modules to the symplectic setting.

Where Pith is reading between the lines

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

  • The same Postnikov analysis might apply to other module categories or to higher-degree Euler class groups if vanishing can be shown there.
  • The isomorphism result could simplify explicit computations of Euler classes on concrete affine varieties of low dimension.
  • Connections between A1-homotopy vanishing and classical Chow groups may extend to related questions in motivic cohomology.

Load-bearing premise

The top cohomology groups with coefficients in the relevant homotopy sheaves vanish for the Postnikov towers of symplectic modules over the smooth affine variety.

What would settle it

A smooth affine variety of dimension d together with a symplectic module in the critical range where either cancellation fails or the (d-1)th Euler class group differs from the (d-1)th Chow group.

read the original abstract

In this paper, we discuss the cancellation and splitting of the symplectic modules. The symplectic cancellation result presented here can be thought of as an analog of the Projective module cancellation result of Fasel. The symplectic splitting is similar to Murthy's splitting theorem. To prove the cancellation and splitting, we carefully analyze the Postnikov towers in the $\mathbb{A}^1$-homotopy category. Then we prove the vanishing of top cohomology with coefficients in some homotopy sheaf. As another application of the vanishing results, we answer partially a question of Mrinal Das about the isomorphism of $(d-1)$-th Euler class group and $(d-1)$-th Chow group, where $d$ is the dimension of the underlying smooth affine variety.

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

1 major / 0 minor

Summary. The paper establishes cancellation and splitting results for symplectic modules over smooth affine varieties of dimension d in the critical range, by analyzing Postnikov towers in the A¹-homotopy category and proving vanishing of top cohomology groups with coefficients in relevant homotopy sheaves. As an application, it partially answers a question of Mrinal Das by establishing an isomorphism between the (d-1)th Euler class group and the (d-1)th Chow group in some cases. The results are positioned as analogs of Fasel's projective module cancellation and Murthy's splitting theorem.

Significance. If the vanishing results hold, the work provides a symplectic analog to established projective module results and advances the theory of Euler class groups for affine varieties. The approach via A¹-homotopy theory and explicit vanishing statements offers a structured method that could extend to related questions in algebraic geometry.

major comments (1)
  1. [Postnikov tower analysis and vanishing results (as described in the abstract and central to the proofs)] The vanishing of the top cohomology group with coefficients in the relevant homotopy sheaf (asserted after the Postnikov tower analysis) is load-bearing for both the cancellation theorem and the partial isomorphism with Chow groups. The manuscript must explicitly show why this vanishing holds in the symplectic setting, as the relevant homotopy sheaves and their cohomology differ from those in Fasel's projective module case; transfer is not automatic.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We are pleased that the significance of the symplectic analogs to Fasel's and Murthy's results is recognized. We address the major comment below.

read point-by-point responses

  1. Referee: The vanishing of the top cohomology group with coefficients in the relevant homotopy sheaf (asserted after the Postnikov tower analysis) is load-bearing for both the cancellation theorem and the partial isomorphism with Chow groups. The manuscript must explicitly show why this vanishing holds in the symplectic setting, as the relevant homotopy sheaves and their cohomology differ from those in Fasel's projective module case; transfer is not automatic.

    Authors: We agree that an explicit demonstration of the vanishing in the symplectic setting is essential, as the homotopy sheaves involved are indeed different from those in the projective module case. Our manuscript provides a proof of this vanishing by carefully analyzing the Postnikov towers adapted to the symplectic modules and using the specific properties of the corresponding homotopy sheaves in the A¹-homotopy category. The argument is self-contained and does not assume a direct transfer from Fasel's work. To strengthen the presentation and ensure clarity on this point, we will revise the relevant sections to include more detailed explanations of the differences in the sheaves and the specific steps in the vanishing proof. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation rests on internal proofs and external citations

full rationale

The paper's central claims rest on an explicit analysis of Postnikov towers in the A¹-homotopy category followed by a proof of vanishing of top cohomology groups with coefficients in the relevant homotopy sheaves. These steps are presented as new results within the manuscript rather than reductions to fitted parameters or prior self-citations. Fasel’s projective-module cancellation and Murthy’s splitting theorem are invoked as independent external benchmarks. No equation or definition is shown to be equivalent to its own input by construction, and the partial isomorphism between Euler class groups and Chow groups is derived as an application of the newly proved vanishing. The derivation is therefore self-contained against the stated external results.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work relies on standard background from A1-homotopy theory and prior cancellation theorems; no free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • standard math Standard properties of the A1-homotopy category and Postnikov towers allow analysis of cohomology vanishing for symplectic modules.
    Invoked to prove the cancellation and splitting results.

pith-pipeline@v0.9.0 · 5653 in / 1240 out tokens · 21913 ms · 2026-05-24T04:49:07.721943+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

33 extracted references · 33 canonical work pages · 1 internal anchor

  1. [1]

    A. Asok, T. Bachmann, and M.J. Hopkins, On P1-stabilization in unstable motivic homotopy theory. https://arxiv.org/pdf/2306.04631.pdf 2, 6

    work page arXiv

  2. [2]

    A. Asok, B. Doran, and J. Fasel, Smooth models of motivic spheres and the clutching construc tion. Int. Math. Res. Not. IMRN 2017, no. 6, 1890–1925. 8, 11

    work page 2017

  3. [3]

    Asok and J

    A. Asok and J. Fasel, Splitting vector bundles outside the stable range and A1-homotopy sheaves of punctured affine spaces. J. Amer. Math. Soc. 28 (2015), no. 4, 1031–1062. 2, 6, 8, 9, 10, 12

    work page 2015

  4. [4]

    Asok and J

    A. Asok and J. Fasel, Toward a meta-stable range in A1-homotopy theory of punctured affine spaces. Ober- wolfach Report: June 2013, https://dornsife.usc.edu/ass ets/sites/1176/docs/PDF/pinanminus0OWR.pdf 8, 12

    work page 2013

  5. [5]

    Asok and J

    A. Asok and J. Fasel, A cohomological classification of vector bundles on smooth a ffine threefolds. Duke Math. J., 163(14):2561-2601, 2014. 2, 8, 9, 10, 16, 20, 21

    work page 2014

  6. [6]

    Asok, A. and J. Fasel. Comparing Euler classes Quart. J. Math. Oxford Ser. (2) 67, no. 4 (2016): 60335. 14

    work page 2016

  7. [7]

    An explicit KO-degree map and applications

    A. Asok and J. Fasel, An explicit KO-degree map and applications. J. Topology, 10(1):268300, 2017. arXiv:1403.4588. 6

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  8. [8]

    A. Asok, M. Hoyois and M. W endt, Affine representability results in A1-homotopy theory, II: Principal bundles and homogeneous spaces. Geom. Topol. 22 (2018), no. 2, 1181–1225. 5, 18

    work page 2018

  9. [9]

    Asok and J

    A. Asok and J. Fasel, Euler class groups and motivic stable cohomotopy. With an ap pendix by Mrinal Kanti Das. J. Eur. Math. Soc. (JEMS) 24 (2022), no. 8, 27752822. 15, 20, 22

    work page 2022

  10. [10]

    Asok and J

    A. Asok and J. Fasel, and M.J. Hopkins, Localization and nilpotent spaces in A1-homotopy theory. Compos. Math., 158(3):654-720, 2022 5

    work page 2022

  11. [11]

    Bass, K-theory and stable algebra

    H. Bass, K-theory and stable algebra. Publ. Math. Inst. Hautes ´Etudes Sci. 22 (1964), 560. 1

    work page 1964

  12. [12]

    Basu and Ravi A

    R. Basu and Ravi A. Rao, Injective stability for K1 of classical modules. J. Algebra323(2010), no.4, 867877

    work page 2010

  13. [13]

    Bhatwadekar, Cancellation theorems for projective modules over a two-di mensional ring and its poly- nomial extensions

    S.M. Bhatwadekar, Cancellation theorems for projective modules over a two-di mensional ring and its poly- nomial extensions. Compositio Math. 128 (2001), no. 3, 339359. 2

    work page 2001

  14. [14]

    Bhatwadekar and R

    S.M. Bhatwadekar and R. Sridharan, On Euler classes and stably free projective modules. Algebra, arithmetic and geometry, Part I, II (Mumbai, 2000), 139–158, Tata Inst. Fund. Res. Stud. Math., 16, Tata Inst. Fund. Res., Bombay, 2002. 20

    work page 2000

  15. [15]

    International Mathematics Research Notices, Volume 2021, Issue 13, July 2021, Pages 10073-10099, https://doi.org/10.1093/imrn/rnz102 2, 3, 19, 20

    Mrinal Kanti Das, A Criterion for Splitting of a Projective Module in Terms of I ts Generic Sections. International Mathematics Research Notices, Volume 2021, Issue 13, July 2021, Pages 10073-10099, https://doi.org/10.1093/imrn/rnz102 2, 3, 19, 20

    work page doi:10.1093/imrn/rnz102 2021

  16. [16]

    Peng Du, Enumerating non-stable vector bundles. Int. Math. Res. Not. IMRN 2022, no. 19, 14797–14864. 2, 12, 13, 15, 17, 19

    work page 2022

  17. [17]

    Fasel, Some remarks on orbit sets of unimodular rows

    J. Fasel, Some remarks on orbit sets of unimodular rows. Comment. Math. Helv. 86 (2011), no. 1, 13–39. 11

    work page 2011

  18. [18]

    Fasel, Mennicke symbols, K-cohomology and a Bass-Kubota theorem

    J. Fasel, Mennicke symbols, K-cohomology and a Bass-Kubota theorem. Trans. Amer. Math. Soc., 367(1):191208, 2015. 6, 11

    work page 2015

  19. [19]

    Fasel, Suslin’s cancellation conjecture in the smooth case

    J. Fasel, Suslin’s cancellation conjecture in the smooth case. arXiv: https://arxiv.org/pdf/2111.13088.pdf 1, 2, 3, 6, 7, 13, 17 CANCELLATION AND SPLITTING OF SYMPLECTIC MODULES IN THE CRI TICAL RANGE AND EULER CLASS GROUP 23

    work page arXiv

  20. [20]

    Fulton, Intersection theory , Springer-Verlag, 1998

    W. Fulton, Intersection theory , Springer-Verlag, 1998. 19

    work page 1998

  21. [21]

    Mohan Kumar, Stably free modules

    N. Mohan Kumar, Stably free modules. Amer. J. Math. 107 (1985), 14391443. 1

    work page 1985

  22. [22]

    Morel, and V

    F. Morel, and V. Voevodsky, A1-homotopy theory of schemes. Publications mathmatiques de lIHES 90 (2001): 45143, 1999 2, 3

    work page 2001

  23. [23]

    Morel, A1-algebraic topology over a field

    F. Morel, A1-algebraic topology over a field. LNM 2052, Springer, Heidelberg, 2012. 2, 4, 14

    work page 2052

  24. [24]

    Murthy, Zero cycles and projective modules

    M.P. Murthy, Zero cycles and projective modules. The Annals of Mathematics, 140(2), 405, 1994. 1

    work page 1994

  25. [25]

    Sarwar and M

    H.P. Sarwar and M. Schlichting, The third homology of symplectic groups and algebraic K-the ory, to appear in American Journal of Mathematics. https://doi.org/10.4 8550/arXiv.2111.01539 5

    work page arXiv

  26. [26]

    Serre, Modules projectifs et espaces fibr´ es ` a fibre vectorielle

    J.-P. Serre, Modules projectifs et espaces fibr´ es ` a fibre vectorielle. 1958 S´ eminaire P. Dubreil, M.-L. Dubreil- Jacotin et C. Pisot, 1957/58, Fasc. 2, Expos´ e 23 18 pp. 1

    work page 1958

  27. [27]

    Suslin On stably free modules , Mat

    A.A. Suslin On stably free modules , Mat. Sb. Novaya Seriya 102, no. 144 (4) (1977): 47991 15, 16

    work page 1977

  28. [28]

    A. A. Suslin, A cancellation theorem for projective modules over algebra s. Dokl. Akad. Nauk SSSR 236 (1977), 808-811 1

    work page 1977

  29. [29]

    Suslin and L.N

    A.A. Suslin and L.N. Vaserstein, Serres problem on projective modules over polynomial rings and algebraic K-theory. Math. USSR, Izvestija. 10 (1976), no 5, 937–1001. 11

    work page 1976

  30. [30]

    Syed, Cancellation of vector bundles of rank 3 with tr ivial Chern classes on smooth affine fourfolds, Journal of Pure and Applied Algebra, Volume 226, Issue 9, 202 2, 107038

    T. Syed, Cancellation of vector bundles of rank 3 with tr ivial Chern classes on smooth affine fourfolds, Journal of Pure and Applied Algebra, Volume 226, Issue 9, 202 2, 107038

    work page

  31. [31]

    Vaserstein, Stabilization for the classical groups over rings

    L.N. Vaserstein, Stabilization for the classical groups over rings. (Russian)Mat. Sb. (N.S.) 93 (135)(1974), 268295, 327

    work page 1974

  32. [32]

    Graduate Studies in Math

    Charles W eibel, The K-book: an introduction to algebraic K-theory. Graduate Studies in Math. vol. 145, AMS, 2013 9

    work page 2013

  33. [33]

    W endt, Variations in A1 on a theme of Mohan Kumar

    M. W endt, Variations in A1 on a theme of Mohan Kumar. Int. Math. Res. Not. IMRN, 2021, Issue 9, pp. 6621–6655. 17 UMPA, ´Ecole normale sup ´erieure de L yon (ENS de L yon), UMR 5669, CNRS, 46 alle e dItalie 69 364 L yon Cedex 07, France Email address : rakesh.pawar@ens-lyon.fr Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur...

    work page 2021