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arxiv: 2009.07225 · v4 · submitted 2020-09-15 · 🧮 math.KT · math.AT

Hermitian K-theory for stable infty-categories III: Grothendieck-Witt groups of rings

Baptiste Calm\`es , Emanuele Dotto , Yonatan Harpaz , Fabian Hebestreit , Markus Land , Kristian Moi , Denis Nardin , Thomas Nikolaus
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Wolfgang Steimle
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Pith reviewed 2026-05-24 14:24 UTC · model grok-4.3

classification 🧮 math.KT math.AT
keywords Grothendieck-Witt groupshermitian K-theoryalgebraic K-theoryL-theoryfibre sequencesDedekind ringsnumber fieldslocalisation sequences
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The pith

A fibre sequence relates Grothendieck-Witt theory of rings to K-theory C2-orbits and symmetric L-theory.

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

The paper proves a fibre sequence that connects the classical Grothendieck-Witt groups of any ring R to the homotopy C2-orbits of its K-theory and to Ranicki's symmetric L-theory. This sequence makes it possible to remove the assumption that 2 is a unit in R from many prior results on these groups. Sympathetic readers would care because the new results include explicit calculations for the integers, finite generation statements for rings of integers in number fields, and a solution to the Berrick-Karoubi conjecture on localisation sequences.

Core claim

We establish a fibre sequence relating the classical Grothendieck-Witt theory of a ring R to the homotopy C2-orbits of its K-theory and Ranicki's original (non-periodic) symmetric L-theory. We use this fibre sequence to remove the assumption that 2 is a unit in R from various results about Grothendieck-Witt groups. For instance, we solve the homotopy limit problem for Dedekind rings whose fraction field is a number field, calculate the various flavours of Grothendieck-Witt groups of Z, show that the Grothendieck-Witt groups of rings of integers in number fields are finitely generated, and that the comparison map from quadratic to symmetric Grothendieck-Witt theory of Noetherian rings of глоб

What carries the argument

The fibre sequence relating Grothendieck-Witt theory, the homotopy C2-orbits of K-theory, and non-periodic symmetric L-theory.

Load-bearing premise

The fibre sequence and its applications rest on the hermitian K-theory for stable infinity-categories developed in the earlier papers of the series and on the specific definitions of the spectra and the C2-action.

What would settle it

Finding a ring R where 2 is not invertible such that the proposed fibre sequence does not hold exactly, or a Dedekind ring whose Grothendieck-Witt groups fail to be finitely generated.

read the original abstract

We establish a fibre sequence relating the classical Grothendieck-Witt theory of a ring $R$ to the homotopy $\mathrm{C}_2$-orbits of its K-theory and Ranicki's original (non-periodic) symmetric L-theory. We use this fibre sequence to remove the assumption that 2 is a unit in $R$ from various results about Grothendieck-Witt groups. For instance, we solve the homotopy limit problem for Dedekind rings whose fraction field is a number field, calculate the various flavours of Grothendieck-Witt groups of $\mathbb{Z}$, show that the Grothendieck-Witt groups of rings of integers in number fields are finitely generated, and that the comparison map from quadratic to symmetric Grothendieck-Witt theory of Noetherian rings of global dimension $d$ is an equivalence in degrees $\geq d+3$. As an important tool, we establish the hermitian analogue of Quillen's localisation-d\'evissage sequence for Dedekind rings and use it to solve a conjecture of Berrick-Karoubi.

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 establishes a fibre sequence GW(R) → K(R)^{hC₂} → L^s(R) relating the classical Grothendieck-Witt spectrum of a ring R, the C₂-homotopy orbits of its algebraic K-theory spectrum, and Ranicki's non-periodic symmetric L-theory spectrum. This sequence is obtained within the hermitian K-theory framework of the preceding papers in the series and is used to remove the hypothesis that 2 is invertible in R from several results, including the homotopy-limit problem for Dedekind rings with number-field fraction field, finite generation of Grothendieck-Witt groups of rings of integers, high-degree comparison maps between quadratic and symmetric theories for Noetherian rings of global dimension d, and a solution to the Berrick-Karoubi conjecture via a hermitian analogue of Quillen's localisation-dévissage sequence for Dedekind rings.

Significance. If the fibre sequence and the comparison identifications hold, the work supplies a direct computational relation between classical invariants that extends prior results (which required 2 invertible) to general rings, with concrete arithmetic consequences such as finite generation over ℤ and number fields. The hermitian dévissage theorem is a notable technical contribution. The paper supplies the necessary comparison maps and C₂-action while relying on the foundational development of hermitian K-theory for stable ∞-categories from the earlier parts of the series.

minor comments (2)
  1. The precise model of the C₂-action on K(R) and the construction of the comparison maps in the fibre sequence are referenced to prior papers; a brief self-contained recap of the relevant spectra and maps in an early section would improve readability for readers who have not followed the entire series.
  2. Notation for the various flavours of Grothendieck-Witt and L-theory spectra (quadratic vs. symmetric, periodic vs. non-periodic) is introduced gradually; a consolidated table or diagram early in the paper would help track the identifications used in the applications.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive and detailed summary of our work, and for recommending acceptance of the manuscript. We are pleased that the fibre sequence and its applications to removing the 2-unit hypothesis were viewed as significant.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The central result is the construction of the fibre sequence relating GW(R), K(R)^{hC₂} and Ranicki's non-periodic symmetric L-theory, obtained by supplying comparison maps and the C₂-action inside the hermitian K-theory framework of the series. No quoted equation or step reduces a claimed prediction or first-principles result to a fitted input, self-definition, or prior self-citation by construction. The listed applications are formal consequences of the sequence once granted. While the work continues a series, the derivation chain remains independent of the target result and does not match any enumerated circularity pattern.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are identifiable from the abstract. The work relies on the stable infinity-category framework and hermitian K-theory constructions developed in prior parts of the series.

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Forward citations

Cited by 2 Pith papers

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  1. The Classification of Pauli Stabilizer Codes: A Lattice and Continuum Treatise

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  2. An equivalence between two frameworks for real algebraic K-theory

    math.KT 2024-10 unverdicted novelty 6.0

    The paper proves that the real K-theory genuine C₂-spectra defined by Calmès et al. for Poincaré ∞-categories coincide with those defined by the authors for Waldhausen ∞-categories with genuine duality.

Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages · cited by 2 Pith papers · 2 internal anchors

  1. [1]

    This paper series. B. Calmès, E. Dotto, Y. Harpaz, F Hebestreit, M. Land, D. Nard in, K. Moi, T. Nikolaus, and W. Steimle, Hermitian K-theory for stable ∞ -categories, [I] Part I: Foundations, arXiv:2009.07223,

    work page arXiv 2009

  2. [2]

    [II] Part II: Cobordism categories and additivity , arXiv:2009.07224,

    work page arXiv 2009

  3. [3]

    [III] Part III: Grothendieck-Witt groups of rings, arXiv:2009.07225,

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  4. [4]

    [Bal05] P

    arXiv:2005.06778. [Bal05] P. Balmer, Witt groups, Handbook of K-theory (2005), 539–576. [Bar15] C. Barwick, On exact ∞ -categories and the Theorem of the Heart , Compositio Mathematica 151 (2015), no. 11, 2160–

    work page arXiv 2005

  5. [5]

    [BK05] A. J. Berrick and M. Karoubi, Hermitian K-theory of the integers , American Journal of Mathematics 127 (2005), no. 4, 785–823. [BKSØ15] A. J. Berrick, M. Karoubi, M. Schlichting, and P. A. Østvær, The homotopy fixed point theorem and the Quillen- Lichtenbaum conjecture in hermitian K-theory, Advances in Mathematics 278 (2015), 34–55. [BGT13] A. J. B...

    work page 2005

  6. [6]

    Burghelea and Z

    [BF85] D. Burghelea and Z. Fiedorowicz, Hermitian algebraic K-theory of simplicial rings and topol ogical spaces , Journal de Mathématiques Pures et Appliquées (9) 64 (1985), no. 2, 175–235. [Cha87] R. Charney, A generalization of a theorem of Vogtmann , Proceedings of the Northwestern conference on cohomology of groups (Evanston, Ill., 1985), 1987, pp. 1...

    work page 1985

  7. [7]

    [CMM18] D

    arXiv:1905.06611. [CMM18] D. Clausen, A. Mathew, and M. Morrow, K-theory and topological cyclic homology of henselian pair s,

    work page arXiv 1905

  8. [8]

    [DO19] E

    arXiv:1803.10897. [DO19] E. Dotto and C. Ogle, K-theory of Hermitian Mackey functors, real traces, and ass embly, Annals of K-Theory 4 (2019), no. 2, 243–316. [DG02] W. G. Dwyer and J. P. C. Greenlees, Complete modules and torsion modules , American Journal of Mathematics 124 (2002), no. 1, 199–220. [FGV20] T. Feng, S. Galatius, and A. Venkatesh, The Galo...

    work page arXiv 2019

  9. [9]

    [FP78] Z

    arXiv:2007.15078. [FP78] Z. Fiedorowicz and S. Priddy, Homology of classical groups over finite fields and their asso ciated infinite loop spaces , Vol. 674, Springer,

    work page arXiv 2007

  10. [10]

    [Fri76] E. M. Friedlander, Computations of K-theories of finite fields , Topology 15 (1976), no. 1, 87–109. [HLN20] F. Hebestreit, M. Land, and T. Nikolaus, On the homotopy types of L-spectra of the integers , Journal of Topology 14 (2020), no. 1, 183–214. [HS21] F. Hebestreit and W. Steimle, Stable moduli spaces of hermitian forms ,

    work page 1976

  11. [11]

    [Hil81] H

    arXiv:2103.13911v1. [Hil81] H. L. Hiller, /u1D706-rings and algebraic K-theory, Journal of Pure and Applied Algebra 20 (1981), no. 3, 241–266. [Hor02] J. Hornbostel, Constructions and dévissage in hermitian K-theory , K-theory 26 (2002), no. 2, 139–170. [HS04] J. Hornbostel and M. Schlichting, Localization in hermitian K-theory of rings, Journal of the Lo...

    work page arXiv 1981

  12. [12]

    [KSS08] L

    arXiv:2002.05255, to appear in Proceedings of the American Mathematical Society. [KSS08] L. A. Kurdachenko, N. N. Semko, and I. Y. Subbotin, Insight into modules over Dedekind domains, Proceedings of Institute of Mathematics of NAS of Ukraine. Mathematics and its Applic ations, vol. 75, Nats ¯ıonal′na Akadem¯ıya Nauk Ukraïni, ¯ Institut Matematiki, Kiev, ...

    work page arXiv 2002

  13. [13]

    Lin, On conjectures of Mahowald, Segal and Sullivan , Mathematical Proceedings of the Cambridge Philosophical Society, 1980, pp

    [Lin80] W.-H. Lin, On conjectures of Mahowald, Segal and Sullivan , Mathematical Proceedings of the Cambridge Philosophical Society, 1980, pp. 449–458. [Lur11] J. Lurie, Lecture notes on algebraic L-theory and surgery ,

    work page 1980

  14. [14]

    group-completion

    Available from the author’s webpage . [MS76] D. McDuff and G. Segal, Homology fibrations and the “group-completion” theorem , Inventiones Mathematicae 31 (1976), no. 3, 279–284. [MH73] J. W. Milnor and D. Husemoller, Symmetric bilinear forms, Vol. 73, Springer,

    work page 1976

  15. [15]

    [NS18] T

    Translated from the 1992 German original. [NS18] T. Nikolaus and P. Scholze, On topological cyclic homology , Acta Mathematica 221 (2018), no. 2, 203–409. [QSS79] H.-G. Quebbemann, W. Scharlau, and M. Schulte, Quadratic and hermitian forms in additive and abelian categ ories, Journal of Algebra 59 (1979), no. 2, 264–289. [Qui72] D. Quillen, On the cohomol...

    work page 1992

  16. [16]

    Schlichting, Hermitian K-theory of exact categories , Journal of K-theory 5 (2010), no

    [Sch10a] M. Schlichting, Hermitian K-theory of exact categories , Journal of K-theory 5 (2010), no. 1, 105–165. [Sch10b] , The Mayer-Vietoris principle for Grothendieck-Witt group s of schemes , Inventiones Mathematicae 179 (2010), no. 2,

    work page 2010

  17. [17]

    7, 1729–1844

    [Sch17] , Hermitian K-theory, derived equivalences and Karoubi’s fu ndamental theorem, Journal of Pure and Applied Algebra 221 (2017), no. 7, 1729–1844. [Sch19a] , Higher K-theory of forms I. from rings to exact categories , Journal of the Institute of Mathematics of Jussieu (2019), 1–69. [Sch19b] , Symplectic and orthogonal K-groups of the integers , Com...

    work page 2017

  18. [18]

    A Grothendieck-Witt space for stable infinity categories with duality

    arXiv:1610.10044. [SP18] Stacks Project authors, Stacks project,

    work page internal anchor Pith review Pith/arXiv arXiv

  19. [19]

    [Tho83] R

    Available at stacks.math.columbia.edu. [Tho83] R. W. Thomason, The homotopy limit problem, Proceedings of the Northwestern homotopy theory conferen ce (Evanston, Ill., 1982), 1983, pp. 407–419. [Wal70] C. T. C. Wall, On the classification of hermitian forms. I. Rings of algebraic integers, Compositio Mathematica 22 (1970), 425–451. HERMITIAN K-THEORY FOR S...

    work page 1982

  20. [20]

    Weiss and B

    [WW14] M. Weiss and B. Williams, Automorphisms of manifolds and algebraic K-theory: Part II I, Memoirs of the American Mathematical Society 231 (2014), no. 1084, vi+110. UNIVERSITÉ D ’ ARTOIS, L ABORATOIRE DE MATHÉMATIQUES DE LENS (LML), UR 2462, L ENS, F RANCE Email address: baptiste.calmes@univ-artois.fr UNIVERSITY OF WARWICK; M ATHEMATICS INSTITUTE ; C...

    work page 2014