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Notes on Tate cohomology

T0 review · 2 major / 4 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read Tate cohomology can be defined in any three-functor formalism, with a universal property that yields its monoidality and functoriality.

desk verdict Clean, honestly-labeled axiomatization of Tate cohomology via three-functor formalisms; genuinely useful examples, but the whole edifice rests on the un-vetted Heyer–Mann preprint and a definition has a typo. read the letter →

arxiv 2601.17677 v2 pith:LXS6HWDA submitted 2026-01-25 math.AT math.AG

Arpon Raksit This is my paper · ORCID
classification math.ATmath.AG
keywords TatecohomologythreefunctorformalismnormmapPoincarédualitylaxsymmetricmonoidaluniversalpropertylocalsystemsquasicoherentsheaves
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

This paper establishes that Tate cohomology—the amalgam of homology and cohomology originally built for finite groups—can be defined in the very general setting of three-functor formalisms, which axiomatize how homology and cohomology behave for spaces, stacks, and other geometric objects. The definition is a cofiber of a norm map constructed from weak adjoints in a category of kernels, and the main theorem gives this construction a universal property: the canonical map from cohomology to Tate cohomology is initial among accessible functors that vanish on a specified class of nilpotent objects. From this universal property, the paper derives that Tate cohomology is automatically lax symmetric monoidal and functorial, generalizing what was previously known for local systems of spectra on spaces. Two families of examples show the scope: C-valued local systems on spaces, recovering and extending known group-theoretic Tate cohomology, and quasicoherent sheaves on prestacks, covering dualizable Hopf algebras and stably dualizable groups. If correct, this provides a uniform mechanism for producing Tate cohomology and its ring structures in any context with a six-functor-like calculus.

What carries the argument

The paper's machinery is the three-functor formalism D: a formal package assigning to each space (or stack) a symmetric monoidal ∞-category of coefficient systems, with ∗-pullback, !-pushforward, and ♯-pushforward functors subject to base-change and projection formulas. The key objects are the (∞,2)-category of kernels KD(S), in which weak adjoints (rather than honest adjoints) are enough to construct a norm map Nm_f:f_♯(-⊗ν_f)→f_* for any D-Tate map f. Tate cohomology f^t is the cofiber of this norm map. The universal property (Theorem 1.8.7) is proved using the left Kan extension / localization machinery relative to the thick tensor ideal of i-nilpotent objects, for i a D-Tate cover.

What would settle it

A concrete refutation would be an example of a stable presentable three-functor formalism with a D-Tate map f and Tate cover i in which the canonical map f_*→f^t is not initial among accessible functors vanishing on i-nilpotent objects—or, more basically, a failure of the cited local-descent lemma that would invalidate the existence of such covers in the examples.

Watch

Extended reading notes

Core claim

The paper defines, for every D-Tate map f:T→S in a three-functor formalism D, a Tate cohomology functor f^t:D(T)→D(S) as the cofiber of a norm map Nm_f:f_♯(-⊗ν_f)→f_*, where ν_f is a twisting object assembled from the diagonal of f. The central theorem is that, when a D-Tate cover i:V→T exists, the canonical map can:f_*→f^t is initial among accessible functors from D(T) to D(S) that vanish on i-nilpotent objects; moreover can and f^t carry canonical lax symmetric monoidal structures making can initial among such maps of lax symmetric monoidal functors. From this universal property the paper derives that Tate cohomology is functorial under maps that are D-Poincaré, and that norm maps are comp

Load-bearing premise

The whole edifice rests on the correctness of the recently introduced three-functor formalism, including the (∞,2)-categorical machinery of kernels and the cited lemmas establishing norm maps, Poincaré duality criteria, and descent; if any of those imported results is incomplete or wrong, the definitions and theorems here inherit the flaw.

Editorial extensions

If this is right

  • In any stable presentable three-functor formalism with a Tate cover, the Tate cohomology functor is automatically lax symmetric monoidal and the canonical map from cohomology respects this structure, so coefficient ring structures pass to Tate cohomology.
  • For C-valued local systems, the universal property endows Tate cohomology of a compact group space with lax symmetric monoidal structures, and yields functoriality of Tate cohomology under finite covering maps, and under the multiplication-by-n maps on the circle when C is 1-semiadditive.
  • For quasicoherent sheaves on prestacks, the same results give Tate cohomology for classifying stacks of dualizable Hopf algebras, providing a systematic account that includes the theory of stably dualizable groups.
  • The norm maps and Tate cohomology are compatible with base change and composition, so the classes of Poincaré, smooth, and terse maps form a genuine six-functor calculus.
  • The universal property gives a practical criterion: to construct a lax symmetric monoidal map out of Tate cohomology, it suffices to construct one out of cohomology and check vanishing on i-nilpotent objects.

Reading between the lines

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

  • If the underlying three-functor formalism is accepted, the same construction would supply Tate cohomology in other six-functor settings (for example p-adic or rigid-analytic geometries), where no group or space-level description is available, giving candidates for new equivariant invariants.
  • The universal property suggests interpreting Tate cohomology as a colocalization or completion of cohomology that kills the homology part; this could be formalized in a general stable ∞-categorical context, yielding a notion of 'Tate objects' independent of any geometric input.
  • The paper's results in the local-systems and quasicoherent-sheaf settings coincide where the two frameworks overlap (for dualizable group spaces), hinting that the 'correct' domain for Tate cohomology is the abstract three-functor formalism itself rather than either example.
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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 introduces a general definition of norm maps and Tate cohomology in the setting of Heyer–Mann three-functor formalisms. For a D-!-able map f:T→S, using weak adjoints in the kernel (∞,2)-category K_D(S), it defines the ∗-norm and ♯-norm maps, then the norm map Nm_f:f_♯(-⊗ν_f)→f_*, whose cofiber is Tate cohomology f^t. It establishes base change and composition formulas for these norms, proves a Browder-style theorem for left-divisible A2-monoids (Thm 1.5.6), and, assuming D is stable and presentable, proves a universal property of Tate cohomology (Thm 1.8.7) together with induced lax symmetric monoidal structures and functoriality (Thm 1.9.3). It then instantiates the formalism in local systems valued in a stable presentable symmetric monoidal ∞-category and in quasicoherent sheaves on prestacks, recovering classical Tate cohomology and giving examples such as S^1 with K(1)-local coefficients.

Significance. If the imported Heyer–Mann results are correct, this is a useful unifying framework: it gives a clean categorical mechanism for norm maps, recovers Nikolaus–Scholze's universal property, and supplies new examples beyond spectra (chromatic local systems and QCoh of classifying stacks). The paper is clearly organized and carefully attributes each step; Proposition 1.8.10 is a nice self-contained categorical localization result that is likely of independent use. Its main limitation is external dependence: the central theorems are conditional on specific lemmas in the recent preprint [13].

major comments (2)
  1. [§1.8, Theorem 1.8.7; §2.1, Example 2.1.11; §2.2, Example 2.2.11] The proof of the main theorem and the examples depend on imported results from [13]: Proposition 1.2.11 (D.2.8), Propositions 1.6.9 and 1.7.11 (Lemmas 4.5.7–4.5.8), and Proposition 2.2.10 (Theorem 3.4.11). These are load-bearing for the existence of D-Tate covers and hence for the universal property and monoidality/functoriality statements. As [13] is a recent unreviewed preprint, the paper is conditional on external results. Please either re-prove the imported statements or explicitly state the dependency and list the exact lemmas, making the theorem explicitly conditional.
  2. [§1.8, Definition 1.8.5 and Lemma 1.8.13] With the paper's convention that f_* denotes the right adjoint of f^*, condition (2) of Definition 1.8.5 and Lemma 1.8.13 should read i^*:D(T)→D(V) (the pullback functor) is conservative, not i_*. This is the hypothesis used in the proof of Theorem 1.8.7 via Lemma 1.8.14. The notation error should be corrected.
minor comments (4)
  1. [§1.5, Proposition 1.5.3(2)] In part (2), the displayed definition after 'δ_f :=' should be ω_f, not δ_f.
  2. [§1.8, proof of Theorem 1.8.7] The step saying 'if fib(γ) preserves colimits, then γ is initial' is compressed; please spell out that fib(γ) is left Kan extended from D(T)_{i-nil}, so it is the localization.
  3. [§1.8, Proposition 1.8.10(3)] The assertion that the localization q of Ind(B^κ) is compatible with the symmetric monoidal structure is stated in one 'hence'; please supply a sentence explaining why the tensor ideal hypothesis suffices.
  4. [§2.2, Example 2.2.9] Typo: 'isomorphim' should be 'isomorphism'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the core universal property is proved from external [13]/[25] results and in-paper lemmas; self-citations are not load-bearing.

full rationale

The claimed derivation chain is not circular. Construction 1.3.8 defines Tate cohomology f^t as the cofiber of the norm map, and Theorem 1.8.7's universal property is obtained by applying Proposition 1.8.10, whose proof is contained in §1.8 and is a variation of Nikolaus–Scholze [25, Thms I.3.3(iii), I.3.6(ii)] — an external source, not the author's own prior work. The background three-functor formalism and the norm/duality criteria are imported from Heyer–Mann [13] (e.g., Prop D.2.8, Lemmas 4.5.5–4.5.8, Thm 3.4.11); these are independent external results, and the paper explicitly states in §0.3 that it is 'reviewing and lightly mixing ideas explained in [13] and the work of Nikolaus–Scholze [25]'. The author's self-citations ([9], [26]) occur only as applications and motivation (Examples 0.2.1–0.2.2, §2.2), not as justification of the universal property. No fitted parameter is renamed as a prediction, and no uniqueness theorem from the author's own work is invoked. The apparent typo in Definition 1.8.5(2) — 'the functor i_*: D(T)→D(V) is conservative' where the proof of Theorem 1.8.7 via Lemmas 1.8.13–1.8.14 requires conservativity of i^* — is a correctness issue for the hypotheses, not a circularity. Thus no specific reduction of the paper's claims to their own inputs can be exhibited.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No numerical parameters are fitted; the paper is a pure mathematics exposition. Its load-bearing inputs are the Heyer–Mann three-functor formalism and the cited theorems from [13] and [25], which are accepted as black boxes. The notion of weak adjoint is defined in the paper but is a mathematical concept, not an invented entity with empirical content.

assumptions (5)
  • domain assumption There exists a three-functor formalism D on a geometric setup (S, S!) as defined by Heyer–Mann [13], including the associated (∞,2)-category of kernels KD(S) with the stated (co)monoidal and Frobenius structures, and the cited lemmas from [13] (e.g., Lemmas 4.5.5–4.5.8, Proposition D.2.8, Theorem 3.
    The entire paper operates inside this framework and inherits all its properties; no independent proof is given.
  • domain assumption D is stable and presentable in §1.8–1.9 (§1.8 opening paragraph).
    The universal property and lax monoidal structure of Tate cohomology require stability for cofibers and presentability for adjoints and accessibility arguments.
  • domain assumption D(S) is idempotent complete in Theorem 1.5.6 (stated in the theorem).
    Used to pass dualizability through retracts in the proof of Theorem 1.5.6.
  • standard math Standard ∞-category theorems (monadicity theorem, left Kan extension, Ind-completions, adjoint functor theorem) as in Lurie's Higher Algebra [22].
    Invoked e.g. in Lemma 1.8.13, Proposition 1.8.10, and throughout §1.1.
  • domain assumption Browder's theorem that any compact connected H-space is Poincaré, and the fact that compact topological manifolds are Poincaré (Spivak/Wall), are imported for the local-system examples (§0.1, Example 2.1.11).
    Needed to show the basepoint inclusion into BG is a Tate cover in Example 2.1.11.

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Pith. "Pith review of Notes on Tate cohomology." pith.science (2026). https://pith.science/paper/LXS6HWDA

@misc{pith2026260117677,
  author       = {Pith},
  title        = {Pith review of: Notes on Tate cohomology},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LXS6HWDA}},
  note         = {Machine review of arXiv:2601.17677}
}
read the original abstract

We formulate a definition of Tate cohomology in the context of three functor formalisms, and we establish basic monoidality and functoriality properties of it in this context. Our approach to these properties is based on the treatment of Nikolaus-Scholze in the setting of local systems of spectra on spaces. We discuss a couple of other specific settings of interest that are accommodated by our generalization.

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

31 extracted references · 4 linked inside Pith

  1. [13]

    6-Functor Formalisms and Smooth Representations

    Claudius Heyer and Lucas Mann. “6-Functor Formalisms and Smooth Representations”. arXiv: 2410.13038 [math.CT] (2024)

    arXiv 2024

  2. [25]

    On topological cyclic homology

    Thomas Nikolaus and Peter Scholze. “On topological cyclic homology”. Acta Mathematica 221.2 (2018)

    2018

  3. [9]

    THH(Z) and the image ofJ

    Sanath K. Devalapurkar and Arpon Raksit. “THH(Z) and the image ofJ”. arXiv: 2505.02218[math.AT] (2025)

    arXiv 2025

  4. [1]

    Cyclotomic synthetic spectra

    Benjamin Antieau and Noah Riggenbach. “Cyclotomic synthetic spectra”. arXiv: 2411.19929[math.KT] (2024)

    arXiv 2024

  5. [2]

    Finite loop spaces are manifolds

    Tilman Bauer, Nitu Kitchloo, Dietrich Notbohm, and Erik Kjær Pedersen. “Finite loop spaces are manifolds”. Acta Mathematica 192.1 (2004)

    2004

  6. [3]

    Topological Hochschild homology and integralp-adic Hodge theory

    Bhargav Bhatt, Matthew Morrow, and Peter Scholze. “Topological Hochschild homology and integralp-adic Hodge theory”. Publications mathématiques de l’IHÉS 129.1 (2019)

    2019

  7. [4]

    Topological cyclic homology of the integers

    Marcel Bökstedt and Ib Madsen. “Topological cyclic homology of the integers”.K-theory - Strasbourg, 1992 . Astérisque 226 (1994)

    1992

  8. [5]

    Torsion inH-spaces

    William Browder. “Torsion inH-spaces”. Annals of Mathematics 74 (1961)

    1961

Show all 31 references
  1. [6]

    Ambidexterity in chromatic homotopy theory

    Shachar Carmeli, Tomer M. Schlank, and Lior Yanovski. “Ambidexterity in chromatic homotopy theory”. Inventiones Mathematicae 228.3 (2022)

    2022

  2. [7]

    Lectures on Analytic Stacks

    Dustin Clausen and Peter Scholze. “Lectures on Analytic Stacks”.url: https://www.youtube.com/playlist? list=PLx5f8IelFRgGmu6gmL-Kf_Rl_6Mm7juZO (2024)

    2024

  3. [8]

    Universality of span 2-categories and the construction of 6-functor formalisms

    Bastiaan Cnossen, Tobias Lenz, and Sil Linskens. “Universality of span 2-categories and the construction of 6-functor formalisms”. arXiv: 2505.19192[math.CT] (2025)

    2025

  4. [10]

    An extension of Tate cohomology to a class of infinite groups

    F. Thomas Farrell. “An extension of Tate cohomology to a class of infinite groups”. Journal of Pure and Applied Algebra 10.2 (1977)

    1977

  5. [11]

    Generalized Tate cohomology

    John P. C. Greenlees and J. Peter May. “Generalized Tate cohomology”. Memoirs of the American Mathematical Society 113.543 (1995)

    1995

  6. [12]

    The Tate spectrum ofvn-periodic complex oriented theories

    John P. C. Greenlees and Hal Sadofsky. “The Tate spectrum ofvn-periodic complex oriented theories”. Mathematische Zeitschrift 222 (1996)

    1996

  7. [14]

    Note on principalS3-bundles

    Peter Hilton and Joseph Roitberg. “Note on principalS3-bundles”. Bulletin of the American Mathematical Society 74 (1968)

    1968

  8. [15]

    Ambidexterity inK(n)-Local Stable Homotopy Theory

    Michael Hopkins and Jacob Lurie. “Ambidexterity inK(n)-Local Stable Homotopy Theory”.url: https: //www.math.ias.edu/~lurie/papers/Ambidexterity.pdf

  9. [16]

    Tate cohomology lowers chromatic Bousfield classes

    Mark Hovey and Hal Sadofsky. “Tate cohomology lowers chromatic Bousfield classes”. Proceedings of the American Mathematical Society 124.11 (1996)

    1996

  10. [17]

    Cyclic homology and equivariant homology

    John D. S. Jones. “Cyclic homology and equivariant homology”. Inventiones Mathematicae 87.2 (1987)

    1987

  11. [18]

    Axioms for generalized Farrell-Tate cohomology

    John R. Klein. “Axioms for generalized Farrell-Tate cohomology”. Journal of Pure and Applied Algebra 172.2-3 (2002)

    2002

  12. [19]

    The dualizing spectrum of a topological group

    John R. Klein. “The dualizing spectrum of a topological group”. Mathematische Annalen 319.3 (2001)

    2001

  13. [20]

    Tate cohomology and periodic localization of polynomial functors

    Nicholas J. Kuhn. “Tate cohomology and periodic localization of polynomial functors”. Inventiones Mathe- maticae 157.2 (2004)

    2004

  14. [21]

    Elliptic Cohomology I: Spectral Abelian Varieties

    Jacob Lurie. “Elliptic Cohomology I: Spectral Abelian Varieties”.url: https://www.math.ias.edu/~lurie/ papers/Elliptic-I.pdf

  15. [22]

    Higher Algebra

    Jacob Lurie. “Higher Algebra”.url: https://www.math.ias.edu/~lurie/papers/HA.pdf

  16. [23]

    Ap-adic 6-Functor Formalism in Rigid-Analytic Geometry

    Lucas Mann. “Ap-adic 6-Functor Formalism in Rigid-Analytic Geometry”. arXiv: 2206.02022[math.AG] (2022). 39

    2022 arXiv

  17. [24]

    A universal Hochschild-Kostant-Rosenberg theorem

    Tasos Moulinos, Marco Robalo, and Bertrand Toën. “A universal Hochschild-Kostant-Rosenberg theorem”. Geometry & Topology 26.2 (2022)

    2022

  18. [26]

    Hochschild homology and the derived de Rham complex revisited

    Arpon Raksit. “Hochschild homology and the derived de Rham complex revisited”. arXiv: 2007.02576 [math.AG] (2020)

    2007

  19. [27]

    Galois extensions of structured ring spectra. Stably dualizable groups

    John Rognes. “Galois extensions of structured ring spectra. Stably dualizable groups”. Memoirs of the American Mathematical Society 192.898 (2008)

    2008

  20. [28]

    Six-Functor Formalisms

    Peter Scholze. “Six-Functor Formalisms”. arXiv: 2510.26269[math.AG] (2025)

    2025

  21. [29]

    Spaces satisfying Poincaré duality

    Michael Spivak. “Spaces satisfying Poincaré duality”. Topology 6.1 (1967)

    1967

  22. [30]

    The higher dimensional cohomology groups of class field theory

    John Tate. “The higher dimensional cohomology groups of class field theory”. Annals of Mathematics 56 (1952)

    1952

  23. [31]

    Poincaré complexes. I

    C. T. C. Wall. “Poincaré complexes. I.” Annals of Mathematics 86.2 (1967). 40

    1967

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