A family completion theorem for tempered cohomology

Leonard Tokic

REVIEW 2 major objections 5 minor 117 references

A family completion theorem for tempered cohomology

T0 review · 2 major / 5 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read Tempered cohomology satisfies a family completion theorem: geometric completion at a family of subgroups agrees with algebraic completion at an ideal.

desk verdict A substantial family completion theorem for tempered cohomology; the main theorem is new and credible, but Proposition 3.7's support classification has a compressed gluing step that should be expanded before I'd call the proof complete. read the letter →

arxiv 2608.02390 v1 pith:C3IKTJKV submitted 2026-08-03 math.AT

Leonard Tokic This is my paper · ORCID

classification math.AT MSC 55N9155P91

keywords temperedcohomologyfamilycompletiontheoremequivariantK-theorycharacterstackssupporttheoryrecollementstopologicalmodularformsP-divisiblegroups

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

The paper establishes a family completion theorem for tempered cohomology, a framework that encompasses equivariant K-theory, equivariant elliptic cohomology, and genuine equivariant topological modular forms. The result says that two different ways of completing modules over the equivariant coefficient ring coincide: completing with respect to a family F of subgroups (the geometric side) agrees with completing at the ideal I_G(F) defined by restrictions to those subgroups (the algebraic side). This recovers the classical completion theorem for equivariant K-theory as a special case and extends the statement to general locally noetherian geometric base stacks. The proof works by studying the tempered character stack G{BG} and proving that its points have support in a single conjugacy class of abelian subgroups.

What carries the argument

The tempered character stack G{BG}, built by left Kan extension from the affinization of tempered cohomology on abelian subgroups, is the central object. A support theory for its points is developed: proposition 3.7 shows that the support of any point is a single conjugacy class of abelian subgroups, and proposition 3.15 identifies the preimage of V(I_G(F)) under the affinization map with the vanishing locus of points supported inside F. The proof of the main theorem uses this to compare restrictions of ideals under change of groups, avoiding a direct comparison of the two ideals, and then assembles the result via recollement basechange.

What would settle it

Produce a finite group G and a family F for which some I_G(F)-complete R(G)^G-module is not F-complete; equivalently, find a module whose algebraic completion at the ideal vanishes but whose geometric family completion does not. The theorem claims this is impossible, and checking a small nonabelian example, such as G of order 8 with F the trivial family, would settle it.

Watch

Extended reading notes

Core claim

The central claim, Theorem 3.20, is an equivalence of recollements: for a finite group G and a family F, the geometric family-completion recollement associated to the idempotent algebra gEF, after tensoring up to modules over R(G)^G, is equivalent to the algebraic ideal-completion recollement associated to I_G(F), the intersection of the kernels of the restriction maps from π0R(G)^G to π0R(G)^H for H in F. In concrete terms, an R(G)^G-module is F-complete exactly when it is I_G(F)-complete. Theorem 3.29 extends this from affine bases to locally noetherian geometric stacks, replacing the ideal by an open substack of the tempered character stack, which yields a family completion theorem for ge

Load-bearing premise

The hard converse direction depends on imported structural facts about tempered cohomology modules—that certain nilpotent modules are generated by modules induced from proper subgroups, and that connected-étale splitting holds after faithful flat base change—so if those facts fail in this setting, the induction collapses.

Editorial extensions

If this is right

The classical completion theorem for equivariant K-theory, and its extension to arbitrary families of subgroups, follow as special cases of one statement.
For genuine equivariant topological modular forms over the moduli stack of elliptic curves, the family completion theorem holds, giving a parallel to the K-theory story.
Bounded-height tempered cohomology is right Kan extended from abelian groups generated by at most n elements; for equivariant TMF, this recovers detection from groups generated by at most two elements.
The support-theoretic description of character stacks gives a new structural handle on the prime-like points of tempered cohomology, in the spirit of the classical description of the complex representation ring.

Reading between the lines

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

Since the theorem is stated as an equivalence of completed objects rather than of filtrations, a natural extension would be to refine the argument to compare the filtrations on the two completions, strengthening the classical pro-group version.
The support stratification by conjugacy classes of abelian subgroups suggests a full 'prime spectrum' description for tempered cohomology of nonabelian groups, going beyond the abelian case treated so far.
Because the base-stack version replaces ideals by open substacks, a concrete computational form for TMF may require a 0-affine descent argument to identify those open substacks with ideal completions.
The statement assumes families are closed under subgroups and conjugation; testing whether the theorem survives for arbitrary families would clarify how essential that hypothesis is.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Tokic proves a real generalization of the classical family completion theorems. Theorem 3.20 identifies geometric F-completion with algebraic I_G(F)-completion for tempered cohomology of finite groups, recovering AHJM for mu over KU and Atiyah-Segal as the trivial-family case. Proposition 3.21 (bounded height detection) and Theorem 3.29 (LNG stacks, including genuine equivariant TMF) are genuinely new. The support theory for the tempered character stack is the right tool, and the paper is honestly written: limitations about filtrations and the qualitative nature of the 0-affine case are stated plainly, and the appendices do real work.

What I'd want a referee to focus on is Proposition 3.7. The stress-test note is on target: after reducing to the faithful flat splitting algebra S, the proof shows |Spec S(G^circ)^V|_n is a point and then asserts that a chosen prime p_V can be transported along a zig-zag of subconjugations to p_W, with compatible lifts through the colimit presentation. That compatibility is not proved. It is load-bearing: if the components attached to non-conjugate V and W were not identified, the support of a point could contain non-conjugate abelian subgroups, and the induction in Theorem 3.20 would collapse. I don't think the step is wrong—the intended argument via colimit finality and careful base change is plausible, and the rest of the paper is coherent—but it is the point where I'd want the author to write out the diagram chase before publication.

The other dependence is external: the abelian case uses [GLP24, prop. 15.14] and the nonabelian step uses [BDL26, prop. 5.4.4]. Those are heavy, not machine-checked, but they are cited as lemmas rather than assumed conclusions, and the paper's own contribution is not circular. The reader's moderate confidence is about right.

Bottom line: this is a serious paper for people in equivariant stable homotopy and chromatic homotopy theory. It deserves a proper peer review, and I'd expect acceptance after the Proposition 3.7 gap is closed. I'd cite it once it's out.

Referee Report

2 major / 5 minor

Summary. The paper proves a family completion theorem for tempered cohomology. For a finite group G, a family F of subgroups, and an oriented P-divisible group G over a noetherian E-infinity ring R, it compares the geometric family-completion recollement R(gE_F) with the algebraic ideal-completion recollement associated to I_G(F), and shows they are equivalent (Theorem 3.20). This recovers the Adams-Haeberly-Jackowski-May theorem for equivariant K-theory and the Atiyah-Segal theorem as special cases. The proof uses a new support theory for points of the tempered character stack G{BG}, and a local-to-global descent argument for locally noetherian geometric stacks, yielding a version for genuine equivariant topological modular forms (Theorem 3.29). The paper is long and contains substantial foundational material on recollements and spectral algebraic geometry in appendices.

Significance. If the main theorem is correct, it is a significant advance: it gives a uniform family completion statement for a large class of equivariant cohomology theories, with the classical AHM theorem as a special case and with new applications to TMF. The support theory for tempered character stacks, modeled on Segal's analysis of representation rings, is a genuinely new and promising tool. The paper is carefully structured, explicitly citing the external inputs [GLP24], [BDL26], and [Lur18a]; no machine-checked proofs are provided, but the proof is presented in enough detail that a referee can identify the exact places needing scrutiny. The appendices, especially Appendix B, are largely self-contained and useful in their own right. However, two load-bearing points in the proof of the main theorem need repair before the paper can be accepted.

major comments (2)

[Section 3.1 (proof of Proposition 3.7)] The final step of Proposition 3.7 rests on an unproved compatibility of chosen lifts. After fixing a prime p_V and a zig-zag of subconjugations connecting x_V and x_W, the text says: 'We then choose lifts and restrict to transport p_V along the zig-zag' and asserts that this ensures y_V and y_W map to the same point. But the proof does not show that these lifts are compatible with the colimit presentation G{BG} ≃ colim over Orb_ab(G) G{BH}, nor with the faithfully flat base change to S. This compatibility is exactly what forces the components attached to non-conjugate V and W to be identified. Since Proposition 3.7 feeds into Propositions 3.13 and 3.15, the induction in Theorem 3.20 inherits this gap. The step should be written out in full, or replaced by an explicit argument using the universal property of the colimit.
[Section 3.2 (Theorem 3.20, nonabelian case)] The assertion 'we have gE_P = gE_P tensor gE_AB' is false. For a proper nonabelian subgroup H of G, (gE_P)^H = S^0, while (gE_P tensor gE_AB)^H = (gE_P)^H tensor (gE_AB)^H = S^0 tensor * = *. The subsequent conclusion that AB-completeness of every R(G)^G-module gives X = 0 is therefore not justified as written. The intended argument can likely be repaired, for instance by showing directly that every R(G)^G-module is P-complete using the AB-nilpotence of R(G)^G in the sense of [MNN17], but the present proof is not valid at this point.

minor comments (5)

[Throughout] The manuscript contains numerous typographical errors: 'isomoprhic', 'constrctions', 'noethrian', 'P-divisable', 'deontet', 'catgeory', 'seperation', 'continiuous', 'surejctions', 'singelton', 'unqiely', and others. These should be corrected before final publication.
[Abstract] The abstract uses the same letter G for the P-divisible group and for the finite group; in the body the former is a different symbol. Harmonize the notation for clarity.
[Section 3.2, proof of Lemma 3.19] The sentence 'If M is an I_P-nilpotent R(G)^A-module, then M^A is an I_P-complete R(G)^A-module' is terse. It would help the reader to see the identification of M^A with the A-fixed points and why the relevant completeness is checked there.
[Section 3.3, Definition 3.22] Minor wording: 'with each R_i noethrian' should be 'noetherian.' Also, the notion of LNG-stack is central, so it may be worth stating explicitly that the transition maps being flat implies the canonical maps Spec R_i to M are flat.
[Appendix A] The appendix is long and some statements are not used later (for example, Example A.20). This is not a defect, but a short indication of which parts are needed for the main results would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the family completion theorem is derived from external results and new support theory, not assumed.

full rationale

The central equivalence (Theorem 3.20) is not an input: the ideal I_G(F) is defined independently as an intersection of restriction ideals, the geometric recollement R(gE_F) is defined from the idempotent algebra gE_F, and the theorem then proves that the two classes of complete objects agree. The easy direction is Proposition 2.20; the converse is an induction using external results [GLP24, prop. 15.14], [BDL26, prop. 5.4.4], and the paper's own Proposition 3.7. These cited results are used as lemmas by other authors, not as restatements of the conclusion, and none is the paper's own prior work. Proposition 3.7's support classification is new and is derived from the colimit presentation of the tempered character stack, external point-counting [Lur18a, prop. 2.3.9], and connected-étale splitting, rather than being assumed. The geometric-stack Theorem 3.29 is deduced by descent from the affine Theorem 3.20. No fitted parameter is renamed as a prediction. Recovering Atiyah–Segal and Adams–Haeberly–Jackowski–May is a consistency check against external benchmarks, not a renaming, since the paper proves a new equivalence of recollements for tempered cohomology. The skeptic's concern about the zig-zag transport in the proof of Proposition 3.7 is a possible gap in proof detail, not a circularity: even if that step were underjustified, it would not make the conclusion equal to an input. No circular step can be exhibited from the text.

Assumptions & free parameters 0 free parameters · 6 assumptions · 1 invented entities

The theorem has no fitted numerical parameters; the inputs (finite group, family, P-divisible group, LNG-stack) are structural hypotheses. The proof imports substantial external results: representability and abelian generation from [GLP24], geometric fixed-point vanishing from [BDL26], and connected P-divisible group input from [Lur18a]/[Lur19]. The only new object is the character stack, which is defined from existing data rather than postulated ad hoc.

assumptions (6)

domain assumption Representability of tempered cohomology: R(G) ∈ CAlg(Sp^gl_fin) represents R•_G ([GLP24, thm. E]).
Theorem 3.20 is stated inside Mod_{R(G)^G}(Sp^G); without this external representability theorem there is no R(G)^G at which to complete.
domain assumption For finite abelian A, every I_G(P)-nilpotent R(G)^A-module lies in the localizing tensor ideal generated by modules induced from proper subgroups ([GLP24, prop. 15.14]).
Used in Lemma 3.19 for the abelian base case of the induction in Theorem 3.20.
domain assumption Connected P-divisible groups have no nontrivial maps from reduced fields ([Lur18a, prop. 2.3.9]) and connected-étale sequences split after a faithful flat extension ([Lur19, prop. 2.7.15]).
These underpin the point-counting claim in Proposition 3.7, which controls supports and makes the induction in Theorem 3.20 work.
domain assumption Geometric fixed points at nonabelian subgroups vanish: (R(G)^G)^{ΦH}=0 for H nonabelian ([BDL26, prop. 5.4.4]).
Yields Lemma 3.17 (R(G)^G is AB-nilpotent), used in the nonabelian case of Theorem 3.20.
domain assumption Mor_Ell is 0-affine ([MM15, thm. 7.2]).
Needed to identify modules over the relative character stack with TMF^G-modules in Example 3.30.
standard math The underlying topological space functor |–| has cartesian lifts of open embeddings (Remark A.32); LNG-presentations glue closed subsets (Lemma 3.25, Proposition 3.26).
Infrastructure for Theorem C; mostly proven in Appendix A but imported as framework from [BDL25].

invented entities (1)

Tempered character stack G{BG} independent evidence
purpose: A stack whose global sections recover R(G)^G and whose points carry the support theory used to compare the two family completions.
It is defined by left Kan extension from affine pieces G{BH} for abelian H and is matched to Spec R(G)^G by affinization; for μ_{P^∞}/KU it recovers the classical character stack picture underlying Atiyah–Segal.

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Pith. "Pith review of A family completion theorem for tempered cohomology." pith.science (2026). https://pith.science/paper/C3IKTJKV

@misc{pith2026260802390,
  author       = {Pith},
  title        = {Pith review of: A family completion theorem for tempered cohomology},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/C3IKTJKV}},
  note         = {Machine review of arXiv:2608.02390}
}

abstract

Let ${\mathbb{G}}$ be an oriented $\mathbb{P}$-divisible group over a noetherian $\mathbb{E}_\infty$-ring $R$, let $G$ be a finite group, and let $\mathcal{F}$ be a family of subgroups of $G$. We show that completion of $R({\mathbb{G}})_G$-modules at $\mathcal{F}$ agrees with algebraic completion at the ideal $I_{\mathbb{G}}(\mathcal{F})=\bigcap_{H\in\mathcal{F}}\mathrm{ker} (\pi_0R({\mathbb{G}})^{ G}\to \pi_0R({\mathbb{G}})^{ H}).$ For ${\mathbb{G}}=\mu_{\mathbb{P}^\infty}$ over $\mathrm{KU}$ this recovers the family completion theorem of Adams, Haeberly, Jackowski, and May, and for the trivial family the classical Atiyah-Segal completion theorem. The main input is a theory of support for points of the tempered character stack ${\mathbb{G}}\{{\mathbb{B}} G\}$, in the spirit of Segal's analysis of the prime spectrum of the complex representation ring: we show that the support of a point is a single conjugacy class of abelian subgroups of $G$, and that the points supported inside $\mathcal{F}$ are exactly the preimage of $V(I_{\mathbb{G}}(\mathcal{F}))$ under the affinization map. We also prove a version over locally noetherian geometric base stacks, in which the ideal is replaced by an open substack of ${\mathbb{G}}({\mathbb{B}} G)$, the analogue over such a base of $\mathrm{Spec}\,R({\mathbb{G}})^{ G}$, and which applies for instance to genuine equivariant topological modular forms.

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Reference graph

Works this paper leans on

117 extracted references · 11 canonical work pages

[1]

and Strickland, N

Hovey, M. and Strickland, N. P. , title =. Mem. Amer. Math. Soc. , fjournal =. 1999 , number =

1999
[2]

Anderson, D. W. and Brown, Jr., E. H. and Peterson, F. P. , title =. Ann. of Math. (2) , fjournal =. 1967 , pages =

1967
[3]

Adams, J. F. , title =. 1995 , pages =

1995
[4]

and Hopkins, M

Ando, M. and Hopkins, M. J. and Rezk, C. , url =. Multiplicative Orientations of. 2010 , urldate =

2010
[5]

, title =

Beaudry, A. , title =. Algebr. Geom. Topol. , fjournal =. 2015 , number =

2015
[6]

, title =

Balderrama, W. , title =. J. Topol. , fjournal =. 2023 , number =

2023
[7]

Dixon, J. D. and du Sautoy, M. P. F. and Mann, A. and Segal, D. , title =. 1999 , pages =

1999
[8]

Anderson, D. W. , journal =. The Real K-Theory of Classifying Spaces , urldate =

Show all 117 references

[9]

Anderson, D. W. and Brown, Jr., E. H. and Peterson, F. P. , title =. Ann. of Math. (2) , fjournal =. 1966 , pages =

1966
[10]

and Hopkins, M

Ando, M. and Hopkins, M. J. and Strickland, N. P. , title =. Amer. J. Math. , fjournal =. 2004 , number =

2004
[11]

Ardakov, K

K. Ardakov, K. A. Brown , journal =. Ring-theoretic properties of Iwasawa algebras: a survey. , url =
[12]

Atiyah, M. F. and Singer, I. M. , title =. Bull. Amer. Math. Soc. , fjournal =. 1963 , pages =

1963
[13]

and Beaudry, A

Barthel, T. and Beaudry, A. , title =. Handbook of homotopy theory , series =. 2020 , mrclass =

2020
[14]

and Beaudry, A

Barthel, T. and Beaudry, A. and Goerss, P. G. and Stojanoska, V. , title =. Math. Z. , fjournal =. 2022 , number =

2022
[15]

and Heard, D

Barthel, T. and Heard, D. , title =. Topology Appl. , fjournal =. 2016 , pages =

2016
[16]

Behrens , title =

M. Behrens , title =. Topological modular forms , series =. 2014 , doi =

2014
[17]

2024 , eprint =

Elliptic cohomology and quantum field theory , author =. 2024 , eprint =

2024
[18]

Representations of compact

Br\". Representations of compact. 1985 , pages =

1985
[19]

2012 , eprint =

Finite subgroups of extended Morava stabilizer groups , author =. 2012 , eprint =

2012
[20]

2022 , eprint =

The Chromatic Nullstellensatz , author =. 2022 , eprint =

2022
[21]

and Chabert, J.-L

Cahen, P.-J. and Chabert, J.-L. , title =. Amer. Math. Monthly , fjournal =. 2016 , number =

2016
[22]

and Olbermann, M

Laures, G. and Olbermann, M. , title =. Manuscripta Math. , fjournal =. 2018 , number =

2018
[23]

2024 , eprint =

The descent spectral sequence for topological modular forms , author =. 2024 , eprint =

2024
[24]

and Hopkins, M

Ando, M. and Hopkins, M. J. and Strickland, N. P. , title =. Invent. Math. , fjournal =. 2001 , number =

2001
[25]

Davies, J. M. , title =. Math. Z. , fjournal =. 2025 , number =

2025
[26]

Davis, D. M. , title =. Symposium on
[27]

Davis, D. M. and Mahowald, M. , title =. Amer. J. Math. , fjournal =. 1982 , number =

1982
[28]

2020 , eprint =

The Ando-Hopkins-Rezk orientation is surjective , author =. 2020 , eprint =

2020
[29]

Devalapurkar, S. K. , title =. Algebr. Geom. Topol. , fjournal =. 2024 , number =

2024
[30]

2005 , doi =

Recognizing Hopf algebroids defined by a group action , journal =. 2005 , doi =

2005
[31]

Devinatz, E. S. and Hopkins, M. J. , title =. Topology , fjournal =. 2004 , number =

2004
[32]

Introduction to the theory of formal groups , series =

Dieudonn\'. Introduction to the theory of formal groups , series =. 1973 , pages =

1973
[33]

, title =

Peterson, E. , title =. 2019 , pages =

2019
[34]

Goerss, P. G. and Hopkins, M. J. , title =. Structured ring spectra , series =. 2004 , mrclass =

2004
[35]

and Kamel, Y

Halladay, Z. and Kamel, Y. , year =. Real spin bordism and orientations of topological. 2405.00963 , archiveprefix =

arXiv
[36]

, title =

Henn, H.-W. , title =. Algebraic topology , series =. 2017 , mrclass =

2017
[37]

, title =

Hirzebruch, F. , title =. Proceedings of the National Academy of Sciences of the United States of America , volume =. 1953 , doi =

1953
[38]

Hopkins, M. J. , title =. Topological modular forms , series =. 2014 , doi =

2014
[39]

Hopkins, M. J. and Hovey, M. , title =. Math. Z. , fjournal =. 1992 , number =

1992
[40]

, title =

Hovey, M. , title =. Homology Homotopy Appl. , fjournal =. 2004 , number =

2004
[41]

, title =

Hovey, M. , title =. Trans. Amer. Math. Soc. , fjournal =. 2008 , number =

2008
[42]

, title =

Hovey, M. , title =. Algebr. Geom. Topol. , fjournal =. 2008 , number =

2008
[43]

Hovey, M. A. and Ravenel, D. C. , title =. Trans. Amer. Math. Soc. , fjournal =. 1995 , number =

1995
[44]

and Sadofsky, H

Hovey, M. and Sadofsky, H. , title =. J. London Math. Soc. (2) , fjournal =. 1999 , number =

1999
[45]

, title =

Beaudry, A. , title =. Adv. Math. , fjournal =. 2017 , pages =

2017
[46]

Katz, N. M. and Mazur, B. , title =. 1985 , pages =

1985
[47]

and Laures, G

Kitchloo, N. and Laures, G. and Wilson, W. S. , title =. Adv. Math. , fjournal =. 2004 , number =

2004
[48]

Kuhn, N. J. , editor =. Morava K-theories and infinite loop spaces , booktitle =. 1989 , publisher =

1989
[49]

Kultze, R. and W\". A note on the algebra. Manuscripta Math. , fjournal =. 1987 , number =

1987
[50]

, title =

Laures, G. , title =. Trans. Amer. Math. Soc. , fjournal =. 2016 , number =

2016
[51]

and Naumann, N

Lawson, T. and Naumann, N. , title =. J. Topol. , fjournal =. 2012 , number =

2012
[52]

, title =

Lazard, M. , title =. Inst. Hautes \'. 1965 , pages =

1965
[53]

, title =

Lurie, J. , title =. Algebraic topology , series =. 2009 , doi =

2009
[54]

and Rezk, C

Mahowald, M. and Rezk, C. , title =. Pure Appl. Math. Q. , fjournal =. 2009 , number =

2009
[55]

, year =

Mathew, A. , year =. The homology of \(. 1305.6100 , archiveprefix =

arXiv
[56]

, title =

McTague, C. , title =. New directions in homotopy theory , series =. 2018 , mrclass =

2018
[57]

and Schuster, B

Laures, G. and Schuster, B. , title =. Proc. Amer. Math. Soc. , fjournal =. 2019 , number =

2019
[58]

and Schmidt, A

Neukirch, J. and Schmidt, A. and Wingberg, K. , title =. 2008 , pages =

2008
[59]

and Segal, D

Nikolov, N. and Segal, D. , title =. Ann. of Math. (2) , fjournal =. 2007 , number =

2007
[60]

Passman, D. S. , title =. 1989 , pages =

1989
[61]

Pengelley, D. J. , title =. Proc. Amer. Math. Soc. , fjournal =. 1983 , number =

1983
[62]

Pengelley, D. J. , title =. Amer. J. Math. , fjournal =. 1982 , number =

1982
[63]

P. Pstr. Finite height chromatic homotopy theory , url =
[64]

Ravenel, D. C. , title =. 1986 , pages =

1986
[65]

Ravenel, D. C. , title =. Ast\'. 1990 , pages =

1990
[66]

Ando and M

M. Ando and M. J. Hopkins and N. P. Strickland , title =
[67]

and Laures, G

Kitchloo, N. and Laures, G. , title =. K -Theory , fjournal =. 2002 , number =

2002
[68]

, title =

Rognes, J. , title =. Mem. Amer. Math. Soc. , fjournal =. 2008 , number =

2008
[69]

2025 , eprint =

The action of the Morava stabilizer group on the coefficients of Morava E-theory at height 2 , author =. 2025 , eprint =

2025
[70]

, title =

Senger, A. , title =. Compos. Math. , fjournal =. 2023 , number =

2023
[71]

Snaith, V. P. , title =. Topics in
[72]

, title =

Snaith, V. , title =. Math. Proc. Cambridge Philos. Soc. , fjournal =. 1981 , number =

1981
[73]

, title =

Laures, G. , title =. Amer. J. Math. , fjournal =. 2003 , number =

2003
[74]

Sweedler, M. E. , title =
[75]

, title =

Thom, R. , title =. Comment. Math. Helv. , fjournal =. 1954 , pages =

1954
[76]

and Olbermann, M

Laures, G. and Olbermann, M. , title =. Math. Z. , fjournal =. 2016 , number =

2016
[77]

2024 , eprint =

K(2) -local splittings of finite Galois extensions of MU 6 and MString , author =. 2024 , eprint =

2024
[78]

Wall, C. T. C. , title =. Ann. of Math. (2) , fjournal =. 1960 , pages =

1960
[79]

2019 , eprint =

The Omega spectrum for Pengelley's BoP , author =. 2019 , eprint =

2019
[80]

, title =

Witten, E. , title =. Elliptic curves and modular forms in algebraic topology (. 1988 , doi =

1988
[81]

, title =

Zhu, Y. , title =. J. Pure Appl. Algebra , fjournal =. 2020 , number =

2020
[82]

Hovey , title =

Mark A. Hovey , title =. Duke Mathematical Journal , number =. 1997 , doi =

1997
[83]

2023 , eprint=

Picard and Brauer groups of K(n) -local spectra via profinite Galois descent , author=. 2023 , eprint=

2023
[84]

, publisher =

Lurie, J. , publisher =. Higher Topos Theory , year =
[85]

, title =

Lurie, J. , title =. 2017 , url =

2017
[86]

Rumynin and J

D. Rumynin and J. Taylor , year=. Real representations of. 2006.09765 , archivePrefix=

2006 arXiv
[87]

2025 , eprint=

Magnetic Equivariant K-theory , author=. 2025 , eprint=

2025
[88]

, title =

Schwede, S. , title =. 2026 , url =

2026
[89]

, TITLE =

Palais, Richard S. , TITLE =. Mem. Amer. Math. Soc. , FJOURNAL =. 1960 , PAGES =

1960
[91]

Linskens, Sil and Nardin, Denis and Pol, Luca , TITLE =. Geom. Topol. , FJOURNAL =. 2025 , NUMBER =. doi:10.2140/gt.2025.29.1345 , URL =

2025 doi
[92]

2018 , PAGES =

Schwede, Stefan , TITLE =. 2018 , PAGES =. doi:10.1017/9781108349161 , URL =

2018 doi
[93]

2022 , eprint=

Equivariant Structure on Smash Powers , author=. 2022 , eprint=

2022
[94]

Degrijse, Dieter and Hausmann, Markus and Luck, Wolfgang and Patchkoria, Irakli and Schwede, Stefan , TITLE =. Mem. Amer. Math. Soc. , FJOURNAL =. 2023 , NUMBER =. doi:10.1090/memo/1432 , URL =

2023 doi
[95]

Gepner, David and Meier, Lennart , TITLE =. Compos. Math. , FJOURNAL =. 2023 , NUMBER =. doi:10.1112/s0010437x23007509 , URL =

2023 doi
[96]

doi:10.48550/arXiv.2407.05124 , url =

Global 2-Rings and Genuine Refinements , author =. doi:10.48550/arXiv.2407.05124 , url =

doi arXiv
[97]

Elliptic

Lurie, Jacob , date =. Elliptic
[98]

Ambidextrous Global Spectra and Tempered Cohomology , author =
[99]

The Stacks project , howpublished =

The. The Stacks project , howpublished =
[100]

Affineness and Reconstruction in Complex-Periodic Geometry , author =
[101]

doi:10.1016/j.aim.2016.09.027 , url =

Nilpotence and Descent in Equivariant Stable Homotopy Theory , author =. doi:10.1016/j.aim.2016.09.027 , url =

2016 doi
[102]

Genuine \

Lin, Ying-Hsuan and Tominaga, Akira and Yamashita, Mayuko , date =. Genuine \. doi:10.48550/arXiv.2510.23602 , url =

doi
[103]

A Note on Stable Recollements , author =
[104]

Hermitian

Calmès, Baptiste and Harpaz, Yonatan and Land, Markus and Nardin, Denis and Steimle, Wolfgang and Dotto, Emanuele and Hebestreit, Fabian and Moi, Kristian and Nikolaus, Thomas , date =. Hermitian. doi:10.4310/ACTA.2025.n235.n2.a1 , url =

2025 doi
[105]

doi:10.4310/ACTA.2018.v221.n2.a1 , url =

On Topological Cyclic Homology , author =. doi:10.4310/ACTA.2018.v221.n2.a1 , url =

2018 doi
[106]

doi:10.48550/arXiv.2410.21537 , url =

Dualizable Presentable \ infty\ -Categories , author =. doi:10.48550/arXiv.2410.21537 , url =

doi arXiv
[107]

What Are Recollements in Stable Homotopy Theory? , author =
[108]

An Elementary Proof of the Naturality of the

Ramzi, Maxime , date =. An Elementary Proof of the Naturality of the. doi:10.1090/proc/16469 , url =

doi
[109]

Carchedi, David , date =. Higher. doi:10.1090/memo/1282 , url =

doi
[110]

Frames and

Picado, Jorge and Pultr, Aleš , date =. Frames and. doi:10.1007/978-3-0348-0154-6 , url =

doi
[111]

doi:10.48550/arXiv.2310.12925 , url =

Derived \ infty\ -Categories as Exact Completions , author =. doi:10.48550/arXiv.2310.12925 , url =

doi arXiv
[112]

Dirac Geometry

Hesselholt, Lars and Pstrągowski, Piotr , date =. Dirac Geometry. doi:10.1017/fms.2024.2 , url =

2024 doi
[113]

doi:10.48550/arXiv.2410.21524 , url =

Locally Rigid \ infty\ -Categories , author =. doi:10.48550/arXiv.2410.21524 , url =

doi
[114]

The Representation-Ring of a Compact

Segal, Graeme , date =. The Representation-Ring of a Compact. doi:10.1007/BF02684592 , url =

doi
[115]

doi:10.1016/0040-9383(88)90002-X , url =

A Generalization of the Atiyah-Segal Completion Theorem , author =. doi:10.1016/0040-9383(88)90002-X , url =

doi
[116]

Atiyah, M. F. and Segal, G. B. , date =. Equivariant \. doi:10.4310/jdg/1214428815 , url =
[117]

doi:10.1112/jtopol/jtv005 , url =

Affineness and Chromatic Homotopy Theory , author =. doi:10.1112/jtopol/jtv005 , url =

doi
[118]

doi:10.1016/0040-9383(88)90003-1 , url =

A Generalization of the Segal Conjecture , author =. doi:10.1016/0040-9383(88)90003-1 , url =

doi

Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.

[1] [1]

and Strickland, N

Hovey, M. and Strickland, N. P. , title =. Mem. Amer. Math. Soc. , fjournal =. 1999 , number =

1999

[2] [2]

Anderson, D. W. and Brown, Jr., E. H. and Peterson, F. P. , title =. Ann. of Math. (2) , fjournal =. 1967 , pages =

1967

[3] [3]

Adams, J. F. , title =. 1995 , pages =

1995

[4] [4]

and Hopkins, M

Ando, M. and Hopkins, M. J. and Rezk, C. , url =. Multiplicative Orientations of. 2010 , urldate =

2010

[5] [5]

, title =

Beaudry, A. , title =. Algebr. Geom. Topol. , fjournal =. 2015 , number =

2015

[6] [6]

, title =

Balderrama, W. , title =. J. Topol. , fjournal =. 2023 , number =

2023

[7] [7]

Dixon, J. D. and du Sautoy, M. P. F. and Mann, A. and Segal, D. , title =. 1999 , pages =

1999

[8] [8]

Anderson, D. W. , journal =. The Real K-Theory of Classifying Spaces , urldate =