The homotopy theory of cyclotomic spectra, 10 years later

Andrew J. Blumberg; Michael A. Mandell

REVIEW 3 major objections 4 minor 27 references

The homotopy theory of cyclotomic spectra, 10 years later

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read This paper proves that geometric fixed points admit a left derived functor on equivariant commutative ring spectra, that the derived functor agrees with the one on the underlying spectra, and that a multiplicative tom Dieck splitting…

desk verdict A serious, mostly detailed foundational paper whose main results hang on a set of unproved model-structure theorems the authors wave at as 'well-known'. read the letter →

arxiv 2412.15928 v1 pith:CWXVJB5P submitted 2024-12-20 math.AT

Andrew J. Blumberg , Michael A. Mandell This is my paper · ORCID

classification math.AT MSC 55P9119D55

keywords geometricfixedpointsequivariantcommutativeringspectramodelcategoriespre-cyclotomictopologicalcyclichomologytomDiecksplittingorthogonalLodayconstructions

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 aims to close a foundational gap in equivariant stable homotopy theory: geometric fixed points were understood homotopically on spectra, but not on equivariant commutative ring spectra, where the multiplicative structure matters for invariants such as topological cyclic homology. The paper's central claim is that, for a compact Lie group $G$ and a finite set of closed subgroups, there is a model structure on commutative ring $G$-spectra in which the point-set geometric fixed point functor preserves weak equivalences between cofibrant objects and agrees with the derived functor on the underlying spectra. For the circle group, the same machinery produces model structures on commutative ring pre-cyclotomic spectra and a formula for their derived mapping spaces as homotopy equalizers. The paper also proves a multiplicative tom Dieck splitting: geometric fixed points of a non-equivariant commutative ring spectrum made $G$-equivariant decompose as a tensor product of factors $R \otimes BW_H$ indexed by conjugacy classes of subgroups. The engine behind these results is a new class of spectra, generalized orbit desuspension spectra, built from equivariant vector bundles with finite quotient group actions.

What carries the argument

The load-bearing construction is the generalized orbit desuspension spectrum $J_\eta$: given a $(\Gamma,Q)$ vector bundle $\eta$ (an equivariant real vector bundle with a finite group $Q$ acting on it) and a $\Gamma$-representation $W$, its $W$th level is the Thom space of the bundle of isometric embeddings of the fiber of $\eta$ into $W$, quotiented by $Q$. These spectra generalize the orbit desuspension spectra $\Gamma/H_+\wedge F_V S^0$ and the norm-style complete cells, and they are flat for the smash product. The paper combines them with $V$-constrained $F$-model structures, whose cells are restricted to chosen families of subgroups and representation constraints, and with variant model structures built from a set $A$ of such cells. The theorems hold under faithfulness hypotheses: $\eta$ is $Q$-faithful when each point stabilizer acts faithfully on the fiber, and the paper isolates further inherited faithfulness conditions that make $\Phi_H$ commute with smash products and symmetric powers up to isomorphism or weak equivalence.

What would settle it

Find a compact Lie group $G$, a finite set $\mathcal H$ of closed subgroups, and a cell $A\in A_W^{(\wedge)}$ from Chapter II for which $A^{(n)}/\Sigma_n$ is not flat, or for which symmetric powers do not send $F$-equivalences from cofibrant approximations to $F$-equivalences; any such $A$ violates the hypotheses of the unproved Theorem II.1.13 and would invalidate the model structure on which Theorem E depends.

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Extended reading notes

Core claim

The paper establishes that the point-set geometric fixed point functor $\Phi_H$ has a left derived functor on commutative ring $G$-spectra, and that this derived functor agrees with the derived geometric fixed point functor on the underlying $G$-spectra. Concretely, Theorem E constructs, for a compact Lie group $G$ and a finite set $\mathcal{H}$ of closed subgroups, a model structure on commutative ring $G$-spectra such that if $R$ is cofibrant in it and $X\to R$ is a cofibrant approximation in the standard model structure, then $\Phi_H X\to \Phi_H R$ is a weak equivalence for every $H\in\mathcal{H}$. In the circle group case, Theorems B, C, and D produce model structures on commutative ring pre-cyclotomic spectra and show the derived mapping space from $A$ to $B$ is the homotopy equalizer of $\mathrm{Com}^T(A,B)\rightrightarrows \mathrm{Com}^T(\Phi A,B)$. Theorem A, the multiplicative tom Dieck splitting, gives a natural weak equivalence $\Phi_G(\epsilon^\ast R)\simeq R\otimes (\coprod_{[H]} BW_H)$ for finite $G$, where $\epsilon^\ast$ is the left derived pushforward from non-equivariant commutative ring spectra.

Load-bearing premise

The results rest on four cell-restricted model-structure theorems that the paper states without proof, calling them well-known; if those model structures do not exist in the stated generality, the derived functor construction and the multiplicative splittings lose their foundation.

Editorial extensions

If this is right

The derived geometric $H$-fixed point functor exists on commutative ring $G$-spectra for any finite prescribed set of subgroups $H$, and it agrees with the derived functor on the underlying spectra.
Geometric fixed points preserve weak equivalences between cofibrant objects in the new model structures, so point-set level computations on cofibrant replacements are homotopically correct.
For the circle group, commutative ring pre-cyclotomic and p-pre-cyclotomic spectra carry topological model structures, with compatible module and algebra model structures; the derived mapping space of commutative ring ppc spectra is the homotopy equalizer of $\mathrm{Com}^T(A,B)\rightrightarrows\mathrm{Com}^T(\Phi A,B)$.
The multiplicative tom Dieck splitting identifies $\Phi_G(\epsilon^\ast R)$ with the tensor product over conjugacy classes $[H]$ of $R\otimes BW_H$, making the multiplicative transfers on geometric fixed points visible and computable.
The cells used for the circle group are closed under smash product, symmetric powers, and the geometric fixed point endofunctors, so the model structures are self-contained and cofibrant objects have flatness and monoidal control.

Reading between the lines

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

Beyond the paper's claims, the A-variant model-structure template should apply to any point-set functor known to be homotopically correct on a set of cells closed under smash products and symmetric powers; the generalized orbit desuspension spectra provide a systematic source of such cell sets.
A testable next step is to compute the factors $R\otimes BW_H$ for specific graded rings $R$ and small groups $G$; the paper reports that these Loday constructions are largely uncalculated, so any explicit computation would either confirm the splitting's utility or expose a missing hypothesis.
One might also expect the mapping-space formula of Theorem B to translate into a spectral-sequence or homotopy-limit comparison for topological cyclic homology of rings, giving a computational route that the paper does not pursue.

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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

The genuine contributions are real. The multiplicative tom Dieck splitting for commutative ring spectra promoted from non-equivariant rings (Theorem A) is new and structurally striking. The construction of a left derived functor for geometric fixed points on commutative ring G-spectra and its comparison to the underlying derived functor (Theorem E) resolves a question that HHR and the authors' earlier cyclotomic work sidestepped. The generalized orbit desuspension spectra are a new class, and the detailed analysis of their behavior under smash products, symmetric powers, and geometric fixed points (Chapters V–VI) is a solid piece of work. Theorem B's mapping space formula for commutative ring cyclotomic spectra is also a genuine payoff.

The soft spot is exactly where the reader and the stress-test put it: Chapter II.1. Theorems II.1.9, II.1.11, and especially II.1.13 are load-bearing, and they are stated without proof. The authors explicitly say they claim no novelty and regard the techniques as well-known enough to omit all proofs. That is a real problem, not because the statements are likely false, but because II.1.13 is the mechanism that turns flatness and symmetric-power hypotheses on the cell set into the existence of the commutative ring model structure. Theorems C, D, and E all rest on it, and Theorem 2.3 verifies its hypotheses using the new generalized orbit desuspension spectra. A gap there would propagate directly into the main conclusions.

That said, this is a missing foundation, not a detected error. The rest of the paper is full of detailed, checkable proofs. The citations to prior work, including the authors' own, are appropriate. The central claims look well supported if the omitted theorems are correct, and those theorems are plausibly standard to an expert in equivariant model categories.

I would send this to a serious referee. The referee should be asked specifically to verify the statements of II.1.9–II.1.13 against the literature, or to demand that the authors supply proofs or precise references with proofs for these theorems before final acceptance. The paper is important enough to deserve that referee time.

Referee Report

3 major / 4 minor

Summary. The paper develops new foundations for multiplicative equivariant stable homotopy theory. It introduces generalized orbit desuspension spectra J_η and studies their behavior under geometric fixed points, smash products, and symmetric powers. On this basis it constructs new model structures on equivariant commutative ring spectra, proves the existence and comparison of a derived geometric fixed point functor on commutative ring G-spectra (Theorem E), constructs model structures for commutative ring (pre-)cyclotomic spectra and derives mapping-space formulas (Theorems B, C, D), and proves a multiplicative tom Dieck splitting for the left derived pushforward of a non-equivariant commutative ring spectrum (Theorem A). The technical core is a series of 'V-constrained' model structure theorems stated in Chapter II.1, especially Theorems II.1.9, II.1.11, and II.1.13.

Significance. If the central results hold, the paper closes a recognized gap in the foundations of geometric fixed points for commutative ring equivariant spectra and provides tools needed for current work on relative topological cyclic homology. The paper's positive contributions are substantial: the new class of generalized orbit desuspension spectra is well motivated, the statements of Theorems A and E are precise and useful, and Chapters V--VI contain detailed, nontrivial proofs of the point-set and homotopical properties of these spectra. The multiplicative tom Dieck splitting is an original and striking structural result. However, the paper's central edifice rests on a set of model-categorical theorems in Chapter II.1 for which no proofs are supplied, and this weakness is load-bearing rather than cosmetic.

major comments (3)

[Chapter II.1, Theorems 1.9, 1.10, 1.11, and 1.13] These theorems are stated without proof, with the authors explicitly writing that they 'claim no novelty on these results and in fact we regard the techniques as well-known enough to omit all proofs.' The omission is load-bearing: Theorem 1.13 is what converts the flatness and symmetric-power hypotheses on A into the existence of the commutative ring model structure, and both Theorem E (via Theorem 2.3 and the 'Model structures for Theorem E' passage) and Theorem C (via Theorem III.1.2) rely directly on it. In particular, Theorem 1.13 asserts that hypotheses (i) and (ii) suffice to create a model structure on commutative ring G-spectra, but no argument or precise reference is given for this sufficiency. The standard route to such a result normally requires a commutative monoid axiom or an equivalent control of symmetric powers of cofibrant replacements, and Theorem 1.13 packages that control in a way that is asserted rather than proved. Because Theorem 2.3 then verifies these hypotheses using Chapter V, an error or missing hypothesis in II.1.13 would propagate directly into the derived geometric fixed point functor on commutative ring spectra. I request that full proofs of Theorems II.1.9, II.1.10, II.1.11, and II.1.13 be supplied, or that the authors give a complete and precise citation to proofs of exactly these statements in the stated generality.
[Chapter II.2, Theorem 2.3 and Corollary 2.4] Theorem 2.3 is the main existence result used to instantiate the hypotheses of Corollary 2.2 and hence to prove Theorem E. Its proof is very compressed: it asserts that every A in the closure A can be expressed as a generalized orbit desuspension spectrum J_η for a bundle of the stated form, and then invokes several results from Chapter V to verify flatness, symmetric power, and geometric fixed point properties. While the induction pattern is plausible and is supported by Lemma III.1.4 in the circle-group case, the general compact Lie group case needs a more explicit verification. In particular, the reader must check that the faithfulness hypotheses of Propositions V.3.2 and V.3.6 hold for all bundles obtained by the iterative closure under smash products and symmetric powers, not only for the initial cells A♯(F,V). I recommend that the proof be expanded to spell out the induction and the precise families of subgroups F_WH for which the Φ_H equivalences are asserted.
[Chapter III.1, Theorem 1.2 and Lemma 1.4] Theorem 1.2, which is the technical heart of Theorem C, depends on the black-box sets A_cyc and A_cyc^p and on Lemma 1.4. The lemma is proved in Chapter III.3, but the proof of Theorem 1.2 also uses Proposition II.1.14 and Theorem II.1.13 without proof. In particular, the assertion that the model structures are enriched over the standard model structure on non-equivariant spectra uses Lemma 1.4(vi) and the unproved Proposition II.1.14. Since Theorem C(ii) and the mapping-space theorems B and D depend on the enrichment statement, the missing proof of Proposition II.1.14 is part of the same foundational gap. This should be addressed together with the Chapter II.1 theorems.

minor comments (4)

[Chapter III.4, Definition 4.1] The text refers to 'Section II.II' in the paragraph after Definition 4.1; this should be 'Section II.1'.
[Chapter II.2, 'Model structures for Theorem E'] The sentence referring to the statement of Theorem 2.3 as 'peripatetic in this regard' is informal and unclear; the authors should state directly which properties of the closure A_W^{(∧)} are being invoked.
[Chapter I, Theorem B and Chapter III.2, Theorem 2.7] The statement of Theorem B in the introduction differs slightly in wording from Theorem 2.7, and the introduction does not define the notation hoEq before using it. Please align the two statements and define hoEq in the introduction.
[Chapter II.1, Definition 1.6] The definition of a regular I_A-cofibration is clear, but the phrase 'pushout along a map in I_α' in the sentence after the display appears to contain a typo: it should read 'I_A'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper's new theorems are derived from prior established equivariant stable homotopy theory, with unproved model-structure results serving as stated assumptions rather than as self-referential inputs.

full rationale

The derivation chain in this paper is self-contained in the relevant sense: the central claims are new mathematical theorems proved from explicit constructions (generalized orbit desuspension spectra), stated hypotheses, and previously established results in equivariant stable homotopy theory. There are no fitted parameters, no quantities that are defined in terms of the quantities they are said to predict, and no instance in which a theorem's conclusion is assumed in its hypotheses. The paper's reliance on the V-constrained model-structure theorems of Chapter II.1, especially Theorems II.1.9, II.1.10, II.1.11, and II.1.13, is substantial, but the authors explicitly state that they 'claim no novelty on these results and in fact regard the techniques as well-known enough to omit all proofs.' Omitting proofs of background model-category results, or deferring them to prior work, is an incompleteness or foundational-risk concern rather than circularity: the theorems are presented as inputs with stated hypotheses, not as consequences of the claims they support. Similarly, the paper's citations to the authors' earlier work [4] supply definitions, the category of pre-cyclotomic spectra, and proof templates for mapping-space results, but the new Theorems C, D, and B are not forced by those citations; rather, the authors generalize and reprove the relevant arguments in the new model structures. The multiplicative tom Dieck splitting (Theorem A) is proved by an explicit cell-by-cell computation using the multiplicative transfers and generalized orbit desuspension spectra, with no step in which the output is equivalent to the input by construction. Overall, the paper presents independent mathematical content; the main risk is the unproved model-structure foundation, which is a correctness or completeness issue, not a circularity issue.

Assumptions & free parameters 0 free parameters · 5 assumptions · 2 invented entities

The paper's central claims depend on a collection of standard results in equivariant stable homotopy theory, on the authors' earlier work on pre-cyclotomic spectra, and on the unproved V-constrained model structure theorems. There are no numerical free parameters or fits to data. The main invented mathematical objects are the generalized orbit desuspension spectra and the Acyc cell sets, both of which are rigorously constructed rather than empirically motivated.

assumptions (5)

ad hoc to paper The V-constrained model structure theorems II.1.9, II.1.10, II.1.11, and II.1.13 exist as stated.
The authors state that the techniques are well known and omit all proofs, but these theorems are used to construct every model structure in the paper.
domain assumption The standard, positive, and complete model structures on equivariant orthogonal spectra exist with the stated weak equivalences and cofibrations.
This is standard background from Mandell-May [22] and is used throughout, for example in the description of cells in Chapter II.
domain assumption The Hill-Hopkins-Ravenel norm and diagonal constructions, including the results cited as [15, B.190] and [15, B.209], are correct.
These are used in the proof of the multiplicative tom Dieck splitting and in the circle group arguments.
domain assumption The model structure for pre-cyclotomic spectra and the mapping spectrum results from the authors' previous paper [4] are correct.
The paper uses [4] for the definition and basic homotopy theory of ppc and pc spectra, and for the proof pattern of Theorem B.
standard math The Peter-Weyl theorem and the representation theory of compact Lie groups provide the required representation constraints.
This is invoked in Chapter II.2 to construct the representation constraint V needed for Theorem E.

invented entities (2)

Generalized orbit desuspension spectra J_eta
purpose: A new class of equivariant orthogonal spectra constructed from equivariant vector bundles, designed to behave well under the geometric fixed point functor, smash products, and symmetric powers.
There is no external empirical handle; the entity is justified entirely by its internal mathematical properties, which are proved in Chapters V and VI.
The sets Acyc and Acyc^p of T-spectra
purpose: Black-box generating sets of cells used to construct the circle-group model structures in Theorem C and the cyclotomic model structures in Theorem D.
These sets are defined in Definition III.3.3 and their properties are established by Lemma III.1.4. They have no existence outside the paper's construction.

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Cite this review

Pith. "Pith review of The homotopy theory of cyclotomic spectra, 10 years later." pith.science (2026). https://pith.science/paper/CWXVJB5P

@misc{pith2026241215928,
  author       = {Pith},
  title        = {Pith review of: The homotopy theory of cyclotomic spectra, 10 years later},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CWXVJB5P}},
  note         = {Machine review of arXiv:2412.15928}
}

read the original abstract

This paper studies the foundations of the geometric fixed point functor in multiplicative equivariant stable homotopy theory. We introduce a new class of equivariant orthogonal spectra called generalized orbit desuspension spectra and analyze their homotopical behavior with respect to the geometric fixed point functor and especially the interaction with smash products and symmetric powers. This analysis leads to several new foundational results, including the construction of the derived functor of geometric fixed points on equivariant commutative ring orthogonal spectra and its comparison to the derived functor on the underlying equivariant orthogonal spectra. In addition this theory provides foundations for the category of commutative ring pre-cyclotomic spectra and a formula for the derived space of maps in this category (a formula needed in the authors' paper with Yuan on relative TC). Finally, we prove a new multiplicative tom Dieck splitting for equivariant commutative ring spectra obtained as the pushforward of non-equivariant commutative ring spectra.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

27 extracted references · 26 canonical work pages

[1]

Blumberg, Teena Gerhardt , Michael A

Vigleik Angeltveit, Andrew J. Blumberg, Teena Gerhardt , Michael A. Hill, Tyler Lawson, and Michael A. Mandell. Topological cyclic homology via the nor m. Doc. Math. , 23:2101–2163, 2018

work page 2018
[2]

Integ ral p-adic Hodge theory

Bhargav Bhatt, Matthew Morrow, and Peter Scholze. Integ ral p-adic Hodge theory. Publ. Math. Inst. Hautes ´Etudes Sci. , 128:219–397, 2018

2018
[3]

Blumberg and Michael A

Andrew J. Blumberg and Michael A. Hill. Operadic multipl ications in equivariant spectra, norms, and transfers. Adv. Math. , 285:658–708, 2015

work page 2015
[4]

Blumberg and Michael A

Andrew J. Blumberg and Michael A. Mandell. The homotopy t heory of cyclotomic spectra. Geom. Topol., 19(6):3105–3147, 2015

work page 2015
[5]

Blumberg, Michael A

Andrew J. Blumberg, Michael A. Mandell, and Allen Yuan. R elative cyclotomic structures and equivariant complex cobordism. Preprint, arXiv:2310. 02348, 2023

work page 2023
[6]

B¨ okstedt, W

M. B¨ okstedt, W. C. Hsiang, and I. Madsen. The cyclotomic trace and algebraic K-theory of spaces. Invent. Math. , 111(3):465–539, 1993

work page 1993
[7]

Equivariant Structure on Smash Powers

Morten Brun, Bjørn Ian Dundas, and Martin Stolz. Equivar iant structure on smash powers. Preprint, arXiv:1604.05939, 2022

work page Pith review arXiv 2022
[8]

Equivariant stable homotopy and Segal ’s Burnside ring conjecture

Gunnar Carlsson. Equivariant stable homotopy and Segal ’s Burnside ring conjecture. Ann. of Math. (2) , 120(2):189–224, 1984

work page 1984

Show all 27 references

[9]

Goodwillie, and Randy McCart hy

Bjørn Ian Dundas, Thomas G. Goodwillie, and Randy McCart hy. The local structure of algebraic K-theory , volume 18 of Algebra and Applications . Springer-Verlag London, Ltd., London, 2013

2013
[10]

Higher Hochschild ho mology is not a stable invariant

Bjørn Ian Dundas and Andrea Tenti. Higher Hochschild ho mology is not a stable invariant. Math. Z. , 290(1-2):145–154, 2018

2018
[11]

Freed, Michael J

Daniel S. Freed, Michael J. Hopkins, and Constantin Tel eman. Loop groups and twisted K-theory III. Ann. of Math. (2) , 174(2):947–1007, 2011

2011
[12]

Freed, Michael J

Daniel S. Freed, Michael J. Hopkins, and Constantin Tel eman. Loop groups and twisted K-theory II. J. Amer. Math. Soc. , 26(3):595–644, 2013

2013
[13]

J. P. C. Greenlees and J. P. May. Localization and comple tion theorems for M U-module spectra. Ann. of Math. (2) , 146(3):509–544, 1997

1997
[14]

On the K-theory of local ﬁelds

Lars Hesselholt and Ib Madsen. On the K-theory of local ﬁelds. Ann. of Math. (2) , 158(1):1– 113, 2003

2003
[15]

M. A. Hill, M. J. Hopkins, and D. C. Ravenel. On the nonexi stence of elements of Kervaire invariant one. Ann. of Math. (2) , 184(1):1–262, 2016

2016
[16]

Hill, XiaoLin Danny Shi, Guozhen W ang, and Zh ouli Xu

Michael A. Hill, XiaoLin Danny Shi, Guozhen W ang, and Zh ouli Xu. The slice spec- tral sequence of a C4-equivariant height-4 Lubin-Tate theory. Mem. Amer. Math. Soc. , 288(1429):v+119, 2023

2023
[17]

Hopkins, Nicholas J

Michael J. Hopkins, Nicholas J. Kuhn, and Douglas C. Rav enel. Generalized group characters and complex oriented cohomology theories. J. Amer. Math. Soc. , 13(3):553–594, 2000

2000
[18]

D. Kaledin. Motivic structures in non-commutative geo metry. In Proceedings of the Interna- tional Congress of Mathematicians. Volume II , pages 461–496, New Delhi, 2010. Hindustan Book Agency

2010
[19]

Involutions on s[ωm ]

Tore August Kro. Involutions on s[ωm ]. Preprint, arXiv:math/0510221, 2005

2005 arXiv
[20]

L. G. Lewis, Jr., J. P. May, M. Steinberger, and J. E. McCl ure. Equivariant stable homotopy theory, volume 1213 of Lecture Notes in Mathematics . Springer-Verlag, Berlin, 1986. With contributions by J. E. McClure

1986
[21]

Gaunce Lewis, Jr

L. Gaunce Lewis, Jr. and Michael A. Mandell. Modules in m onoidal model categories. J. Pure Appl. Algebra, 210(2):395–421, 2007. 107 108 BIBLIOGRAPHY

2007
[22]

M. A. Mandell and J. P. May. Equivariant orthogonal spec tra and S-modules. Mem. Amer. Math. Soc. , 159(755):x+108, 2002

2002
[23]

M. A. Mandell, J. P. May, S. Schwede, and B. Shipley. Mode l categories of diagram spectra. Proc. London Math. Soc. (3) , 82(2):441–512, 2001

2001
[24]

J. P. May. Equivariant homotopy and cohomology theory , volume 91 of CBMS Regional Conference Series in Mathematics . Published for the Conference Board of the Mathematical Sciences, W ashington, DC, 1996. With contributions by M. Cole, G. Comeza˜ na, S. Costenoble, A. D. Elme...

1996
[25]

Stefan Schwede and Brooke E. Shipley. Algebras and modu les in monoidal model categories. Proc. London Math. Soc. (3) , 80(2):491–511, 2000

2000
[26]

A convenient model category for commut ative ring spectra

Brooke Shipley. A convenient model category for commut ative ring spectra. In Homotopy theory: relations with algebraic geometry, group cohomolo gy, and algebraic K-theory, volume 346 of Contemp. Math. , pages 473–483. Amer. Math. Soc., Providence, RI, 2004

2004
[27]

Equivariant structure on smash powers of commutative ring spectra

Martin Stolz. Equivariant structure on smash powers of commutative ring spectra. University of Begen thesis, 2011

2011

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

Blumberg, Teena Gerhardt , Michael A

Vigleik Angeltveit, Andrew J. Blumberg, Teena Gerhardt , Michael A. Hill, Tyler Lawson, and Michael A. Mandell. Topological cyclic homology via the nor m. Doc. Math. , 23:2101–2163, 2018

work page 2018

[2] [2]

Integ ral p-adic Hodge theory

Bhargav Bhatt, Matthew Morrow, and Peter Scholze. Integ ral p-adic Hodge theory. Publ. Math. Inst. Hautes ´Etudes Sci. , 128:219–397, 2018

2018

[3] [3]

Blumberg and Michael A

Andrew J. Blumberg and Michael A. Hill. Operadic multipl ications in equivariant spectra, norms, and transfers. Adv. Math. , 285:658–708, 2015

work page 2015

[4] [4]

Blumberg and Michael A

Andrew J. Blumberg and Michael A. Mandell. The homotopy t heory of cyclotomic spectra. Geom. Topol., 19(6):3105–3147, 2015

work page 2015

[5] [5]

Blumberg, Michael A

Andrew J. Blumberg, Michael A. Mandell, and Allen Yuan. R elative cyclotomic structures and equivariant complex cobordism. Preprint, arXiv:2310. 02348, 2023

work page 2023

[6] [6]

B¨ okstedt, W

M. B¨ okstedt, W. C. Hsiang, and I. Madsen. The cyclotomic trace and algebraic K-theory of spaces. Invent. Math. , 111(3):465–539, 1993

work page 1993

[7] [7]

Equivariant Structure on Smash Powers

Morten Brun, Bjørn Ian Dundas, and Martin Stolz. Equivar iant structure on smash powers. Preprint, arXiv:1604.05939, 2022

work page Pith review arXiv 2022

[8] [8]

Equivariant stable homotopy and Segal ’s Burnside ring conjecture

Gunnar Carlsson. Equivariant stable homotopy and Segal ’s Burnside ring conjecture. Ann. of Math. (2) , 120(2):189–224, 1984

work page 1984