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Modular fixed points in equivariant homotopy theory

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Pith's one-line read The derived infinity-category of permutation modules is equivalent to modules over a constant Mackey spectrum in G-spectra; this yields modular fixed points and the Picard classification for p-groups.

desk verdict Strong on the finite-group core; the Picard classification proof has a genuine gap. read the letter →

arxiv 2506.21413 v2 pith:LZQLEZDK submitted 2025-06-26 math.AT math.RT

Yorick Fuhrmann This is my paper · ORCID
classification math.ATmath.RT MSC 55P9155P4219A22
keywords equivarianthomotopytheorypermutationmodulesMackeyfunctorsmodularfixedpointsgeometricweightstructuresPicardgroupBorel-Smithconditions
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's central project is to show that the derived $\infty$-category of permutation modules of a finite or profinite group and the category of modules over an equivariant Eilenberg-MacLane spectrum of the constant Mackey functor are the same symmetric monoidal $\infty$-category. It proves $\mathrm{Mod}_{HR}(\mathrm{Sp}_G) \simeq \mathrm{DPerm}(G;R)$ by two routes: through cohomological Mackey functors, and through a new bounded weight structure on compact $HR$-modules whose weight heart is the idempotent completion of finitely generated permutation modules. On this common ground the paper constructs an equivariant modular fixed point functor $\Psi_H$ by applying geometric fixed points and then changing scalars back to $HR$, and proves that this functor agrees with the modular fixed point functor previously defined on derived permutation modules. As a payoff, when $G$ is a $p$-group and $k$ is a field of characteristic $p$, the invertible objects in $\mathrm{Mod}_{Hk}(\mathrm{Sp}_G)$ are classified by the group $\mathrm{CF}^b(G)$ of integral class functions satisfying the Borel-Smith conditions.

What carries the argument

The argument is carried by three linked objects. The constant Mackey functor $\underline{R}$ has identity restriction maps and induction maps that multiply by the index, and its Eilenberg-MacLane spectrum $HR$ has the property that modules over it in $G$-spectra form the derived category of cohomological Mackey functors. The geometric fixed point functor $\Phi_H \colon \mathrm{Sp}_G \to \mathrm{Sp}_{G//H}$ is a symmetric monoidal left adjoint built by localising away from orbits that do not contain $H$; it satisfies $\Phi_H(\Sigma^\infty X) \simeq \Sigma^\infty X^H$, but it does not preserve Eilenberg-MacLane spectra. The paper shows $\pi_0(\Phi_H(HR)) \cong R$ when $H$ is a $p$-subgroup and $p=0$ in $R$, and uses the resulting ring map $\Phi_H(HR) \to HR$ as a base change, producing a functor that is $HR$-linear and weight exact. The third piece is a bounded weight structure, a categorical filtration by degrees analogous to a t-structure, on the compact part of $\mathrm{Mod}_{HR}(\mathrm{Sp}_G)$; its weight heart is identified with $\mathrm{perm}(G;R)^\natural$, the idempotent completion of finitely generated permutation modules. That identification gives $\mathrm{Mod}^{\omega}_{HR}(\mathrm{Sp}_G) \simeq K^b(\mathrm{perm}(G;R)^\natural)$ and is the mechanism by which facts about permutation modules are transported into genuine equivariant spectra.

What would settle it

Compute the mapping spectra between $HR \otimes \Sigma^\infty G/H_+$ and $HR \otimes \Sigma^\infty G/K_+$ for a small group such as $C_2 \times C_2$; if any higher homotopy group $\pi_n$ for $n \geq 1$ is nonzero, or if the endomorphism ring is not the permutation-module hom ring described in Lemma 5.11, the weight-heart identification fails. Alternatively, compute $\pi_0(\Phi_H(Hk))$ for a non-normal $p$-subgroup $H$; the modular fixed point functor of Definition 6.13 is only well-defined if every section is $k$, so a single vanishing section would refute Lemma 6.11 and the construction.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the derived $\infty$-category of permutation modules is exactly the module category of the Eilenberg-MacLane spectrum associated to the constant Mackey functor in $G$-spectra: $\mathrm{DPerm}(G;R) \simeq \mathrm{Mod}_{HR}(\mathrm{Sp}_G)$, obtained through the intermediate identification $\mathrm{DPerm}(G;R) \simeq D(\mathrm{Mack}^{\mathrm{coh}}_R(G))$. On compact objects the equivalence is made concrete by a bounded weight structure whose heart is the idempotent completion of the finitely generated permutation modules, $\mathrm{Mod}^{\omega}_{HR}(\mathrm{Sp}_G) \simeq K^b(\mathrm{perm}(G;R)^\natural)$. The paper then defines, for a $p$-subgroup $H$ with $p=0$ in $R$, the equivariant modular fixed point functor $\Psi_H$ as geometric fixed points followed by extension of scalars along $\Phi_H(HR) \to HR$, and proves that it matches the modular fixed point functor on derived permutation modules. The consequence for the Picard group is that for a finite $p$-group $G$ and a field $k$ of characteristic $p$, $\mathrm{Pic}(\mathrm{Mod}_{Hk}(\mathrm{Sp}_G)) \cong \mathrm{CF}^b(G)$, with surjectivity supplied by dimension functions of real representations; this also extends to pro-$p$-groups.

Load-bearing premise

The whole bridge rests on the claim that a certain filtration layer of compact modules over the constant Mackey spectrum is exactly built from finitely generated permutation modules; if that identification fails, the equivalence between the two categories fails, and with it the comparison of modular fixed points and the Picard classification.

Editorial extensions

If this is right

  • The equivalence $\mathrm{Mod}_{HR}(\mathrm{Sp}_G) \simeq \mathrm{DPerm}(G;R)$ gives permutation modules an $\infty$-categorical, symmetric monoidal refinement inside genuine equivariant spectra, so constructions such as base change along group homomorphisms, norms, and fixed-point adjunctions become available for them.
  • For every $p$-subgroup $H$ there is an $HR$-linear, weight exact functor $\Psi_H$ that is compatible with nesting and restriction and recovers the classical Brauer quotient on permutation modules: $R(X) \mapsto R(X^H)$.
  • The family of functors $\bar{\Psi}_H$ (modular fixed points followed by restriction to the trivial group) is jointly conservative, so an invertible $Hk$-module is detected by the single degree in which each $\bar{\Psi}_H(X)$ is a copy of $k$.
  • For $p$-groups, $\mathrm{Pic}(\mathrm{Mod}_{Hk}(\mathrm{Sp}_G)) \cong \mathrm{CF}^b(G)$, so invertible objects are enumerated by Borel-Smith class functions, and the same classification extends to pro-$p$-groups.
  • The surjectivity of the Picard isomorphism is proved topologically, from representation spheres and dimension functions, so the classification does not require the representation-theoretic theory of endotrivial complexes.

Reading between the lines

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

  • A natural reading of the equivalence is that $\mathrm{DPerm}(G;R)$ secretly carries a stable $\infty$-categorical structure with both a t-structure and a weight structure inherited from $G$-spectra; this extra structure is not visible in the tensor-triangular formulation and may be useful for classification problems beyond the Picard group.
  • The same base-change template may define modular fixed points for other Green functors $B$ whenever $\pi_0(\Phi_H(HB))$ identifies with $B$; the paper restricts to the constant Mackey functor, but its Lemma 6.11 suggests the locus of such $B$ is determined by vanishing of certain indices.
  • It seems plausible that joint conservativity of the $\Psi_H$ family, combined with the tensor-triangular stratification of permutation modules, could compute the Balmer spectrum of $\mathrm{Mod}_{HR}(\mathrm{Sp}_G)$ by descent over $p$-subgroups; the paper does not spell this out.
  • A concrete testable extension is to replace the field $k$ by a ring $R$ in which $p=0$; the topological proof should still produce a map into Borel-Smith class functions, while the representation-theoretic interpretation may need adjustment.
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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

3 major / 4 minor

Summary. The paper develops an ∞-categorical framework for modular representation theory in equivariant spectra. It proves that the category of modules over the constant Mackey functor Eilenberg–MacLane spectrum HR in G-spectra is equivalent to the derived ∞-category of permutation modules, by two routes: through cohomological Mackey functors and through a bounded weight structure whose heart is the idempotent completion of finitely generated permutation modules. It then defines an equivariant modular fixed point functor Ψ_H by geometric fixed points and extension of scalars, and identifies it with the Balmer–Gallauer modular fixed point functor on derived permutation modules. The final application claims an isomorphism Pic(ModHk(SpG)) ≅ CFb(G) for p-groups, with a formal extension to pro-p-groups.

Significance. If the main equivalences and the comparison theorem hold, the paper gives a substantial bridge between equivariant stable homotopy theory and the modular representation theory of permutation modules. The two independent routes to ModHR(SpG) ≃ DPerm(G;R), the explicit weight structure, and the topological construction of modular fixed points are genuinely useful and go beyond a mere citation of [BG23]. The paper also gives a new-looking topological route to Miller's Picard group classification. However, the Picard application is not established as written: the injectivity argument in Proposition 7.4 is incorrect, and the group structure on the auxiliary group Λ(G;k) is not clearly defined. Since the advertised topological proof of Theorem 7.8 relies on that injectivity, the headline classification is currently supported only on the surjectivity side.

major comments (3)
  1. [Section 7, Proposition 7.4] The injectivity proof is invalid. The paper argues that if all Ψ_H(X) are concentrated in degree 0 then X lies in the weight heart, and then says that 'k(G/G) is the only element of perm(G;R)^♮ which has k-dimension 1', concluding X ≃ Hk. The condition that the total fixed-point module Ψ_G(X) has k-dimension 1 does not force X itself to have k-dimension 1. For G = C_p, take X = k(G/G) ⊕ k(G/1) in K^b(perm(G;k)^♮), or its image under the equivalence of Theorem 5.12/6.36. Then Ψ_1(X) ≅ k ⊕ kG is nonzero only in degree 0, and Ψ_G(X) ≅ k is 1-dimensional in degree 0, so X satisfies Definition 7.3. But λ_X(H) = 0 for every H while X ≇ Hk, since its k-dimension is p+1. Thus λ is not injective as stated, and the proof of Theorem 7.8, which proves only surjectivity and gives no separate injectivity argument, leaves the isomorphism Pic(ModHk(SpG)) ≅ CFb(G) unsupported.
  2. [Section 7, Definition 7.3] The object Λ(G;k) is not an abelian group with the asserted group structure. The tensor product on ModHk(SpG) is symmetric monoidal, but an object satisfying the spectral conditions of Definition 7.3 need not be invertible; for instance the object X = k(G/G) ⊕ k(G/1) in the previous comment has no tensor inverse. Consequently the phrase 'group structure is induced by the symmetric monoidal structure' is not meaningful without either restricting to invertible objects or to capped V-endosplit-trivial complexes as in Miller's work. This affects the statement that λ is a 'surjective group homomorphism' in Proposition 7.4.
  3. [Section 7, Theorem 7.8] The theorem is presented as the paper's own topological proof of the Picard group classification, but its proof only establishes surjectivity of θ via the dimension homomorphism RO(G) → CFb(G). Injectivity is not proved there and instead is inherited from Proposition 7.4, whose injectivity claim is false as noted above. Unless the injectivity of θ is proved directly or explicitly imported from [Mil24c], Theorem 7.8 should not be stated as proved.
minor comments (4)
  1. [Recollection 5.1] There is a typo in the definition of the shifted weight structures: the clause 'Cw≥n ..= ΣnCw≤0' should read 'Cw≥n ..= ΣnCw≥0'.
  2. [Lemma 6.11] In the last paragraph of the proof, the inequality concluding that all indices vanish is written backwards: it should be k > l, not l > k.
  3. [Lemma 6.17 and Section 7] The paper moves between pointed G-spaces and unpointed G-sets without always making the basepoint convention explicit. In particular, Lemma 6.17 writes X^H for pointed fixed points, while Definition 7.3 and Proposition 7.4 use permutation-module fixed points; clarifying this convention would avoid ambiguity in the counterexample discussed above.
  4. [Proposition 7.4] The equality 'perm(G; R)^♮ = perm(G; R)' in the last paragraph is at least misleading: the left side is the idempotent completion, which is not literally equal to perm(G;R) in general.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central equivalences and the Picard theorem are proved by independent weight-structure, presheaf, and topological arguments rather than by assuming their conclusions.

full rationale

The paper's derivation chain does not reduce to its inputs by construction. The equivalence ModHR(SpG) ≃ DPerm(G;R) is obtained in two independent ways: via cohomological Mackey functors and Yoshida's equivalence (Corollary 3.15, Theorem 4.7, Corollary 4.8), and via a separately constructed bounded weight structure with heart perm(G;R)^♮ (Proposition 5.10, Lemma 5.11, Theorem 5.12). The latter uses Sosnilo's theorem-of-the-heart machinery rather than assuming the target equivalence. Theorem 6.36, identifying the newly defined equivariant modular fixed point functor with Balmer-Gallauer's, is proved by checking the weight hearts and the generators of morphisms, using the explicit geometric fixed point formula of Lemma 6.17 and the cited behaviour of the Brauer quotient; it is a verification, not a definitional identity. The Picard theorem is proved with a new topological surjectivity argument via tom Dieck's dimension-function theorem, while injectivity is attempted through Proposition 7.4 and the Bouc–Yalçın identification of Borel-Smith functions with the kernel of the Bouc homomorphism; Miller's classification is cited as context, not used as the proof. The author's frequent citations to Balmer–Gallauer and Binda–Gallauer–Vezzani are real external results with published proofs, and none of them is invoked as an unproved uniqueness theorem to force the conclusion. The skeptic's objection to Proposition 7.4 is a mathematical correctness concern about the claimed injectivity, not a circularity: it alleges a false assertion, not an equation that is equivalent to its own input. Because no 'prediction' is fitted and no load-bearing claim is justified solely by a self-citation, the circularity score is 0.

Assumptions & free parameters 0 free parameters · 7 assumptions · 2 invented entities

No free parameters are fitted. The proof rests on a network of established results in equivariant homotopy theory, Mackey functors, and weight structures. The principal new object is the equivariant modular fixed point functor, which is constructed from geometric fixed points and benchmarked against the modular fixed point functor of Balmer-Gallauer.

assumptions (7)
  • domain assumption Equivalence of derived Mackey functors with modules over HA for a finite group (Patchkoria-Sanders-Wimmer, Theorem 5.10), extended to profinite groups in Theorem 3.9.
    This is the backbone of Section 3, allowing the paper to identify D(Mack(G)) with ModHA(SpG). The profinite extension is proven in Theorem 3.9 using [BBB24] and [MNN17].
  • domain assumption Yoshida's equivalence between cohomological Mackey functors and R-linear presheaves on permutation modules (Lemma 2.5, citing [BG23, Corollary 4.22]).
    Used to identify the weight heart and to pass between Mackey functors and permutation modules throughout.
  • standard math Theorems of Sosnilo and Aoki on weight structures, in particular [Sos19, Corollary 3.4] and [Aok20, Theorem 4.3].
    These provide the weight complex functor and its symmetric monoidal enhancement used in Theorems 5.12 and 6.36.
  • domain assumption The partially lax limit framework of Linskens-Nardin-Pol [LNP24] and the existence of a highly coherent geometric fixed point natural transformation.
    Construction 6.28 uses this framework to extend modular fixed points from finite to profinite groups; the paper only sketches the construction of the natural transformation.
  • standard math Bouc-Yalcin theorem: the kernel of the Bouc homomorphism is the group of Borel-Smith class functions [BY07, Theorem 1.2].
    Used in Lemma 7.7 to prove that the Picard group maps into CFb(G).
  • standard math tom Dieck's result that the dimension homomorphism RO(G) to CFb(G) is surjective for p-groups [Die87, Theorem III.5.4], relying on fixed point results of Smith and Borel.
    Used in Theorem 7.8 to prove surjectivity of the Picard map.
  • domain assumption Balmer-Gallauer's definition and properties of the modular fixed point functor on DPerm(G;R) and their stratification theorem.
    The paper's equivariant modular fixed point functor is compared to this functor, and the joint conservativity in Proposition 7.1 uses [BG22a, Theorem 5.12].
invented entities (2)
  • Equivariant modular fixed point functor ΨH on ModHR(SpG) independent evidence
    purpose: Provides an HR-linear, symmetric monoidal left adjoint that linearizes geometric fixed points and lands in HR-modules for the Weyl group.
    It is a new construction (Definition 6.13) built from geometric fixed points and extension of scalars. It is benchmarked externally by the free-module property ΨH(HR⊗Σ∞X) ≃ HR⊗Σ∞(X^H) and by the comparison with Balmer-Gallauer's functor in Theorem 6.36.
  • Ring map ψ_H: ΦH(HR) to HR independent evidence
    purpose: Defines the base change that makes geometric fixed points land in HR-modules; its choice fixes the HR-linearity of ΨH.
    The map corresponds to the identity on π0 after the isomorphism π0(ΦH(HR)) ≅ R of Lemma 6.11. It is constrained by Proposition 6.22 once the HR-linear structure is fixed.

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Pith. "Pith review of Modular fixed points in equivariant homotopy theory." pith.science (2026). https://pith.science/paper/LZQLEZDK

@misc{pith2026250621413,
  author       = {Pith},
  title        = {Pith review of: Modular fixed points in equivariant homotopy theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LZQLEZDK}},
  note         = {Machine review of arXiv:2506.21413}
}
abstract

We show that the derived $\infty$-category of permutation modules is equivalent to the category of modules over the Eilenberg-MacLane spectrum associated to a constant Mackey functor in the $\infty$-category of equivariant spectra. On such module categories we define a modular fixed point functor using geometric fixed points followed by an extension of scalars and identify it with the modular fixed point functor on derived permutation modules introduced by Balmer-Gallauer. As an application, we show that the Picard group of such a module category for a $p$-group is given by the group of class functions satisfying the Borel-Smith conditions. In the language of representation theory, this result was first obtained by Miller.

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Works this paper leans on

25 extracted references · 18 canonical work pages · cited by 2 Pith papers

  1. [1]

    ‘The weight complex functor is symmetric monoidal’

    [Aok20] Ko Aoki. ‘The weight complex functor is symmetric monoidal’. In: Adv. Math.368 (2020), pp. 107145,

    work page 2020

  2. [5]

    [BG22a] Paul Balmer and Martin Gallauer

    arXiv:2104.03870. [BG22a] Paul Balmer and Martin Gallauer. ‘Permutation modules and cohomological singu- larity’. In:Comment. Math. Helv.97.3 (2022), pp. 413–430. [BG22b] Paul Balmer and Martin Gallauer. The tt-geometry of permutation modules. Part I: Stratification

    arXiv 2022

  3. [7]

    The spectrum of Artin motives

    arXiv: 2401.02722. [BGV24] Federico Binda, Martin Gallauer and Alberto Vezzani. Motivic monodromy and p- adic cohomology theories

    work page Pith review arXiv

  4. [8]

    [BH21] Tom Bachmann and Marc Hoyois

    arXiv:2306.05099. [BH21] Tom Bachmann and Marc Hoyois. ‘Norms in motivic homotopy theory’. In: As- térisque 425 (2021), pp. ix+207. [Blu+19] Andrew J. Blumberg, Teena Gerhardt, Michael A. Hill and Tyler Lawson. ‘The Witt vectors for Green functors’. In:J. Algebra537 (2019), pp. 197–244. [Bon10] Mikhail V. Bondarko. ‘Weight structures vs. t-structures; wei...

    arXiv 2021

  5. [9]

    [Cno+24] Bastiaan Cnossen, Rune Haugseng, Tobias Lenz and Sil Linskens.Normed equivari- ant ring spectra and higher Tambara functors

    arXiv:2011.08233. [Cno+24] Bastiaan Cnossen, Rune Haugseng, Tobias Lenz and Sil Linskens.Normed equivari- ant ring spectra and higher Tambara functors

    arXiv 2011

  6. [10]

    ‘L’algèbre de Hopf et le groupe de Galois motiviques d’un corps de caractéristique nulle, I’

    [Ayo14] Joseph Ayoub. ‘L’algèbre de Hopf et le groupe de Galois motiviques d’un corps de caractéristique nulle, I’. In:J. Reine Angew. Math.693 (2014), pp. 1–149. [Bac16] Tom Bachmann. Invertible objects in motivic homotopy theory. PhD thesis

    work page 2014

  7. [11]

    [ČS24] Kęstutis Česnavičius and Peter Scholze

    arXiv:2407.08399. [ČS24] Kęstutis Česnavičius and Peter Scholze. ‘Purity for flat cohomology’. In: Ann. of Math. (2)199.1 (2024), pp. 51–180. [Dad78] Everett C. Dade. ‘Endo-Permutation Modules over p-groups, I’. In: Annals of Math- ematics 107.3 (1978), pp. 459–494. [DG22] Brad Drew and Martin Gallauer. ‘The universal six-functor formalism’. In: Ann. K-Th...

    arXiv 2024

  8. [12]

    Walter de Gruyter & Co., Berlin, 1987, pp

    De Gruyter Studies in Mathem- atics. Walter de Gruyter & Co., Berlin, 1987, pp. x+312. [Fau08] Halvard Fausk. ‘Equivariant homotopy theory for pro-spectra’. In: Geom. Topol. 12.1 (2008), pp. 103–176. [GGN15] David Gepner, Moritz Groth and Thomas Nikolaus. ‘Universality of multiplicative infinite loop space machines’. In:Algebr. Geom. Topol.15.6 (2015), pp...

    work page 2008

Show all 25 references
  1. [15]

    [Lüc05] Wolfgang Lück

    arXiv:2206.01556. [Lüc05] Wolfgang Lück. ‘Survey on classifying spaces for families of subgroups’. In: Infinite groups: geometric, combinatorial and dynamical aspects. Vol

    arXiv

  2. [19]

    [Mil24c] Sam K

    arXiv:2402.08042. [Mil24c] Sam K. Miller. The classification of endotrivial complexes

  3. [20]

    [MM02] M

    arXiv:2403.04088. [MM02] M. A. Mandell and J. P. May. ‘Equivariant orthogonal spectra and S-modules’. In: Mem. Amer. Math. Soc.159.755 (2002), pp. x+108. [MNN17] Akhil Mathew, Niko Naumann and Justin Noel. ‘Nilpotence and descent in equivari- ant stable homotopy theory’. In:Ad...

    2002

  4. [21]

    [Nar17] Denis Nardin

    arXiv:1608.07704. [Nar17] Denis Nardin. Stability and distributivity over orbital ∞-categories. PhD thesis

    arXiv

  5. [34]

    Cambridge University Press, Cambridge, 2018, pp

    New Mathematical Monographs. Cambridge University Press, Cambridge, 2018, pp. xviii+828. [Shi07] Brooke Shipley. ‘ HZ-Algebra Spectra Are Differential Graded Algebras’. In:Amer- ican Journal of Mathematics129.2 (2007), pp. 351–379. [Sos19] Vladimir Sosnilo. ‘Theorem of the hea...

    2007

  6. [91]

    Cambridge University Press, Cambridge, 2018, pp

    Lon- don Mathematical Society Student Texts. Cambridge University Press, Cambridge, 2018, pp. x+515. [LNP24] Sil Linskens, Denis Nardin and Luca Pol. Global homotopy theory via partially lax limits

    2018

  7. [170]

    Prin- ceton University Press, Princeton, NJ, 2009, pp

    Annals of Mathematics Studies. Prin- ceton University Press, Princeton, NJ, 2009, pp. xviii+925. [Lur17] Jacob Lurie. Higher Algebra. https://www.math.ias.edu/~lurie/

    2009

  8. [248]

    Progr. Math. Birkhäuser, Basel, 2005, pp. 269–322. [Lur09] Jacob Lurie. Higher topos theory. Vol

    2005

  9. [1213]

    With contributions by J

    Lecture Notes in Mathematics. With contributions by J. E. McClure. Springer-Verlag, Berlin, 1986, pp. x+538. [Lin18] Markus Linckelmann. The block theory of finite group algebras. Vol. I. Vol

    1986

  10. [2016]

    ‘Spectral Mackey functors and equivariant algebraic K-theory (I)’

    [Bar17] Clark Barwick. ‘Spectral Mackey functors and equivariant algebraic K-theory (I)’. In: Adv. Math.304 (2017), pp. 646–727. [BBB24] Scott Balchin, David Barnes and Tobias Barthel. Profinite equivariant spectra and their tensor-triangular geometry

    2017

  11. [2017]

    ‘Synthetic spectra and the cellular motivic category’

    [Pst23] Piotr Pstrągowski. ‘Synthetic spectra and the cellular motivic category’. In: Inven- tiones mathematicae232 (2023), pp. 553–681. [PSW22] Irakli Patchkoria, Beren Sanders and Christian Wimmer. ‘The spectrum of derived Mackey functors’. In:Trans. Amer. Math. Soc.375.6 (2...

    2023

  12. [2018]

    Yorick Fuhrmann, W ar wick Mathematics Institute, Coventry CV4 7AL, UK Email address: yorick.fuhrmann@warwick.ac.uk URL: https://warwick.ac.uk/fac/sci/maths/people/staff/fuhrmann

    arXiv: 1710.01769. Yorick Fuhrmann, W ar wick Mathematics Institute, Coventry CV4 7AL, UK Email address: yorick.fuhrmann@warwick.ac.uk URL: https://warwick.ac.uk/fac/sci/maths/people/staff/fuhrmann

    arXiv

  13. [2020]

    [TW95] Jacques Thévenaz and Peter Webb

    arXiv: 2001.07404. [TW95] Jacques Thévenaz and Peter Webb. ‘The structure of Mackey functors’. In: Trans. Amer. Math. Soc.347.6 (1995), pp. 1865–1961. [Zen18] Mingcong Zeng. Equivariant Eilenberg-Mac Lane spectra in cyclicp-groups

    1995 arXiv

  14. [2022]

    [BG23] Paul Balmer and Martin Gallauer

    arXiv:2210.08311. [BG23] Paul Balmer and Martin Gallauer. ‘Permutation modules, Mackey functors, and Artin motives’. In: Representations of algebras and related structures. EMS Ser. Congr. Rep. EMS Press, Berlin, 2023, pp. 37–75. 50 REFERENCES [BG24] Paul Balmer and Martin Gal...

    2023 arXiv

  15. [2023]

    [Kal11] Dmitry Kaledin

    arXiv:2310.11183. [Kal11] Dmitry Kaledin. ‘Derived Mackey functors’. In: Mosc. Math. J.11.4 (2011), pp. 723–

    2011 arXiv

  16. [2024]

    [BCN24] Lukas Brantner, Ricardo Campos and Joost Nuiten

    arXiv:2401.01878. [BCN24] Lukas Brantner, Ricardo Campos and Joost Nuiten. PD Operads and Explicit Par- tition Lie Algebras

    arXiv

  17. [2025]

    [Mil24a] Sam K. Miller. ‘Endotrivial complexes’. In: J. Algebra650 (2024), pp. 173–218. [Mil24b] Sam K. Miller. Relatively endotrivial complexes

    2024

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