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The Balmer spectrum and telescope conjecture for infinite groups

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Pith's one-line read For type-FP∞ groups the dualisable-module spectrum is the prime-ideal space of group cohomology, and for p-rank-one free products it is the Stone-Čech compactification of N; the paper also settles the telescope conjecture in both settings.

desk verdict Genuinely new Balmer spectra for infinite group stable categories and telescope counterexamples, but the FP∞ spectrum theorem has a load-bearing finiteness assertion that needs a real proof or citation. read the letter →

arxiv 2504.16602 v1 pith:O6PTQLU6 submitted 2025-04-23 math.RT math.CT

Gregory Kendall This is my paper · ORCID
classification math.RTmath.CT MSC 20C2020J0618G80
keywords BalmerspectrumstablemodulecategorytelescopeconjectureH1FgroupsFP-infinityStone-Čechcompactificationtensortriangulargeometrydualisableobjects
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 computes the Balmer spectrum — the space that classifies thick tensor ideals — of dualisable objects in the stable module category for two families of infinite groups, where the category is not rigidly compactly generated and few spectra were previously known. For an $\mathrm{H}_1\mathfrak{F}$ group of type $\mathrm{FP}_\infty$ over a field of characteristic $p > 0$ — a group with a finite-dimensional classifying space for proper actions and a finitely generated projective resolution of the trivial module — the spectrum is homeomorphic to $\mathrm{Proj}(H^*(G,k))$, the space of homogeneous prime ideals of the group's cohomology. For an infinite free product of finite groups of $p$-rank one, the spectrum is instead the Stone-Čech compactification of $\mathbb{N}$. The paper also settles the telescope conjecture — whether every smashing tensor ideal is generated by dualisable objects — in these settings: it holds for the $\mathrm{FP}_\infty$ groups and fails for the free-product examples, giving counterexamples in the infinite-group setting. Along the way it shows that the stable category of an infinite free product of cyclic groups is not stratified by the spectrum of its dualisable objects: the spectrum has $2^{2^{\aleph_0}}$ points, and there are only $2^{\aleph_0}$ localising tensor ideals, so no bijection can exist.

What carries the argument

The argument runs through the lattice-theoretic description of the Balmer spectrum via Stone duality: the spectrum of a tensor-triangulated category is the spectral space attached to the distributive lattice $\mathrm{Thickid}_f$ of finitely generated thick tensor ideals, and restriction to finite elementary abelian $p$-subgroups assembles a map $\mathrm{Thickid}_f(\underline{\mathrm{Mod}}(kG)^d) \to \lim_{E \in \mathcal{A}_E(G)} \mathrm{Thickid}_f(\underline{\mathrm{Mod}}(kE)^d)$, which the paper proves to be an isomorphism in both settings. For $\mathrm{H}_1\mathfrak{F}$ groups of type $\mathrm{FP}_\infty$, the decisive input is the paper's Theorem 6.4: a uniform $F$-isomorphism $H^*(G,k) \to \lim_E H^*(E,k)$, finite generation of $H^*(G,k)$, and finitely many conjugacy classes of elementary abelian $p$-subgroups; this yields the homeomorphism of projective spectra and, crucially, finite generation of each $H^*(E,k)$ as a module over $H^*(G,k)$, so that unions of supports over the conjugacy classes are closed. For free products, the decisive tool is the $\xi_M$ criterion of [4]: $\mathrm{Thick}(M) = \{X \mid \xi_M^{\otimes n} \otimes X = 0 \text{ for some } n \geq 0\}$, where $\xi_M$ fills the coevaluation triangle $C_M \to k \to M \otimes M^*$; for a finite group of $p$-rank one a uniform bound $C$ satisfies $\xi_M^{\otimes C} = 0$ for every $M$, and this uniform nilpotence lets the restriction map to the product of the factor lattices be inverted. The telescope-conjecture dichotomy has the same root: for $\mathrm{FP}_\infty$ groups every smashing ideal is generated by dualisable objects, while in the free-product case the tensor unit is not compact, and distinct thick ideals of dualisable objects can generate the same smashing subcategory.

What would settle it

For $G = \mathbb{Z}^n$, an $\mathrm{H}_1\mathfrak{F}$ group of type $\mathrm{FP}_\infty$, and $E = (C_p)^n$ its maximal elementary abelian $p$-subgroup, verify whether $H^*(E,k)$ is finitely generated as a module over $H^*(G,k)$ under restriction: Proposition 6.9 requires this for every such pair, and one failure would break the surjectivity step and with it the identification of the spectrum with $\mathrm{Proj}(H^*(G,k))$. On the free-product side, the load-bearing claim is that for $H = C_p$ every non-projective finitely generated module $M$ has $\xi_M = 0$ in the stable category; computing the coevaluation morphism $\xi_M$ for one such module, for instance the augmentation ideal, either confirms or refutes the uniform nilpotence bound on which Theorem 5.14 rests.

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Extended reading notes

Core claim

The central assertion is Theorem 1.1. Let $k$ be a field of characteristic $p > 0$. If $G$ is an $\mathrm{H}_1\mathfrak{F}$ group of type $\mathrm{FP}_\infty$ — a group with a finite-dimensional model for the classifying space for proper actions whose trivial module has a resolution by finitely generated projectives — then the Balmer spectrum $\mathrm{Spc}(\underline{\mathrm{Mod}}(kG)^d)$ of dualisable objects in the stable module category is homeomorphic to $\mathrm{Proj}(H^*(G,k))$, the homogeneous prime spectrum of group cohomology; if instead $G$ is an infinite free product of finite groups of $p$-rank one, the same spectrum is the Stone-Čech compactification of $\mathbb{N}$. These are the first Balmer spectra for infinite-group stable module categories that are not rigidly compactly generated. The paper further proves that the telescope conjecture holds in the $\mathrm{FP}_\infty$ case — every smashing localising tensor ideal is generated by dualisable objects — and fails for the free-product examples, where distinct thick tensor ideals of dualisable objects generate the same smashing subcategory. In the rank-one free-product case the stable category is not stratified by the spectrum of its dualisable objects, since $\beta\mathbb{N}$ has $2^{2^{\aleph_0}}$ points but there are only $2^{\aleph_0}$ localising tensor ideals.

Load-bearing premise

The FP∞ spectrum theorem rests on the assertion, made in a single sentence inside the proof of Proposition 6.9, that for every elementary abelian $p$-subgroup $E$, the cohomology $H^*(E,k)$ is finitely generated as a module over $H^*(G,k)$ via restriction, because only this finite generation makes the support unions over the finitely many conjugacy classes closed subsets of $\mathrm{Proj}(H^*(G,k))$ and lets the restriction map on lattices surject; if that premise failed, the spectrum could be strictly larger than $\mathrm{Proj}(H^*(G,k))$.

Editorial extensions

If this is right

  • For an $\mathrm{H}_1\mathfrak{F}$ group of type $\mathrm{FP}_\infty$, the Balmer spectrum of dualisable objects is $\mathrm{Proj}(H^*(G,k))$ and the telescope conjecture holds, so every smashing localising tensor ideal is generated by dualisable objects and distinct thick ideals of dualisable objects generate distinct smashing ideals.
  • For an infinite free product of finite $p$-groups, the telescope conjecture fails: $\mathrm{Thick}^b(\{k\uparrow^G_{H_n}\})$ and $\mathrm{Thick}^b(k)$ are distinct thick tensor ideals of dualisable objects, yet both generate the whole stable category as a localising tensor ideal.
  • For an infinite free product of cyclic groups, the stable category is not stratified by the Balmer spectrum of dualisable objects: the spectrum $\beta\mathbb{N}$ has $2^{2^{\aleph_0}}$ points and there are $2^{\aleph_0}$ localising tensor ideals.
  • More generally, a countable product of tensor-triangular fields has spectrum $\beta\mathbb{N}$ and is not stratified by the spectrum of its dualisable objects.
  • When the free factors have $p$-rank at least two, the restriction functors to the factors are jointly conservative yet do not induce a jointly surjective map on the Balmer spectrum, unlike what happens for rigidly compactly generated categories.

Reading between the lines

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

  • If the spectrum computation is right, the same uniform-nilpotence mechanism ($\xi_M^{\otimes C} = 0$ for $p$-rank-one factors) should compute the spectrum of dualisable objects for other graphs of finite groups with cyclic edge groups — amalgamated free products and HNN extensions — as an appropriate compactification of the vertex or edge set, with the Stone-Čech compactification of $\mathbb{N}$ a
  • The cardinality mismatch behind the non-stratification result points to a systematic limitation: for non-rigid categories, the spectrum of dualisable objects is too coarse to stratify the category, and a support theory defined on all compact objects — which here must exist without compacts being tensor-closed — would be needed to recover a bijection with localising tensor ideals.
  • The paper's dichotomy separates two hypotheses that often travel together: the tensor unit being compact and full cohomological finiteness of type $\mathrm{FP}_\infty$; a natural test is whether the telescope conjecture continues to hold for $\mathrm{H}_1\mathfrak{F}$ groups whose tensor unit is compact but whose cohomology is not finitely generated, which would show the $\mathrm{FP}_\infty$ hypot
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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 / 5 minor

Summary. The paper determines the Balmer spectrum of the subcategory of dualisable objects in the stable module category for two classes of infinite groups: H1F groups of type FP∞ over a field of characteristic p>0, and infinite free products of finite groups of p-rank one. In the FP∞ case the spectrum is shown to be Proj(H*(G,k)), extending the finite-group theorem of Benson, Carlson, and Rickard, and the telescope conjecture is proved. In the free-product case the spectrum is the Stone-Čech compactification of N, while for more general free products the restriction map from the spectral coproduct is shown not to be surjective. The paper also gives examples where the stable category is not stratified by the spectrum of dualisable objects and where the telescope conjecture fails.

Significance. If the arguments are correct, these are the first computations of Balmer spectra for non-rigidly-compactly generated stable module categories, and they provide a meaningful extension of the telescope conjecture to a setting where compact and dualisable objects diverge. The paper is clearly written and uses a coherent lattice-theoretic framework; it builds on the author's prior classification of localising tensor ideals, and the free-product counterexamples are explicit and instructive. The main theorems are substantial and would be of considerable interest to the tensor-triangular geometry and modular representation theory communities.

major comments (2)
  1. [§6, Proposition 6.9] The proof of surjectivity of Thickid^f(G^d) → lim_E Thickid^f(E^d) relies on the assertion that V = ∪_E res*_{G,E}(Supp_E(N_E)) is closed in Proj(H*(G,k)). The text states that 'H*(E,k) is finitely generated as a module over lim_{F∈AE(G)} H*(F,k) and hence by the F-isomorphism of Theorem 6.4 ... finitely generated as a module over H*(G,k)'. The second step is not a consequence of uniform F-isomorphism: a uniform F-isomorphism A → B does not imply that a B-module is finite over A. The finiteness of H*(E,k) over H*(G,k) via restriction is a separate, non-obvious assertion; it is needed to conclude that res*_{G,E} is a closed map, hence that V is closed, and hence that Proposition 6.7 applies. If this finiteness fails, the surjectivity proof collapses, and with it the spectrum computation of Corollary 6.11 and the injectivity part of Corollary 6.15. Please provide a proof or a precise reference (for example, a result in [12]) for this module finiteness.
  2. [§6, Proposition 6.8] In the proof of injectivity, the text passes from the vanishing (ξ_N^{⊗n}⊗X)↓_E = 0 for each elementary abelian E to the statement (ξ_N^{⊗m}⊗X^{⊗m})↓_E = 0 for some m, and then invokes Proposition 6.2. This transition is not justified as written: one must first tensor the zero morphism (ξ_N^{⊗n}⊗X) with X^{⊗(n-1)} to get (ξ_N^{⊗n}⊗X^{⊗n})↓_E = 0, and then tensor with suitable copies of X and ξ_N to obtain the claimed exponent m on both factors. Moreover, the application of Proposition 6.2 requires the hypothesis that a single morphism f has f↓_E = 0 for all E, which should be stated explicitly. The gap appears repairable with a short explanation, but it is load-bearing for the injectivity of the lattice map and hence for the spectrum theorem.
minor comments (5)
  1. [Abstract] There are typographical errors in the abstract: 'W e' should be 'We', and 'FP 8' appears where 'FP∞' is clearly intended.
  2. [§2, Definition 2.6 and elsewhere] The tensor unit is rendered as '/BD' (likely a LaTeX artifact) in several places, for example in Proposition 2.5 and Definition 2.6; this should be replaced by a proper symbol such as 𝟙.
  3. [§4, Lemma 4.7] In the proof, the notation 'M òG_EÓG_F' in the first sentence is confusing; the subscript in the restriction should likely be E rather than F, or the sentence should be rewritten for clarity.
  4. [§6, Proposition 6.8] The sentence 'Since there are only finite many conjugacy classes ... we may take some m > 0 such that (ξ_N^{⊗m}⊗X^{⊗m})↓_E = 0' would benefit from an explicit explanation of how the exponent m on X is obtained, as discussed in the corresponding major comment.
  5. [§5, Theorem 5.14] In the proof, the phrase 'If N is projective, and Y ∈ Thick(N), then Y is projective and so ξ_N ⊗ Y = 0' should be expanded to clarify that this covers the case where M↓_{H_n} is projective, and that the conclusion needed is (ξ_M^{⊗C}⊗X)↓_{H_n} = 0.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the spectrum and telescope-conjecture theorems are derived from prior independent constructions and external results, not assumed as inputs.

full rationale

Walked the claimed derivation chain from the construction of the stable category [27,28] and Henn's F-isomorphism [24] to the Balmer spectra (Theorem 5.14, Corollary 6.11) and telescope results (Corollaries 5.7, 5.9, 6.15). The main theorems are not used as inputs: the spectrum is obtained by proving that Thickid^f(G^d) -> lim Thickid^f(E^d) is injective (Proposition 6.8) and surjective (Proposition 6.9); the telescope conjecture is proved via Proposition 4.13 and Corollary 6.13, not assumed. Citations to the author's own [27] and [28] are to separate, earlier classifications (localising tensor ideals and construction of the stable category) that do not contain the target spectrum statements. The unsupported finite-generation claim in Proposition 6.9 is a gap in justification, not circularity: an unproved premise does not make the conclusion identical to an input. No fitted parameters, no renaming of known results, no imported uniqueness theorem, and no ansatz smuggled via citation were found. Hence no significant circularity.

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

The central claims rest on a substantial stack of prior results: the construction and properties of the stable category for LH𝔉 groups, the classification of its localising tensor ideals, Henn's Quillen stratification, and the Neeman-Thomason and finite-group telescope theorems. No new free parameters or invented entities are introduced; the paper is a proof-based derivation within established mathematics.

assumptions (6)
  • domain assumption The stable module category ĆMod(kG) for LH𝔉 groups is a closed tensor-triangulated category with the properties listed in [28, Section 4].
    Used throughout; the entire paper operates inside this category, whose construction is cited to the author's earlier preprint [28] and not reproduced.
  • domain assumption The classification of localising tensor ideals and colocalising hom-closed subcategories of ĆMod(kG) by the Quillen category of elementary abelian subgroups [27, Theorems 3.24 and 4.10].
    Invoked in Theorem 3.6, Corollary 3.7, Proposition 3.8, Proposition 3.12 and heavily in Sections 5 and 6. This is the classification that makes support a bijection.
  • domain assumption Henn's Quillen stratification for H1𝔉 groups of type FP∞: H*(G,k) to lim H*(E,k) is a uniform F-isomorphism, there are finitely many conjugacy classes of elementary abelian p-subgroups, and H*(G,k) is finitely generated [24, Appendix].
    Theorem 6.4 is quoted and used to identify Proj H*(G,k) with the colimit (Corollary 6.5) and to prove finiteness in Proposition 6.9.
  • domain assumption Chouinard's theorem, in the form that a kG-module M is zero in the stable category if and only if M↓E is zero for every finite elementary abelian p-subgroup E≤G, extended to LH𝔉 groups.
    Used in Proposition 3.3 and Proposition 3.10(i) to reduce detection and support to elementary abelian subgroups.
  • domain assumption The Neeman-Thomason localisation theorem and the telescope conjecture for finite groups: for a finite group F, smashing localising tensor ideals of ĆMod(kF) are generated by compact objects [16, Theorem 11.12].
    Used in Proposition 4.10 to lift generation by compact objects from finite subgroups to G, and in Proposition 6.9 to pass from localising to thick generation.
  • domain assumption The Mazza-Symonds construction producing a kG-module with prescribed restrictions M↓Hn to each free factor [33, Section 7].
    Used in Proposition 5.10, Theorem 5.19, and Proposition 6.9 to build global modules from local data.

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Pith. "Pith review of The Balmer spectrum and telescope conjecture for infinite groups." pith.science (2026). https://pith.science/paper/O6PTQLU6

@misc{pith2026250416602,
  author       = {Pith},
  title        = {Pith review of: The Balmer spectrum and telescope conjecture for infinite groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/O6PTQLU6}},
  note         = {Machine review of arXiv:2504.16602}
}
abstract

We determine the Balmer spectrum of dualisable objects in the stable module category for $\mathrm{H}_1\mathfrak{F}$ groups of type $\mathrm{FP}_{\infty}$ and show that the telescope conjecture holds for these categories. We also determine the spectrum of dualisable objects for certain infinite free products of finite groups. Using this, we give examples where the stable category is not stratified by the spectrum of dualisable objects and where the telescope conjecture does not hold.

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Works this paper leans on

36 extracted references · 34 canonical work pages

  1. [12]

    D. J. Benson, Complexity and varieties for infinite groups. I, II , J. Algebra 193 (1997), no. 1, 260–287, 288–317

    work page 1997

  2. [27]

    Gregory Kendall, (Co)stratification of stable categories for infinite groups , 2024, arXiv preprint arXiv:2410.23941

    work page Pith review arXiv 2024

  3. [28]

    , The stable module category and model structures for hierarc hically defined groups , 2024, arXiv preprint arXiv:2409.16094

    arXiv 2024

  4. [1]

    Shawn Baland, Alexandru Chirvasitu, and Greg Stevenson, The prime spectra of relative stable module categories, Trans. Amer. Math. Soc. 371 (2019), no. 1, 489–503

    work page 2019

  5. [2]

    Reine Angew

    Paul Balmer, The spectrum of prime ideals in tensor triangulated categor ies, J. Reine Angew. Math. 588 (2005), 149–168

    work page 2005

  6. [3]

    Volume II, Hindustan Book Agency, New Delhi, 2010, pp

    , Tensor triangular geometry , Proceedings of the International Congress of Mathematici ans. Volume II, Hindustan Book Agency, New Delhi, 2010, pp. 85–112

    work page 2010

  7. [4]

    , On the surjectivity of the map of spectra associated to a tens or-triangulated functor , Bull. Lond. Math. Soc. 50 (2018), no. 3, 487–495

    work page 2018

  8. [5]

    Paul Balmer and Giordano Favi, Generalized tensor idempotents and the telescope conjectu re, Proc. Lond. Math. Soc. (3) 102 (2011), no. 6, 1161–1185

    work page 2011

Show all 36 references
  1. [6]

    (N.S.) 25 (2019), no

    Paul Balmer, Henning Krause, and Greg Stevenson, Tensor-triangular fields: ruminations , Selecta Math. (N.S.) 25 (2019), no. 1, Paper No. 13, 36

    2019

  2. [7]

    871, Springer-Verlag, Berlin-New York, 1981

    Bernhard Banaschewski and Rudolf-Eberhard Hoffmann (eds .), Continuous lattices , Lecture Notes in Math- ematics, vol. 871, Springer-Verlag, Berlin-New York, 1981

    1981

  3. [8]

    Tobias Barthel, Nat` alia Castellana, Drew Heard, Niko Na umann, Luca Pol, and Beren Sanders, Descent in tensor triangular geometry , Triangulated Categories in Representation Theory and Bey ond (Cham) (Pet- ter Andreas Bergh, Steffen Oppermann, and Øyvind Solberg, ed s.), Spri...

    2024

  4. [9]

    24 GREGORY KENDALL

    Tobias Barthel, Natalia Castellana, Drew Heard, and Bere n Sanders, Cosupport in tensor triangular geometry , 2023, arXiv preprint arXiv:2303.13480. 24 GREGORY KENDALL

    2023 arXiv

  5. [10]

    Tobias Barthel, Nat` alia Castellana, Drew Heard, and Be ren Sanders, On surjectivity in tensor triangular geometry, Math. Z. 308 (2024), no. 4, Paper No. 65, 7

    2024

  6. [11]

    Tobias Barthel, Drew Heard, and Beren Sanders, Stratification in tensor triangular geometry with applicat ions to spectral Mackey functors , Camb. J. Math. 11 (2023), no. 4, 829–915

    2023

  7. [13]

    D. J. Benson, Jon F. Carlson, and J. Rickard, Complexity and varieties for infinitely generated modules, II, Math. Proc. Cambridge Philos. Soc. 120 (1996), no. 4, 597–615

    1996

  8. [14]

    Dave Benson, Jon Carlson, and Jeremy Rickard, Thick subcategories of the stable module category , Fund. Math. 153 (1997), no. 1, 59–80

    1997

  9. [15]

    Dave Benson, Srikanth Iyengar, and Henning Krause, Local cohomology and support for triangulated cate- gories, Ann. Sci. ´Ec. Norm. Sup´ er. (4) 41 (2008), no. 4, 575–621. 16. , Stratifying modular representations of finite groups , Ann. of Math. (2) 174 (2008), 1643–1684

    2008

  10. [17]

    , Localising subcategories for cochains on the classifying s pace of a finite group , C. R. Math. Acad. Sci. Paris 349 (2011), no. 17, 953–956

    2011

  11. [18]

    , Stratifying triangulated categories , J. Topol. 4 (2011), no. 3, 641–666

    2011

  12. [19]

    Benson, Representations and cohomology: Volume 2, cohomology of gr oups and modules , Cambridge Univ

    David J. Benson, Representations and cohomology: Volume 2, cohomology of gr oups and modules , Cambridge Univ. Press, 1991

    1991

  13. [20]

    Benson and Jon F

    David J. Benson and Jon F. Carlson, Nilpotence and generation in the stable module category , J. Pure Appl. Algebra 222 (2018), no. 11, 3566–3584

    2018

  14. [21]

    Max Dickmann, Niels Schwartz, and Marcus Tressl, Spectral spaces, New Mathematical Monographs, Cam- bridge University Press, 2019

    2019

  15. [22]

    Juan Omar G´ omez, On the Picard group of the stable module category for infinite groups, Int. Math. Res. Not. IMRN 2024 (2024), no. 16, 11514–11539

    2024

  16. [23]

    Jack Hall and David Rydh, The telescope conjecture for algebraic stacks , J. Topol. 10 (2017), no. 3, 776–794

    2017

  17. [24]

    63 of Proc

    Hans-W erner Henn, Unstable modules over the Steenrod algebra and cohomology o f groups , Group represen- tations: cohomology, group actions and topology (Seattle, W A, 1996), vol. 63 of Proc. Sympos. Pure Math., Amer. Math. Soc. Providence, 1998, pp. 277–300

    1996

  18. [25]

    Hochster, Prime ideal structure in commutative rings , Trans

    M. Hochster, Prime ideal structure in commutative rings , Trans. Amer. Math. Soc. 142 (1969), 43–60

    1969

  19. [26]

    Palmieri, and Neil P

    Mark Hovey, John H. Palmieri, and Neil P. Strickland, Axiomatic stable homotopy theory , Mem. Amer. Math. Soc. 128 (1997), no. 610, x+114

    1997

  20. [29]

    Joachim Kock and W olfgang Pitsch, Hochster duality in derived categories and point-free reco nstruction of schemes, Trans. Amer. Math. Soc. 369 (2017), no. 1, 223–261

    2017

  21. [30]

    Krause and G

    H. Krause and G. Stevenson, The derived category of the projective line , Spectral structures and topological methods in mathematics, EMS Ser. Congr. Rep., EMS Publ. Hous e, Z¨ urich, 2019, pp. 275–297

    2019

  22. [31]

    Henning Krause, Localization theory for triangulated categories , London Math. Soc. Lecture Note Ser. 375 (2008)

    2008

  23. [32]

    Kropholler, On groups of type pFPq8, J

    Peter H. Kropholler, On groups of type pFPq8, J. Pure Appl. Algebra 90 (1993), no. 1, 55–67

    1993

  24. [33]

    Nadia Mazza and Peter Symonds, The stable category and invertible modules for infinite grou ps, Adv. Math. 358 (2019), 1–26

    2019

  25. [34]

    I, II , Ann

    Daniel Quillen, The spectrum of an equivariant cohomology ring. I, II , Ann. of Math. (2) 94 (1971), no. 3, 573–602

    1971

  26. [35]

    Johan Steen and Greg Stevenson, Strong generators in tensor triangulated categories , Bull. Lond. Math. Soc. 47 (2015), no. 4, 607–616

    2015

  27. [36]

    Luke W olcott, Variations of the telescope conjecture and Bousfield lattic es for localized categories of spectra, Pacific J

    F. Luke W olcott, Variations of the telescope conjecture and Bousfield lattic es for localized categories of spectra, Pacific J. Math. 276 (2015), no. 2, 483–509

    2015

  28. [37]

    Fei Xu, Spectra of tensor triangulated categories over category al gebras, Arch. Math. (Basel) 103 (2014), no. 3, 235–253

    2014

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