Decompositions and Obstructions for the Stable Module ∞-Category

Decompositions and obstructions for the stable module ∞-category

Joshua Edward Hunt

This thesis has been submitted to the PhD School of the Faculty of Science, University of Copenhagen. Joshua Edward Hunt Department of Mathematical Sciences University of Copenhagen Universitetsparken 5 DK-2100 København Ø Denmark [email protected] https://joshua-hunt.net

Date of submission: 31st July 2020 Date of defence: 21st September 2020 Advisor: Jesper Grodal University of Copenhagen, Denmark Assessment committee: Jon F. Carlson University of Georgia, USA Jesper Michael Møller (chair) University of Copenhagen, Denmark Stefan Schwede Universität Bonn, Germany

ISBN 978-87-7078-898-4 © Joshua Edward Hunt 3

Abstract. This thesis consists of two research papers prefaced by an introduction. Both papers relate to the stable module ∞-category of a finite group 퐺 over a field 푘 of characteristic 푝. In the first, joint with Tobias Barthel and Jesper Grodal, we study the Picard group ofthe stable module ∞-category, more commonly known as the group of endotrivial modules 푇 (퐺). We describe the whole group of endotrivial modules in terms of 푝-local data, using the celebrated classification of endotrivial modules on its Sylow 푝-subgroup as input. In particular, we find an explicit obstruction to lifting a 퐺-stable endotrivial 푆-module up to 퐺 itself, which vanishes when 퐺 has a normal 푝-subgroup or when 푝 = 2 but does not vanish in general. As a consequence, we see that conjectures of Carlson–Mazza–Thévenaz about torsion-free endotrivial modules hold in certain cases, but not always. We illustrate the computability of our description by calculating concrete cases, such as the symmetric groups. In the second, I show that the stable module ∞-category of a finite group 퐺 decomposes in three different ways as a limit of the stable module ∞-categories of certain subgroups of 퐺. Analogously to Dwyer’s terminology for homology decompositions, I call these the centraliser, normaliser, and subgroup decompositions. I construct centraliser and normaliser decompositions and extend the subgroup decomposition (constructed by Mathew) to more collections of subgroups. The methods used are not specific to the stable module ∞-category, so may alsobe applicable in other settings where an ∞-category depends functorially on 퐺.

Resumé. Denne ph.d.-afhandling består af to forskningsartikler samt en kort indledning. Begge artikler handler om den stabile modul-∞-kategori for en endelig gruppe 퐺 over et legeme 푘 af karakteristik 푝. I den første artikel, som er skrevet i samarbejde med Tobias Barthel og Jesper Grodal, stud- erer vi Picard-gruppen af den stabile modul-∞-kategori, også kaldet gruppen af endotrivielle moduler 푇 (퐺). Vi beskriver hele gruppen 푇 (퐺) med 푝-lokal information ved at brug den fejrede klassificering af endotrivielle moduler påen 푝-Sylowundergruppe som input. Særligt finder vi en konkret forhindring for at løfte en 퐺-stabil endotriviel modul til 퐺 selv, der forsvinder når 퐺 har en normal 푝-undergruppe eller når 푝 = 2, men som ikke generelt forsvinder. Vi viser som følge af dette, at formodninger af Carlson–Mazza–Thévenaz om torsionsfri endotrivielle moduler gælder i visse tilfælde, men ikke altid. Vi illustrerer beregneligheden af vores model ved at anvende den i konkrete tilfælde, f.eks. de symmetriske grupper. I den anden artikel viser jeg, at den stabile modul-∞-kategori kan dekomponeres på tre forskel- lige måder som et invers limes af de stabile modul-∞-kategorier af visse undergrupper af 퐺. I for- længelse af Dwyers terminologi kaldes disse dekompositioner for centralisatordekompositionen, normalisatordekompositionen og undergruppedekompositionen. Jeg konstruerer centralisatordekompositionen og normalisatordekompositionen og udvider undergruppedekompositionen (først konstrueret af Mathew) til flere samlinger af undergrupper. De anvendte metoder erikke specifikke for den stabile modul-∞-kategori, så de forventes at være anvendelige i andresitua- tioner, hvor en ∞-kategori afhænger funktorielt af 퐺. 4

Acknowledgements. First and foremost, I would like to thank my advisor, Jesper Grodal, for all his advice and support throughout my PhD studies, for guiding my research to interesting places, and for all the mathematical wisdom he shared. I would like to thank all the past and present members of the maths department of UCPH who contributed to such a lively, sociable, and invigorating work environment over the last four years. Special thanks go to those who time and again had the patience to answer mathematical questions, shared their ideas and insights, or gave feedback on my work: Tobias Barthel, Benjamin Böhme, Rune Haugseng, Markus Hausmann, Kaif Hilman, Mikala Ørsnes Jansen, Markus Land, Malte Leip, Espen Nielsen, and David Sprehn. I would also like to thank everyone outside the maths department who has made living in Copenhagen so enjoyable, especially Julie Courraud, Peter James, and My Tran. I am grateful to the Danish National Research Foundation for their financial support via the UCPH Centre for Symmetry and Deformation (DNRF92) and the Copenhagen Centre for Geometry and Topology (DNRF151). I owe a great deal to Christina Hedges and Peter James for their help and advice through difficult times. Last but far from least, I would like to thank Marie and my family, for all their support. Contents

General introduction 7 Torsion-free endotrivial modules via homotopy theory 15 Decompositions of the stable module ∞-category 75

Part I

General introduction

This thesis consists of two papers: Paper A: Torsion-free endotrivial modules via homotopy theory, and Paper B: Decompositions of the stable module ∞-category. The first of these is joint with Tobias Barthel and Jesper Grodal. In this general introduction, I will attempt to explain how the papers are related, with a less specialised audience in mind than the introductions to the papers themselves. To avoid too much overlap with the other introductions, I omit mention of many results from the papers, as well as background that is irrelevant to the exposition here. For more details, including a more extensive bibliography, I refer to the introductions of Papers A and B. The common theme of the papers is the study of the stable module ∞-category using decompositions of ∞-categories. By a decomposition of an ∞-category C, I mean an equivalence C ∼ C −−→ lim 푖 푖∈퐼 C where 퐼 is some indexing diagram and the 푖 are in some way simpler or easier to understand C C C than . Such a decomposition allows us to study by first studying each 푖 and then attempting to “glue” our results together to say something about C. This is far from a new idea: the analogous concept for classifying spaces is a classical tool of homotopical group theory. Let 퐺 be a finite group and 푝 be a prime dividing the order of 퐺. A homology decomposition of the classifying space B퐺 is a diagram of spaces 퐹 ∶ 퐷 → S such that for every 푑 ∈ 퐷 there is some 퐻 ≤ 퐺 for which 퐹(푑) ≃ B퐻, together with a map hocolim 퐹 → B퐺 that induces an isomorphism on mod 푝 homology. Here we have a homotopy colimit instead of a limit of ∞-categories, but the principle is the same: we can now attempt to understand 퐺 (or at least the mod 푝 homology of its classifying space) inductively by first understanding its subgroups and then gluing this information together to study 퐺. Dwyer [Dwy97] identifies three types of homology decompositions, each with theirown indexing category. In different ways, each of the indexing categories describes how thesub- groups in C behave under conjugation. Let C be a collection of subgroups of 퐺, i.e. a set of subgroups of 퐺 that is closed under conjugation by elements of 퐺. Firstly, we have the orbit category OC (퐺), which is equivalent to the category whose objects are the left 퐺-sets 퐺/퐻 with 퐻 ∈ C and whose morphisms are 퐺-equivariant maps. Secondly, we have the fusion category FC (퐺), whose objects are the subgroups in C and whose morphisms are the group homorphisms that are induced by conjugation in 퐺. Finally, we have the orbit simplex category 푠푆C (퐺), which is the poset of the 퐺-conjugacy classes of non-empty chains of subgroups C 퐻0 < … < 퐻푛 in , ordered by refinement. These give us (potential) decompositions associated with C : Subgroup decomposition: There is a functor 퐹 ∶ OC (퐺) → S with 퐹(퐺/퐻) ≃ B퐻. F op S Centraliser decomposition: There is a functor 퐹 ∶ C (퐺) → with 퐹(퐻) ≃ B퐶퐺(퐻). Normaliser decomposition: There is a functor 퐹 ∶ 푠푆C (퐺)op → S with

퐹(퐻0 < … < 퐻푛) ≃ B ( ⋂ 푁퐺(퐻푖)) . 1≤푖≤푛 Dwyer shows that these three are indeed decompositions of B퐺 provided that C is an “ample” collection. There are many standard examples of ample collections, such as the collection S푝(퐺) 9 10 Joshua Hunt

of all non-trivial 푝-subgroups of 퐺. In the cases that are of interest to us, C will be a subset of S푝(퐺). We let OS (퐺) denote the orbit category on S푝(퐺). In my work, I am interested not in the classifying space B퐺, but in the stable module category of 푘퐺, where 푘 is a field of characteristic 푝. Since the characteristic of 푘 divides the order of 퐺, we are in the realm of “modular” representation theory, and in general the module category Mod푘퐺 is poorly behaved: unless the Sylow 푝-subgroup of 퐺 is a member of a small list of fami- lies, 푘퐺 has “wild representation type”, and classifying the indecomposable 푘퐺-modules would in some sense entail the simultaneous classification of the indecomposable representations of all finite dimensional 푘-algebras. (Specifically, the cases where 퐺 does not have wild representation type are when its Sylow is a cyclic, Klein four, dihedral, semi-dihedral, or generalised quaternion group; see Theorem 4.4.4 of [Ben98].) Such a classification is generally viewed as being unreasonably difficult. Dade [Dad78a] remarks that,

“There are just too many modules over 푝-groups! Faced with this vast, literally incomprehensible family of modules, we are reduced to looking for subfamilies which are, at the same time, small enough to be classified and large enough tobe useful.”

In this spirit, the stable module category is of interest to representation theorists because it contains important information about 푘퐺-modules while still being simple enough to study. Informally speaking, the stable module category discards the “characteristic zero” information in the module category, leaving only the behaviour that is specific to characteristic 푝. More precisely, to obtain the stable module category of 퐺, we declare a morphism 푓 ∶ 푀 → 푁 in Mod푘퐺 to be equivalant to the zero morphism if it factors through a projective module. We have therefore forced all projective modules to be equivalant to the zero module in the stable module category. (Recall that if the characteristic of 푘 is zero then all modules are projective, and so the stable module category over such a field is itself zero; this is what was meant by “discarding the characteristic zero information” above.) Traditionally, the stable module category has been considered as a triangulated category, for example in Section 5 of [Car96]. However, in order to be able to get a decomposition of the stable module category, we will need to consider it as an ∞-category. In Section 2.1 of [Mat15], Mathew gives a construction of the stable module ∞-category StMod푘퐺, a stable ∞-category whose homotopy category is the aforementioned triangulated category. In both the triangulated and the stable ∞-category settings, the stable module (∞-)category inherits a symmetric monoidal structure from Mod푘퐺, which is given by 푀 ⊗푘 푁 with a diagonal 퐺-action. It is unfortunately difficult to give anything more than a cursory explanation of therole that ∞-categories [Lur09; Lur17] play here. Informally speaking, an ∞-category behaves analogously to a category where, instead of having a unique composition of morphisms, there is only a composition that is “unique up to homotopy”. An ∞-category encodes not only all the homotopies between the different compositions, but also the higher homotopies between the homotopies, and so on. This causes them in some ways to behave more like topological spaces, which is important when attempting to construct decompositions. The analogous decomposition statements for the 1-categorical stable module category are false: we have forgotten too much information by passing to the homotopy category. Corollary 9.16 of [Mat16] proves an analogue of the subgroup decomposition for the stable module ∞-category, showing that

(1.1) StMod −−→∼ lim StMod 푘퐺 op 푘푃 퐺/푃∈OS (퐺) General introduction 11 is an equivalence of symmetric monoidal ∞-categories. (Note that Mathew’s result covers more collections than just S푝(퐺).) This equivalence further justifies the earlier statement that StMod푘퐺 only sees the behaviour of Mod푘퐺 that is specific to the prime 푝: each of the subgroups 푃 appearing on the right hand side is a 푝-group, and OS (퐺) only contains data about 푝-subgroups of 퐺 and their normalisers (that is, the “푝-local” group theory of 퐺). In Paper B, I prove the existence of a subgroup decomposition for more collections than those covered by Corollary 9.16 of [Mat16] and show that there are also centraliser and normaliser decompositions for many collections: C S A B I Z Theorem. Let be one of the collections 푝(퐺), 푝(퐺), 푝(퐺), 푝(퐺), or 푝(퐺). There is a subgroup decomposition ∼ StMod푘퐺 −−→ lim StMod푘푃 OC (퐺)op and a normaliser decomposition StMod −−→∼ lim StMod . 푘퐺 푘푁퐺 (휎) 푠푆C (퐺) C S A Z If is 푝(퐺), 푝(퐺), or 푝(퐺), then there is additionally a centraliser decomposition StMod −−→∼ lim StMod . 푘퐺 푘퐶퐺 (푃) FC (퐺) The definitions of the collections in the statement of the theorem can all be found inthein- troduction to Paper B. For the subgroup decomposition, the collections that are not covered B I Z by Mathew’s result are 푝(퐺), 푝(퐺), and 푝(퐺). The proof of the theorem uses 퐺-spaces to encode decompositions, building on the ideas in Section 3 of Dwyer’s [Dwy98], and as a key input uses work of Grodal–Smith [GS06] that provides equivariant equivalences between these encoding 퐺-spaces. Paper A gives an example of how to use such decompositions of ∞-categories in practice. It uses Mathew’s subgroup decomposition (1.1) to study the Picard group of the stable module ∞-category: this group consists of the equivalence classes of modules 푀 ∈ StMod푘퐺 that have an inverse under the tensor product, i.e. for which there exist a module 푁 ∈ StMod푘퐺 and an equivalence 푀 ⊗푘 푁 ≃ 푘. It has been studied by representation theorists as the group of endotrivial modules and is commonly denoted by 푇 (퐺). The name “endotrivial” arises because whenever a finite-dimensional module 푀 has an inverse under the tensor product, the inverse must be equivalent to the 푘-linear dual of 푀; that is, ∗ End푘(푀) ≃ 푀 ⊗푘 푀 ≃ 푘, so the 푘-linear endomorphisms of 푀 are equivalant to 푘 as a stable 퐺-module. Picard groups are often interesting objects to study and endotrivial modules are no exception, appearing in several places in modular representation theory. A lot of information is already known about 푇 (퐺): when 퐺 is a finite 푝-group this group was classified in a number of important papers, starting in the 1970swith[Dad78a; Dad78b] and culminating in the early 2000s with [CT05; CT04]. For an arbitrary finite group 퐺, we therefore want to understand the image and kernel of the restriction homomorphism 퐺 res푆 ∶ 푇 (퐺) → 푇 (푆) where 푆 is a Sylow 푝-subgroup of 퐺. The kernel is called the group of Sylow-trivial modules and was described in terms of the local group theory of 퐺 by Balmer in [Bal13] and by Grodal in [Gro18]. By earlier work of Carlson, Mazza, Nakano, and Thévenaz ([CMN06], [MT07], and [CMT13]), we also know what the image of the restriction homomorphism is as an abstract group; therefore, the aim of Paper A is to identify the image as a subgroup of 푇 (푆). In the cases 12 Joshua Hunt

of interest, it turns out that 푇 (푆) is a free abelian group, so we wish to identify the sublattice of 푇 (푆) that corresponds to modules restricted from 푇 (퐺). In order to use the subgroup decomposition to study the Picard group 푇 (퐺), we need to replace the Picard group by the Picard space: that is, the space of tensor-invertible objects in StMod푘퐺 together with the equivalences between them, and homotopies between those equivalences, and the homotopies between those homotopies, and so on. This is a Kan complex whose set of connected components is (by definition) isomorphic to the Picard group. Are- sult of Mathew–Stojanoska [MS16, Proposition 2.2.3] tells us that the subgroup decomposition induces a decomposition of the Picard space as a homotopy limit: Pic(StMod ) −−→∼ holim Pic(StMod ). 푘퐺 op 푘푃 OS (퐺) Our algebraic question has become topological, and standard techniques from algebraic topology give an obstruction that determines the image of the restriction: Theorem. There is an exact sequence 훼 0 → H1(OS (퐺); 푘×) → 푇 (퐺) → 푇 퐺-stable(푆) −→ H2(OS (퐺); 푘×), and we can explicitly describe a 2-cocycle representing the obstruction 훼. Here 푇 퐺-stable(푆) denotes the group of 퐺-stable endotrivial 푆-modules, which is the subgroup of op 푇 (푆) that is invariant under 퐺-conjugation. It is isomorphic to limOS (퐺) 푇 (−), which is how we denote it throughout Paper A, and it is straightforward to show that it contains the image 퐺 of the restriction morphism res푆 . The rest of Paper A involves manipulating the obstruction 훼 in order to obtain an algebraic model for 푇 (퐺), expressed in terms of the 푝-subgroups of 퐺 and the one-dimensional characters of their normalisers. Since the groups of characters in question have often already been computed, this model is useful for calculations. Along the way, we provide an algebraic criterion for showing that a given 퐺-stable endotrivial 푆-module lifts to 푇 (퐺): a module lifts if and only if it is possible to specify a one-dimensional character of 푁퐺(푃) for each 푝-subgroup 푃 ≤ 퐺 such that these characters satisfy a compatibility condition on the cyclic 푝-subgroups. We carry out these calculations for the group 퐺 = PSL3(푝), where 푝 ≡ 1 (mod 3), showing that the image of the restriction is a subgroup of 푇 (푆) of index three. This allows us to settle two conjectures of Carlson–Mazza–Thévenaz [CMT14, Conjectures 9.2 and 10.1], partially positively and partially negatively.

References [Bal13] P. Balmer. “Modular representations of finite groups with trivial restriction to Sylow subgroups”. In: J. Eur. Math. Soc. (JEMS) 15.6 (2013), pp. 2061–2079. issn: 1435-9855 (cit. on p. 11). [Ben98] D. J. Benson. Representations and cohomology. I. 2nd ed. Vol. 30. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1998. isbn: 0- 521-63653-1 (cit. on p. 10). [Car96] J. F. Carlson. Modules and group algebras. Lectures in Mathematics ETH Zürich. Notes by Ruedi Suter. Birkhäuser Verlag, Basel, 1996. isbn: 3-7643-5389-9. doi: 10. 1007/978-3-0348-9189-9 (cit. on p. 10). [CMN06] J. F. Carlson, N. Mazza, and D. K. Nakano. “Endotrivial modules for finite groups of Lie type”. In: J. Reine Angew. Math. 595 (2006), pp. 93–119. issn: 0075-4102 (cit. on p. 11). General introduction 13

[CMT13] J. Carlson, N. Mazza, and J. Thévenaz. “Endotrivial modules over groups with quaternion or semi-dihedral Sylow 2-subgroup.” In: Journal of the European Mathe- matical Society 15.1 (2013), pp. 157–177. issn: 1435-9855. doi: 10.4171/JEMS/358 (cit. on p. 11). [CMT14] J. F. Carlson, N. Mazza, and J. Thévenaz. “Torsion-free endotrivial modules”. In: J. Algebra 398 (2014), pp. 413–433. issn: 0021-8693. doi: 10 . 1016 / j . jalgebra . 2013.01.020 (cit. on p. 12). [CT04] J. F. Carlson and J. Thévenaz. “The classification of endo-trivial modules”. In: Invent. Math. 158.2 (2004), pp. 389–411. issn: 0020-9910 (cit. on p. 11). [CT05] J. F. Carlson and J. Thévenaz. “The classification of torsion endo-trivial modules”. In: Ann. of Math. (2) 162.2 (2005), pp. 823–883. issn: 0003-486X. doi: 10. 4007 / annals.2005.162.823 (cit. on p. 11). [Dad78a] E. C. Dade. “Endo-permutation modules over 푝-groups. I”. In: Ann. of Math. (2) 107.3 (1978), pp. 459–494. issn: 0003-486X. doi: 10.2307/1971125 (cit. on pp. 10, 11). [Dad78b] E. Dade. “Endo-permutation modules over 푝-groups, II”. In: Ann. of Math. (2) 108.2 (1978), pp. 317–346. issn: 0003-486X. doi: 10.2307/1971169 (cit. on p. 11). [Dwy97] W. G. Dwyer. “Homology decompositions for classifying spaces of finite groups”. In: Topology 36.4 (1997), pp. 783–804. issn: 0040-9383. doi: 10.1016/S0040-9383(96) 00031-6 (cit. on p. 9). [Dwy98] W. G. Dwyer. “Sharp homology decompositions for classifying spaces of finite groups”. In: Group representations: cohomology, group actions and topology (Seattle, WA, 1996). Vol. 63. Proc. Sympos. Pure Math. Amer. Math. Soc., Providence, RI, 1998, pp. 197–220. doi: 10.1090/pspum/063/1603159 (cit. on p. 11). [Gro18] J. Grodal. Endotrivial modules for finite groups via homotopy theory. December 9, 2018. arXiv: 1608.00499v2 [math.GR]. (Visited on November 20, 2019) (cit. on p. 11). [GS06] J. Grodal and S. D. Smith. “Propagating sharp group homology decompositions”. In: Advances in Mathematics 200.2 (2006), pp. 525–538. issn: 0001-8708. doi: 10.1016/ j.aim.2005.01.006 (cit. on p. 11). [Lur09] J. Lurie. Higher topos theory. Vol. 170. Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, 2009. isbn: 978-0-691-14049-0. doi: 10 . 1515 / 9781400830558 (cit. on p. 10). [Lur17] J. Lurie. Higher Algebra. September 18, 2017. url: https://www.math.ias.edu/ ~lurie/papers/HA.pdf (visited on November 22, 2019) (cit. on p. 10). [Mat15] A. Mathew. Torus actions on stable module categories, Picard groups, and localizing subcategories. December 6, 2015. arXiv: 1512 . 01716v1 [math.RT]. (Visited on November 20, 2019) (cit. on p. 10). [Mat16] A. Mathew. “The Galois group of a stable homotopy theory”. In: Adv. Math. 291 (2016), pp. 403–541. issn: 0001-8708 (cit. on pp. 10, 11). [MS16] A. Mathew and V. Stojanoska. “The Picard group of topological modular forms via descent theory”. In: Geom. Topol. 20.6 (2016), pp. 3133–3217. issn: 1465-3060 (cit. on p. 12). [MT07] N. Mazza and J. Thévenaz. “Endotrivial modules in the cyclic case”. In: Arch. Math. (Basel) 89.6 (2007), pp. 497–503. issn: 0003-889X. doi: 10 . 1007 / s00013 - 007 - 2365-2 (cit. on p. 11).

Part II

Torsion-free endotrivial modules via homotopy theory

TORSION-FREE ENDOTRIVIAL MODULES VIA HOMOTOPY THEORY

TOBIAS BARTHEL, JESPER GRODAL, AND JOSHUA HUNT

Abstract. The group of endotrivial 푘퐺-modules 푇 (퐺), for 퐺 a finite group and 푘 a field of characteristic 푝, is by definition the Picard group of the stable module category of 푘퐺-modules. The quest to understand this group has a long history, as it occurs in many parts of representation theory. In this paper, we describe the whole group of endotrivial modules of an arbitrary finite group in terms of computable 푝-local data, using the celebrated classification of endotrivial modules on its Sylow 푝-subgroup 푆 as input. We accomplish this by combining techniques from higher algebra with recent homotopy methods of the second-named author, which he used to describe the subgroup of torsion endotrivial modules. In particular, we find an explicit obstruction to lifting a 퐺-stable endotrivial 푆-module up to 퐺 itself, which vanishes when 퐺 has a normal 푝-subgroup or when 푝 = 2 but does not vanish in general. As a consequence, we see that conjectures of Carlson–Mazza–Thévenaz about torsion-free endotrivial modules hold in certain cases, but not always. We illustrate the computability of our description by calculating concrete cases, such as the symmetric groups.

Contents 1. Introduction 18 2. Notation and preliminaries 23 3. Decomposition of the stable module ∞-category over the orbit category 27 4. Obstruction theory for homotopy limits 28 5. An obstruction in Bredon cohomology 32 6. Compatible actions in the homotopy category 38 7. Lifting via orientations 43 8. An algebraic description of the group of endotrivial modules 47 9. A local description of the group of endotrivial modules 50 10. Endotrivial modules for PSL3(푝) 53 11. Torsion-free endotrivial modules 57 12. Endotrivial modules for the alternating and symmetric groups 60 Appendix A. Comparison with an obstruction due to Balmer 65 References 68

Date: 2020-07-31. The first-named author was funded by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 751794 (at the University of Copenhagen) and the Max Planck Society. All three authors were supported by the Danish National Research Foundation Grant DNRF92 through the UCPH Centre for Symmetry and Deformation, and also by the Isaac Newton Institute (EPSRC Grant number EP/R014604/1) during the Autumn 2018 Homotopy Harnessing Higher Structures Programme. Addition- ally, the second- and third-named authors were supported by the Danish National Research Foundation through the Copenhagen Centre for Geometry and Topology (DNRF151). 17 18 Tobias Barthel, Jesper Grodal, and Joshua Hunt

1. Introduction

Endotrivial modules, i.e. 푘퐺-modules such that End푘(푀) ≅ 푘 ⊕ (proj), for 퐺 a finite group and 푘 a field of characteristic 푝, play an important role in many parts of representation theory, which is surveyed in [Thé07; Car12; Car17]. They form an abelian group under tensor product, denoted 푇 (퐺) or 푇푘(퐺). This is the group of tensor-invertible objects in the stable module category of 퐺, and it is known to be finitely generatedCMN06 [ , Corollary 2.5]. When 푃 is a finite 푝-group, 푇 (푃) was classified in a number of important papers, starting in the 1970s with [Dad78a; Dad78b] and culminating in the early 2000s with [CT05; CT04]. For 퐺 a general finite group, computable formulas for the torsion partof 푇 (퐺) in terms of 푝-local group theory were provided recently by the second-named author in [Gro18]. The goal of this paper is to combine the homotopy methods from [Gro18] with additional tools from higher algebra to obtain a computable model for the whole of 푇 (퐺). In particular, we obtain information about the torsion-free generators, settling conjectures of Carlson–Mazza–Thévenaz [CMT14] both in the positive and in the negative. We briefly recall the outline of the classification of 푇 (푃); more details on the known structure of 푇 (퐺) are provided in Section 2.6. Dade showed in [Dad78a; Dad78b] that when 푃 is a finite abelian 푝-group, 푇 (푃) consists of shifts of the trivial module, i.e. it is infinite cyclic unless 푃 itself is cyclic, in which case it is 0 if |푃| ≤ 2 and ℤ/2 otherwise. Carlson–Thévenaz showed in [CT05] that when 푃 is an arbitrary finite 푝-group, 푇 (푃) is torsion-free except in the exceptional cases when the 푝-group is cyclic, a semi-dihedral 2-group, or a generalised quaternion 2-group. In all the exceptional cases, we know explicit generators for 푇 (푃). In [CT04], they also established generators in the cases where 푇 (푃) is torsion-free, proving that a set of elements given earlier by Alperin [Alp01] that was known to form a rational basis is in fact an integral basis.

1.1. Obstruction theory. Since we understand the group of endotrivial modules for 푝-groups, we can aim to understand 푇 (퐺) for arbitrary finite 퐺 by choosing a Sylow 푝-subgroup 푆 of 퐺 and considering the exact sequence

res (1.2) 0 → 푇 (퐺, 푆) → 푇 (퐺) −−→푇 (푆).

Here 푇 (퐺, 푆) is defined as the kernel of the restriction map, i.e. the modules 푀 such that 퐺 res푆 푀 ≅ 푘 ⊕ (proj), referred to as Sylow-trivial modules. In [Gro18, Theorem A], the second- named author established an isomorphism 푇 (퐺, 푆) ≅ H1(OS (퐺); 푘×), building on earlier ideas of Balmer [Bal13]. Here OS (퐺) denotes the orbit category of 퐺 on the collection S푝(퐺) of non- trivial 푝-subgroups of 퐺. This cohomological description is very amenable to manipulation via standard methods from homotopy theory, and [Gro18, Theorem D] says that 푇 (퐺, 푆) can alternatively be described as a collection of one-dimensional characters of normalisers of chains 푁퐺(푃0 < … < 푃푛), compatible under refinement. This implies the Carlson–Thévenaz conjecture [Gro18, Theorem F], which gives an algorithmic way of calculating 푇 (퐺, 푆) from 푝-local group theory information. In terms of homotopy theory, the statement about one-dimensional 0 S 1 × ∗ characters can again be rewritten as 푇 (퐺, 푆) ≅ H퐺( 푝(퐺); H (−; 푘 )), where here H퐺(푋; 퐹) denotes the 퐺-equivariant Bredon cohomology of a 퐺-space 푋 with coefficients in the functor 퐹; see [Gro18, §5]. Understanding the exact image of the restriction 푇 (퐺) → 푇 (푆) is subtle, as we now explain. Carlson–Mazza–Nakano [CMN06, Theorem 3.1] show that the rank of TF (퐺) can be described Torsion-free endotrivial modules via homotopy theory 19 by a formula similar to Alperin’s formula for 푝-groups [Alp01, Theorem 4]:

0 if rk푝(퐺) = 1, (1.3) dimℚ(푇 (퐺) ⊗ ℚ) = rk(TF (퐺)) = { 푛퐺 if rk푝(퐺) = 2, and 푛퐺 + 1 if rk푝(퐺) ≥ 3.

Here rk푝(퐺) is the 푝-rank of 퐺, i.e. the rank 푟 of the largest elementary abelian 푝-subgroup 푟 (ℤ/푝) of 퐺, and 푛퐺 is the number of conjugacy classes of rank two maximal elementary abelian subgroups of 퐺. This formula determines TF (퐺) as an abstract group. Apart from the group theoretic question of actually calculating 푛퐺 for classes of groups (see Section 2.6), the above rank formula for TF (퐺) also still leaves open the important question of describing torsion-free generators for 푇 (퐺), for example via their image in 푇 (푆), and pinning down its representation theoretic structure—a status in some ways analogous to the status in the 푝-group case prior to [CT04]. This has been the subject of a number of works, both in the abstract and for concrete collections of finite groups; the paper [CMT14] summarises many of the pre-existing approaches to this problem. An obvious condition on the image of the restriction 푇 (퐺) → 푇 (푆) is that it must land in the subgroup limOop 푇 (−) of 퐺-stable modules in 푇 (푆), i.e. the endotrivial 푆-modules that restrict S to compatible elements under restriction and conjugation in 퐺. Using methods from higher algebra we find a computable obstruction for this being the case. Extending the resultsin [Gro18] “to the right” we get the following sequences describing 푇 (퐺) in terms of 푇 (푆) and 푝-local information. Theorem A. There are exact sequences res 훼 0 → H1(OS (퐺); 푘×) → 푇 (퐺) −−→ lim 푇 (−) −→ H2(OS (퐺); 푘×) op OS (퐺) and res 훽 0 → H0 (C ; H1(−; 푘×)) → 푇 (퐺) −−→ lim 푇 (−) −→ H1 (C ; H1(−; 푘×)), 퐺 op 퐺 OS (퐺) where C is any collection of non-trivial 푝-subgroups of 퐺 for which C ↪ S푝(퐺) is a 퐺-homotopy equivalence (Theorem 4.5 and Corollary 5.10). We can explicitly describe cocycles representing the obstructions 훼 and 훽 (Theorem 4.5 and Proposition 5.11). The two left-most terms in the exact sequences are descriptions of 푇 (퐺, 푆) from [Gro18], as explained above. All of the cohomology groups mentioned are finite abelian 푝′-groups described in terms of the 푝-local structure of 퐺. We provide information on cases when 훼 and 훽 are known either to vanish or to be non-zero in Theorem C; in particular, when 푝 = 2 the obstruction classes vanish, but at odd primes they do not vanish in general. Previously, Balmer [Bal15] provided an obstruction to lifting modules from a subgroup 퐻 (of 푝′-index) to 퐺, lying in a Čech cohomology group with respect to a “sipp topology” that he introduced. While obviously fundamental and an inspiration for this work, the theory was not immediately amenable to calculations of the obstruction group or obstruction class. In Appendix A—logically independent of the rest of the paper—we go back and show that, when 퐻 is a Sylow 푝-subgroup, Balmer’s obstruction group is in fact isomorphic to H2(OS ; 푘×) and his obstruction corresponds to 훼 under our constructed isomorphism. 1.4. An algebraic model. As mentioned previously, the obstruction class in Theorem A may 0 C 1 × be non-zero when 푝 is odd. Just as an element of the cohomology group H퐺( ; H (−; 푘 )) can × be thought of as a set of one-dimensional characters 푁퐺(푃) → 푘 that agree as 푃 varies, we 1 C 1 × can view the obstruction group H퐺( ; H (−; 푘 )) as encoding the obstruction to the existence 20 Tobias Barthel, Jesper Grodal, and Joshua Hunt of such a set of characters. It therefore makes sense to build these characters into a model for 푇 (퐺), which is the content of our Theorem B. Given an endotrivial module 푀 ∈ 푇 (퐺), the most obvious invariant that we can assign to it is the data of its restrictions to elementary abelian 푝-subgroups. We let A푝(퐺) denote the poset of non-trivial elementary abelian 푝-subgroups of 퐺. Recall that Dade’s theorem implies that, for 푉 ∈ A푝(퐺), every endotrivial 푉 -module is a shift of the trivial module. We can therefore choose a type function 푛 ∶ A푝(퐺) → ℤ that satisfies 퐺 푛(푉 ) (1.5) res푉 푀 ≃ Ω 푘 for every 푉 ∈ A푝(퐺). This function determines the image of 푀 in TF (퐺) but not 푀 itself [CMT14, Theorem 2.2]. Having chosen such a type function, we can assign further invariants to 푀, which de- ′ A scribe the action of 푝 -elements of 퐺: for any 푉 ∈ 푝(퐺), we consider the 푁퐺(푉 )-module ̂ 푛(푉 ) 1 × H (푉 ; 푀), which is one-dimensional because of (1.5). We let 휑푉 ∈ H (푁퐺(푉 ); 푘 ) denote the one-dimensional character of 푁퐺(푉 ) that corresponds to the aforementioned module. These invariants satisfy compatibility conditions because they come from a 퐺-module. Since the restriction of 푀 to 푇 (푆) is an endotrivial 푆-module, the type function 푛 must satisfy conditions imposed by the classification of endotrivial modules on 푇 (푆), as in [CMT14, §3]. Fur- thermore, since the restriction of 푀 to 푇 (푆) is 퐺-stable, the type function must be constant on 퐺-conjugacy classes of elementary abelian subgroups. The compatibility conditions on the one-dimensional characters relate to the action of 푝′- elements on cyclic subgroups: for a cyclic 푝-subgroup 푉 ≤ 푆, the fact that 푇 (푉 ) ≅ ℤ/2 imposes a compatibility condition on the type function 푛 and hence on the possible structures of a module in 푇 (푆). However, when we lift to 퐺, we might have a 푝′-element 푥 that acts non-trivially on 푉 , in which case the order of Ω푘 in 푇 (푉 ⋊ ⟨푥⟩) is greater than two. The compatibility condition on the one-dimensional characters arises from this larger periodicity, and this phenomenon is also exactly what our obstructions in Theorem A are measuring. We say that a set of characters satisfying this compatibility condition is an orientation for 푛; see Definition 7.6. We work through the concrete example 퐺 = PSL3(푝) in Example 10.2, where this phenomenon is clearly visible; that example does not use any of the obstruction theory that we develop in the first part of the paper, and some readers may find it useful to skip ahead to the example before reading the intervening sections. Our algebraic model for 푇 (퐺) says that the compatibility conditions in the above two para- graphs determine a unique endotrivial 퐺-module: A Theorem B. The group 푇 (퐺) is isomorphic to the abelian group of tuples (푀푆, 푛 ∶ 푝(퐺) → A ℤ, { 휑푉 ∶ 푉 ∈ 푝(퐺) }), where 푀푆 is a 퐺-stable endotrivial 푆-module, 푛 is a type function for 푀푆, and {휑푉 } is an orientation for 푛, under an explicit equivalence relation (Theorem 8.3). There is also an isomorphism

푇 (퐺) ≅ lim 푇 (푁퐺(푃0 < … < 푃푛)), [푃0<…<푃푛] where the indexing of the limit is over the poset of conjugacy classes of non-empty chains of 푝- subgroups of 퐺, ordered by refinement (Theorem 9.4). In particular, 푇 (퐺) is an invariant of the 푝-local structure of 퐺. The conditions for being a type function and an orientation, as well as the equivalence relation, are all straightforward to verify for explicit tuples, so we obtain a computable model for 푇 (퐺). The description of 푇 (퐺) in terms of the endotrivial modules of normalisers of chains of subgroups 푇 (푁퐺(푃0 < … < 푃푛)) generalises the normaliser decomposition of [Gro18, Theorem D]. Torsion-free endotrivial modules via homotopy theory 21

1.6. (Non)-vanishing of the obstructions. We study the behaviour of the obstructions 훼 and 훽 appearing in Theorem A using a range of techniques from homotopy theory. Here we collect our results on when they vanish or not: Theorem C. If 푝 = 2 or 퐺 has a non-trivial normal 푝-subgroup, then 훼 and 훽 are zero (Corol- lary 7.12 and Proposition 5.8); in this case, we obtain an exact sequence res 0 → 푇 (퐺, 푆) → 푇 (퐺) −−→ lim 푇 (−) → 0. op OS (퐺) If 훼 or 훽 is zero and 푝 is odd, then we obtain that ≅ TF (퐺) −→ lim TF (−) op OS (퐺) via restriction (Proposition 11.1), where the right-hand side denotes the 퐺-stable elements in TF (푆). In general, 훼 and 훽 are non-zero: for example, when 퐺 = PSL3(푝) and 푝 ≡ 1 mod 3, we have 1 C 1 × that H퐺( ; H (−; 푘 )) ≅ ℤ/3 and 훽 is surjective (Example 10.2), so we obtain an exact sequence 0 → 푇 (퐺) → lim 푇 (−) → ℤ/3 → 0. op OS (퐺) Even though our theorem says that our obstructions may be non-zero in general when 푝 is odd, the cases where this occurs seem to be very limited, for a range of reasons. It may be possible to classify exactly when this happens for many classes of groups, for example the simple groups, by using our model from Theorem B in combination with the well-developed theory for calculating the obstruction groups that is demonstrated in [Gro18]. As an example, we carry this out for symmetric groups in Section 12: the image of the restriction is a proper op subgroup of limOS (퐺) 푇 (−), and we show that a a rational basis constructed by Carlson– Hemmer–Mazza in [CHM10] is also an integral basis. Theorem C sheds light on some open questions in the literature about 푝-fusion invariance of TF (퐺) and choice of generators: Corollary D. Suppose that 휙 ∶ 퐺 → 퐺′ is a group homomorphism that controls 푝-fusion. The induced homomorphism 휙∗ ∶ TF (퐺′) → TF (퐺) is an isomorphism if 푝 = 2. If 푝 is odd, this is not necessarily the case; for example, when 푝 ≡ 1 mod 3 the quotient map 휙 ∶ SL3(푝) → ∗ PSL3(푝) preserves fusion but the induced map 휙 ∶ TF (PSL3(푝)) ↪ TF (SL3(푝)) is the inclusion of a subgroup of index three. This answers [CMT14, Conjecture 10.1] in the positive when 푝 = 2, but in the negative in general; it is proved as Theorem 11.3. A related question is what qualitative properties the generators of TF (퐺) possess: the previously known methods for building generators have all involved constructing them from shifts of the trivial module, which led to the conjecture [CMT14, Conjecture 9.2] that these might all be chosen to lie in the principal block of 푘퐺, i.e. the block containing the trivial module. As not all endotrivial modules for SL3(푝) are inflated from PSL3(푝), this conjecture was too optimistic:

Corollary E. When 푝 ≡ 1 mod 3, a generating set of TF (SL3(푝)) cannot be chosen in the principal block. This is proved as Theorem 11.4. 1.7. Outline of proofs. Let us now sketch how we prove these theorems, and give an outline of the paper. The key starting point for our approach is to combine the viewpoint in [Gro18] with the power of ∞-categories, which enables us to set up the obstruction theory for lifting 22 Tobias Barthel, Jesper Grodal, and Joshua Hunt a compatible family of endotrivial 푘푃-modules, for 푃 ranging over 푝-subgroups of 퐺, to an endotrivial 푘퐺-module. More precisely, using a result of Mathew [Mat16], we obtain that

StMod −−→∼ holim StMod 푘퐺 op 푘푃 퐺/푃∈OS (퐺) as symmetric monoidal ∞-categories (see Proposition 3.1). The Picard space Pic(−) is a functor from symmetric monoidal ∞-categories to pointed spaces, which commutes with homotopy limits [MS16]. Applying this functor produces a homotopy decomposition of spaces

Pic(StMod ) −−→∼ holim Pic(StMod ). 푘퐺 op 푘푃 퐺/푃∈OS (퐺)

Once in the world of spaces, there is a classical spectral sequence and obstruction theory, developed by Bousfield–Kan [BK72] and others, for calculating the homotopy groups of a homotopy limit of a diagram 퐹 ∶ 퐼 → S of spaces in terms of the derived functors of the inverse limit of the homotopy groups of the spaces 퐹(푖). The spectral sequence has the slight complication that it is “fringed”, which is related to problems with strictifying basepoints; indeed, determining whether or not a basepoint lifts is our main concern! However, spectral sequences of thisform have been intensively studied in unstable homotopy theory, for example in connection with calculating maps between classifying spaces, and there is also an extensive literature on calculating the derived limits that appear (see e.g. [Oli98; Gro02; Gro10; Gro18]). As the homotopy groups of the Picard space Pic(StMod푘푃 ) identify in positive degrees with Tate cohomology, we can use a vanishing result of Jackowski–McClure for derived limits with values in 푝-local Mackey functors to show that only one “existence” and one “uniqueness” group might be non- trivial, arriving at the first exact sequence in Theorem A. This is done in Section 3 and Section 4, with preliminaries in Section 2. We then use the isotropy spectral sequence of a collection C to relate the obstruction class 훼 ∈ H2(OS (퐺); 푘×) to Bredon cohomology, as in [Gro18], and derive the second exact sequence in Theorem A. To deduce Theorem B and Theorem C from this, we have to analyse the obstruction class 1 C 1 × 훽 ∈ H퐺( ; H (−; 푘 )), and understand it in terms of representation theory. As alluded to prior to the statement of Theorem B, the key point is that for a 푘퐺-module 푀, the normaliser 푆 푁퐺(푃) acts on res푃 푀 in the homotopy category of StMod푘푃 , giving rise to a one-dimensional character of 푁퐺(푃) when 푃 is abelian. Since the restrictions of 푀 to non-conjugate abelian 푝-subgroups might be equivalent to different shifts of the trivial module, we obtain a relation between the one-dimensional characters on these subgroups that is more complicated than the relation in the Sylow-trivial case in [Gro18, Theorem D]. For a 퐺-stable endotrivial 푆-module, 1 C 1 × the obstruction in H퐺( ; H (−; 푘 )) vanishes if these relations can be simultaneously satisfied. In Section 6, we carry out an analysis of actions on objects in the homotopy category of StMod푘푃 , as in the above paragraph, and use it in Section 7 to reinterpret 훽 in terms of type functions and orientations. We thereby see that the obstruction to lifting a 퐺-stable endotrivial 푆-module can be seen directly, via representation theory, without using the cohomological obstruction theory. This reinterpretation is used in Sections 8 and 9 to prove the two parts of Theorem B. The remaining sections give other consequences and computations, and the appendix establishes the relationship between the obstruction class 훼 and Balmer’s work. We end this discussion of proofs by pointing out that the method of proof is rather general, combining ∞-categorical techniques with algebraic obstruction group calculations as they oc- cur in classical homotopy theory, and hence should apply to other to other symmetric monoidal ∞-categories in algebra and topology. Torsion-free endotrivial modules via homotopy theory 23

Acknowledgements. We would like to thank Paul Balmer, Benjamin Böhme, Rune Haugseng, and Markus Hausmann for useful conversations about the material of this paper. The third- named author would like to thank Paul Balmer for his hospitality during a visit to UCLA. The hospitality of the University of Copenhagen, the Max Planck Institute Bonn, and the Isaac Newton Institute Cambridge is also gratefully acknowledged.

2. Notation and preliminaries

Throughout the paper, we let 퐺 be a finite group, 푝 be a prime dividing its order, 푘 be a field of characteristic 푝, and 푆 be Sylow 푝-subgroup of 퐺. We use the unadorned word “space” to mean “simplicial set”. Since we will need to work with both 1-categorical and ∞-categorical (co)limits, we will call the former “(co)limits” and the latter “homotopy (co)limits”.

2.1. Group theory. Let S푝(퐺) denote the poset of non-trivial 푝-subgroups of 퐺, let A푝(퐺) denote its subposet of non-trivial elementary abelian 푝-subgroups, and let B푝(퐺) denote the S poset of non-trivial 푝-radical subgroups of 퐺 (i.e. subgroups 푃 ∈ 푝(퐺) with 푃 = 푂푝(푁퐺(푃)), where 푂푝(퐻) is the largest normal 푝-subgroup of 퐻). By a collection of subgroups we mean a set C of subgroups of 퐺 that is closed under conjugation: for example, the posets A푝(퐺), B푝(퐺), and S푝(퐺) introduced in the previous paragraph. For any such collection C , we write TC (퐺) for its transport category, i.e. the Grothendieck construction of the left action of 퐺 on C . More explicitly, TC (퐺) is the category whose objects are the elements of C and whose morphisms are given by Mor(푃, 푄) ≔ { 푔 ∈ 퐺 ∶ 푔푃 ≤ 푄 }, where 푔푃 denotes 푔푃푔−1. When referring to the transport category of a named collection such as S푝(퐺), we will drop the decorations and write TS (퐺) (similarly for the categories OS (퐺) and FS (퐺) defined below). We have an orbit category OC (퐺) associated with such a collection, namely the category of transitive 퐺-sets with isotropy groups in C and 퐺-equivariant maps between them. We can again be more explicit: this category is equivalent to the category with objects 퐺/푃 for 푃 ∈ C and morphisms given by Mor(퐺/푃, 퐺/푄) ≔ { 푔 ∈ 퐺 ∶ 푃푔 ≤ 푄 }/푄, where 푃푔 denotes 푔−1푃푔. There is a canonical quotient functor TC (퐺) → OC (퐺) that sends 푃 to 퐺/푃 and sends a morphism 푔 ∈ Mor(푃, 푄) to the coset 푔−1푄 ∈ Mor(퐺/푃, 퐺/푄). We will sometimes need to refer to the orbit category on all 푝-subgroups, not just the non-trivial 푝-subgroups, and we will denote this by O푝(퐺). This is the same as adding the free 퐺-set to OS (퐺). We also have a fusion category FC (퐺) associated with such a collection. The objects of FC (퐺) are the elements of C and the morphisms are the group homomorphisms that are induced by conjugation in 퐺: 푔 Mor(푃, 푄) ≅ { 푔 ∈ 퐺 ∶ 푃 ≤ 푄 }/퐶퐺(푃). There is again a natural quotient functor TC (퐺) → FC (퐺) that is the identity on objects and the natural quotient on morphism sets. Finally, we have an orbit simplex category 푠푆C (퐺), which is the poset of 퐺-conjugacy classes of chains of subgroups in C : we let 휎 ≤ 휏 if 휎 is a face of 휏 for some representatives 휎 ∈ 휎 and 휏 ∈ 휏. (This is the opposite of the category with this name in [Dwy98, §1.7].) This category is 24 Tobias Barthel, Jesper Grodal, and Joshua Hunt equivalent to the category whose objects are chains of subgroups in C , and where there is a unique morphism 휎 → 휏 when there is an element of 퐺 conjugating 휎 to a face of 휏.

2.2. The stable module ∞-category of a finite group. We write Mod푘퐺 for the category of 푘퐺-modules, with the symmetric monoidal structure (⊗푘, 푘) coming from the Hopf algebra structure of 푘퐺. A map 푓 ∶ 푀 → 푁 is called a stable equivalence if there exists 푔 ∶ 푁 → 푀 such that 푓 푔 − id푁 and 푔푓 − id푀 both factor through a projective 푘퐺-module. Inverting the stable equivalences on Mod푘퐺 yields the stable module category, which is a triangulated category that has been studied by representation theorists, for example as a way of classifying 푘퐺-modules when Mod푘퐺 has wild representation type; see [BIK11]. In order to study decompositions of the stable module category over the orbit category of 퐺, we need to work within a suitable higher categorical setting [Lur09; Lur17]. Therefore, we consider primarily the stable module ∞-category StMod푘퐺, which is defined as the ∞-categorical localisation of Mod푘퐺 at the stable equivalences. We will use Hom퐺(푀, 푁 ) to denote the mapping space between objects 푀, 푁 ∈ StMod푘퐺. The homotopy category of StMod푘퐺 is the stable module category as defined in the previous paragraph. Its objects are 푘퐺-modules and its hom sets are given by

휋0 Hom퐺(푀, 푁 ) ≅ Hom퐺(푀, 푁 )/(푓 ∼ 0 if 푓 factors through a projective). Every stable ∞-category C comes equipped with a shift or desuspension functor Ω∶ C −−→∼ C. (See [Lur17, §1] for a general introduction to the theory of stable ∞-categories.) For C = StMod푘퐺, this identifies with the Heller shift, which can be computed by taking the kernel of a projective cover. The salient features of StMod푘퐺 are summarised in the following theorem:

2.3. Theorem. The stable module ∞-category StMod푘퐺 has the structure of a symmetric monoidal stable ∞-category with unit 푘, where the monoidal structure is inherited from the symmetric monoidal structure on Mod푘퐺. Moreover, StMod푘퐺 is rigidly compactly generated, i.e. it admits a set of compact generators and the compact objects coincide with the dualisable ones. See [Car96, §5] for a discussion of the triangulated category and [Mat15, Definition 2.2] for a construction of StMod푘퐺 as a symmetric monoidal stable ∞-category. 2.4. Remark. More explicitly, the stable module ∞-category may be constructed as the underlying ∞-category of a model structure on Mod푘퐺 for which: • weak equivalences are given by stable equivalences, • cofibrations are precisely the monomorphisms, and • fibrations are precisely the epimorphisms in Mod푘퐺. We refer to [Hov99, §2.2] for a more detailed discussion of this model structure. Given a subgroup 퐻 → 퐺 and an element 푔 ∈ 퐺, restricting along the right-conjugation map 푔 푐푔 ∶ 퐻 → 퐻 gives a functor

푔 ⊗퐻 −∶ StMod푘퐻 → StMod푘(푔퐻). −1 Explicitly, 푔 ⊗퐻 푀 has the twisted action 푔ℎ푔 ⋅(푔 ⊗퐻 푚) ≔ 푔 ⊗퐻 (ℎ⋅푚). Note that 푔 ⊗퐻 푀 can be identified with a submodule of the induced module 푘퐺 ⊗퐻 푀. More generally, restriction to a subgroup 푖∶ 퐻 → 퐺 gives rise to a symmetric monoidal functor 퐺 ∗ res퐻 = 푖 ∶ StMod푘퐺 → StMod푘퐻 which preserves all (homotopy) limits and (homotopy) colimits. This implies that 푖∗ admits a left adjoint 푖! and a right adjoint 푖∗, usually referred to as induction and coinduction, respectively. Since 퐺/퐻 is finite, there is in fact a natural equivalence 푖∗ ≃ 푖!. Since 푖∗ inherits a Torsion-free endotrivial modules via homotopy theory 25

∗ lax monoidal structure from 푖 , it follows that 푖∗푘 is a commutative algebra object in StMod푘퐺, whose underlying 푘퐺-module can be identified with ∏퐺/퐻 푘 with its permutation action, and whose multiplication is given by pulling back along the diagonal map 퐺/퐻 → 퐺/퐻 ×퐺/퐻. We 퐺 will denote this algebra object by 퐴퐻 . Balmer [Bal15, Theorem 1.2] proves an equivalence be- 퐺 tween StMod푘퐻 and the category of modules over 퐴퐻 internal to StMod푘퐺; see [Mat16, Propo- sition 9.12] for the ∞-categorical version of this theorem. 2.5. Theorem (Balmer, Mathew). For any subgroup 퐻 ≤ 퐺, there is a natural symmetric monoidal equivalence ∼ 퐺 StMod푘퐻 −−→ ModStMod푘퐺 (퐴퐻 ), ⟵ 퐺 induced by coinduction, under which the free/forget adjunction StMod푘퐺 ⟶ ModStMod푘퐻 (퐴퐻 ) ⟵ corresponds to the restriction/coinduction adjunction StMod푘퐺 ⟶ StMod푘퐻 . We can extend the construction of the stable module ∞-category to an arbitrary finite 퐺-set 푋 by setting

StMod(푋) ≔ ModStMod푘퐺 (∏푋 푘) , where the algebra structure on ∏푋 푘 is again given by pullback along the diagonal map 푋 → 푋 × 푋. (More formally, this is a homotopy Kan extension of StMod푘− along the inclusion of the orbit category into all finite 퐺-sets.) With this notation, Theorem 2.5 provides an equivalence

StMod(∐푖∈퐼 퐺/퐻푖) ≃ ∏푖∈퐼 StMod푘퐻푖 for any finite set of transitive 퐺-sets { 퐺/퐻푖 ∶ 푖 ∈ 퐼 }.

2.6. Endotrivial modules. Our primary object of study is the group of endotrivial modules of 퐺, denoted 푇 (퐺).A 푘퐺-module 푀 is said to be endotrivial if ∗ 푀 ⊗푘 푀 ≃ 푘 ⊕ (proj) as 푘퐺-modules, where 푀∗ is the 푘-linear dual of 푀 and (proj) denotes some projective 푘퐺- module. In particular, a finitely generated module 푀 is endotrivial if and only if its 푘-endomorphism ring is stably equivalent to the trivial module. We write [푀] for the stable equivalence class of 푀 in StMod푘퐺, and define 푇 (퐺) ≔ { [푀] ∶ 푀 endotrivial }, which is an abelian group under the operation [푀] + [푁 ] ≔ [푀 ⊗푘 푁 ]. (We will henceforth ∗ omit the subscript 푘 in ⊗푘.) The inverse of [푀] is its 푘-linear dual [푀 ]. For any finite group 퐺, the abelian group 푇 (퐺) is finitely generated: see [CMN06, Corollary 2.5] and earlier work of Puig [Pui90, Corollary 2.4] for the case of finite 푝-groups. A fact about endotrivial modules that we will use repeatedly is that their endomorphisms are all homotopic to multiplication by a scalar:

2.7. Lemma. If 푀 is an endotrivial 퐺-module, then 휋0 End퐺(푀) ≅ 푘 as 푘-algebras.

Proof. Since 푀 is necessarily a compact object of StMod푘퐺, we have ∗ Hom퐺(푀, 푀) ≃ Hom퐺(푘, 푀 ⊗ 푀) ≃ Hom퐺(푘, 푘), □ so up to homotopy any endomorphism of 푀 is given by 푀 ⊗ 휑 for some 휑 ∈ Hom퐺(푘, 푘). One key property of the shift functor Ω introduced previously is that Ω푛푁 ⊗ Ω푚푀 ≃ Ω푚+푛(푁 ⊗ 푀). In particular, for every finite group 퐺, the shifts {Ω푛푘 ∶ 푛 ∈ ℤ } of the trivial module form a cyclic subgroup of 푇 (퐺) generated by Ω푘. A classical theorem of Dade says that for abelian 푝-groups, this is the entire group of endotrivial modules: 26 Tobias Barthel, Jesper Grodal, and Joshua Hunt

2.8. Theorem ([Dad78b, Proposition 9.16, Theorem 10.1]). If 푃 is a non-trivial abelian 푝-group then 푇 (푃) is cyclic, generated by Ω푘푃 . Furthermore:

⎧0 if 푃 ≅ ℤ/2, 푇 (푃) ≅ ℤ/2ℤ if 푃 ≅ ℤ/푛 with 푛 > 2, and ⎨ ⎩ℤ otherwise. A lot is already known about the structure of 푇 (퐺). In addition to Dade’s theorem above, Carlson–Thévenaz showed in [CT05] that when 푃 is an arbitrary finite 푝-group, 푇 (푃) is torsion- free except in the cases where the 푝-group is cyclic, a semi-dihedral 2-group (where the torsion subgroup is ℤ/2), or a generalised quaternion 2-group (where it is either ℤ/4 or ℤ/4 ⊕ ℤ/2, depending on whether or not 푘 has a primitive third root of unity). Carlson–Mazza–Nakano [CMN06, Theorem 3.1] explicitly describe the torsion-free rank of 푇 (퐺):

0 if rk푝(퐺) = 1, (2.9) dimℚ(푇 (퐺) ⊗ ℚ) = rk(TF (퐺)) = { 푛퐺 if rk푝(퐺) = 2, and 푛퐺 + 1 if rk푝(퐺) ≥ 3, where rk푝(퐺) is the 푝-rank of 퐺, i.e. the rank 푟 of the largest elementary abelian 푝-subgroup 푟 (ℤ/푝) of 퐺, and 푛퐺 is the number of conjugacy classes of rank two maximal elementary abelian 푝-subgroups of 퐺. This generalises Alperin’s formula for 푝-groups [Alp01, Theorem 4]. The number 푛퐺 is related to many questions in classical group theory, and there has been a lot of research into determining its value; see e.g. [Mac70; GM10; CMN06; CGMN20]. In particular, unless 퐺 itself has low rank, the number 푛퐺 is zero and the torsion free part of 푇 (퐺) is of rank one, generated by shifts of the trivial module. Specifically, by “low rank”we mean less than or equal to 푝 if 푝 is odd [GM10, Theorem A] and less than or equal to four if 푝 = 2, by [Mac70, “Four generator theorem”] together with [GM10, Lemma 2.4]. The origin of the formula in (2.9) is that it computes the number of connected components of the poset of conjugacy classes of elementary abelian 푝-subgroups of 퐺 of rank at least two; on these subgroups the torsion-free rank is known to be one by Dade’s theorem. (We sketch a reproof of (2.9) in Remark 4.8, using the obstruction theory of Section 4.) In light of (2.9), to determine 푇 (퐺) as an abstract group we only need to calculate its torsion subgroup. Since 푇 (푆) is torsion-free except in the few exceptional cases listed above, this is in most cases equal to the kernel 푇 (퐺, 푆) of the restriction map 푇 (퐺) → 푇 (푆), also known as the group of Sylow trivial modules. More explicitly, these are modules 푀 ∈ 푇 (퐺) such that 퐺 res푆 푀 ≅ 푘 ⊕ (free). In [Gro18], the second-named author established several descriptions of 푇 (퐺, 푆) in terms of local group theory, building on earlier ideas of Balmer [Bal13]. Here we list the descriptions that we will use later: • There is an isomorphism 푇 (퐺, 푆) ≅ H1(OS (퐺); 푘×).[Gro18, Theorem A] • Let C be a collection of 푝-subgroups of 퐺 such that the inclusion C ↪ S푝(퐺) is a 퐺-homotopy equivalence. There is an isomorphism 1 × 푇 (퐺, 푆) ≅ lim H (푁퐺(푃0 < … < 푃푛); 푘 ), [푃0<…<푃푛] where the limit is indexed over chains of subgroups in C , ordered by refinement. [Gro18, Theorem D] C ∗ • Let be as above and let H퐺(푋; 퐹) denote the 퐺-equivariant Bredon cohomology of a 퐺-space 푋 with coefficients in the functor 퐹 (see Section 5). There is an isomorphism 0 C 1 × 푇 (퐺, 푆) ≅ H퐺( ; H (−; 푘 )).[Gro18, §5] Torsion-free endotrivial modules via homotopy theory 27

What happens in the exceptional cases, when 푇 (푆) has torsion, is also completely understood: Carlson, Mazza, and Thévenaz [MT07; CMT13] show that the restriction map 푇 (퐺) → 푇 (푆) is a surjection; in particular, all endotrivial 푆-modules lift to 푇 (퐺) and hence are 퐺-stable. Fur- thermore, apart from the (well-understood) cyclic case, the restriction map is split, so TT (퐺) ≅ 푇 (퐺, 푆) ⊕ TT (푆). This allows for a full description of TT (퐺) from that of 푇 (퐺, 푆), using the classification of 푇 (푆).

2.10. Picard spaces. The group 푇 (퐺) is in fact the Picard group of StMod푘퐺, i.e. the group of equivalence classes of ⊗-invertible objects. The Picard group construction provides a functor from symmetric monoidal categories to abelian groups. One unfortunate property of the Picard group is that it does not preserve (homotopy) limits of categories, so is hard to calculate using descent methods. This can be remedied by considering instead the Picard space Pic(C) of a symmetric monoidal ∞-category C: this is the ∞-groupoid (i.e. space) of ⊗-invertible objects in C and equivalences between them. Its set of connected components can be identified with the Picard group of C, but due to its extra structure, Pic commutes with homotopy limits as a functor to pointed spaces. More information can be found in [MS16, §2]. For any symmetric monoidal ∞-category (C, ⊗, ퟙ), we have a description of the higher homotopy groups of Pic(C). These are given by PicGp(C) when 푖 = 0, C × 휋푖 Pic( ) ≅ { (휋0 Hom(ퟙ, ퟙ)) when 푖 = 1, and 휋푖−1 Hom(ퟙ, ퟙ) when 푖 ≥ 2, where PicGp denotes the Picard group. In the case of StMod푘퐺, we have 푇 (퐺) when 푖 = 0, × (2.11) 휋푖 Pic(StMod푘퐺) ≅ { 푘 when 푖 = 1, and Ĥ 1−푖(퐺; 푘) when 푖 ≥ 2, ̂ 푛 푛 where H (퐺; 푀) ≅ 휋0(Hom퐺(Ω 푘, 푀)) denotes the Tate cohomology of 퐺.

3. Decomposition of the stable module ∞-category over the orbit category

We continue to let 퐺 be a finite group. A key input for deriving our obstruction willbea symmetric monoidal decomposition of the stable module ∞-category as a homotopy limit over ⊗ the orbit category. Let Cat∞ be the ∞-category of symmetric monoidal ∞-categories. There are O op ⊗ various ways to construct the required functor S (퐺) → Cat∞; one possibility is to follow the approach of [Mat16, §9.5] and use the composite of the functors O op (퐺) → CAlg(StMod푘퐺) 퐺/퐻 ↦ ∏ 푘 퐺/퐻 and ⊗ Mod(−)∶ CAlg(StMod푘퐺) → Cat∞

퐴 ↦ ModStMod푘퐺 (퐴) along with Theorem 2.5. Here CAlg(C) denotes the ∞-category of commutative algebra objects in a symmetric monoidal ∞-category C. Note that, by [Lur17, Proposition 3.2.2.1], the ∞-category CAlg(C) has all homotopy limits which exist in C and these are preserved by the forgetful functor CAlg(C) → C. 28 Tobias Barthel, Jesper Grodal, and Joshua Hunt

3.1. Proposition. Let C 푒 be a collection of subgroups of 퐺 that is closed under intersection and such that every elementary abelian 푝-subgroup of 퐺 is contained in a subgroup in C 푒. Let C denote the non-trivial subgroups in C . There is an equivalence of symmetric monoidal ∞-categories ∼ StMod푘퐺 −−→ holim StMod푘퐻 , 퐺/퐻∈OC (퐺)op where the homotopy limit is taken in the ∞-category of symmetric monoidal ∞-categories. Proof. Corollary 9.16 in [Mat16] implies that

StMod푘퐺 ≃ holim StMod푘퐻 . OC 푒 (퐺)op

We note that StMod푘{1} is trivial, so we can remove 퐺/{1} from the diagram without changing O op ⊗ the value of the homotopy limit: indeed, if we let 퐹 ∶ C 푒 (퐺) → Cat∞ denote the functor op op constructed above and 푖∶ OC (퐺) → OC 푒 (퐺) denote the inclusion, then 퐹 is equivalent to ∗ ∗ the homotopy right Kan extension along 푖 of the restriction 푖 퐹. The homotopy limit of Ran푖 푖 퐹 ∗ ∗ agrees with the homotopy limit of 푖 퐹, because Ran푖 is right adjoint to 푖 and taking homotopy limits is right adjoint to the constant diagram functor. Therefore, we get ∗ ∗ □ StMod푘퐺 ≃ holim Ran푖 푖 퐹 ≃ holim 푖 퐹. OC 푒 (퐺)op OC (퐺)op The Picard space functor commutes with homotopy limits [MS16, Proposition 2.2.3], so we get a decomposition of pointed simplicial sets

Pic(StMod푘퐺) ≃ holim Pic(StMod푘퐻 ). 퐺/퐻∈OC (퐺)op

The homomorphism 푇 (퐺) → limOC (퐺)op 푇 (−) is exactly the induced map

휋0 Pic(StMod푘퐺) → lim 휋0 Pic(StMod푘퐻 ). 퐺/퐻∈OC (퐺)op

4. Obstruction theory for homotopy limits

In [Bou89, §5.2], Bousfield gives an obstruction theory for cosimplicial spaces that we can apply to the problem of lifting from Olim C (퐺)op 휋0 Pic(StMod푘푃 ) to 휋0 holimOC (퐺)op Pic(StMod푘푃 ), i.e. from limOC (퐺)op 푇 (−) to 푇 (퐺). We recall this obstruction theory in the case of a diagram of Kan complexes with trivial action of their fundamental group on their higher homotopy groups, then specialise to the case of interest. Let sSet be the category of simplicial sets and let csSet be the category of cosimplicial spaces. Given a diagram of Kan complexes 퐹 ∶ 퐼 op → sSet, let 푋 • denote the the fibrant cosimplicial space given by 푛 (4.1) 푋 ≔ ∏ 퐹(푖0), 푖0→…→푖푛 where the product is indexed by the 푛-simplices in the nerve of 퐼. The space Tot 푋 • is a model • • • for the homotopy limit of 퐹, where Tot 푋 denotes the simplicial set MapcsSet(Δ , 푋 ). This model is itself a limit of the tower of fibrations • • • … → Tot푛 푋 → Tot푛−1 푋 → … → Tot0 푋 , • • • where Tot푛 푋 denotes the simplicial set MapcsSet(sk푛 Δ , 푋 ). 푠 푠 푠 Assume that 휋1(푋 , 푥) acts trivially on 휋푛(푋 , 푥) for all 푛 ≥ 1, 푠 ≥ 0 and 푥 ∈ 푋 . If we • • choose a vertex 푣푟 ∈ Tot푟 푋 that lifts to Tot푟+1 푋 , then Bousfield’s obstruction theory gives Torsion-free endotrivial modules via homotopy theory 29

푟+2 • • an obstruction class in 휋 휋푟+1푋 that vanishes if and only if 푣푟 is liftable to Tot푟+2 푋 (this 푟 • th is explained in more detail in the proof of the following lemma). Here 휋 휋푡 푋 denotes the 푟 • th cohomotopy group of 휋푡 푋 , defined to be the 푟 cohomology group of the cochain complex given by the normalised homotopy groups

•−1 • 푗 • •−1 푗 푁 휋푡 (푋 , 푣) ≔ ⋂ ker(푠 ∶ 휋푡 (푋 , 푣) → 휋푡 (푋 , 푠 푣)) 푖=0 and the differential induced by the alternating sum of the face maps. 4.2. Lemma. Let C be a collection for which we have a decomposition ∼ StMod푘퐺 −−→ holim StMod푘퐻 OC (퐺)op and let (푀퐻 )퐺/퐻∈OC (퐺) ∈ limOC (퐺)op 휋0 Pic(StMod푘−). There are obstructions to lifting (푀퐻 ) to 푟+2 O an element of 휋0 Pic(StMod푘퐺) that lie in H ( C (퐺); 휋푟+1 Pic StMod푘−) for 푟 ≥ 0. When 푟 ≥ 1, these obstruction groups identify with H푟+2(OC (퐺); Ĥ −푟 (−; 푘)).

Proof. The Picard space is by construction a Kan complex, and the actions of its fundamental group on its higher homotopy groups are trivial since it has an 퐸∞-space structure. We apply • Bousfield’s obstruction theory: by hypothesis, we start with avertex 푣0 ∈ Tot0 푋 that lifts to • • Tot1 푋 , where 푋 is the cosimplicial space defined in (4.1) associated with the functor O op Pic(StMod푘−)∶ C (퐺) → sSet, whose totalisation computes the homotopy limit holimOC (퐺)op Pic(StMod푘−). The trivial action of the fundamental group implies that this case is covered by the spectral sequence of [Bou89, 2 • §2.6], and [Bou89, §5.2] gives an obstruction class in 휋 휋1푋 that vanishes if and only if 푣0 lifts • • • further to Tot2 푋 . Now let 푟 ≥ 1 and 푣푟 ∈ Tot푟 푋 be a vertex liftable to Tot푟+1 푋 .[Bou89, 푟+2 • • §5.2] gives an obstruction in 휋 휋푟+1푋 that vanishes if and only if 푣푟 lifts further to Tot푟+2 푋 . (In particular, we do not require any hypotheses on the vanishing of Whitehead products in this case.) 푟+2 • 푟+2 O The obstruction group 휋 휋푟+1푋 is isomorphic to H ( C (퐺); 휋푟+1 Pic StMod푘−) by [GJ09, VIII.2 (2.18)], so it remains for us to check that the homotopy groups of Pic(StMod푘퐻 ) are as we claimed earlier in Equation (2.11). Since Ω Pic(StMod푘퐻 ) ≃ Aut(푘), we have

휋푟+1 Pic(StMod푘퐻 ) ≅ 휋푟 Aut(푘), × and hence 휋1 Pic(StMod푘퐻 ) ≅ 푘 . For 푟 ≥ 1 we have −푟 −푟 □ 휋푟+1 Pic(StMod푘퐻 ) ≅ 휋푟 Hom(푘, 푘) ≅ 휋0(Hom(Ω 푘, 푘)) ≅ Ĥ (퐻; 푘). Since 푘 is a field of characteristic 푝, a result of Jackowski and McClure shows that all but the first of these obstruction groups are zero when C = S푝(퐺): 푗 O op 4.3. Lemma. Let 푗 ∈ ℤ. The functor Ĥ (−; 푘)∶ S (퐺) → Mod푘 is acyclic, i.e.

H푖(OS (퐺); Ĥ 푗(−; 푘)) = 0 for any 푖 ≥ 1.

Proof. We again let O푝(퐺) denote the orbit category on all 푝-subgroups of 퐺, and consider the 푗 O op extended functor 퐹 ≔ Ĥ (−; 푘)∶ 푝(퐺) → Mod푘. Proposition 5.14 in [JM92] says that any 30 Tobias Barthel, Jesper Grodal, and Joshua Hunt

O op proto-Mackey functor 푝(퐺) → Modℤ(푝) is acyclic, and we observe that 퐹 is indeed a proto- Mackey functor. This is well-known, with the covariant part of the Mackey functor sending a morphism 푔 ∶ 퐺/푃 → 퐺/푄 to the transfer map ̂ 푗 푗 푗 H (푃; 푘) ≅ 휋0 Hom푃 (res푔 Ω 푘, res푔 푘) ≅ 휋0 Hom푄(Ω 푘, ind푔 res푔 푘) 휀 ∗ 푗 ̂ 푗 −−→휋0 Hom푄(Ω 푘, 푘) ≅ H (푄; 푘). The Mackey decomposition formula for 퐹 follows from the Mackey decomposition theorem [Ben98, Theorem 3.3.4] applied to 푘. ′ 푗 O op Next we need to deduce acyclicity of 퐹 ≔ Ĥ (−; 푘)∶ S (퐺) → Mod푘 from acyclicity of 퐹: that is, we need to remove the trivial subgroup again. Since right Kan extension along op op OS (퐺) → O푝(퐺) extends by the zero group on the coset 퐺/푒, it follows that 퐹 is the right Kan extension of 퐹 ′. It also follows that taking right Kan extensions preserves epimorphisms. 푖 ′ 푖 ′ By [JM92, Lemma 3.1], we have H (OS (퐺); 퐹 ) ≅ H (O푝(퐺); 퐹) for all 푖, so 퐹 is also acyclic. □ 4.4. Remark. Note that the functors considered in the vanishing result [JM92, Proposition 5.14] are implicitly contravariant. It is also important that we consider the orbit category on all 푝- subgroups OS (퐺) rather than, for example, restricting to the orbit category on elementary abelian 푝-subgroups OA(퐺), where in general the higher limits do not vanish; see [Gro02, Re- mark 3.4]. (As in the above proof, the inclusion or exclusion of the trivial subgroup is irrelevant here.) We can illustrate the latter remark by the example 퐺 = 퐷8, making forward reference to some results later in the paper. There are two non-conjugate elementary abelian subgroups of op 퐺, which we will call 퐸1 and 퐸2. The limit limOA(퐺) 푇 (−) then identifies with 푇 (퐸1) ⊕ 푇 (퐸2) ≅ ℤ2. By Corollary 7.12, the obstruction appearing in Theorem 4.5 below vanishes when 푝 = 2. Since 퐺 is itself a 2-group, 푇 (퐺, 푆) = 0. Therefore, if all the higher obstructions vanished, ∼ we would have an isomorphism 푇 (퐺) −−→푇 (퐸1) ⊕ 푇 (퐸2) induced by restriction. Since the re- 2 striction from 푇 (퐺) → 푇 (퐸1) ⊕ 푇 (퐸2) ≅ ℤ lands only in the “even” part of the lattice [CT00, Theorem 5.4], one of the groups H푟+2(OA(퐺); Ĥ −푟 (−; 푘)) must be non-zero. We can nevertheless hope to extract useful information from the decomposition over OA(퐺); see Remark 4.8 for an example. 4.5. Theorem. There is an exact sequence 훼 0 → 푇 (퐺, 푆) → 푇 (퐺) → lim 푇 (−) −→ H2(OS (퐺); 푘×), op OS (퐺) and we can explicitly describe a 2-cocycle that represents 훼: given an element (푀푃 )퐺/푃∈OS (퐺) ∈ op limOS (퐺) 푇 (−), choose an equivalence 휆푔 ∶ res푔 푀푄 → 푀푃 in StMod푘푃 for every morphism 푔 ∶ 퐺/푃 → 퐺/푄 in OS (퐺). We obtain a potentially non-commutative diagram of equivalences

res푔 resℎ 푀푅 res푔 휆ℎ 휆푔ℎ

휆푔 res푔 푀푄 푀푃

푔 ℎ O × for every 퐺/푃 −→퐺/푄 −→퐺/푅 in S (퐺). We define 훼푀 (푔, ℎ) to be the element of 푘 that corresponds to the automorphism

휆푔ℎ 휆푔 res푔 휆ℎ res푔 resℎ 푀푅 −−−→푀푃 ←−− res푔 푀푄 ←−−−−− res푔 resℎ 푀푅 Torsion-free endotrivial modules via homotopy theory 31

× determined by the above diagram. (Recall from Lemma 2.7 that 휋0 Aut(푀) ≅ 푘 for any endotrivial module 푀.) The cohomology class 훼((푀푃 )) is represented by 훼푀 .

4.6. Notation. When the 퐺-stable endotrivial 푆-module (푀푃 ) is clear from context, we will drop the subscript on 훼푀 and use 훼 to refer to both the homomorphism and the cocycle. Proof. Lemma 4.3 shows that all the obstructions given in Lemma 4.2 vanish with the possible exception of the obstruction in H2(OS (퐺); 푘×). This establishes the existence of the exact sequence, so we just need to show that 훼푀 actually represents 훼. Let 푋 • be the cosimplicial space defined in (4.1) associated with the functor O op Pic(StMod푘−)∶ S (퐺) → sSet. Section 5.1 in [Bou89] describes the obstruction in H2(OS (퐺); 푘×): if we have chosen a vertex • 2 2 2 푏 ∈ Tot1 푋 then we obtain a map 푐(푏)∶ 휕Δ = sk1 Δ → 푋 in the normalised homotopy 2 2 • 푁 휋1(푋 , 푏) that represents the obstruction in 휋 휋1(푋 , 푏). • Choosing a vertex in Tot1 푋 is the same as choosing endotrivial modules 푀푃 ∈ StMod푘푃 O for all 퐺/푃 ∈ S (퐺) along with equivalences 휆푔 ∶ res푔 푀푄 → 푀푃 for all 푔 ∶ 퐺/푃 → 퐺/푄. The obstruction 푐(푏) corresponding to this choice of vertex is then the 2-cocycle described in the statement of the theorem. □ 4.7. Remark. In Appendix A, we compare the obstruction 훼 constructed above with an obstruction that appeared in work of Balmer [Bal15]. 4.8. Remark. We outline how Proposition 3.1 should allow us to re-derive Alperin’s formula for the torsion-free rank of 푇 (퐺). Let 푛퐺 denote the number of conjugacy classes of rank 2 maximal elementary abelian subgroups of 퐺. Recall that the torsion-free rank of 푇 (퐺) is given by 0 if the 푝-rank of 퐺 is one, dimℚ(푇 (퐺) ⊗ ℚ) = { 푛퐺 if the 푝-rank of 퐺 is two, and 푛퐺 + 1 if the 푝-rank of 퐺 is three or greater. This was proved for finite groups by Carlson–Mazza–Nakano [CMN06, Theorem 3.1] and for 푝-groups by Alperin [Alp01, Theorem 4]. O We use Proposition 3.1 to decompose StMod푘퐺 over A(퐺): StMod −−→∼ holim StMod . 푘퐺 op 푘푉 OA(퐺)

O op Since A(퐺) is a finite 1-category, Alperin’s formula holds for limOA(퐺) 푇 (−): rationalisa- tion commutes with finite limits, and the group-theoretic argument given prior to Lemma4 of [Alp01] shows that any two elementary abelian subgroups of 푝-rank at least three are connected by a zig-zag of elementary abelian subgroups of 푝-rank at least two. Therefore, op limOA(퐺) (푇 (−) ⊗ ℚ) has one copy of ℚ for each conjugacy class of rank two maximal elementary abelian subgroups, plus an extra copy if there are any elementary abelian subgroups of op 푝-rank at least three. Lemma 4.2 implies that 푇 (퐺) is obtained from limOA(퐺) 푇 (−) by repeat- 푟+2 O edly taking the kernel of a homomorphism to H ( A(퐺); 휋푟+1 Pic StMod푘−), so it is enough to show that this procedure preserves the torsion-free rank. We now come to a gap in the argument: it is not clear that the obstructions given in [Bou89, §5.2] arise as the differentials in a spectral sequence of the form constructed in[Bou89, §§2.4– 2.6]. Assuming that this is nevertheless the case, the analogue for the stable module ∞-category of [MNN19, Theorem 2.25] shows that this spectral sequence has a horizontal vanishing line at a finite stage. Note that the nilpotency condition of that theorem is satisfied for anymodule 퐺 퐺 in StMod푘퐺 because the algebra object ∏푉 ∈A푝(퐺) 퐴푉 is descendable. (Recall that 퐴푉 denotes 32 Tobias Barthel, Jesper Grodal, and Joshua Hunt

퐺 coind푉 (푘); see Theorem 2.5.) It seems likely that the domain of definition of the spectral sequences can be expanded as claimed, at least if one is willing to construct a less general spectral sequence than in [Bou89]. Therefore, there are only finitely many obstruction groups to consider, and it is enoughto check that each of them is torsion. For the obstruction in H2(OA(퐺); 푘×), we have rational equivalences O ≅ T ≅ A H∗( A(퐺)) ⊗ ℚ ←−− H∗( A(퐺)) ⊗ ℚ −−→ H∗( 푝(퐺)/퐺) ⊗ ℚ by [Gro18, Proposition 4.33]. Symonds’ theorem [Sym98] says that S푝(퐺)/퐺 is contractible, A O so 푝(퐺)/퐺 is as well. Since H푖( A(퐺)) is finitely generated, we deduce that it is finite when 푖 > 0. The universal coefficient theorem then implies thatH2(OA(퐺); 푘×) is also finite provided 푘 is algebraically closed. The argument for the higher obstruction groups is simpler: each H푟+2(OA; Ĥ −푟 (−; 푘)) is a 푘-vector space and hence is annihilated by multiplication by 푝.

5. An obstruction in Bredon cohomology

While useful conceptually, the obstruction 훼 of Theorem 4.5 can be difficult to work with in practice. The category OS (퐺) contains the Weyl group of every 푝-subgroup of 퐺, so it can be hard to compute its cohomology for specific choices of 퐺 that we are interested in. In this section, we show that the obstruction 훼 in H2(OS (퐺); 푘×) vanishes if and only if another ob- 1 S 1 × struction 훽 in the Bredon cohomology group H퐺( 푝(퐺); H (−; 푘 )) vanishes (Proposition 5.9). This latter obstruction has several advantages for practical computations: for example, S푝(퐺) is a finite poset, so computing the obstruction group is much easier. We can also replace S푝(퐺) with a smaller 퐺-homotopy equivalent poset, such as the poset B푝(퐺) of non-trivial 푝-radical subgroups or the poset A푝(퐺) of non-trivial elementary abelian 푝-subgroups. It will later allow us to provide an alternative viewpoint on the question of lifting: given a 퐺-stable endotrivial 푆-module (푀푃 )퐺/푃∈OS (퐺), there is a lift of 푀푆 to 푇 (퐺) if and only if we can specify one-dimensional characters, one for every elementary abelian 푝-subgroup 푃 ∈ A푝(퐺), such that they satisfy a certain compatibility condition; see Section 7. We start by recalling the isotropy spectral sequence as defined in [Bro82, VII (5.3)] or [Dwy98, §2.3]. Given a 퐺-space 푋 and an abelian group 퐴, we have a quasi-isomorphism • • • (5.1) C (푋ℎ퐺; 퐴) ≃ Tot (Hom퐺(C•(E퐺), C (푋; 퐴)). • Here C (푋; 퐴) denotes the cochains on 푋 with coefficients in 퐴; similarly, C•(푋; 퐴) denotes the chains on 푋 with coefficients in 퐴. We can filter the right-hand side of Equation (5.1) by ∗ skeleta of 푋, giving rise to a spectral sequence calculating the cohomology H (푋ℎ퐺; 퐴). More explicitly, we use the filtration • • • 푛 푠 … ⊆ 퐹1 ⊆ 퐹0 = C (푋ℎ퐺; 퐴) with 퐹푟 ≔ ⨁ Hom퐺(C푡 (E퐺), C (푋; 퐴)). 푠+푡=푛 푠≥푟 The corresponding filtered complex spectral sequence is isomorphic to the double complex spectral sequence where one first takes the differential inthe 퐶•(E퐺) direction. Therefore, the •• spectral sequence has 퐸1 page given by 푠,푡 푡 푠 퐸1 = H (퐺; C (푋; 퐴)) with differential induced by the differential inC•(푋; 퐴). Torsion-free endotrivial modules via homotopy theory 33

•• 푠 푡 The rows of the 퐸1 page identify with the cochain complexes C퐺(푋; H (−; 퐴)), where • C퐺(푋; 퐹) denotes the 퐺-equivariant Bredon cochains of 푋 with coefficients in some functor 퐹 ∶ O(퐺)op → Ab. This is defined by 퐺 푠 (5.2) C퐺(푋; 퐹) ≔ ( ∏ 퐹(퐺휎 )) , 휎∈푋푠 where 퐺휎 is the stabiliser of 휎 in 퐺. The 퐺-equivariant Bredon cohomology of 푋 with coefficients ∗ in 퐹, denoted by H퐺(푋; 퐹), is then defined to be the cohomology of the above cochain complex, which is an invariant of the 퐺-equivariant homotopy type of 푋; see [Bre67, §I.6]. In this way, we obtain the isotropy spectral sequence 푠,푡 푠 푡 푠+푡 (5.3) 퐸2 ≅ H퐺(푋; H (−; 퐴)) ⇒ H (푋ℎ퐺; 퐴). Let C be a collection. Thomason’s theorem [Tho79, Theorem 1.2] implies that TC is homotopy C C equivalent to ℎ퐺. Therefore, when applied to , the isotropy spectral sequence computes the cohomology of TC in terms of the Bredon cohomology of C . The canonical quotient functor TS (퐺) → OS (퐺) induces an injective map

(5.4) H2(OS (퐺); 푘×) → H2(TS (퐺); 푘×) that identifies H2(OS (퐺); 푘×) with a direct summand of H2(TS (퐺); 푘×), by [Gro18, Corol- C 1 C 1 × lary 4.36]. For certain collections , we can also identify H퐺( ; H (−; 푘 )) with a subgroup of H2(TS (퐺); 푘×): 5.5. Lemma. There is a natural injective map 1 C 1 × 2 T × H퐺( ; H (−; 푘 )) → H ( S (퐺); 푘 ) for any collection C of non-trivial 푝-subgroups such that C ↪ S푝(퐺) is a 퐺-homotopy equivalence. Proof. Consider the isotropy spectral sequence (5.3) for 푋 = C . Symonds’ theorem [Sym98] says that S푝(퐺)/퐺 is contractible, and hence so is C /퐺. Therefore for 푠 > 0 we have 푠,0 푠 C 0 × 푠 C × 퐸2 ≅ H퐺( ; H (−; 푘 )) ≅ H ( /퐺; 푘 ) = 0, and the bottom-left corner of the 퐸2 page of the spectral sequence is as follows: 0 C 2 × H퐺( ; H (−; 푘 ))

1 C 1 × (5.6) H퐺( ; H (−; 푘 ))

푘× 0 0 1,1 1 C 1 × 2 T × C We deduce that 퐸2 ≅ H퐺( ; H (−; 푘 )) injects into H ( C (퐺); 푘 ). Since is 퐺-homotopy equivalent to S푝(퐺), Thomason’s theorem implies that TC (퐺) is homotopy equivalent to TS (퐺). □ 2 T × 5.7. Remark. We can describe the image 훾푀 of 훼푀 in H ( S (퐺); 푘 ) analogously to the de- −1 T scription of 훼푀 in Theorem 4.5: for every 푔 ∶ 푃 → 푄 in S (퐺), we choose an equivalence 휆푔 ∶ 푔 ⊗푄 푀푄 → 푀푃 in StMod푘푃 , and then for every 2-chain

푔−1 ℎ−1 푃 −−−→푄 −−−→푅 34 Tobias Barthel, Jesper Grodal, and Joshua Hunt

−1 −1 × we define 훾푀 (푔 , ℎ ) to be the element of 푘 that corresponds to the automorphism

푔⊗휆ℎ 휆푔 휆푔ℎ 푔ℎ ⊗푅 푀푅 −−−−→푔 ⊗푄 푀푄 −−→푀푃 ←−−−푔ℎ ⊗푅 푀푅

2 T × of 푔ℎ⊗푅푀푅. That 훾푀 represents the same cohomology class as the image of 훼푀 in H ( S (퐺); 푘 ) follows from the commutativity of the diagram

res 퐺 푔 퐺 ModStMod푘퐺 (퐴푄) ModStMod푘퐺 (퐴푃 ) ∼ ∼

푔⊗푃 − StMod푘푄 StMod푘푃 for every 푔−1 ∶ 푃 → 푄 in TS (퐺). This can be checked using the description of the vertical maps 퐺 퐺 given in Construction 5.23 of [MNN17]. Here 퐴푃 denotes the algebra object coind푃 푘 (recall Theorem 2.5). Intuitively, there is a canonical choice of equivalence 푥 ⊗푃 푀푃 → 푀푃 for every 푥 ∈ 푃, given by the action of 푥 on 푀푃 , so the difference between the orbit category and the transport category is not relevant for our obstruction class.

The injectivity of the homomorphism H2(OS (퐺); 푘×) → H2(TS (퐺); 푘×) combined with The- orem 4.5 implies that we also have an exact sequence

훾 0 → 푇 (퐺, 푆) → 푇 (퐺) → lim 푇 (−) −→ H2(TS (퐺); 푘×). op OS (퐺)

We use this to show that we can always lift in the following special case: 5.8. Proposition. Suppose that 퐺 has a non-trivial normal 푝-subgroup. Any 퐺-stable endotrivial 푆-module lifts to a module in 푇 (퐺).

Proof. Let 푃 be a non-trivial normal 푝-subgroup of 퐺. We consider the following commutative diagram: 훾 op 2 T × 푇 (퐺) limOS (퐺) 푇 (−) H ( S (퐺); 푘 ) ≅ 훾 ′ 푇 (푍(푃)) H2(퐺; 푘×)

The right-hand map is restriction to the full subcategory on the object 푍(푃). It is an isomor- T S S phism because S (퐺) is homotopy equivalent to 푝(퐺)ℎ퐺 and 푝(퐺) is 퐺-equivariantly contractible to the object 푍(푃). The map 훾 ′ is defined similarly to 훾, but again restricted to automorphisms of the object 푍(푃). Since neither 훾 nor 훾 ′ depend on the choices of equivalences that appear in their respective constructions, the right-hand square commutes. By Dade’s Theorem 2.8, every module in 푇 (푍(푃)) is equivalent to Ω푛푘 for some 푛, and hence the diagonal map from 푇 (퐺) is surjective. Since the top composite is zero, this implies that 훾 ′ is also zero. The proposition now follows from exactness of the top row. □

5.9. Proposition. Let C be any collection of non-trivial 푝-subgroups such that C ↪ S푝(퐺) is a 퐺-homotopy equivalence. There is a homomorphism 훽 making the diagram Torsion-free endotrivial modules via homotopy theory 35

훼 lim 푇 (−) H2(OS (퐺); 푘×) op OS (퐺)

훽 1 C 1 × 2 T × H퐺( ; H (−; 푘 )) H ( S (퐺); 푘 ) commute. C Proof. Recall the form of the 퐸2-page of the isotropy spectral sequence for , as shown in 2 T × 1 C 1 × Diagram 5.6. To show that a class 휃 ∈ H ( S (퐺); 푘 ) lies in the subgroup H퐺( ; H (−; 푘 )), it 0 C 2 × is enough to show that it maps to zero in H퐺( ; H (−; 푘 )). Corollary 7.2 in [Gro02] shows that 0 C 2 × 2 × C H퐺( ; H (−; 푘 )) identifies with lim푠푆C (퐺) H (푁퐺(−); 푘 ), where 푠푆 (퐺) denotes the poset of 퐺-conjugacy classes of chains of subgroups of C with the order relation given by refinement, 2 × so it is enough to check that 휃 maps to the zero class in H (푁퐺(휎); 푘 ) for every 휎 ∈ 푠푆C (퐺). 2 × If 휎 = [푃0 < … < 푃푛] then the image of 휃 in H (푁퐺(휎); 푘 ) is given by restriction to 푁퐺(휎) ≤ AutTS (퐺)(푃0). 2 T × For the class 훾푀 that denotes the image of 훼푀 in H ( S (퐺); 푘 ), we can immediately reduce to the case where 퐺 = 푁퐺(휎) by considering the diagram 훾 op 2 T × limOS (퐺) 푇 (−) H ( S (퐺); 푘 )

훾 2 × limO op 푇 (−) H (TS (푁 (휎)); 푘 ) S (푁퐺 (휎)) 퐺 2 T × 2 × and observing that the restriction H ( S (퐺); 푘 ) → H (푁퐺(휎); 푘 ) factors through the right- hand map of this diagram. By Proposition 5.8, the bottom map of the square is zero, and hence 2 × so is the restriction of 훾푀 to H (푁퐺(휎); 푘 ). 2 O × We have now shown that the obstruction class 훼푀 ∈ H ( S (퐺); 푘 ) maps to zero in the 0 C 2 × zeroth Bredon cohomology group H퐺( ; H (−; 푘 )). Therefore, there is a homomorphism 훽 as claimed. □ 5.10. Corollary. There is an exact sequence 훽 0 → 푇 (퐺, 푆) → 푇 (퐺) → lim 푇 (−) −→ H1 (C ; H1(−; 푘×)) op 퐺 OS (퐺) for any collection C of non-trivial 푝-subgroups such that C ↪ S푝(퐺) is a 퐺-homotopy equivalence. We can give an explicit description of the obstruction in Bredon cohomology:

5.11. Proposition. Let C be a collection of non-trivial 푝-subgroups such that C ↪ S푝(퐺) is a 퐺-homotopy equivalence. The homomorphism 훽 ∶ lim 푇 (−) → H1 (C ; H1(−; 푘×)) op 퐺 OS (퐺) has the following explicit description. Let (푀푃 )퐺/푃∈OC (퐺) be a 퐺-stable endotrivial 푆-module. 푃 C Choose equivalences 휆푔 ∶ 푔 ⊗푃 푀푃 → 푀푃 in StMod푘푃 for every 푃 ∈ and 푔 ∈ 푁퐺(푃), such that

푃 푔⊗푃 휆ℎ 푔ℎ ⊗푃 푀푃 푔 ⊗푃 푀푃 (5.12) 푃 푃 휆푔ℎ 휆푔 푀푃 36 Tobias Barthel, Jesper Grodal, and Joshua Hunt commutes in the homotopy category of StMod푘푃 . Additionally, choose an equivalence

푄 휆푃<푄 ∶ res푃 푀푄 → 푀푃 in StMod푘푃 for every 푃 < 푄. We insist that both of these sets of choices be invariant under conjugation. For any 푔 ∈ 푁퐺(푃 < 푄) we obtain a potentially non-commutative diagram

푄 휆푔 푔 ⊗푃 푀푄 푀푄

푔⊗푃 휆푃<푄 휆푃<푄 푃 휆푔 푔 ⊗푃 푀푃 푀푃 1 C 1 × in the homotopy category of StMod푘푃 . We define a cocycle 훽푀 ∈ C퐺( ; H (−; 푘 )) by letting × 훽푀 (푃 < 푄)(푔) be the element in 푘 corresponding to the automorphism of 푀푃 given by the above diagram. The class 훽((푀푃 )) is represented by 훽푀 .

5.13. Notation. When the 퐺-stable endotrivial 푆-module (푀푃 ) is clear from context, we will drop the subscript on 훽푀 and use 훽 to refer to both the homomorphism and the cocycle.

5.14. Remark. The condition in Diagram 5.12 is necessary to ensure that 훽푀 (푃 < 푄) is actually a character. It can always be satisfied: this is exactly what we proved in Proposition 5.9 in 1 C 1 × order to show that 훾 lies in H퐺( ; H (−; 푘 )). 푃 The conjugation invariance condition that we imposed on the 휆푔 and 휆푃<푄 is necessary to ensure that 훽푀 is a Bredon 1-cochain, i.e. lies in the 퐺-fixed points in (5.2). It can always be satisfied by choosing such data for representatives of conjugacy classes of 0- and 1-simplices in C and extending by conjugation. 5.15. Remark. Note that in many cases we have to specify significantly less data in order to compute 훽 than we do to compute 훼: for the latter we need to choose

휆푔 ∶ res푔 푀푄 → 푀푃 for every 푔 ∶ 퐺/푃 → 퐺/푄 in OS (퐺), whereas for the former we need only choose such an equivalence when 푔 is an isomorphism in TS (퐺) or an inclusion 푃 ↪ 푄. Furthermore, if it C C C happens that /퐺 is isomorphic to a subposet 0 ⊆ then we only need to specify 휆푔 when C C 푔 is an automorphism of a subgroup in 0 or an inclusion of subgroups in 0, which again makes computations much simpler in practice. Examples where this occurs include the case 2 2 퐺 = PSL3(푝) with 푝 ≡ 1 mod 3 and the case 퐺 = Σ푛 with 푝 ≤ 푛 < 푝 + 푝; see Sections 10 and 12 respectively.

Proof of Proposition 5.11. This is a computation working through the construction of the isotropy 푃 spectral sequence. We first extend our choices of 휆푔 and 휆푃<푄 to the rest of the transport category, i.e. we make choices of equivalences

(5.16) 휆푔 ∶ 푔 ⊗푄 푀푄 → 푀푃

−1 T in StMod푘푃 for every remaining 푔 ∶ 푃 → 푄 in S (퐺). As in Remark 5.7, these choices determine a cocyle 훾 ∈ C2(TS (퐺); 푘×) that represents the cohomology class of the image of 훼 in H2(TS (퐺); 푘×). Torsion-free endotrivial modules via homotopy theory 37

We will now check that the class represented by 훾 in H2(TS (퐺); 푘×) restricts to the class of the cocycle 훽 defined in the statement of the proposition. Let 푛 푠 C × 퐹푟 ≔ ⨁ Hom퐺(C푡 (E퐺), C ( ; 푘 )) 푠+푡=푛 푠≥푟 • • be the filtration 퐹0 ⊇ 퐹1 ⊇ … that gives rise to the isotropy spectral sequence. The inclusion 1 C 1 × 2 T × H퐺( ; H (−; 푘 )) → H ( S (퐺); 푘 ) of Lemma 5.5 is induced by the composition 1 C 1 × 1 1 C × 2 • • 2 • 2 • ≅ 2 T × C퐺( ; H (−; 푘 )) ≅ H (퐺; C ( ; 푘 )) ≅ H (퐹1/퐹2) ← H (퐹1) → H (퐹0) ←−− H ( S (퐺); 푘 ), where a left-facing arrow means choosing a lift along that map (the image is independent of the choice of lift). The right-most isomorphism 2 T × ≅ 2 • 2 • • C × H ( S (퐺); 푘 ) −−→ H (퐹0) = H (Tot Hom퐺(C•(E퐺), C ( ; 푘 )) is given by pulling back along some choice of chain homotopy inverse to the Alexander– Whitney map, then applying the tensor-hom adjunction. Under one such choice of isomor- 2 • C × phism, 훾 corresponds to the element of Tot Hom퐺(C•(E퐺), C ( ; 푘 )) that sends 1 ↦ [푃 ≤ 푄 ≤ 푅 ↦ 훼(푃 ≤ 푄 ≤ 푅)] 푔 푔 훼(푃 ≤ 푄 −→ 푔푄) 1 −→푔 ↦ [푃 ≤ 푄 ↦ ] 푔 훼(푃 −→ 푔푃 ≤ 푔푄) 푔 ℎ 푔 ℎ (5.17) 1 −→푔 −→ℎ푔 ↦ [푃 ↦ 훼(푃 −→ 푔푃 −→ ℎ푔푃)]. 0 C × The component of 훾 labelled (5.17), living in Hom퐺(C2(E퐺), C ( ; 푘 )), is equivalent to zero 푃 because we chose the 휆푔 to make Diagram 5.12 commute. (More precisely, the subcategory of the transport category on conjugates of 푃 is a connected groupoid, and hence is equivalent to 푃 the full subcategory on the object 푃. We chose the 휆푔 compatibly on 푃, so when we extended these choices at the start of the proof, we can do so in such a way that 훾 is equal to zero.) 2 • Therefore we can lift 훾 to the class in H (퐹1) that sends 1 ↦ [푃 ≤ 푄 ≤ 푅 ↦ 훼(푃 ≤ 푄 ≤ 푅)] 푔 푔 훼(푃 ≤ 푄 −→ 푔푄) (5.18) 1 −→푔 ↦ [푃 ≤ 푄 ↦ ] . 푔 훼(푃 −→ 푔푃 ≤ 푔푄) 2 • 2 • • 1 1 C × Under the map H (퐹1) → H (퐹1/퐹2) ≅ H (퐺; C ( ; 푘 )), this class is sent to the homomorphism labelled (5.18). By comparing to the formula of 훼 given in Theorem 4.5, we see that this homomorphism represents a class that identifies with 훽 under the isomorphism 1 1 C × 1 C 1 × H (퐺; C ( ; 푘 )) ≅ C퐺( ; H (−; 푘 )), thereby completing the proof. □ 38 Tobias Barthel, Jesper Grodal, and Joshua Hunt

6. Compatible actions in the homotopy category

1 C 1 × The obstruction 훽 in the Bredon cohomology group H퐺( ; H (−; 푘 )) demonstrates a connection between lifting endotrivial modules and one-dimensional characters: the coefficients of the obstruction group take a subgroup 푃 to the group of one-dimensional characters of 푁퐺(푃). (The normaliser of 푃 appears here because this is the stabiliser in 퐺 of 푃, considered as a 0-simplex of the 퐺-space C .) In the next section, we will expand on this connection, giving a necessary and sufficient condition for a 퐺-stable endotrivial 푆-module to lift that is completely algebraic and makes no reference to obstruction groups; the condition involves only specifying one-dimensional characters of 푁퐺(푃) for varying 푃. The current section lays the groundwork for this by analysing the relationship between the definition of 훽 and one-dimensional characters. Throughout this section, we work in the stable module category (i.e. the homotopy category of the stable module ∞-category) unless stated otherwise. We start by giving a name to a concept that has already appeared several times in Section 5. It captures the part of the structure of an 푁퐺(푃)-module that can still be seen in StMod푘푃 . Closely related objects were considered in [Bal15, Theorem 9.9].

6.1. Definition. Let 푁 be a finite group with a normal 푝-subgroup 푃 and let 푀 ∈ StMod푘푃 . We say 푀 is 푁 -stable if 푔 ⊗푃 푀 ≃ 푀 for every 푔 ∈ 푁 .A compatible 푁 -action on an 푁 -stable module 푀 is a set of equivalences ∼ { 휆푔 ∶ 푔 ⊗푃 푀 −−→푀 | 푔 ∈ 푁 } such that

푔⊗푃 휆ℎ 푔ℎ ⊗푃 푀 푔 ⊗푃 푀 (6.2) 휆푔ℎ 휆푔 푀 ′ commutes in the homotopy category of StMod푘푃 for every 푔, ℎ ∈ 푁 . A map 푓 ∶ 푀 → 푀 in StMod푘푃 between modules with compatible 푁 -actions is 푁 -equivariant if it respects the 푁 -actions. 6.3. Remark. We have been using the term “퐺-stable endotrivial 푆-module” to mean an element op of limOS (퐺) 푇 (−), i.e. an element of 푇 (푆) that is invariant under restriction and conjugation by elements of 퐺. Since 푃 is normal in 푁 , if 푀 is an 푁 -stable 푃-module in the sense of Definition 6.1 O that is also endotrivial, then 푀 is an element of limO op 푇 (−), where S (푁 )≤푃 denotes the S (푁 )≤푃 full subcategory of OS (푁 ) on objects 퐺/푄 with 푄 ≤ 푃. 6.4. Remark. Let 푖∶ 푃 → 푁 be the inclusion exhibiting 푃 as a normal subgroup of 푁 and ∗ consider the induction/restriction monad 푖 푖! on StMod푘푃 . By normality, the underlying functor of this monad can be identified with 푘[푁 /푃]⊗푘 (−). The resulting algebra structure on 푘[푁 /푃] is then determined by the formula [푔] ⋅ [ℎ] = [푔ℎ] for cosets [푔], [ℎ] ∈ 푁 /푃. A compatible 푁 -action on 푀 ∈ StMod푘푃 is the same data as an action of 푘[푁 /푃] on 푀 in the homotopy category of StMod푘푃 . We could consequently construct an ∞-categorical version of ∗ the above definition, namely the ∞-category of algebras over the monad 푖 푖! in StMod푘푃 : Alg (푖∗푖 ) ≃ Mod (푘[푁 /푃]). StMod푘푃 ! StMod푘푃 However, for our purposes it is sufficient to work with the hands-on definition given above. Compatible actions are more rigid than they appear at first sight, and the next lemma shows that their behaviour is controlled on subgroups of 푃: Torsion-free endotrivial modules via homotopy theory 39

6.5. Lemma. Let 푁 be a finite group with a normal 푝-subgroup 푃, and let 푍 ≤ 푃 be a subgroup that is normal in 푁 . Let 푀 ∈ StMod푘푃 be an 푁 -stable endotrivial module. The group of one- dimensional characters H1(푁 ; 푘×) acts freely and transitively on the set of compatible 푁 -actions on 푀, and restriction induces an H1(푁 ; 푘×)-equivariant bijection ≅ 푃 { compatible 푁 -actions on 푀 } −−→{ compatible 푁 -actions on res푍 푀 }. ′ Here we consider two compatible actions 휆푔 and 휆푔 on 푀 to be the same if id푀 is 푁 -equivariant, ′ i.e. if 휆푔 is homotopic to 휆푔 for every 푔 ∈ 푁 .

1 × Proof. Let 휑 ∈ H (푁 ; 푘 ) act on the set of compatible 푁 -actions on 푀 by sending 휆푔 to 휑(푔)휆푔. By Lemma 2.7, this action is free and transitive. It is clear that a compatible 푁 -action on 푀 푃 gives rise to one on res푍 푀 by restriction, so we need to provide the inverse map. Suppose that 푃 푃 {휆푔 ∶ 푔 ⊗푍 res푍 푀 → res푍 푀} is a compatible 푁 -action. Since 푔 ⊗푃 푀 and 푀 are equivalent endotrivial 푃-modules, restriction induces an isomorphism of one-dimensional 푘-vector spaces 푃 ≅ 푃 푃 res푍 ∶ 휋0 Hom푃 (푔 ⊗푃 푀, 푀) −−→휋0 Hom푍 (푔 ⊗푍 res푍 푀, res푍 푀).

We define 휇푔 ∶ 푔 ⊗푃 푀 → 푀 to be the unique (up to homotopy) map that restricts to 휆푔. Since □ the 휆푔 make Diagram 6.2 commute, so too do the 휇푔. 6.6. Remark. Any compatible 푁 -action on an endotrivial 푃-module 푀 necessarily has the property that for every 푥 ∈ 푃, the map 휆푥 ∶ 푥 ⊗푃 푀 → 푀 is homotopic to the natural map given by the 푃-module structure of 푀. Since 푃 is a 푝-group, the group H1(푃; 푘×) is trivial. Lemma 6.5 therefore implies that all compatible actions of 푃 on 푀 are homotopic, and the 푃-module structure of 푀 is one such compatible action. 6.7. Remark. Let 푀 and 푀′ be endotrivial 푃-modules with compatible 푁 -actions. Since they are endotrivial, either all equivalences 푀 −−→푀∼ ′ as stable 푃-modules are 푁 -equivariant or none of them are. Similarly, suppose that 푀 is an endotrivial 푃-module with a compatible 푁 -action and that ′ ′ 푀 is a 푃-module equivalent to 푀 in StMod푘푃 . There is a unique compatible 푁 -action on 푀 induced by the existence of an equivalence 푀 ≃ 푀′, i.e. the induced 푁 -action does not depend on the choice of equivalence.

The Heller shift Ω on StMod푘푃 induces a shift functor Ω̃ on stable 푃-modules with compatible 푁 -actions: Ω푀̃ has Ω푀 as its underlying stable 푃-module and a compatible 푁 -action given by

Ω휆푔 푔 ⊗푃 Ω푀 ≃ Ω(푔 ⊗푃 푀) −−−→Ω푀. When 푃 is cyclic, the periodicity of Ω̃ may differ from that of Ω, since the 푁 -action is altered by the functor Ω̃ . To understand this difference in periodicity, we need to determine the action of 푁퐺(푃) on Tate cohomology. 6.8. Lemma. Let 푍 be a non-trivial cyclic 푝-group that is normal in a finite group 푁 . Let 푀 be a stable 푍-module with a compatible 푁 -action such that 푀 ≃ Ω푛푘 as stable 푍-modules. Let 푙 denote the periodicity of a projective resolution of 푘 as a 푍-module (i.e. 푙 = 1 if 푍 is cyclic of order two and 푙 = 2 otherwise). There is a natural isomorphism Ĥ 푙(푍; 푘) ⊗ Ĥ 푛(푍; 푀) −−→≅ Ĥ 푛+푙(푍; 푀) as one-dimensional characters of 푁 , which is induced by composition of morphisms in the stable module category of 푍. 40 Tobias Barthel, Jesper Grodal, and Joshua Hunt

Proof. Since 푀 ≃ Ω푛푘 as stable 푍-modules, both sides of the morphism induced by composition are one-dimensional 푘-vector spaces. The interesting part of the statement is that composition is 푁 -equivariant. We will work in the homotopy category of the stable module ∞-category of 푍, and temporarily introduce the notation [푀, 푁 ] for the morphism set 휋0 Hom푍 (푀, 푁 ) in that homotopy category. Recall that Ĥ 푛(푍; 푀) ≅ [Ω푛푘, 푀]. The action of 푔 ∈ 푁 on Ĥ 푛(푍; 푀), considered as a 푘-vector space, is given by 푔⊗ − 푛 푍 푛 푛 푛 푔 ⋅ −∶ [Ω 푘, 푀] −−−−−→[푔 ⊗푍 Ω 푘, 푔 ⊗푍 푀] → [Ω 푘, 푔 ⊗푍 푀] → [Ω 푘, 푀], 푛 푛 푛 where the middle map is induced by the the natural equivalence 푔 ⊗푍 Ω 푘 ≃ Ω (푔 ⊗푍 푘) ≃ Ω 푘 and the last map by composing with the action 푔 ⊗푍 푀 → 푀 of 푔 on 푀. It suffices to show that each of these maps respects composition Ω푛⊗id ∘ [Ω푙푘, 푘] ⊗ [Ω푛푘, 푀] −−−−−→[Ω푛+푙푘, Ω푛푘] ⊗ [Ω푛푘, 푀] −→[Ω푛+푙푘, 푀]. □ This is a straightforward check using the fact that 푔 ⊗푍 − commutes with Ω. 6.9. Lemma. Let 푍 be a non-trivial cyclic 푝-group that is normal in a finite group 푁 . When 푝 is odd, let 푌 be the unique subgroup of 푍 of order 푝 (which will also be normal in 푁 ), and let 휈 denote the twisting character × × 휈 ∶ 푁 → Aut(푌 ) ≅ 픽푝 ≤ 푘 given by the right action of 푁 on 푌 . We have Ĥ 2(푍; 푘) = 휈 as one-dimensional characters of 푁 . When 푝 = 2, let 푙 denote the periodicity of a projective resolution of 푘 as a 푍-module (i.e. 푙 = 1 if 푍 is cyclic of order two and 푙 = 2 otherwise). The one-dimensional character Ĥ 푙(푍; 푘) is trivial. Proof. We deal first with the case where 푝 is odd. Since 푘 ≃ Ω2푘, the Tate cohomology Ĥ 2(푍; 푘) is one-dimensional, and we wish to determine the 푁 -action on it. Let [푀, 푁 ] again denote ̂ 2 휋0 Hom푍 (푀, 푁 ). The action of 푔 ∈ 푁 on H (푍; 푘) is given by 푔⊗ − 2 푍 2 2 (6.10) 푔 ⋅ −∶ [Ω 푘, 푘] −−−−−→[푔 ⊗푍 Ω 푘, 푔 ⊗푍 푘] ≅ [Ω 푘, 푘]. We will choose a convenient model for Ω2푘 and compute this action explicitly. Let 푥 be a generator of 푍. We have strict pullback squares in the model category Mod푘푍 푘 푘푍 ⌟ 푥−1

0 Ω푘 ⌟ 푘푍

0 푘 that are also homotopy pullbacks, which witness that 푘 is a model for Ω2푘. These pullbacks are completely determined by the resolution of 푘 as a 푘푍-module 푥−1 (6.11) 푘 → 푘푍 −−−→푘푍 → 푘 that appears along the top and right edges of the above diagram. Torsion-free endotrivial modules via homotopy theory 41

With this identification, we can consider id∶ 푘 → 푘 as an element of Ĥ 2(푍; 푘) and compute where it is sent to by the action of 푔 given in (6.10). The crucial point here is that in the isomorphism [푔 ⊗푍 푘, 푔 ⊗푍 푘] ≅ [푘, 푘], the equivalence between 푔 ⊗푍 푘 and 푘 is different in the domain and codomain: in the domain, we are treating 푘 as a model for Ω2푘 and so need to take the data of the resolution (6.11) into account, whereas in the codomain the equivalence is induced by the (trivial) action of 푔 on 푘. Therefore, the action of 푔 on Ĥ 2(푍; 푘) is given by multiplication by the scalar 휆 that makes the following diagram commute in Mod푘푍 :

푘 푘푍 푥−1 푘푍 푘

휆 푓2 푓1 id

푘 푔 ⊗푍 푘푍 푔 ⊗푍 푘푍 푘 푔⊗푍 (푥−1) This amounts to lifting the identity to a comparison map between the two resolutions of 푘. 푔 We can choose 푓1 to send 1 to 푔 ⊗푍 1, i.e. send 푧 to 푔 ⊗푍 푧 . We need to choose 푓2 such that 푔 (푔 ⊗푍 (푥 − 1)) ⋅ 푓2(1) = 푓1(푥 − 1) = 푔 ⊗푍 (푥 − 1). Write |푍| = 푝푟 . Expanding with respect to the basis given by powers of (푥 − 1), we have a unique expression 푔 푖 (6.12) 푥 − 1 = ∑ 푎푖(푥 − 1) 1≤푖<푝푟 푖−1 for some scalars 푎푖 ∈ 푘. We can therefore choose 푓2(1) ≔ ∑1≤푖<푝푟 푎푖푔 ⊗푍 (푥 − 1) . The map 푘 → 푔 ⊗푍 푘푍 is given by sending 1 to 푔 ⊗푍 푁푍 , where 푁푍 denotes the norm element of 푘푍. By definition, 휆 satisfies

휆푔 ⊗푍 푁푍 = 푓2(푁푍 ) = 푎1푔 ⊗푍 푁푍 , so 휆 = 푎1. To determine 푎1, we observe that 푥푚 = ((푥 − 1) + 1)푚 ≡ 푚(푥 − 1) + 1 mod (푥 − 1)2 for any integer 푚, and hence for an appropriate choice of 푚 we have 푥푔 − 1 = 푥푚 − 1 ≡ 푚(푥 − 1) mod (푥 − 1)2.

Comparing with Equation (6.12), we see that 푎1, and hence also 휆, is equal to the image of 푚 in 푘 whenever 푚 satisfies 푥푚 = 푥푔. We learnt the above trick of expanding modulo (푥 − 1)2 from [MT07]. 푟−1 Since 푌 is generated by 푥푝 , if 푥푚 = 푥푔 then we have 푟−1 푟−1 푟−1 (푥푝 )휈(푔) ≔ (푥푝 )푔 = (푥푝 )푚 and so 휈(푔) is also equal to the image of 푚 in 푘. Therefore, 휆 = 휈(푔) and Ĥ 2(푍; 푘) = 휈 as one-dimensional characters of 푁 . When 푝 = 2 and 푙 = 2, one can carry out the same computation to get 휆 = 휈(푔). However, in this case the character 휈 is trivial: the image of 푚 in 푘 is always 1. For 푝 = 2 and 푙 = 1, we ̂ 1 □ have 푔 ⊗퐶2 푘퐶2 ≅ 푘퐶2, so the action of 푔 on H (푍; 푘) is again trivial. Combining these two lemmas, we obtain: 42 Tobias Barthel, Jesper Grodal, and Joshua Hunt

6.13. Corollary. Let 푍 be a non-trivial cyclic 푝-group that is normal in a finite group 푁 , and let 휈 be the twisting character defined in Lemma 6.9. Let 푀 be a stable 푍-module with a compatible 푁 -action such that 푀 ≃ Ω푛푘 as stable 푍-modules. When 푝 is odd we have Ĥ 푛+2(푍; 푀) ≅ 휈 ⊗ Ĥ 푛(푍; 푀) as one-dimensional 푁 -modules, and when 푝 = 2 we have Ĥ 푛+푙(푍; 푀) ≅ Ĥ 푛(푍; 푀), where 푙 is the periodicity of a projective resolution of 푘 as a 푍-module (i.e. 푙 = 1 if 푍 is cyclic of order two and 푙 = 2 otherwise).

6.14. Notation. Let 푍 be a normal subgroup of 푁 . For any 푀 ∈ StMod푘푁 , we get an induced 푁 compatible 푁 -action on the stable 푍-module res푍 푀. When we want to emphasise the fact 푁 푁 that we wish to consider res푍 푀 with its compatible 푁 -action, we will denote it by res̃푍 푀. We can now use Corollary 6.13 to show several useful facts about stable modules with compatible actions: 6.15. Proposition. Let 푍 be a non-trivial 푝-group that is normal in a group 푁 . In the case where 푍 is cyclic, let 휈 denote the twisting character defined in Lemma 6.9. 푁 푛 ̃ 푛 푁 (i) For any 푀 ∈ StMod푘푁 we have an equivalence res̃푍 (Ω 푀) ≃ Ω res̃푍 푀 as stable 푍- modules with compatible 푁 -actions. (ii) If 푍 is abelian, then every endotrivial 푍-module with a compatible 푁 -action is equivalent 푁 푛 1 × to res̃푍 (Ω 휑) for some 푛 ∈ ℤ and 휑 ∈ H (푁 ; 푘 ). (iii) Let 휑, 휓 ∈ H1(푁 ; 푘×) be one-dimensional characters. There is an equivalence 푁 푛 푁 푛 res̃푍 (Ω 휑) ≃ res̃푍 (Ω 휓) of stable 푍-modules with compatible 푁 -actions if and only if 휑 = 휓 as characters. 푁 2 푁 휈−1 (iv) If 푍 is cyclic and 푝 is odd, then res̃푍 (Ω 휑) ≃ res̃푍 ( 휑) as stable 푍-modules with compatible 푁 -actions. 푁 푁 (v) If 푍 is cyclic and 푝 = 2, then res̃푍 (Ω휑) ≃ res̃푍 휑 as stable 푍-modules with compatible 푁 -actions.

Proof. (i) It is enough to prove this for 푛 = ±1. We will prove it for 푛 = 1, since the 푛 = −1 case is just a dual argument. We need to show that, for any 푔 ∈ 푁 , the action of 푔 on the 푁 -module Ω푀 is given by

Ω(act푔 ) 푔 ⊗푁 Ω푀 ≃ Ω(푔 ⊗푁 푀) −−−−−−→Ω푀

up to homotopy. Let 푃푀 denote the projective cover of 푀. We have a commutative diagram in the 1-category Mod푘푁

0 Ω(푔 ⊗푁 푀) 푔 ⊗푁 푃푀 푔 ⊗푁 푀 0 ≅

0 푔 ⊗푁 Ω푀 푔 ⊗푁 푃푀 푔 ⊗푁 푀 0

act푔 act푔 act푔

0 Ω푀 푃푀 푀 0 Torsion-free endotrivial modules via homotopy theory 43

since the horizontal maps are 푁 -equivariant. This shows that the left-hand composite is homotopic to Ω(act푔), proving the claim. (ii) This is a consequence of Dade’s Theorem 2.8: the underlying stable 푍-module must be equivalent to Ω푛푘 for some 푘, so if we apply Ω̃ −푛 then we get a compatible action on 푘. These naturally biject with one-dimensional characters of 푁 . ̃ −푛 푁 (iii) We can apply Ω to reduce to the case where 푛 = 0. The underlying modules res푍 휑 푁 푁 푁 and res푍 휓 are both 푘, and res̃푍 휑 ≃ res̃푍 휓 if and only if the diagram 휑(푔) 푔 ⊗푁 휑 휑

푔⊗푁 푓 푓 휓(푔) 푔 ⊗푁 휓 휓 commutes up to homotopy for all 푔 ∈ 푁 , which in turn is if and only if 휑 = 휓 as characters. 푁 2 (iv) & (v) In the case where 푝 is odd, the module res푍 (Ω 휑) is equivalent to 푘, and hence 푁 2 푁 res̃푍 (Ω 휑) ≃ res̃푍 휓 for some one-dimensional character 휓. To determine 휓, we can apply zeroth Tate cohomology, and Corollary 6.13 tells us that Ĥ 0(푍; Ω2휑) ≅ Ĥ −2(푍; 휑) ≅ 휈−1 ⊗ 휑. A similar argument works when 푝 is even, in which case 휈 is the trivial character. □ 6.16. Remark. Despite Proposition 6.15(i), Ω and Ω̃ can have quite different behaviour. In particular, even if Ω̃ is periodic, it need not be the case that Ω be periodic. One example where this happens is when 푍 is the trivial group and 푁 is an elementary abelian subgroup of rank at least two: then Ω is not periodic but the category of stable 푍-modules with compatible 푁 -action is trivial, so Ω̃ is the identity. A less trivial example occurs when 푍 is a cyclic group and 푁 contains an elementary abelian subgroup of rank at least two: Ω is again not periodic, but the periodicity of Ω̃ is at most twice 1 × the order of 휈푍 in H (푁 ; 푘 ).

7. Lifting via orientations

In this section, we give a more explicit condition for lifting a 퐺-stable endotrivial 푆-module C A (푀푃 )퐺/푃∈OS (퐺). For a suitable collection , such as 푝(퐺), the module 푀푆 lifts to 푇 (퐺) if C and only if we can specify a one-dimensional character of 푁퐺(푃) for every 푃 ∈ such that the characters satisfy the compatibility condition stated in Proposition 7.5. We will refer to such a set of compatible characters as an orientation. This reduces the question of lifting 퐺- stable endotrivial modules to local group theory, since we just need to understand the one- dimensional characters of normalisers of 푝-subgroups and how they behave under restriction. Viewing the obstruction to lifting in terms of characters makes clear exactly “what cango wrong” when trying to lift: as explained in the introduction, when we attempt tolifta 퐺-stable endotrivial 푆-module to 퐺, we have to take into account the possibility that there is a 푝′-element 푥 that acts non-trivially on a cyclic subgroup 푍, in which case the order of Ω푘 in 푇 (푍 ⋊ ⟨푥⟩) is larger than its order in 푇 (푍). This puts extra compatibility conditions on type functions of modules in 푇 (퐺) that are not visible for modules in limOop 푇 (−). S 44 Tobias Barthel, Jesper Grodal, and Joshua Hunt

Recall that by Dade’s Theorem 2.8, for abelian 푝-groups, every endotrivial module is equivalent to a Heller shift of the unit 푘. With this in mind, we give a variation of a definition from [CMT14, §2]:

op 7.1. Definition. Let (푀푃 ) ∈ limOS (퐺) 푇 (−) be a 퐺-stable endotrivial 푆-module. A type func- C C tion for (푀푃 ) on a collection is a function 푛 ∶ → ℤ that is invariant under 퐺-conjugation and that satisfies res푃 푀 ≃ Ω푛(푃)푘 for all 퐺/푃 ∈ OC (퐺). 푍(푃) 푃

7.2. Remark. All type functions for a fixed 퐺-stable endotrivial 푆-module (푀푃 ) agree when restricted to the subset of C consisting of subgroups with non-cyclic centre. In other words, the only flexibility one has when choosing a type function 푛 for (푀푃 ) is to change its value on a subgroup 푃 with cyclic centre: if 푍(푃) ≅ 퐶2, then there are no constraints on 푛(푃), and otherwise 푛(푃) is determined modulo two. For the same reason, the restriction that 푛 be conjugation- invariant will automatically hold whenever 푍(푃) is non-cyclic. 7.3. Remark. The definition of the type of a module in[CMT14, §2], which we will briefly recall, is closely related to our definition of a type function on S푝(퐺). When the 푝-rank of 퐺 is one, the type of a module is undefined. When the 푝-rank is two, we choose representatives 퐸1, … , 퐸푟 for the 퐺-conjugacy classes of maximal elementary abelian subgroups of 퐺. The type 퐺 푛 of a 퐺-module 푀 is defined to be the 푟-tuple (푛1, … , 푛푟 ), where res 푀 ≃ Ω 푖 푘. When the 푝- 퐸푖 rank is greater than two, let 퐸2, … , 퐸푟 be representatives for the 퐺-conjugacy classes of rank two maximal elementary abelian subgroups of 퐺 and let 퐸1 be a maximal elementary abelian subgroup of rank greater than two. We define the type of 푀 to be (푛1, … , 푛푟 ) where the 푛푖 are defined as before. S A type function 푛 ∶ 푝(퐺) → ℤ for (푀푃 ) determines the type of 푀푆 by restricting to a subset of S푝(퐺) consisting of a maximal elementary abelian subgroup of rank at least three (if 퐺 has such a subgroup) together with a representative of each 퐺-conjugacy class of rank two maximal elementary abelian subgroups. Conversely, the restriction homomorphism

푇 (푆) → ∏ 푇 (퐸푖) 1≤푖≤푟 has kernel equal to the torsion subgroup of 푇 (푆), so specifying the type of a 퐺-stable endotrivial module determines it up to torsion. Therefore, in the absence of torsion, the type of a 퐺-stable endotrivial 푆-module determines its type function on S푝(퐺) up to the ambiguity mentioned in Remark 7.2. When there is torsion in 푇 (푆), a type function can contain more information than the type: when 푆 is generalised quaternion or cyclic of order at least three, there is no definition of the type of a module, yet type functions on S푝(퐺) can distinguish between Ω푘 and 푘. Similarly, when 푆 is the semi-dihedral 2-group with presentation ⟨ 푥, 푦 | 푥2푚 = 푦2 = 1, 푦푥푦 = 푥푚−1 ⟩, 1 the generator Ω 퐿 of the torsion part of 푇 (푆퐷4푚) that was constructed in [CT00, Theorem 7.1] has a non-trivial restriction to 푇 (⟨푦⟩) but has a trivial restriction to every elementary abelian 1 subgroup. Hence type functions on S푝(퐺) can distinguish between Ω 퐿 and 푘 despite these modules having the same type. 7.4. Definition. Let 푍 be a non-trivial normal 푝-subgroup of a finite group 푁 . In order to avoid the need to split into cases depending on whether or not 푍 is cyclic, we will make the following convention: the twisting character 휈 for the right action of 푁 on 푍 is defined to be Torsion-free endotrivial modules via homotopy theory 45

• the trivial character of 푁 , when 푍 is non-cyclic, and • the character defined in Lemma 6.9, when 푍 is cyclic.

7.5. Proposition. Let C be a collection of non-trivial 푝-subgroups such that C ↪ S푝(퐺) is a 퐺-homotopy equivalence and for every 푃 < 푄 in C , there is an abelian 푝-subgroup 푉 ≤ 푄 that is normalised by 푁퐺(푃 < 푄) and that contains both 푍(푃) and 푍(푄) as subgroups. S Let (푀푃 )퐺/푃∈OS (퐺) be a 퐺-stable endotrivial 푆-module, and let 푛 ∶ 푝(퐺) → ℤ be any type function for (푀푃 ). The module 푀푆 lifts to an endotrivial module in 푇 (퐺) if and only if it is possible 1 × C to specify one-dimensional characters { 휑푃 ∈ H (푁퐺(푃); 푘 ) ∶ 푃 ∈ } satisfying the following conditions for every 푃 < 푄 in C :

(i) If 푝 = 2, then 휑푃 = 휑푄 as one-dimensional characters of 푁퐺(푃 < 푄). (ii) If 푝 is odd, then (푛(푉 )−푛(푃))/2 (푛(푉 )−푛(푄))/2 휈푃 휑푃 = 휈푄 휑푄 as one-dimensional characters of 푁퐺(푃 < 푄), where 휈푃 is the twisting character for the right action of 푁퐺(푃) on 푍(푃), as in Definition 7.4, and 푉 is any abelian 푝-subgroup of 푄 that is normalised by 푁퐺(푃 < 푄) and that contains both 푍(푃) and 푍(푄) as subgroups. (iii) For every 푔 ∈ 퐺 we have 푔 ⊗ 휑 = 휑푔 . 푁퐺 (푃) 푃 푃 1 × C 7.6. Definition. We will say that a choice of characters { 휑푃 ∈ H (푁퐺(푃); 푘 ) ∶ 푃 ∈ } satisfying the conditions in Proposition 7.5 is an orientation for 푛. 7.7. Remark. When 푝 is odd, we necessarily have 푛(푉 ) ≡ 푛(푃) ≡ 푛(푄) mod 2, so the exponents appearing in condition (ii) are integers. Additionally, condition (ii) does not depend on the choice of 푉 : let 푊 be the intersection of all abelian 푝-subgroups 푉 ≤ 푄 that are normalised by 푁퐺(푃 < 푄) and that contain both 푍(푃) and 푍(푄). If 푊 is non-cyclic, then 푛(푉 ) is independent of 푉 . If 푊 is cyclic, then the twisting characters for the right actions of 푁 on 푊 , on 푍(푃), and on 푍(푄) all coincide.

7.8. Remark. We emphasise that the choice of type function 푛 ∶ S푝(퐺) → ℤ does not affect whether or not a module lifts: as in Remark 7.2, the type function is determined by (푀푃 ) away from 푝-subgroups with cyclic centres. Any change in the type function on a 푝-subgroup with a cyclic centre can be corrected for by multiplying the orientation characters by the appropriate power of a twisting character such that condition (ii) still holds. Proof of Proposition 7.5. Suppose that we had an endotrivial module 푀 ∈ 푇 (퐺) that restricted to S C 푀푆. We wish to construct an orientation for the type function 푛 ∶ 푝(퐺) → ℤ. For every 푃 ∈ 푛(푃) we get a one-dimensional 푁퐺(푃)-module Ĥ (푍(푃); 푀), corresponding to a one-dimensional character 휑푃 . We check that these satisfy the three conditions for being an orientation: (i) If 푝 = 2, then 푛(푃) ≡ 푛(푉 ) modulo the periodicity of a projective resolution of 푘 as a 푍(푃)-module, and Corollary 6.13 implies that Ĥ 푛(푃)(푍(푃); 푀) ≅ Ĥ 푛(푉 )(푍(푃); 푀) ≅ Ĥ 푛(푉 )(푉 ; 푀)

as 푁퐺(푃 < 푄)-modules; similarly for 푄 instead of 푃. (ii) If 푝 is odd, then 푛(푃) ≡ 푛(푉 ) mod 2, and Corollary 6.13 implies that (푛(푉 )−푛(푃))/2 ̂ 푛(푃) ̂ 푛(푉 ) 휈푃 ⊗ H (푍(푃); 푀) ≅ H (푉 ; 푀) as 푁퐺(푃 < 푄)-modules; similarly for 푄 instead of 푃. (iii) This follows since we require type functions to be constant on conjugacy classes. C Conversely, suppose that we had an orientation { 휑푃 ∶ 푃 ∈ } for (푀푃 ). We wish to show that we can lift 푀푆 to 푇 (퐺). By the description of the obstruction 훽 in Proposition 5.11, it 46 Tobias Barthel, Jesper Grodal, and Joshua Hunt

C is enough to construct a compatible 푁퐺(푃)-action on 푀푃 for every 푃 ∈ , invariant under 푄 conjugation in 퐺, such that some (and hence every) equivalence res푃 푀푄 → 푀푃 is 푁퐺(푃 < 푄)- equivariant. The orientation gives us a stable 푍(푃)-module with compatible 푁퐺(푃) action, namely 퐿 ≔ res̃푁 (푃)(Ω푛(푃)휑 ). 푃 푍(푃) 푃 Since res푃 푀 ≃ 퐿 as stable 푍(푃)-modules, we get an induced compatible 푁 (푃)-action on 푍(푃) 푃 푃 퐺 res푃 푀 , and hence also one on 푀 by Lemma 6.5. Recall from Remark 6.7 that the induced 푍(푃) 푃 푃 푁 (푃)-action on res푃 푀 is independent of the choice of equivalence as 푍(푃)-modules, be- 퐺 푍(푃) 푃 cause 푀푃 is endotrivial. Condition (iii) in the statement of the proposition implies that the resulting 푁퐺(푃)-actions are invariant under conjugation, as required. C Let 푃 < 푄 be subgroups in , and write 푁 ≔ 푁퐺(푃 < 푄). In order to complete the proof of the proposition, we check that the conditions imposed on 휑푃 and 휑푄 imply that 푀푃 ≃ 푀푄 as stable 푃-modules with a compatible 푁 -action. By Lemma 6.5 again, it is enough to check that they are equivalent as stable 푍(푃)-modules with a compatible 푁 -action. By assumption, there is a subgroup 푉 ≤ 푄 that is normalised by 푁 and for which there exists a zig-zag 푍(푃) ≤ 푉 ≥ 푍(푄). We will use this zig-zag to compare the 푁 -actions on 퐿푃 and 퐿푄 (and thereby the 푁 -actions on 푀푃 and 푀푄). Note that 푁 normalises all three subgroups of the zig-zag, so it does make sense to talk about compatible 푁 -actions for these subgroups. We deal first with the case where 푝 is odd. Using Lemma 6.5 applied to the 푉 -module Ω푛(푉 )푘, we see that it has a unique compatible 푁 -action that restricts to res̃푁 퐿 . We will denote 푍(푄) 푄 this stable 푉 -module with its compatible 푁 -action by 퐿푉 . The 푁 -action on 푀푄 agrees with the 푁 -action on 퐿푉 : both were defined such that they restrict to 퐿푄, and 푍(푄) ≤ 푉 ≤ 푄 by assumption. Therefore, the restriction of 푀 to a stable 푍(푃)-module is equivalent to res̃푁 퐿 . 푄 푍(푃) 푉 By Proposition 6.15, we must have 푁 푛(푉 ) 휈(푛(푉 )−푛(푄))/2 퐿푉 ≃ res̃푉 Ω ( 푄 ⋅ 휑푄) and on restriction to 푍(푃) this is equivalent to res̃푁 Ω푛(푃) (휈(푛(푃)−푛(푉 ))/2 ⋅ 휈(푛(푉 )−푛(푄))/2 ⋅ 휑 ) . 푍(푃) 푃 푄 푄

By condition (ii), this is in turn equivalent to 퐿푃 and we are done. Finally, we deal with the case where 푝 is even. This is identical to the previous case, except that the twisting character is always trivial and we do not necessarily have that 푛(푃) ≡ 푛(푉 ) mod 2. As before, we deduce that 푁 푛(푉 ) 퐿푉 ≃ res̃푉 Ω 휑푄 and consequently that the restriction of 푀푄 to a stable 푍(푃)-module with a compatible 푁 -action □ is equivalent to 퐿푃 . 7.9. Remark. Let 푀 ∈ 푇 (퐺) and let 푃 ∈ C . In the above proof of Proposition 7.5, we defined ̂ 푛(푃) 휑푃 ≔ H (푍(푃); 푀). We could equally well have defined 휑푃 to be the unique one-dimensional character such that 푁퐺 (푃) 퐺 푁퐺 (푃) 푛(푃) res̃ res 푀 ≃ res̃ (Ω 휑푃 ). 푍(푃) 푁퐺 (푃) 푍(푃) 7.10. Remark. One can prove an analogous proposition with a slightly different condition on C : instead of asking that 푍(푃) ≤ 푉 ≥ 푍(푄), we could instead ask that 푉 be non-trivial and 푍(푃) ≥ 푉 ≤ 푍(푄). The proof of the proposition goes through essentially unchanged. Torsion-free endotrivial modules via homotopy theory 47

7.11. Example. Suppose that C = A푝(퐺) and 푝 is odd. Since 푍(푃) = 푃 for every 푃 ∈ A푝(퐺), choosing 푉 ≔ 푄 shows that A푝(퐺) satisfies the conditions on C in Proposition 7.5. The com- C patibility conditions on { 휑푃 ∶ 푃 ∈ } reduce to: (1) if 푃 is not cyclic, then 휑푃 and 휑푄 agree on restriction to 푁퐺(푃 < 푄), (2) if 푃 is cyclic, then (푛(푃)−푛(푄))/2 휑푃 = 휈푃 ⋅ 휑푄 as 푁퐺(푃 < 푄)-characters, where 휈푃 is the twisting character induced by the right action of 푁퐺(푃) on 푃, and (3) for every 푔 ∈ 퐺, we have 푔 ⊗ 휑 = 휑푔 . 푁퐺 (푃) 푃 푃 In order to be more explicit, we have split condition (ii) into two conditions here: since 푛(푃) = 푛(푄) when 푃 is not cyclic, the two formulations are equivalent. 7.12. Corollary. When 푝 = 2, any 퐺-stable endotrivial 푆-module lifts to a module in 푇 (퐺). C A Proof. We apply Proposition 7.5 to = 푝(퐺), and observe that letting each 휑푃 be the trivial character gives an orientation for any type function. □

8. An algebraic description of the group of endotrivial modules

In this section, we use the obstruction 훽 of Section 5 to provide an algebraic description of the group of endotrivial modules of 퐺 along the lines of the viewpoint from [Gro18].

Recall that, by Remark 7.2, a 퐺-stable endotrivial 푆-module (푀푃 )퐺/푃∈OS (퐺) together with A a type function for (푀푃 ) on 푝(퐺) uniquely determines an extension of that type function S to 푝(퐺). We will implicitly identify (푀푃 ) with the module 푀푆 on the Sylow subgroup, from which we can recover the other modules by restriction.

8.1. Definition. We define an equivalence relation ∼ on the set of tuples (푀푆, 푛, {휑푉 }), where A A 푀푆 is a 퐺-stable endotrivial 푆-module, 푛 ∶ 푝(퐺) → ℤ is a type function for 푀푆 on 푝(퐺), A S and { 휑푉 ∶ 푉 ∈ 푝(퐺) } is an orientation for the unique extension of 푛 to 푝(퐺). We say that ′ ′ ′ ′ A (푀푆, 푛, {휑푉 }) ∼ (푀푆, 푛 , {휑푉 }) if 푀푆 ≃ 푀푆 and for every 푉 ∈ 푝(퐺) we have ′ 휑푉 if 푝 = 2 or 휑푉 = { ′ (푛(푉 )−푛′(푉 ))/2 휑푉 ⋅ 휈푉 if 푝 odd, where 휈푉 denotes the twisting character for the right action of 푁퐺(푉 ) on 푉 . (Recall from Definition 7.4 that 휈푉 is trivial by convention when 푉 is not cyclic.) We define 퐴(퐺) to be the set of tuples (푀푆, 푛, {휑푉 }) up to the equivalence relation ∼. Note that for a given equivalence class of ∼, the choice of type function determines the orientation; that is, there is a bijection between type functions of 푀푆 and elements of the equivalence class. 8.2. Remark. It is possible to remove the equivalence relation from the definition of 퐴(퐺) if one is willing to put a restriction on 푛, for example that 0 ≤ 푛(푍) < |푇 (푍)| for every cyclic 푍 ∈ A푝(퐺). This picks out a distinguished representative in each equivalence class. Alternatively, one could omit the type function altogether and instead provide an infinite sequence of characters, one for each possible value of 푛(푍). We can put an abelian group structure on 퐴(퐺) by defining ′ ′ ′ ′ ′ ′ (푀푆, 푛, {휑푉 }) ⋅ (푀푆, 푛 , {휑푉 }) ≔ (푀푆 ⊗ 푀푆, 푛 + 푛 , {휑푉 ⋅ 휑푉 }). ′ ′ This multiplication respects the equivalence relation, and {휑푉 ⋅ 휑푉 } is an orientation for 푛 + 푛 ′ because both {휑푉 } and {휑푉 } are orientations for their respective type functions. 48 Tobias Barthel, Jesper Grodal, and Joshua Hunt

8.3. Theorem. There is an isomorphism of groups Θ∶ 푇 (퐺) → 퐴(퐺) that sends an endotrivial 퐺-module 푀 to the equivalence class of the tuple 퐺 ̂ 푛(푉 ) (res푆 푀, 푛, 휑푉 ≔ H (푉 ; 푀)), where 푛 is any type function 푛 ∶ A푝(퐺) → ℤ for 푀. Proof. We compare the exact sequence in Corollary 5.10 to a similar sequence containing 퐴(퐺), which we will show to be exact below:

훽 1 1 × 0 푇 (퐺, 푆) 푇 (퐺) limOop 푇 (−) H (A푝(퐺); H (−; 푘 )) S 퐺 ≅ Θ

0 1 × 푓0 푓1 훽 1 1 × 0 H (A푝(퐺); H (−; 푘 )) 퐴(퐺) limOop 푇 (−) H (A푝(퐺); H (−; 푘 )). 퐺 S 퐺

The leftmost vertical map is the description of 푇 (퐺, 푆) given in [Gro18, §5]. The map 푓0 is the A inclusion map that sends a Bredon 0-cocycle { 휑푉 ∶ 푉 ∈ 푝(퐺) } to the tuple (푘, 푧, {휑푉 }), where A 푧 is the constant function on 푝(퐺) with value 0. The characters {휑푉 } define an orientation for 푛 because, for any 푉 < 푊 , the characters 휑푉 and 휑푊 are compatible upon restriction to 푁퐺(푉 < 푊 ), by definition of Bredon cohomology. The map 푓1 is the projection map onto the first factor, remembering only the 퐺-stable endotrivial 푆-module. The only square whose commutativity needs justification is the leftmost one. This amounts to checking that themap 0 A 1 × 푇 (퐺, 푆) → H퐺( 푝(퐺); H (−; 푘 )) is given by sending a Sylow-trivial module 푀 to the cocycle whose value on a subgroup 푉 is the 0 one-dimensional 푁퐺(푉 )-character given by Ĥ (푉 ; 푀). This is implicit in [Gro18], combining the description of the isomorphism in Theorem A with the proof of Proposition 5.3 (dualised to cohomology). We now justify the exactness of the lower sequence; the theorem will then follow from the 5-lemma. The injectivity of 푓0 is clear: the tuple (푘, 푧, {휑푉 }) represents the zero class only when all 휑푉 are trivial. Exactness at 퐴(퐺) is also straightforward: if 푓1(푀푆, 푛, {휑푉 }) ≃ 푘 then by changing the representative of the equivalence class in 퐴(퐺), we can take 푛 to be equal to 푧, and hence {휑푉 } forms a Bredon 0-cocycle. Finally, we consider exactness at the limit term. That 훽푓1 = 0 follows from Proposition 7.5: we asked that {휑푉 } be an orientation for 푛, and so 훽(푀푆) = 0. Conversely, if 훽(푀푆) = 0, we know that we can lift 푀푆 to some 푀 ∈ 푇 (퐺), by exactness of the top sequence. Applying Θ to 푀 gives an equivalence class in 퐴(퐺) that lifts □ 푀푆. 8.4. Remark. If 푇 (푆) does have torsion, then one might worry that it could be difficult in practice to verify whether a given module in 푇 (푆) is 퐺-stable, since the type does not detect torsion. Fortunately, in this case [MT07, Theorem 3.6] and [CMT13, Theorems 4.5 and 6.4] show that 푇 (퐺) → 푇 (푆) is surjective, and hence that all endotrivial 푆-modules are 퐺-stable.

In the case where 푝 is odd, we do not need to specify the module 푀푆 in 퐴(퐺), because this is recoverable from the type function 푛. In order to make this simplification, we need to be able to recognise which type functions are realisable by a 퐺-stable endotrivial 푆-module. This is the content of Proposition 8.5 below, which we state after recalling some notation from [CMT14, §3]. Let 푝 be odd. Without loss of generality, we can assume that the torsion-free rank of 푇 (푆) is at least two: otherwise, 푇 (푆) is cyclic (and so detected by 푛) or trivial. This implies that the 푝-rank of 퐺 is at least two. If the 푝-rank of 퐺 is two, then let 퐸1, … , 퐸푟 be representatives for the 푆-conjugacy classes of rank two maximal elementary abelian subgroups of 푆. If the 푝-rank Torsion-free endotrivial modules via homotopy theory 49 of 퐺 is three or greater, then let 퐸2, … , 퐸푟 be representatives for the 푆-conjugacy classes of rank two maximal elementary abelian subgroups of 푆, and let 퐸1 be a maximal elementary abelian subgroup of rank greater than two. As in [CMT14, Theorem 4.1], 푇 (푆) is generated by Ω푘 and 2푝 modules 푁2, … , 푁푟 , where each 푁푖 restricts trivially to 퐸푗 unless 푗 = 푖 and restricts to Ω 푘 on 퐸푖. A 8.5. Proposition. Let 푝 be odd, and consider the set of pairs (푛, { 휑푉 ∶ 푉 ∈ 푝(퐺) }) where A 1 × 푛 ∶ 푝(퐺) → ℤ, each 휑푉 ∈ H (푁퐺(푉 ); 푘 ), and 푛 and {휑푉 } satisfy the following conditions: (i) For every 푉 < 푊 in A푝(퐺) with 푉 non-cyclic, we have 푛(푉 ) = 푛(푊 ). (ii) For every 푉 < 푊 in A푝(퐺) with 푉 cyclic, we have 푛(푉 ) ≡ 푛(푊 ) mod 2. (iii) For every 푖 with 1 < 푖 ≤ 푟, we have 푛(퐸푖) ≡ 푛(퐸1) mod 2푝. (iv) 푛 is constant on 퐺-conjugacy classes. (v) For every 푉 < 푊 in A푝(퐺), we have

(푛(푊 )−푛(푉 ))/2 휑푊 = 휈푉 휑푉

as one-dimensional characters of 푁퐺(푉 < 푊 ), where 휈푉 denotes the twisting character for the right action of 푁퐺(푉 ) on 푉 as in Definition 7.4. ′ ′ We put an equivalence relation on such pairs by defining (푛, {휑푉 }) ∼ (푛 , {휑푉 }) if for every non- ′ cyclic 푉 ∈ A푝(퐺), we have 푛(푉 ) = 푛 (푉 ), and for every 푉 ∈ A푝(퐺) we have

′ (푛(푉 )−푛′(푉 ))/2 휑푉 = 휑푉 ⋅ 휈푉 . Let 퐵(퐺) denote the set of equivalence classes for ∼, and give 퐵(퐺) the abelian group structure ′ ′ ′ ′ (푛, {휑푉 }) ⋅ (푛 , {휑푉 }) ≔ (푛 + 푛 , {휑푉 ⋅ 휑푉 }).

There is an isomorphism of groups 퐴(퐺) → 퐵(퐺) induced by forgetting the 푀푆 factor.

Proof. From the description of 퐴(퐺) given in Theorem 8.3, it is enough to show that a function 푛 ∶ A푝(퐺) → ℤ is the type function of a 퐺-stable endotrivial 푆-module if and only if 푛 satisfies conditions (i)–(iv) in the statement of the proposition, in which case there is a unique 퐺-stable endotrivial 푆-module whose type function is 푛. We start with the uniqueness statement: for a 푝-group 푃, if 푃 is cyclic or 푇 (푃) has no torsion, then the restriction (8.6) 푇 (푃) → ∏ 푇 (푉 ) A 푉 ∈ 푝(푃) is injective. (In the cyclic case, we are using the assumption that 푝 is odd, and otherwise the injectivity is [Pui90, Theorem 2.2].) Therefore, for any type function 푛, there can be at most one 퐺-stable endotrivial 푆-module that realises it. Using [CMT14, Theorem 4.1], we see that the type function of any endotrivial 푆-module must satisfy conditions (i)–(iii) when 푊 ≤ 푆. If (푀푃 ) is 퐺-stable, then condition (iv) follows from the equivalences

푛(푔푉 ) 푛(푉 ) Ω 푘 ≃ 푀푔푉 ≃ 푔 ⊗푉 푀푉 ≃ 푔 ⊗푉 Ω 푘, and this implies conditions (i)–(iii) for all 푉 < 푊 in A푝(퐺). Conversely, given a function 푛 satisfying (i)–(iv), we define an endotrivial 푆-module

푛(퐸1) 푒2 푒푟 푀푆 ≔ Ω 푘 ⊗ 푁2 ⊗ … ⊗ 푁푟 , 50 Tobias Barthel, Jesper Grodal, and Joshua Hunt where 푒푖 ≔ (푛(퐸푖) − 푛(퐸1))/2푝. This has type function 푛, so we just need to check that 푀푆 is 퐺-stable. For every 푔 ∈ 퐺 and 푉 ≤ 푔푆 ∩ 푆, we have 푔푆 푛(푉 푔 ) 푛(푉 ) 푆 res푉 (푔 ⊗푆 푀푆) ≃ Ω 푘 ≃ Ω 푘 ≃ res푉 푀푆. Either the subgroup 푔푆 ∩ 푆 is cyclic or 푇 (푔푆 ∩ 푆) is torsion-free. In either case, the injectivity of 푔 □ (8.6) shows that 푔 ⊗푆 푀푆 and 푀푆 are equivalent as 푆 ∩ 푆 modules.

9. A local description of the group of endotrivial modules

We can provide another link between compatible actions and endotrivial modules by giving a description of 푇 (퐺) purely in terms of local information, i.e. in terms of 푁 -stable elements of 푇 (푃0), where 푁 is the normaliser in 퐺 of some chain of 푝-subgroups 푃0 < … < 푃푛. We first need to slightly expand the definition of Bredon cohomology that wasgivenin Section 5 to cover more general coefficient systems. 9.1. Definition. Let Δ푋 denote the category of simplices of a 퐺-space 푋, whose objects are simplices 휎 ∶ [푛] → 푋 and whose morphisms 휎 → 휎′ are commutative diagrams

[푛] 휎 푋. ′ [푛′] 휎

Let (Δ푋)퐺 denote the Grothendieck construction for the left action of 퐺 on Δ푋, whose objects are simplices 휎 ∈ Δ푋 and whose morphisms 휎 → 휏 are pairs (푔 ∈ 퐺, 푓 ∶ 푔휎 → 휏), where 푓 is a morphism in Δ푋.A (cohomological) 퐺-local coefficient system on 푋 is a functor 퐹 ∶ (Δ푋)퐺 → Ab. Given a 퐺-local coefficient system 퐹 on 푋, we define the non-equivariant cochains C푠(푋; 퐹) ≔ ∏ 퐹(휎) 휎∈푋푠 푖 푖 with differential given by 훿(푓 )(휎) ≔ ∑(−1) 퐹(1, 푑 )(푓 (푑푖휎)). We can define a 퐺-action on C푠(푋; 퐹) where 푔 acts by 퐹(푔, id푔휎 )∶ 퐹(휎) → 퐹(푔휎) on the component 퐹(휎). The 퐺-equivariant Bredon cohomology of 푋 with coefficients in 퐹 is 푠 푠 퐺 then defined to be the cohomology of the cochain complex퐺 C (푋; 퐹) ≔ C (푋; 퐹) . We have a functor C O op (Δ )퐺 → (퐺) , 휎 ↦ 퐺/퐺휎 where 퐺휎 denotes the stabiliser of the simplex 휎, and a morphism (푔, 푓 ∶ 푔휎 → 휏) is sent to the composite 푔 푓 퐺/퐺휎 ←−퐺/퐺푔휎 ←−퐺/퐺휏 . C In Section 5, we considered only 퐺-isotropy coefficient systems, where 퐹 ∶ (Δ )퐺 → Ab is O op C O op induced from a functor (퐺) → Ab by pulling back along the above functor (Δ )퐺 → (퐺) . Torsion-free endotrivial modules via homotopy theory 51

Given a functor 푠푆S → Ab, we can pull back along C (Δ )퐺 → 푠푆C 휎 ↦ [휎] to obtain a 퐺-local coefficient system. Bredon cohomology with 퐺-isotropy coefficients is an invariant of the 퐺-equivariant homotopy type of 푋, but this is no longer the case for a general 퐺-local coefficient system.

9.2. Lemma. Let 퐶 denote the functor 푠푆S (퐺) → Ab that takes a chain 푃0 < … < 푃푛 to the group of isomorphism classes of endotrivial 푃0-modules with a compatible 푁퐺(푃0 < … < 푃푛)-action. There is an isomorphism 푇 (퐺) −−→≅ lim 퐶(휎) 휎∈푠푆S (퐺) induced by the restriction maps res̃푁퐺 (휎) res퐺 ∶ 푇 (퐺) → 퐶(휎). 푃0 푁퐺 (휎) 푁 푁 Recall from Notation 6.14 that res̃푃 (푀) denotes the module res푃 (푀) with a compatible 푁 -action given by its forgotten 푁 -module structure.

Proof. We introduce two other functors from 푠푆S (퐺). We will use 휎 to refer to a chain of 1 × subgroups 푃0 < … < 푃푛. Firstly, let 퐻 ∶ 푠푆S (퐺) → Ab take 휎 to H (푁퐺(휎); 푘 ). Secondly, let st 푇 ∶ 푠푆S (퐺) → Ab take 휎 to the group of 푁퐺(휎)-stable elements of 푇 (푃0). We have a natural transformation 퐻 → 퐶 that takes a one-dimensional character 휑 to res̃푁 (휎)휑. We also have a natural transformation 퐶 → 푇 st that forgets the compatible action 푃0 and remembers only the underlying endotrivial 푃0-module. These functors assemble into a short exact sequence of functors 0 → 퐻 → 퐶 → 푇 st → 0. Indeed, such a sequence is exact if and only if it is exact pointwise. Proposition 6.15(iii) implies that the first map is injective. The second map is surjective because we can equipany 푁퐺(휎)-stable 푃0-module 푀 with a compatible action: by Lemma 6.5 it is enough to provide a compatible action on res푃0 푀, which is equivalent to res퐺 Ω푛푘 for some 푛. Finally, exact- 푍(푃0) 푍(푃0) ness at the middle term follows because the kernel of the second map is precisely the group of compatible 푁퐺(휎)-actions on 푘. We therefore get an exact sequence 0 → lim 퐻 → lim 퐶 → lim 푇 st → lim1 퐻, 푠푆S (퐺) 푠푆S (퐺) 푠푆S (퐺) 푠푆S (퐺) which we compare to the exact sequence from Corollary 5.10:

훽 op 1 S 1 × 0 푇 (퐺, 푆) 푇 (퐺) limOS (퐺) 푇 H퐺( 푝(퐺); H (−; 푘 )) ≅ ≅ ≅

st 1 0 lim푠푆S (퐺) 퐻 lim푠푆S (퐺) 퐶 lim푠푆S (퐺) 푇 lim푠푆S (퐺) 퐻.

The first vertical map is the isomorphism described by[Gro18, Theorem D]. It is induced by the 0 maps sending a Sylow-trivial module 푀 to the one-dimensional character given by Ĥ (푃0; 푀), considered as a one-dimensional character of 푁퐺(휎). The second vertical map is induced by 52 Tobias Barthel, Jesper Grodal, and Joshua Hunt

푁퐺 (휎) 퐺 the composition res̃ res . The third vertical map is induced by projection onto 푇 (푃0), 푃0 푁퐺 (휎) the image of which lands inside 푇 st(휎); its inverse is induced by

st op lim푠푆S (퐺) 푇 (−) limOS (퐺) 푇 (−)

푇 st([푃]) 푇 (푃). The fourth vertical map is the isomorphism described by [Gro02, Proposition 7.1], which spe- cialises to the statement that ∗ ∗ S (9.3) lim 퐹 ≅ H퐺( 푝(퐺); 퐹) 푠푆S (퐺) for any functor 퐹 ∶ 푠푆S (퐺) → Ab. The rightmost square is the only one whose commutativity is not straightforward to check. Equation (9.3) implies that it is enough to check that the description of 훽 given in Proposi- tion 5.11 agrees with the boundary map arising from the short exact sequence of cochain complexes • S • S • S st 0 → C퐺( 푝(퐺); 퐻) → C퐺( 푝(퐺); 퐶) → C퐺( 푝(퐺); 푇 ) → 0. We omit this check. The lemma then follows from the five lemma. □

The above lemma implies a similar statement for the functor 푇 (푁퐺(−)): 9.4. Theorem. There is an isomorphism ≅ 푇 (퐺) −−→ lim 푇 (푁퐺(휎)) 휎∈푠푆S (퐺) induced by restriction. Proof. We can factorise the map in Lemma 9.2 as ≅ 푇 (퐺) lim푠푆S (퐺) 퐶(휎)

res퐺 푓 푁퐺 (휎) lim푠푆S (퐺) 푇 (푁퐺(휎)) so it is sufficient to show that the map 푓 is injective. Suppose that (푀휎 ) ∈ lim푠푆S (퐺) 푇 (푁퐺(휎)) mapped to zero in lim푠푆S (퐺) 퐶(휎). We will start by showing that each 푀휎 is Sylow-trivial. Let 휎 ∈ 푠푆S (퐺), and let 푅 denote a Sylow subgroup of 푁퐺(휎). We claim that there is a zig-zag though simplices normalised by 푅 that starts at 휎 and ends S at a simplex in 푝(퐺) all of whose subgroups contain 푅. Let 휎 = (푃0 < … < 푃푛) and let 푖 be maximal such that 푅 ≰ 푃푖. Since 푅 ≤ 푁퐺(휎), the group 푅푃푖 is still a 푝-subgroup of 퐺 with 푅 ≤ 푁퐺(푅푃푖). We can now replace 푃푖 with 푅푃푖 by means of the following zig-zag in 푠푆S (퐺): ′ 휎 = (… < 푃푖−1 < 푃푖 < 푃푖+1푅 < …) (… < 푃푖−1 < 푃푖푅 ≤ 푃푖+1푅 < …) ≕ 휎

(… < 푃푖−1 < 푃푖 < 푃푖푅 ≤ 푃푖+1푅 < …). Note that 휎′ has strictly fewer vertices that do not contain 푅 (and is of a smaller dimension if 푃푖푅 = 푃푖+1푅). Since all of the simplices in the diagram are normalised by 푅, the modules 푀휎 and 푀휎 ′ agree after restriction to 푅. By repeating this procedure, we find a simplex 휏 = (푄0 < Torsion-free endotrivial modules via homotopy theory 53

… < 푄푚) with 푅 ≤ 푄0 such that 푀휎 and 푀휏 agree on restriction to 푅. By assumption, 푀휏 maps to zero in 퐶(휏) and hence restricts trivially to 푅. Therefore 푀휎 is Sylow-trivial, as claimed. Each 푁퐺(휎) has a non-trivial normal 푝-subgroup, so all of its Sylow-trivial modules are one- dimensional [MT07, Lemma 2.6]. Finally, by Proposition 6.15, if a one-dimensional character of 푇 (푁퐺(휎)) maps to zero in 퐶(휎), then it must have been the trivial character. Therefore, 푓 is injective as claimed. □

10. Endotrivial modules for PSL3(푝)

In this section, we apply our obstruction theory to the example of 퐺 = PSL3(푝), for 푝 ≡ 1 mod 3, in which the obstruction to lifting 퐺-stable endotrivial 푆-modules can be seen directly without reference to the cohomological obstruction classes. We will compute the obstruc- 1 S 1 × tion group H퐺( 푝(퐺); H (−; 푘 )) and use orientations to determine exactly which 퐺-stable endotrivial 푆-modules lift to 푇 (퐺). We first collect some group-theoretic facts about the groups SL3(푝) and PSL3(푝) that we will need below. Recall that 푂푝(퐻) denotes the 푝-core of 퐻, i.e. its largest normal 푝-subgroup, and that a 푝-radical subgroup of 퐺 is a 푝-subgroup 푃 ≤ 퐺 such that 푃 = 푂푝(푁퐺(푃)). We denote the collection of non-trivial 푝-radical subgroups of 퐺 by B푝(퐺).

10.1. Lemma. Let 퐺̃ ≔ SL3(푝) and 퐺 ≔ PSL3(푝), for any prime 푝. (i) The non-trivial 푝-radical subgroups of 퐺 and 퐺̃ are (up to conjugacy) the Sylow subgroup 푆, consisting of the unipotent upper-triangular matrices, along with the rank 2 elementary abelian subgroups 1 ∗ ∗ 1 0 ∗ 푉 ≔ ( 1 0) and 푊 ≔ ( 1 ∗) . 1 1

(ii) 푁퐺̃ (푉 ), 푁퐺̃ (푊 ) and 푁퐺̃ (푆) are respectively given by the subgroups consisting of matrices with determinant one that are of the form 푣 ∗ ∗ ∗ ∗ ∗ ∗ A ( ) , ( ∗ ) , and ( ∗ ∗ ) . A 푤 ∗ (iii) There are isomorphisms ≅ 푁퐺̃ (푉 )/푉 −−→ GL2(푝) 푣 0 0 ( ) A 푉 ↦ 퐴 and ≅ 푁퐺̃ (푊 )/푊 −−→ GL2(푝), 0 A ( 0 ) 푊 ↦ 퐴 푤 as well as an isomorphism ≅ 푁퐺̃ (푆)/푆 −−→ GL1(푝) × GL1(푝). 푣 0 0 ( (푣푤)−1 0) 푆 ↦ (푣, 푤) 푤 54 Tobias Barthel, Jesper Grodal, and Joshua Hunt

(iv) Suppose 푝 ≡ 1 mod 3, and let 휔 be a primitive third root of unity in 픽푝. The Weyl groups ′ 푁퐺(푉 )/푉 and 푁퐺(푊 )/푊 both fit into a central 푝 -extension

1 → SL2(푝) → ? → GL1(푝)/휔 → 1 induced by the isomorphisms of (iii). The quotient map is given by sending each of the specified coset representatives to det 퐴. Proof. (i) & (ii) Since we are working at the characteristic of 퐺̃, we can apply a corollary of the Borel– Tits theorem [GLS98, Corollary 3.1.5] to deduce that if 푃 is 푝-radical then 푁퐺̃ (푃) is a parabolic subgroup of 퐺̃. The parabolic subgroups are classified up to conjugacy by [GLS98, Theorem 1.13.2]: we have a maximal torus given by the diagonal matrices and a Borel subgroup 퐵 given by the upper-triangular matrices, with respect to which the standard parabolics are 퐺̃, 퐵, 푣 ∗ ∗ ∗ A 푃 ≔ ( ) and 푃 ≔ ( ∗ ) . 푉 A 푊 푤

The corresponding 푝-radical subgroups are 1 = 푂푝(퐺)̃ , 푆 = 푂푝(퐵), 푉 = 푂푝(푃푉 ), and 푊 = 푂푝(푃푊 ). This establishes the claim for 퐺̃. We now transfer the above results down to 퐺. The kernel of the map 휋 ∶ 퐺̃ → 퐺 is a central 푝′-group, so 휋−1(푃) has a unique Sylow 푝-subgroup (isomorphic to 푃) for any 푝-subgroup 푃 ≤ 퐺. This implies that 휋 induces a 퐺̃-equivariant isomorphism of posets S S 휋∗ ∶ 푝(퐺)̃ → 푝(퐺) and that 푂푝(휋(퐻))̃ = 휋(푂푝(퐻))̃ for any 퐻̃ ≤ 퐺̃. It follows that 휋∗ B B restricts to an isomorphism 휋∗ ∶ 푝(퐺)̃ → 푝(퐺), completing the proof. (iii) This is a straightforward calculation using (ii). (iv) This follows from (iii) and the observation that the preimages of SL2(푝) in 푁퐺̃ (푉 ) and □ 푁퐺̃ (푊 ) intersect ker 휋 trivially.

Using the above computations, we give a necessary condition for a PSL3(푝)-stable endotrivial 푆-module to lift to 푇 (PSL3(푝)). Afterwards, we use Proposition 7.5 to show that the condition is also sufficient.

10.2. Example. Let 푝 ≡ 1 mod 3 and 퐺 = PSL3(푝). Let 푉 , 푊 and 푆 be as in Lemma 10.1. Let 푍 ≔ 푍(푆) = 푉 ∩ 푊 , a cyclic subgroup of order 푝. We are interested in the image of the restriction map 푇 (퐺) → lim 푇 (−). op OS (퐺)

op 푚 First consider (푀푃 ) ∈ limOS (퐺) 푇 (−). By Dade’s Theorem 2.8, we have 푀푉 ≃ Ω 푘 and 푛 푀푊 ≃ Ω 푘 for some integers 푚 and 푛. Since 푀푉 and 푀푊 are equivalent after restriction to 푍 and 푇 (푍) ≅ ℤ/2, these integers satisfy 푚 ≡ 푛 mod 2. 퐺 푚 퐺 푛 Now instead consider 푀 ∈ 푇 (퐺). We similarly have res푉 푀 ≃ Ω 푘 and res푊 푀 ≃ Ω 푘 for some integers 푚 and 푛, but we will show that in this case 푚 ≡ 푛 mod 6. This difference arises ′ from the non-trivial action of 푝 -elements in 푁퐺(푍). Since 푝 ≡ 1 mod 3, there is a primitive third root of unity 휔 in 픽푝. Let 1 푥 ≔ ( 휔 ) 휔2 and note that 푥 normalises 푍, 푉 and 푊 . Torsion-free endotrivial modules via homotopy theory 55

퐺 푚 We claim that res푉 ⋊푥 푀 ≃ Ω 푘푉 ⋊푥 , where the notation 푉 ⋊푥 is shorthand for the semi-direct product 푉 ⋊ ⟨푥⟩. Indeed, we have a short exact sequence res 0 → 푇 (푉 ⋊ 푥, 푉 ) → 푇 (푉 ⋊ 푥) −−→푇 (푉 ) → 0 −푚 퐺 which is split by Ω푘푉 ↦ Ω푘푉 ⋊푥 . This implies that Ω res푉 ⋊푥 푀 ∈ 푇 (푉 ⋊ 푥, 푉 ). Since 푉 ⋊ 푥 has a normal 푝-subgroup, 푇 (푉 ⋊ 푥, 푉 ) is isomorphic to the group of one-dimensional characters H1(푉 ⋊ 푥; 푘×). Using the description of 푇 (푉 ⋊ 푥, 푉 ) found in [Gro18, Theorem A], we see −푚 퐺 Ω res푉 ⋊푥 푀 corresponds to the one-dimensional (푉 ⋊푥)-character given by Tate cohomology Ĥ 0(푉 ; Ω−푚푀) ≅ Ĥ 푚(푉 ; 푀). 푚 However, this (푉 ⋊ 푥)-character is the restriction of the 푁퐺(푉 )-character Ĥ (푉 ; 푀), and by Lemma 10.1(iv) the image of 푥 in 푁퐺(푉 )/푉 lies inside the subgroup SL2(푝). Since SL2(푝) is ̂ 푚 퐺 푚 perfect, the (푉 ⋊ 푥)-character H (푉 ; 푀) is necessarily trivial, and res푉 ⋊푥 푀 ≃ Ω 푘 as claimed. The analogous statement holds for 푊 and 푛. 푚 푛 We now consider the restriction of 푀 to 푍 ⋊ 푥. We must have Ω 푘푍⋊푥 ≃ Ω 푘푍⋊푥 , i.e. 푚 and 푛 must be congruent modulo the periodicity of a (푍 ⋊ 푥)-resolution of the trivial module. Identifying 푍 with the additive group of 픽푝, the action of 푥 on 푍 is given by multiplication by 휔2, so this periodicity is six and we have 푚 ≡ 푛 mod 6 as claimed.

op Conversely, we now show that if 푚 ≡ 푛 mod 6, then we can lift (푀푃 ) ∈ limOS (퐺) 푇 (−) to 푇 (퐺):

10.3. Theorem. Let 푝 ≡ 1 (mod 3) and 퐺 = PSL3(푝). We have an exact sequence 0 → 푇 (퐺) → lim 푇 (−) → ℤ/3 → 0. op OS (퐺)

Proof. Let 푉 , 푊 , 푆 and 휔 be as in Lemma 10.1. In Example 10.2, we saw that every type function 푛 ∶ S푝(퐺) → ℤ of a 퐺-module satisfies 푛(푉 ) ≡ 푛(푊 ) mod 6. We will show that this condition is also sufficient for lifting a 퐺-stable endotrivial 푆-module to 푇 (퐺). We apply Proposition 7.5 to B푝(퐺). We have already computed the 푝-radical subgroups of 퐺 up to 퐺-conjugacy. The 푝-radical subgroup that has a cyclic centre is the Sylow, whose centre is 1 0 ∗ ( 1 0) . 1

We have that 푁퐺(푆)/푆 ≅ (GL1(푝) × GL1(푝))/⟨(휔, 휔)⟩ with coset representatives 푣 0 0 ( (푣푤)−1 0) . 푤

If we identify 푍(푆) with 픽푝, then right conjugation by the above coset representative induces −1 1 × multiplication by 푣 푤. This determines the twisting character 휈푆 ∈ H (푁퐺(푆); 푘 ). Since 푁퐺(푉 < 푆) = 푁퐺(푆), to specify an orientation, it is sufficient to give characters 휑푉 ∈ 1 × 1 × H (푁퐺(푉 ); 푘 ) and 휑푊 ∈ H (푁퐺(푊 ); 푘 ) such that (푛(푉 )−푛(푊 ))/2 (10.4) 휑푉 = 휈푆 휑푊 1 × 1 × × in H (푁퐺(푆); 푘 ). We have H (푁퐺(푉 ); 푘 ) ≅ Hom(GL1(푝)/휔, 푘 ), and we let 휓푉 denote the image of 휑푉 under this isomorphism. Similarly, let 휓푊 denote the image of 휑푊 under the 56 Tobias Barthel, Jesper Grodal, and Joshua Hunt analogous isomorphism for 푁퐺(푊 ). When we restrict 휑푉 and 휑푊 to 푁퐺(푆), we get 푣 0 0 푣 0 0 −1 −1 −1 −1 휑푉 (( (푣푤) 0)) = 휓푉 (푣 ) and 휑푊 (( (푣푤) 0)) = 휓푊 (푤 ). 푤 푤 Write 푛(푉 ) − 푛(푊 ) = 6휆 for some 휆 ∈ ℤ. We choose 3휆 × 3휆 × 휓푉 (푥) ≔ 푥 ∈ 푘 and 휓푊 (푥) ≔ 푥 ∈ 푘 , noting that both characters send 휔 to 1 as required. We have 푣 0 0 3휆 −1 −1 3휆 −3휆 (휈푆 ⋅ 휑푊 ) (( (푣푤) 0)) = (푣 푤) ⋅ 푤 푤 = 푣−3휆 푣 0 0 −1 = 휑푉 (( (푣푤) 0)) , 푤 so Equation (10.4) is satisfied by these choices. We have now shown that the congruence condition 푛(푉 ) ≡ 푛(푊 ) mod 6 is both necessary and sufficient for a 퐺-stable endotrivial 푆-module to lift to 푇 (퐺), i.e. we have shown that the op op image of 푇 (퐺) has index three inside limOS (퐺) 푇 (−). Therefore the image of limOS (퐺) 푇 (−) 1 B 1 × inside the obstruction group H퐺( 푝(퐺); H (−; 푘 )) is of order three. It remains to show that the obstruction group itself is of order three and to compute 푇 (퐺, 푆), 0 B 1 × which is isomorphic to H퐺( 푝(퐺); H (−; 푘 )) by [Gro18, §5]. This is a direct computation: the • 1 × cochain complex C (B푝(퐺); H (−; 푘 )) is isomorphic to the complex

1 × H (푁퐺(푉 ); 푘 ) − res푁 (푉 ) 푁 (푆) ⊕

1 × 1 × H (푁퐺(푊 ); 푘 ) H (푁퐺(푆); 푘 ) res푁 (푊 ) 푁 (푆) ⊕ ⊕

1 × id 1 × H (푁퐺(푆); 푘 ) H (푁퐺(푆); 푘 )

0 B 1 × and so splits off an acyclic summand (the bottom row). To compute H퐺( 푝(퐺); H (−; 푘 )) and 1 B 1 × H퐺( 푝(퐺); H (−; 푘 )) we therefore wish to determine the kernel and cokernel of the map to 1 × the top copy of H (푁퐺(푆); 푘 ). We choose generators for 푁퐺(푆)/푆: 휁 0 0 휁 0 0 −2 −1 훾0 ≔ ( 휁 0) and 훾1 ≔ ( 휁 0) 휁 1

× 1 × where 휁 is a generator for 픽푝 . The cohomology group H (푁퐺(푆); 푘 ) is then generated by the ∗ 3 ∗ character 훾0 , defined to send 훾0 to 휁 and 훾1 to 1, and the character 훾1 , defined to send 훾0 to 1 and 훾1 to 휁 . Torsion-free endotrivial modules via homotopy theory 57

1 × By Lemma 10.1, the generator of H (푁퐺(푊 ); 푘 ) sends an element 0 A ( 0 ) 푤 of 푁 (푊 ) to (det 퐴)3. This implies that the image of res푁 (푊 ) is the subgroup generated by 훾 ∗. 퐺 푁 (푆) 0 1 × Similarly, the generator of H (푁퐺(푉 ); 푘 ) sends an element 푣 0 0 ( ) A of 푁 (푉 ) to (det 퐴)3, so the image of res푁 (푉 ) is the subgroup generated by −훾 ∗ − 3훾 ∗. 퐺 푁 (푆) 0 1 0 B 1 × 1 B 1 × ∗ Therefore, H ( 푝(퐺); H (−; 푘 )) = 0 and H ( 푝(퐺); H (−; 푘 )) ≅ ℤ/3, generated by 훾1 . 퐺 퐺 □

10.5. Remark. If 푝 ≢ 1 (mod 3), then PSL3(푝) = SL3(푝) and an analogous calculation shows 1 B 1 × that H퐺( 푝(퐺); H (−; 푘 )) = 0, so the obstruction vanishes in this case. Similarly, if 푛 ≤ 2, then PSL푛(푝) has a cyclic Sylow subgroup, while if 푛 ≥ 4, then PSL푛(푝) has no rank two maximal elementary abelian subgroups. In either case, we can see directly that the restriction 푇 (퐺) → 푇 (푆) is surjective. Therefore, the example of PSL3(푝) with 푝 ≡ 1 (mod 3) is the only case where we get interesting behaviour.

11. Torsion-free endotrivial modules

Let TF (퐺) denote the torsion-free quotient of 푇 (퐺), i.e. the quotient by its torsion subgroup TT (퐺). When 푝 is odd and our obstructions vanish, we obtain an expression for the torsion-free endotrivial modules of 퐺 as a limit over the orbit category: 11.1. Proposition. Let 푝 be odd and suppose that there is a short exact sequence 0 → 푇 (퐺, 푆) → 푇 (퐺) → lim 푇 (−) → 0, op OS (퐺) as is for example the case when 퐺 has a non-trivial normal 푝-subgroup. Restriction induces an isomorphism TF (퐺) −−→≅ lim TF (−). op OS (퐺) Proof. Since 푇 (퐺, 푆) is finite [CMT14, Proposition 2.3], the short exact sequence given in the statement of the proposition implies that TF (퐺) is isomorphic to the torsion-free quotient of op op O limOS (퐺) 푇 (−). We have a short exact sequence of functors from S (퐺) 0 → TT → 푇 → TF → 0 that induces an exact sequence 푓 (11.2) 0 → lim TT (−) → lim 푇 (−) −→ lim TF (−) → lim1 TT (−). op op op op OS (퐺) OS (퐺) OS (퐺) OS (퐺) Since the first term of (11.2) is torsion and the third term is torsion-free, the image of the map op 푓 is isomorphic to the torsion-free quotient of limOS (퐺) 푇 (−), which by the above argument is isomorphic to TF (퐺). It therefore remains to show that 푓 is surjective. 58 Tobias Barthel, Jesper Grodal, and Joshua Hunt

We have a commutative diagram

op limOS (퐺) 푇 (−) 푇 (푆)

푓

op limOS (퐺) TF (−) TF (푆).

Since the right-hand vertical map is surjective, let 푀푆 be a lift to 푇 (푆) of a 퐺-stable element in TF (푆). It suffices to check that 푀푆 is still 퐺-stable. Let 푔 ∈ 퐺. Since the map 푇 (푃) → ∏ 푇 (푉 ) A 푉 ∈ 푝(푃) is injective when 푃 is cyclic (and 푝 is odd) or when 푇 (푃) is torsion free, it is enough to check A 푔 that 푔 ⊗푆 푀푆 and 푀푆 restrict to the same element of 푇 (푉 ) for every 푉 ∈ 푝( 푆 ∩ 푆). If 푉 is cyclic, then this follows because A푝(퐺)/퐺 is contractible, and consequently either the restriction of 푀푆 to every cyclic 푝-subgroup is equivalent to Ω푘 or its restriction to every cyclic 푝-subgroup is equivalent to 푘. Otherwise, TF (푉 ) = 푇 (푉 ) and so 푔푆 푆 res푉 (푔 ⊗푆 푀푆) ≃ res푉 푀푆, □ because 푀푆 is a 퐺-stable element of TF (푆). In [CMT14], Carlson–Mazza–Thévenaz asked several questions about the behaviour of the torsion-free quotient of 푇 (퐺). We are able to answer these questions at certain primes using Theorem 10.3, which computed the obstruction group for PSL3(푝). A homomorphism 휙 ∶ 퐺 → 퐺′ is said to preserve 푝-fusion if and only if it restricts to an isomorphism between some Sylow 푝-subgroups 푆 ≤ 퐺 and 푆′ ≤ 퐺′ and induces an equivalence FS (퐺) → FS (퐺′) of fusion categories. In [CMT14, Conjecture 10.1], Carlson–Mazza–Thévenaz asked whether 휙 controlling fusion implies that 휙∗ ∶ TF (퐺′) → TF (퐺) is an isomorphism. The following theorem shows that this is true for 푝 = 2 but not in general: 11.3. Theorem. Suppose that 휙 ∶ 퐺 → 퐺′ is a group homomorphism that controls 푝-fusion. The induced homomorphism 휙∗ ∶ TF (퐺′) → TF (퐺) is an isomorphism if 푝 = 2. If 푝 is odd, this is not necessarily the case; for example, when 푝 ≡ 1 mod 3 the quotient map 휙 ∶ SL3(푝) → ∗ PSL3(푝) preserves fusion but the induced map 휙 ∶ TF (PSL3(푝)) ↪ TF (SL3(푝)) is the inclusion of a subgroup of index three. Proof. When 푝 = 2, we have a short exact sequence 0 → 푇 (퐺, 푆) → 푇 (퐺) → lim 푇 (−) → 0 op OS (퐺) by Corollary 7.12. Since 푇 (퐺, 푆) is finite [CMT14, Proposition 2.3], we have that TF (퐺) is iso- op O morphic to the torsion-free part of limOS (퐺) 푇 (−). There is a zig-zag of functors S (퐺) ← T F op op S (퐺) → S (퐺), which induces an isomorphism limOS (퐺) 푇 (−) ≅ limFS (퐺) 푇 (−). Since 휙 preserves 푝-fusion, it induces an equivalence on fusion categories, so we deduce that lim 푇 (−) ≅ lim 푇 (−). op op OS (퐺′) OS (퐺) Therefore 휙∗ ∶ TF (퐺′) → TF (퐺) is an isomorphism, completing the proof of the case where 푝 = 2. Torsion-free endotrivial modules via homotopy theory 59

Now let 퐺̃ = SL3(푝) and 퐺 = PSL3(푝) for 푝 ≡ 1 mod 3. One can directly calculate that the obstruction group H1 (B (퐺);̃ H1(−; 푘×)) vanishes and that 푇 (퐺,̃ 푆) = 0 by using the method 퐺̃ 푝 that was used in Theorem 10.3 to carry out the analogous computation for 퐺; the normalisers that appear were computed in Lemma 10.1. We obtain a diagram with exact rows:

op 0 푇 (퐺) limOS (퐺) 푇 (−) ℤ/3 0 ∗ 휙 ≅ ≅ ̃ op 0 푇 (퐺) limOS (퐺)̃ 푇 (−) 0

The middle vertical map is an isomorphism because 휙 preserves 푝-fusion. We see that Im(휙∗) is a subgroup of index three. □

In [CMT14, Conjecture 9.2], Carlson–Mazza–Thévenaz asked whether there is a torsion- free subgroup 퐹 ≤ 푇 (퐺) such that 퐹 consists of modules lying in the principal block of 퐺 and 푇 (퐺) = TT (퐺) ⊕ 퐹. The following theorem shows that this is not true when 퐺 = SL3(푝) and 푝 ≡ 1 (mod 3):

11.4. Theorem. When 푝 ≡ 1 mod 3, the torsion subgroup TT (SL3(푝)) is zero, but the subgroup of 푇 (퐺) consisting of modules lying in the principal block has index at least three.

Proof. We again let 퐺̃ = SL3(푝), 퐺 = PSL3(푝), and 휙 be the quotient map SL3(푝) → PSL3(푝). ′ For any finite group 퐻, the 푝 -core 푂푝′ (퐻) acts trivially on 푘퐻-modules lying in the principal block [HB82, Lemma 13.1], and here 푂푝′ (퐺)̃ = 푍(퐺)̃ . The kernel of 휙 is also equal to 푍(퐺)̃ , being the group generated by 휔 ( 휔 ) 휔 for 휔 a primitive third root of unity in 픽푝. Therefore, if 푂푝′ (퐺)̃ acts trivially on a 퐺̃-module, then that module is inflated from 퐺. Since not every module in 푇 (퐺)̃ is inflated from 퐺, there must be endotrivial 퐺̃-modules that do not lie in the principal block. □

11.5. Remark. Theorem 4.5 shows that a weaker form of the conjecture relating to fusion systems does hold. If instead we ask that 휙 induce an equivalence on orbit categories OS (퐺) → OS (퐺′), then the five lemma applied to the exact sequence

0 → H1(OS (퐺); 푘×) → 푇 (퐺) → lim 푇 (−) → H2(OS (퐺); 푘×) op OS (퐺) shows that 휙 induces an isomorphism on 푇 (퐺). 11.6. Remark. The fact that all 퐺-stable endotrivial 푆-modules lift to 푇 (퐺) when 푝 = 2 does not seem to say anything positive about the conjecture relating to principal blocks. Since 푝-groups only have one block, the information about the block of an indecomposable 푘퐺-module is some- op how contained in the coherence data of the corresponding object of holimOS (퐺) StMod푘푃 , not its restrictions to StMod푘푃 . 60 Tobias Barthel, Jesper Grodal, and Joshua Hunt

12. Endotrivial modules for the alternating and symmetric groups

Here we use orientations in order to compute the image of 푇 (퐺) in 푇 (푆) where 퐺 is either 2 2 the symmetric group Σ푛 or the alternating group 퐴푛, the degree 푛 satisfies 푝 ≤ 푛 < 푝 + 푝, and 푝 is odd. Note that if 푛 < 푝2, then TF (퐺) = 0, while when 푝2 + 푝 ≤ 푛, we have TF (퐺) ≅ ℤ generated by Ω푘. The case 푝2 ≤ 푛 < 푝2 + 푝 is considered in [CHM10], where they calculate that TF (퐺) ≅ ℤ2 and provide bounds on the type of the generators. We provide precise values for the type of the generators in Theorem 12.1. The software package [GAP] was invaluable for working out the details of the computations below. 2 2 When 푝 ≤ 푛 < 푝 + 푝 and 푝 is odd, both Σ푛 and 퐴푛 have two maximal elementary abelian subgroups (up to conjugacy), which we denote 퐸1 and 퐸2. We describe these subgroups after the statement of the theorem.

12.1. Theorem. Let 퐺 denote either the symmetric group Σ푛 or the alternating group 퐴푛, where 푝2 ≤ 푛 < 푝2 + 푝 and 푝 is odd. We have TF (퐺) ≅ ℤ2, generated by [Ω푘] and [푀] for some module 푀 satisfying res퐺 푀 ≃ 푘 and res퐺 푀 ≃ Ω2푝(푝−1)푘. 퐸1 퐸2 Theorem 6.1 in [CHM10] proves that

res퐺 푀 ≃ 푘 and res퐺 푀 ≃ Ω2푝푟 푘 퐸1 퐸2 for some 푟 dividing 푝 − 1. Our contribution is to show that 푟 = 푝 − 1. All of the work in the proof of the theorem is for the case 퐺 = 퐴푛, which we deal with in Proposition 12.10; the case 퐺 = Σ푛 follows as a corollary. To prove the proposition, we require some understanding of the local structure of Σ푛 and 퐴푛, which we discuss first (following the description in[CHM10, §6]). 12.2. Assumption. Without further comment, we assume throughout the remainder of this section that 푝2 ≤ 푛 < 푝2 + 푝 and that 푝 is odd.

Define elements in Σ푛 for 푖 = 1, … , 푝 by 2 2 푥푖 = (푖푝 − 푝 + 1, … , 푖푝) and 푦 = (1, 푝 + 1, … , 푝 − 푝 + 1) … (푝, 2푝, … , 푝 ), where we consider Σ푛 as acting on {1, … , 푛} on the right. These 푝 + 1 elements generate the Sylow subgroup 푆 ≅ 퐶푝 ≀ 퐶푝 of both Σ푛 and 퐴푛. Let 퐸1 be the base subgroup of the wreath product, i.e. 퐸1 ≔ ⟨푥1, … , 푥푝⟩, and let 퐸2 ≔ ⟨푥1 … 푥푝, 푦⟩. The subgroups 퐸1 and 퐸2 are the maximal elementary abelian subgroups of 푆 up to conjugacy, and 퐸1 is clearly a maximal elementary abelian subgroup of 퐴푛 (and hence Σ푛). 퐸2 cannot be subconjugate to 퐸1, since all non-identity elements of 퐸2 are a product of 푝 푝-cycles and 퐸1 has a subgroup 퐸1 ∩ Σ푝2−푝 that is of index 푝 and has no such elements. Therefore 퐸2 is also a maximal elementary abelian subgroup of 퐴푛 (and hence Σ푛). th We will also use the following notation: there is a subgroup Σ푝 ≀ Σ푝 ≤ Σ푛, where the 푖 base factor is viewed as acting on the set {푖푝 − 푝 + 1, … , 푖푝}. We let 훽푖 ∶ Σ푝 → Σ푝 ≀ Σ푝 denote the th inclusion of the 푖 base factor and 휏 ∶ Σ푝 → Σ푝 ≀ Σ푝 denote the inclusion of the twisting factor. With this notation, 푥푖 = 훽푖((1, … , 푝)) and 푦 = 휏((1, … , 푝)). Note that sgn 휏(휎) = sgn 훽(휎) = sgn 휎. 12.3. Example. In the smallest possible case, we have 푝 = 3 and 푛 = 9. The elements defined above are 푥1 = (123), 푥2 = (456), 푥3 = (789), and 푦 = (147)(258)(369). The action of 푦 cyclically permutes the 푥푖. Torsion-free endotrivial modules via homotopy theory 61

We will work with the collection B푝(퐺) of 푝-radical subgroups, so start by determining this collection up to conjugacy:

12.4. Lemma. Up to 퐺-conjugacy, the 푝-radical subgroups of 퐺 = Σ푛 and 퐺 = 퐴푛 are: (i) the Sylow subgroup 푆, (ii) the rank two elementary abelian subgroup 퐸2, and (iii) the rank 푖 elementary abelian subgroup 푅푠 ≔ ⟨푥1, … , 푥푠⟩ for 1 ≤ 푖 ≤ 푝, except when 푝 = 3 and 푛 = 9, in which case 푅2 is not 푝-radical.

12.5. Remark. Note that 푅푝 = 퐸1.

Proof of Lemma 12.4. We use the classification of 푝-radical subgroups of Σ푛 found in [AF90, (2A)], which we now summarise: for 푐 ≥ 1, let 퐵푐 denote the (unique up to conjugacy) transitive 푐 subgroup of Σ푝푐 that is isomorphic to the elementary abelian group of order 푝 . This is given by the permutation representation arising from the right regular action of the elementary abelian group on itself. Note that 퐵1 is conjugate to ⟨푥1⟩ and 퐵2 is conjugate to 퐸2.

For a tuple of positive integers (푐1, … , 푐푡 ), let 퐵(푐1,…,푐푡 ) denote the iterated wreath product 푑 퐵푐1 ≀…≀퐵푐푡 . This group embeds uniquely up to conjugacy as a transitive subgroup of Σ푝 , where 푑 푑 푑 = 푐1 + … + 푐푡 . The subgroup 퐵(푐1,…,푐푡 ) ≤ Σ푝 is said to be a basic subgroup, whose degree is 푝 and whose length is 푡. Any 푝-radical subgroup 푅 of Sym(푉 ) splits as

(12.6) 푅 ≅ 푇0 × … × 푇푠 corresponding to a partition 푉 = 푉0 ⨿ … ⨿ 푉푠. 푇0 is the trivial subgroup of Sym(푉0), and each 푇푖 for 푖 > 0 is a basic subgroup of Sym(푉푖). 푝 푝 푝2+푝+1 푝+1 We note that 퐵(1,1,1) has order (푝 ⋅푝) ⋅푝 = 푝 , which is strictly greater than |푆| = 푝 , so any basic subgroup in a decomposition of a 푝-radical subgroup of Σ푛 has length at most 2. The basic subgroups we need to consider are therefore 2 (i) 퐵(1,1) ≅ 퐶푝 ≀ 퐶푝 ≅ 푆 of degree 푝 , (ii) 퐵(1) ≅ 퐶푝 ≅ ⟨푥1⟩ of degree 푝, and 2 (iii) 퐵(2) ≅ 퐶푝2 ≅ 퐸2 of degree 푝 . 2 Σ푛 itself has degree strictly less than 푝 + 푝, so the possible candidates for 푝-radical subgroups of Σ푛 are 퐵(1,1), 퐵(2), and a direct product of copies of 퐵(1). It is not hard to deduce directly that the 푝-radical subgroups of Σ푛 are as claimed in the statement of the lemma; however, any 푝-radical 푃 in 퐴푛 is a 푝-radical in Σ푛, since

푃 ≤ 퐴푛 ∩ 푂푝(푁Σ푛 (푃)) ≤ 푂푝(푁퐴푛 (푃)) = 푃 and a 푝-group has no subgroups of index two. Therefore, it is sufficient to check that each of our candidates is radical in 퐴푛 and that none of their Σ푛-conjugacy classes splits in two in 퐴푛. We first show the statement about conjugacy classes. For the Sylow subgroup there isnoth- ing to check. By the orbit-stabiliser theorem it is enough to show that the stabiliser in Σ푛 of each other 푝-radical subgroup 푃 is strictly larger than its stabiliser in 퐴푛. That is, we seek an element of Σ푛 that normalises 푃 and does not lie in 퐴푛. Let 휉 ∈ 푁Σ푝 (퐶푝) represent a generator of the quotient 푁Σ푝 (퐶푝)/퐶푝. Note that 휉 is an odd permutation, since the conjugacy class 퐶푝 ⧵ {1} splits into two conjugacy classes in 퐴푝. The element 훽1(휉) … 훽푝(휉) normalises ⟨푦⟩, ⟨푥1 … 푥푝⟩, and each ⟨푥푖⟩. Moreover, it does not lie in 퐴푛. Therefore, the Σ푛-conjugacy class of each of the subgroups listed above is the same as its 퐴푛-conjugacy class. 62 Tobias Barthel, Jesper Grodal, and Joshua Hunt

It remains to check which subgroups in the above list are radical in 퐴푛. For the Sylow subgroup there is nothing to check. For 푅푠 we claim that

(12.7) 푁Σ(푛)(푅푠) ≅ (푁Σ(푝)(퐶푝) ≀ Σ푠) × Σ푛−푝푠.

Indeed, since conjugation preserves cycle type, any 휎 ∈ Σ푝푠 that normalises 푅푠 must satisfy 휎 푛 −1 푥푖 = 푥휂(푖) 푖 for some permutation 휂 ∈ Σ푠. The element 휎 ⋅ 휏(휂) normalises each ⟨푥푖⟩, proving the claim. Write 푊퐺(퐻) ≔ 푁퐺(퐻)/퐻 for the Weyl group of 퐻 in 퐺. We deduce from Equation (12.7) that × 푊Σ(푛)(푅푠) ≅ (픽푝 ≀ Σ푠) × Σ푛−푝푠, and we note that 푊퐴푛 (푅푠) is an index two subgroup of this. The order of 푊Σ푛 (푅푠) is not divis- 2 ible by 푝 , so if 푂푝(푊퐴푛 (푅푠) were non-trivial then it would be a Sylow subgroup of order 푝; in particular, 푊퐴푛 (푅푠) would have a unique Sylow 푝-subgroup, and hence so would 푊Σ푛 (푅푠).

Therefore, to show that 푅푠 is 푝-radical in 퐴푛, it is enough to show 푊Σ푛 (푅푠) does not have a unique subgroup of order 푝. We prove this case-by-case: (i) If 푝 > 3 and 푠 = 푝, then there are (푝 − 2)! subgroups of order 푝 in the Σ푠 of the wreath product. (ii) If 푝 > 3 and 푠 < 푝, then the Σ푛−푝푠 factor of the direct product contains a copy of Σ푝. −1 (iii) If 푝 = 3 and 푠 = 푝, then ⟨푦⟩ and ⟨훽1(휉) 훽2(휉)푦⟩ are different subgroups of order 푝 in

푊Σ푛 (푅3). (iv) If 푝 = 3, 푠 < 푝, and 푛 > 3푠 + 3, then the Σ푛−푝푠 factor of the direct product contains a copy of Σ4. (The only case not covered is when 푝 = 3, 푠 = 2, and 푛 = 9.) It follows that in all of the above cases, 푂푝(푊퐴(푛)(푅푠)) = 1 and 푅푠 is 푝-radical in 퐴푛. In the exceptional case, 푊퐴(9)(푅2) ≅ 퐷24 and 푊Σ(9)(푅2) ≅ 퐷8 × Σ3, both of which have a normal Sylow 3-subgroup.

Finally, we check that 퐸2 is 푝-radical in 퐴푛. By [AF90, (2.1)], the Weyl group of 퐵(푐1,…,푐푡 ) in Σ푝푑 is given by

푑 (12.8) 푊Σ(푝 )(퐵(푐1,…,푐푡 )) ≅ GL푐1 (푝) × GL푐2 (푝) × … × GL푐푡 (푝). Since 퐸2 = 퐵(2), we have 푊Σ(푛)(퐸2) ≅ GL2(푝) × Σ푛−푝2 . ′ The Σ푛−푝2 factor is a 푝 -group, and 푂푝(GL2(푝)) = 1 because the upper-triangular unitary matrices and lower-triangular unitary matrices are two Sylow subgroups whose intersection is trivial. □ 푝′ For any group 퐺, let 퐺푝′ denote the quotient of 퐺 by the smallest normal subgroup 푂 (퐺) 푝′ ′ such that 퐺/푂 (퐺) is a 푝 -group; equivalently, 퐺푝′ is the quotient of 퐺 by the normal subgroup generated by all elements of 푝-power order. × 12.9. Lemma. Let 푝 ≥ 5. The group H1(푁퐴(푝2)(퐸2))푝′ is isomorphic to a subgroup of 픽푝 . With this identification the canonical map × 푁퐴(푝2)(퐸2) → H1(푁퐴(푝2)(퐸2))푝′ ≤ 픽푝 takes an element to the determinant of the matrix that represents it with respect to the basis × {푥1 … 푥푝, 푦} of 퐸2, and the image of the canonical map is the subgroup of squares in 픽푝 .

Proof. We have 푊Σ(푝2)(퐸2) ≅ GL2(푝) by Equation (12.8). An element of 푁Σ(푝2)(퐸2) is sent to the matrix representing it with respect to the basis {푥1 … 푥푝, 푦}. The functors H1(−) and (−)푝′ commute with each other (up to natural isomorphism), so H1(푁퐴(푝2)(퐸2))푝′ ≅ H1(푊퐴(푝2)(퐸2)) via an isomorphism that commutes with the quotient maps from 푁퐴(푝2)(퐸2). The Weyl group Torsion-free endotrivial modules via homotopy theory 63

푊퐴(푝2)(퐸2) identifies with a subgroup 푊 ≤ GL2(푝) of index two, and it is enough to show that the derived subgroup of 푊 is SL2(푝) ⊴ We will first show that SL2(푝) ≤ 푊 . We have |SL2(푝) ∶ 푊 ∩SL2(푝)| ≤ 2, so 1 ≠ 푊 ∩ SL2(푝) SL2(푝). We have a subnormal series of SL2(푝) given by ⊴ ⊴ 1 {±퐼} SL2(푝). 퐶2 PSL2(푝)

We see that {±퐼} ≤ 푊 , so (푊 ∩ SL2(푝))/{±퐼} ≤ PSL2(푝) is a subgroup of index at most two. However, PSL2(푝) is simple (since 푝 ≥ 5) so has no index two subgroups. Therefore, 푊 ∩ SL2(푝) = SL2(푝), i.e. SL2(푝) ≤ 푊 . × We deduce that 푊 / SL2(푝) ≤ 픽푝 is an index two subgroup, and hence that 푊 is the preimage × (under the determinant homomorphism) of the set of squares in 픽푝 . The group SL2(푝) is perfect, since 푝 ≥ 5, so is contained in the derived subgroup of 푊 . Therefore, it is equal to the derived × □ subgroup, and we see that H1(푊퐴(푝2)(퐸2)) is the set of squares in 픽푝 .

12.10. Proposition. If 푀 ∈ 푇 (퐴푛) is of type (0, 2푝푟), then 푝 − 1 divides 푟. B C Proof. Write 퐺 ≔ 퐴푛 for brevity. Note that 푝(퐺) satisfies the condition on found in Propo- sition 7.5: for a 1-simplex 푃 < 푄 in B푝(퐺), if 푄 is elementary abelian then we can set 푉 ≔ 푄. Otherwise, by conjugating 푃 < 푄 we can assume that 푄 = 푆. If 푃 is conjugate to 퐸2, then we can set 푉 ≔ 푃. The remaining case is where 푃 is a 푝-radical subgroup contained in 푆 and subconjugate to 퐸1, and by comparing cycle types we see it must actually be a subgroup of 퐸1. In this case, we claim that we can take 푉 ≔ 퐸1. It is clear that 퐸1 contains both 푍(푃) = 푃 and 푍(푆) = 퐸1 ∩ 퐸2, so we just need to check that 푁퐺(푃 < 푆) normalises 퐸1. In fact, the normaliser of 푆 is contained in the normaliser of 퐸1, because 퐸1 is generated by the elements of 푆 that are 푝-cycles. Therefore, B푝(퐺) satisfies the condition in Proposition 7.5. S Let 푛 ∶ 푝(퐺) → ℤ be a type function for 푀 with 푛(푅푠) = 푛(퐸1) = 푛(푆) = 0 and 푛(퐸2) = 2푝푟. The strategy of the proof will be to compare the characters of an orientation on four 푝-radical subgroups: 푅푝−2, 퐸1, 푆, and 퐸2. We will show that it is impossible to specify an orientation for 푛 unless 푝 − 1 divides 푟. Note that when 푝 = 3 and 푛 = 9, the fact that 푅2 is not 푝-radical does not cause any issues, since we only use 푅1 and 푅3 (which is equal to 퐸1). × Let 휉 ∈ 푁Σ푝 (퐶푝) represent a generator of the quotient 푁Σ푝 (퐶푝)/퐶푝, and let 휆 ∈ 픽푝 be the × generator corresponding to it under the canonical isomorphism 푁Σ푝 (퐶푝)/퐶푝 ≅ 픽푝 given by right conjugation. When 푝 = 3, we will specifically choose 휉 = (2, 3); this choice avoids problems later in the proof due to 퐴3 not being perfect. As argued for previously, 휉 is an odd permutation. We give names to some elements of 퐺:

2 푎 ≔ ( ∏ 훽푖(휉)) ⋅ 휏(휉), 푏 ≔ ∏ 훽푖(휉) , 1≤푖≤푝 1≤푖≤푝 ′ ′ 2 푎 ≔ 훽푝(휉) ⋅ 휏((푝 − 1, 푝)), and 푏 ≔ 훽푝(휉) . × An overview of the proof is given in Figure 1. All boxes represent an element of 픽푝 , and arrows indicate equality; the arrow is labelled by the compatibility condition that we are using in that step. The argument depends on several computations in homology, which we postpone until we have described the outline. We start at the top of the diagram, with the observation ′ ′ that [푎] = [푏] in H1(푁퐺(퐸2))푝′ , where as before the subscript 푝 denotes the 푝 -quotient. This implies that the value the character 휑 takes on the elements 푎 and 푏 must agree. Next we use 퐸2 the condition from Proposition 7.5 that 휑 = 휈푝푟 휑 퐸2 푆 푆 64 Tobias Barthel, Jesper Grodal, and Joshua Hunt

휑 (푎) = 휑 (푏) 퐸2 퐸2 푝푟 휑 = 휈푝푟 휑 휑 = 휈 휑 퐸2 푆 푆 퐸2 푆 푆

푟 2푟 휆 휑푆(푎) 휆 휑푆(푏)

휑 = 휑 휑 = 휑 푆 퐸1 푆 퐸1

푟 푟 ′ 휆 휑퐸 (푎) = 휆 휑퐸 (푎 ) 휆2푟 휑 (푏) = 휆2푟 휑 (푏′) 1 1 퐸1 퐸1

휑퐸 = 휑푅 휑 = 휑 1 푝−2 퐸1 푅푝−2

휆푟 휑 (푎′) = 휆푟 2푟 ′ 2푟 푅푝−2 휆 휑 (푏 ) = 휆 푅푝−2

휆푟 = 휆2푟

Figure 1. Overview of the comparison of characters for the alternating group.

as characters of 푁퐺(퐸2 < 푆), but crucially the twisting character 휈푆 takes different values on 푎 and 푏: the homology classes [푎] and [푏] do not agree in H1(푁퐺(푆))푝′ . We have 휈푆(푎) = 휆 and 휈 (푏) = 휆2. We now use the compatibility condition for orientation characters again: 휑 = 휑 푆 푆 퐸1 as characters of 푁퐺(퐸1 < 푆), so 휑푆(푎) = 휑퐸 (푎) and similarly for 푏. In H1(푁퐺(퐸1)푝′ we have ′ ′ 1′ ′ [푎] = [푎 ] and [푏] = [푏 ], so 휑퐸 (푎) = 휑퐸 (푎 ) and similarly for 푏 and 푏 . Finally, we use the 1 1 ′ ′ compatibility condition one last time to pass to 푅푝−2. Here, however, we have [푎 ] = [푏 ] = 0 푟 2푟 × in H1(푁퐺(푅푝−2)). We have therefore shown that 휆 = 휆 ; since 휆 is a generator for 픽푝 , this can only be the case if 푝 − 1 divides 푟. It remains to justify all the claims made in the above argument. We start with an implicit ′ ′ claim, namely that 푎, 푏 ∈ 푁퐺(퐸1) ∩ 푁퐺(푆) ∩ 푁퐺(퐸2) and 푎 , 푏 ∈ 푁퐺(푅푝−2) ∩ 푁퐺(퐸1). This is a straightforward check, since we know how 훽푖(휉), 휏(휉), and 휏((푝 − 1, 푝)) act on the elements 푥푖 and 푦. Secondly, we show that [푎] = [푏] in H1(푁퐺(퐸2))푝′ . When 푝 ≥ 5, we use Lemma 12.9: 푎 and 푏 represent the same class in H1(푁퐴(푝2)(퐸2))푝′ if they correspond to matrices with the same determinant. With respect to the basis {푥1 … 푥푝, 푦}, the elements 푎 and 푏 act by the matrices 휆 휆2 (12.11) ( ) and ( ) 휆 1 respectively. When 푝 = 3, it can be checked that 푏 = 1 and 푎 = (2, 3)(4, 7)(5, 9)(6, 8) = [(2, 4, 3, 7)(5, 6, 9, 8), (2, 5, 3, 9)(4, 8, 7, 6)] where both of the elements appearing in the commutator normalise 퐸2. The matrices in (12.11) also tell us the value of the twisting character 휈푆 on the elements 푎 and 푏. The twisting character is determined by the right action of 푁퐺(푆) on 푍(푆) = ⟨푥1 … 푥푝⟩, so its value is given by the top- 2 left corner of the matrix. We have 휈푆(푎) = 휆 and 휈푆(푏) = 휆 . Torsion-free endotrivial modules via homotopy theory 65

′ ′ Thirdly, we show that [푎 ] = [푎] and [푏 ] = [푏] in H1(푁퐺(퐸1))푝′ . We have −1 훽푖(휉) ⋅ 훽푗(휉) = [훽푝(휉)훽푖(휉), 훽푝(휉)휏((푖, 푗))] ∈ [푁퐺(퐸1), 푁퐺(퐸1)] 푝−1 for any distinct 푖 and 푗 with 푖, 푗 < 푝, so [훽푖(휉)] = [훽푗(휉)] in H1(푁퐺(퐸1))푝′ . Since 휉 ∈ 퐶푝, this implies that [푏′] = [푏] and [푎′−1푎] = [휏((푝 − 1, 푝) ⋅ 휉)] once we have passed to the 푝′-quotient. When 푝 = 3, we chose 휉 = (2, 3), while when 푝 ≥ 5 we have 휏(퐴푝) = [휏(퐴푝), 휏(퐴푝)] ≤ ′ [푁퐺(퐸1), 푁퐺(퐸1)]. In either case, [푎 ] = [푎] as claimed. ′ ′ ′ ′ Finally, we observe that [푎 ] = [푏 ] = 0 in H1(푁퐺(푅푝−2)). Indeed, both 푎 and 푏 are con- 2 2 □ tained in Alt({푝 − 2푝 + 1, … , 푝 }) ≅ 퐴2푝, which is a perfect subgroup of 푁퐺(푅푝−2). Proof of Theorem 12.1. As noted in the introduction to this section, all that is needed in addition to [CHM10, Theorem 6.1] is to rule out the existence of modules of type (0, 2푝푟) for 0 < 푟 < 푝 −1. Proposition 12.10 does this for the alternating group, and restricting from 푇 (Σ푛) to 푇 (퐴푛) preserves the type of a module so the result follows for the symmetric group too. □

Appendix A. Comparison with an obstruction due to Balmer

Let 퐻 ≤ 퐺 be a subgroup whose index in 퐺 is prime to 푝. In Theorem 10.7 of [Bal15], Balmer provides an abstract obstruction for lifting a 푘퐻-module to a 푘퐺-module, which lives in the second Čech cohomology group of a certain cover U in his “sipp” topology on finite 퐺-sets. The goal of this section is to relate his obstruction to our class 훼 ∈ H2(OS (퐺); 푘×). ∗ U We recall the setup of [Bal15] and show that the Čech cohomology groups Ȟ ( ; 픾푚) are isomorphic to the cohomology groups H∗(OS (퐺); 푘×) in the case where 퐻 = 푆. We also show that our obstruction class 훼푀 is sent to the obstruction of [Bal15, Theorem 10.7] under this isomorphism. Let C be a category, let 퐹 ∶ Cop → Ab be a presheaf of abelian groups, and let 푈 → 푋 be a C morphism for which the iterated pullbacks 푈 ×푋 … ×푋 푈 exist in ; for example, we can take 푈 → 푋 to be a cover in a Grothendieck topology on C. We get a simplicial diagram → ⋯ → 푈 × 푈 × 푈 → 푈 × 푈 → 푈 → 푋 푋 → 푋 → in C, with the face maps induced by projections and degeneracies induced by the diagonal map. Applying 퐹 to the above diagram gives a cosimplicial abelian group and hence a cochain complex (whose differential is given by the alternating sum of face maps). Wedefinethe Čech cohomology of U ≔ {푈 → 푋} with coefficients in 퐹, written Ȟ ∗(U; 퐹), to be the cohomology of this cochain complex. The sipp-topology is the Grothendieck topology on the category 퐺-Set of finite 퐺-sets that is defined as follows: a family of 퐺-maps {훼푖 ∶ 푈푖 → 푋}푖∈퐼 is a covering if, for every 푥 ∈ 푋, there −1 exist an 푖 ∈ 퐼 and 푢 ∈ 훼푖 (푥) with the index |Stab퐺(푥) ∶ Stab퐺(푢)| being prime to 푝. (“Sipp” is an acronym for either “stabilisers of index prime to 푝” or “stabilisateurs d’indice premier à 푝”.) We U are interested in the cover = {퐺/푆 → 퐺/퐺} and the constant sipp-sheaf 픾푚 associated with × op the abelian group 푘 . More explicitly, 픾푚 ∶ 퐺-Set → Ab is the finite-coproduct-preserving functor that takes values 푘× if 푝 divides |퐻| and 픾 (퐺/퐻) ≅ { 푚 0 otherwise, all of whose restrictions are componentwise either the identity or zero. Balmer’s obstruction 2 U lies in Ȟ ( ; 픾푚), and we recall its definition in Remark A.3. 66 Tobias Barthel, Jesper Grodal, and Joshua Hunt

op A.1. Lemma. Let 퐹 ∶ OS (퐺)op → Ab be a functor and let 퐹̃ ∶ 퐺-Set → Ab denote the right Kan extension of 퐹 along the opposite of the inclusion 푖∶ OS (퐺) ↪ 퐺-Set. There is a natural isomorphism H∗(OS (퐺); 퐹) ≅ Ȟ ∗(U; 퐹).̃

A.2. Remark. If 퐹 ∶ OS (퐺)op → Ab is the constant functor with value 푘×, then 퐹̃ is isomorphic 2 O × 2 U to 픾푚 and the lemma implies that H ( S (퐺); 푘 ) ≅ Ȟ ( ; 픾푚). To see this, note that for any 퐺/퐻 we have op O op Ran푖(퐹)(퐺/퐻) ≃ lim ((푖 / (퐺/퐻)) → S (퐺) → Ab) and that 푖/(퐺/퐻) ≃ OS (퐻). This means that if 푝 does not divide |퐻|, then 퐹(퐺/퐻)̃ = 0, while if 푝 divides |퐻|, then 퐹(퐺/퐻)̃ ≅ 푘× because OS (퐻) is a connected category. As 퐹̃ also preserves finite coproducts, it is isomorphic to 픾푚. Proof of Lemma A.1. We can compute a Kan extension along 푖 in two steps: we first extend to 퐺-SetS , the full subcategory of 퐺-Set on objects whose isotropy subgroups are all non-trivial 푝-groups, and from there to 퐺-Set itself. The category 퐺-SetS is the free completion of OS (퐺) under finite coproducts, so byJM92 [ , Corollary 4.4] we have

H∗(OS (퐺); 퐹) ≅ H∗(퐺-SetS ; 퐹).̃

Observe that for every 푛 ≥ 1 the product (퐺/푆)푛 exists in 퐺-SetS , given by discarding all free orbits from the “ordinary” product (퐺/푆)푛 taken in 퐺-Set. This follows from the fact that there are no morphisms 퐺/푃 → 퐺/{1} for 푃 ∈ S푝(퐺). We get a simplicial object → 퐺 퐺 퐺 → 퐺 퐺 퐺 ⋯ → × × → × → → 푆 푆 푆 → 푆 푆 푆 in 퐺-SetS ; write 푋 • ∶ Δop → 퐺-SetS for the functor that classifies it. Observe that every object 푌 in 퐺-SetS admits a map 푌 → 퐺/푆, so pulling back along the functor 푋 • preserves homotopy colimits [MNN17, Proposition 6.28]. This implies that we have

H∗(퐺-SetS ; 퐹)̃ ≅ H∗(Δ; 퐹̃ ∘ (푋 •)op). It remains to identify H∗(Δ; 퐹̃ ∘ (푋 •)op) with Ȟ ∗(U; 퐹)̃ . To this end, let 퐴• ∶ Δ → Ab denote the cosimplicial abelian group 퐹̃ ∘ (푋 •)op and write ℤ for the constant cosimplicial abelian group with value ℤ. The functor category [Δ, Ab] is an abelian category [Mac98, p. 199] and we have ∗ • ∗ • H (Δ; 퐴 ) ≅ Ext[Δ,Ab](ℤ, 퐴 ). Consider ℤΔ(−, −)∶ Δop × Δ → Ab, the ℤ-linearisation of the morphism sets in the category Δ. We claim that … → ℤΔ([2], −) → ℤΔ([1], −) → ℤΔ([0], −) → ℤ is a projective resolution of ℤ in [Δ, Ab]. Each of the functors ℤΔ([푛], −) is projective, since Yoneda’s lemma implies that Hom[Δ,Ab](ℤΔ([푛], −), −) preserves epimorphisms. We can check exactness pointwise: the chain complex … → ℤΔ([2], [푚]) → ℤΔ([1], [푚]) → ℤΔ([0], [푚]) → 0 푚 identifies with the simplicial chain complex C•(Δ ), whose homology is concentrated in degree zero since Δ푚 is contractible. Therefore we get a projective resolution as claimed. Finally, we consider the cochain complex that computes Ext: by another application of Yoneda’s lemma, the chain complex • • 0 → Hom[Δ,Ab](ℤΔ([0], −), 퐴 ) → Hom[Δ,Ab](ℤΔ([1], −), 퐴 ) → … Torsion-free endotrivial modules via homotopy theory 67 identifies with 0 → 퐴0 → 퐴1 → 퐴2 → … and we observe that this complex is precisely the Čech complex Č •(U; 퐹)̃ . □

A.3. Remark. We briefly recall the definition of the obstruction defined in[Bal15, Theorem 10.7]. We consider the cosimplicial diagram of ∞-categories → 2 → StMod(퐺/푆) → StMod((퐺/푆) ) → … appearing in the proof of [Mat16, Proposition 9.13] (c.f. the definition of StMod(푋) for a 퐺-set 푋 in Section 3). This is the counterpart on the categorical level of the cosimplicial abelian group 퐴•; one can recover 퐴• from this diagram by composing with the functor that takes a symmetric monoidal ∞-category to the homotopy classes of automorphisms of its unit. Let 푀 ∈ StMod(퐺/푆) be such that 푑0푀 ≃ 푑1푀 in StMod((퐺/푆)2); this condition is precisely saying that the equivalence class of 푀 is an element of Ȟ 0(U; Picst), in Balmer’s notation. Choose an equivalence 휉 ∶ 푑0푀 −−→푑∼ 1푀 and define 휁 ≔ (푑1휉)−1 ∘ 푑2휉 ∘ 푑0휉. This is an auto- 0 0 3 morphism of 푑 푑 푀, which we can canonically identify with an element of 픾푚((퐺/푆) ). The 2 U 2 U morphism 휁 gives a 2-cocycle in Č ( ; 픾푚) whose equivalence class in Ȟ ( ; 픾푚) depends only on the equivalence class of 푀, not on the choice of 휉. This is Balmer’s obstruction class.

A.4. Lemma. Under the isomorphism given by Lemma A.1, the obstruction [훼] ∈ H2(OS (퐺); 푘×) 2 U defined in Theorem 4.5 identifies with the obstruction in [휁 ] ∈ Ȟ ( ; 픾푚) recalled in Remark A.3. Proof. We will trace [훼] through the isomorphisms in Lemma A.1 and show that it agrees with ′ 2 [휁 ]. The first isomorphism in Lemma A.1 extended [훼] to [훼 ] ∈ H (퐺-SetS ; 픾푚).A 퐺-stable endotrivial module 푀 gives us an element of lim op 휋 Pic StMod(−), i.e. an element 푀 ∈ 푆 퐺-SetS 0 푋 ∗ StMod(푋) for every finite 퐺-set 푋, such that 푓 (푀푌 ) ≃ 푀푋 for every morphism 푓 ∶ 푋 → 푌 in 퐺-SetS . (Note that although we have made a choice of representative 푀푋 here, the equivalence ∗ ∼ class of 푀푋 is canonical.) For every such 푓 , we choose a specific equivalence 휆푓 ∶ 푓 (푀푌 ) −−→ ′ 푀푋 . The 2-cocycle 훼 is then given by sending 푓 푔 푋 −→푌 −→푍 to the element of 픾푚(푋) determined by the automorphism 휆 휆 푓 ∗휆 ∗ ∗ 푔푓 푓 ∗ 푔 ∗ ∗ 푓 푔 푀푍 −−−→푀푋 ←−−푓 푀푌 ←−−−−푓 푔 푀푍 2 O × ′ 2 since this restricts to the correct class [훼] ∈ H ( S (퐺); 푘 ). The class [훼 ] ∈ H (퐺-Set; 픾푚) depends only on the equivalence class of 푀푆 in StMod푘푆, not on 푀푆 itself, nor on the choices of 푀푋 and 휆푓 . The next isomorphism in Lemma A.1 was induced by restriction along the simplicial object 푋 • ∶ Δop → 퐺-SetS given by → 퐺 퐺 퐺 → 퐺 퐺 퐺 ⋯ → × × → × → . → 푆 푆 푆 → 푆 푆 푆 Let 훼″ denote the restriction of 훼′ along 푋 •. We can describe 훼″ ∈ C2(Δ; 퐴) by choosing 푛+1 ∼ representatives 푀[푛] ∈ StMod((퐺/푆) ) and equivalences 휆휃 ∶ 휃∗(푀[푛]) −−→푀[푚] for every 휃 ∶ [푛] → [푚]; the resulting cocycle 훼″ is given by sending

휃 휓 [푛] −→[푚] −→[푙] 68 Tobias Barthel, Jesper Grodal, and Joshua Hunt

푙+1 to the element of 픾푚((퐺/푆) ) determined by the automorphism

휓∗휆휃 휆휓 휆휓휃 (A.5) 휓∗휃∗푀[푛] −−−−→휓∗푀[푚] −−→푀[푙] ←−−−휓∗휃∗푀[푛] ″ of 휓∗휃∗푀[푛]. As before, the cohomology class [훼 ] is independent of all the choices we made. The third and final isomorphism of Lemma A.1 sent H∗(Δ; 퐴) to H∗(퐴), which amounts to removing a subdivision. The cochain complexes 퐴• and C•(Δ; 퐴) arise respectively from the projective resolutions of ℤ ∈ [Δ, Ab] given by ℤΔ([•], −) and C•(Δ/−). We can define a comparison map between these resolutions by sending the identity in ℤΔ([푛], [푛]) to the alternating sum 푑푖1 푑푖2 푑푖푛 id (A.6) ∑(−1)푖1+…+푖푛 [0] −−→[1] −−→… −−→[푛] −−→[푛] in C푛(Δ/[푛]). The sum is indexed over all possible choices of (푖1, 푖2, … , 푖푛). The map in (A.6) determines the natural transformation in every degree by Yoneda’s lemma, and the fact that it is a chain map follows from the cosimplicial identity 푑푗 ∘ 푑푖 = 푑푖 ∘ 푑푗−1 for 푖 < 푗. Since this natural transformation lives over the identity ℤ → ℤ, it is the unique-up-to-homotopy quasi-isomorphism between the resolutions. We deduce that the image in 퐴2 of 훼″ ∈ C2(Δ; 퐴) under the above quasi-isomorphism is 푑푖1 푑푖2 ∑(−1)푖1+푖2 훼″([0] −−→[1] −−→[2]). We can draw this alternating sum in a diagram:

0 0 0 1 2 0 2 1 1 1 1 0 푑 푑 푀[0] 푑 푑 푀[0] = 푑 푑 푀[0] 푑 푑 푀[0] = 푑 푑 푀[0] 푑 푑 푀[0]

0 0 2 2 1 1 푑 휆푑0 푑 휆푑1 푑 휆푑0 푑 휆푑1 푑 휆푑1 푑 휆푑0 0 2 1 푑 푀[1] 푑 푀[1] 푑 푀[1]

휆푑0 휆푑2 휆푑1

푀[2] 푀[2] 푀[2] Here the solid arrows come from the terms in the alternating sum, expanded out using the description of 훼″ in (A.5); there are six maps omitted, namely the backwards-facing arrows in (A.5), which cancel out in pairs. The alternating sum itself is equal to the automorphism of 0 0 1 0 푑 푑 푀[0] = 푑 푑 푀[0] given by the dashed maps. If we choose 휉 to be given by 휆 휆 0 푑0 푑1 1 푑 푀[0] −−−→푀[1] ←−−−푑 푀[0] then these dashed maps are exactly the maps 푑0(휉), 푑2(휉), and 푑1(휉). Comparing with Re- mark A.3, we see that the image of the class [훼″] is equal to [휁 ]. □ A.7. Remark. Under the identification of Lemma A.4, Balmer’s [Bal15, Theorem 10.7(e)] is op equivalent to the exactness at limOS (퐺) 푇 (−) of the sequence appearing in Theorem 4.5.

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Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany

Email address: [email protected] Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 København Ø, Denmark

Email address: [email protected]

Part III

Decompositions of the stable module ∞-category

DECOMPOSITIONS OF THE STABLE MODULE ∞-CATEGORY

JOSHUA HUNT

Abstract. We show that the stable module ∞-category of a finite group 퐺 decomposes in three different ways as a limit of the stable module ∞-categories of certain subgroups of 퐺. Analo- gously to Dwyer’s terminology for homology decompositions, we call these the centraliser, normaliser, and subgroup decompositions. We construct centraliser and normaliser decompositions and extend the subgroup decomposition (constructed by Mathew) to more collections of subgroups. The key step in the proof is extending the stable module ∞-category to be defined for any 퐺-space, then showing that this extension only depends on the 푆-equivariant homotopy type of a 퐺-space. The methods used are not specific to the stable module ∞-category, somay also be applicable in other settings where an ∞-category depends functorially on 퐺.

1. Introduction

Let 퐺 be a finite group and 푝 be a prime dividing the order of 퐺. A homology decomposition of the classifying space B퐺 is a diagram of spaces 퐹 ∶ 퐷 → S such that, for every 푑 ∈ 퐷, the space 퐹(푑) has the homotopy type of B퐻 for some 퐻 ≤ 퐺, together with a map hocolim 퐹 → B퐺 that induces an isomorphism on mod 푝 homology. Homology decompositions have a long history in algebraic topology, with an early success being their use in [JMO92] to classify self- maps of classifying spaces of compact, connected, simple Lie groups. They also played an important role in the classification of 푝-compact groups; see [Gro10] for a survey. More recently, similar decomposition techniques have found applications in modular representation theory, for example in [Mat16], [Gro18], and [BGH]. In [Dwy97], Dwyer was able to give a unified treatment of three different types of homology decompositions for a fixed collection C of subgroups of 퐺 (where by the term collection we always mean a set of subgroups that is closed under conjugation by elements of 퐺): Subgroup: let 퐷 = OC (퐺), the orbit category, and let 퐹 ∶ OC (퐺) → S take 퐺/퐻 to B퐻. F F op S Centraliser: let 퐷 = C (퐺), the fusion category, and let 퐹 ∶ C (퐺) → take 퐻 to B퐶퐺(퐻). Normaliser: let 퐷 = 푠푆C (퐺), the orbit simplex category, and let 퐹 ∶ 푠푆C (퐺)op → S take a simplex 휎 = (퐻0 < … < 퐻푛) to B푁퐺(휎), where 푁퐺(휎) denotes ⋂0≤푖≤푛 푁퐺(퐻푖). We recall the definition of the indexing categories below, in Section 2. Dwyer showed that in all of these cases, 퐹 provides a mod 푝 homology decomposition of 퐺 if and only if the natural C map h퐺 → (∗)h퐺 ≃ B퐺 induces an isomorphism on mod 푝 homology. Dwyer calls a collection C satisfying this condition ample. The stable module category StMod푘퐺 of 퐺 over a field 푘 of characteristic 푝 is obtained by “quotienting” the module category Mod푘퐺 by the projective modules. In this paper, we show that analogues of the subgroup, centraliser and normaliser decompositions exist for StMod푘퐺,

Date: 2020-07-31. This work was supported by the Danish National Research Foundation through the Centre for Symmetry and Deformation (DNRF92) and the Copenhagen Centre for Geometry and Topology (DNRF151). 77 78 Joshua Hunt viewed as an ∞-category, describing it in three different ways as a limit of ∞-categories StMod푘퐻 for subgroups 퐻 ≤ 퐺. In this setting, Mathew has already shown the existence of the subgroup decomposition for certain collections: Theorem ([Mat16, Corollary 9.16]). Let C be a collection of subgroups of 퐺 that is closed under intersection and such that every elementary abelian 푝-subgroup of 퐺 is contained in a subgroup in C . There is an equivalence of symmetric monoidal ∞-categories ∼ StMod푘퐺 −−→ lim StMod푘퐻 . 퐺/퐻∈OC (퐺)op Note that the change from a homotopy colimit (in Dwyer’s result) to a homotopy limit (in Mathew’s result) is due to the differing variances of the homotopy orbits functor (−)h퐺 and the stable module ∞-category functor StMod(−). Following ideas of Dwyer and others, we use 퐺-spaces to encode decompositions of StMod(−): we formally Kan extend the functor O op StMod(−)∶ (퐺) → Cat̂ ∞ to a functor defined on any 퐺-space Sop ̂ (1.1) StMod(−)∶ 퐺 → Cat∞. This extended functor takes small homotopy colimits of 퐺-spaces to homotopy limits of ∞- S categories, so for any diagram of 퐺-spaces 퐹 ∶ 퐷 → 퐺, the canonical map hocolim(퐹) → ∗ induces a comparison map

(1.2) StMod푘퐺 ≃ StMod(∗) → lim StMod(퐹(푑)). 퐷op Dwyer constructed 퐺-spaces (depending on the collection C ) that encode the three homology decompositions listed above. For convenience, we will temporarily refer to these 퐺-spaces as encoding 퐺-spaces. Applying StMod(−) to the encoding 퐺-spaces for C gives functors as in (1.2) that are candidates for the three decompositions of the stable module ∞-category. To show that we do get a decomposition of the stable module ∞-category, we need to prove that the comparison functor (1.2) is an equivalence. For this we use work of Grodal–Smith [GS06], which lists cases when certain canonical maps between the encoding 퐺-spaces are 푆- equivalences, where 푆 is a Sylow 푝-subgroup. (Recall that that a map 푓 ∶ 푋 → 푌 of 퐺-spaces is an 푆-equivalence if it induces a homotopy equivalence 푋 푃 −−→푌∼ 푃 on 푃-fixed points for every 푝- subgroup 푃 ≤ 퐺.) In Section 5, we prove that the extended functor (1.1) inverts 푆-equivalences. In fact, we prove a more general result (Theorem 4.8): Theorem A. Let A and B be small ∞-categories, C be an ∞-category with all small colimits, and 푖 퐹 A −→ B −→ C be functors with 푖 fully faithful. Let 퐹̃ ∶ P(B) → C denote the left Kan extension of 퐹 along the Yoneda embedding 푦B. There is a factorisation ̃ P(B) 퐹 C

푖∗ 퐹̃′ P(A) if and only if 퐹 is the left Kan extension of its restriction to A, i.e. the natural map 퐹푖 colim(푖/푏 → A −−→ C) → 퐹(푏) is an equivalence for every 푏 ∈ B. Decompositions of the stable module ∞-category 79

From this criterion, we deduce Theorem 5.3: O op Theorem B. The right Kan extension of StMod(−)∶ (퐺) → Cat̂ ∞ along the opposite of the O S Yoneda embedding (퐺) → 퐺 factors through the restriction map Sop P O op P O op 퐺 ≃ ( (퐺)) → ( 푝(퐺)) . S In particular, StMod(−) sends 푆-equivalences in 퐺 to equivalences of ∞-categories. We note that the only property of StMod(−) used to deduce Theorem B from Theorem A is the existence of a subgroup decomposition, as in Mathew’s result above. The approach therefore applies whenever an ∞-category depends functorially on 퐺 and satisfies an analogous descent condition. Given Theorem B, it is enough to find a zig-zag of 푆-equivalences from the encoding 퐺-space for Mathew’s subgroup decomposition to the encoding 퐺-space for the decomposition that we are interested in. This problem was studied by Grodal–Smith in [GS06] and we use their results in Section 6 to obtain our main theorem (Theorem 6.4):

Theorem C. Let C be one of the collections S푝(퐺), A푝(퐺), B푝(퐺), I푝(퐺), or Z푝(퐺). There is a subgroup decomposition ∼ StMod푘퐺 −−→ lim StMod푘푃 퐺/푃∈OC (퐺)op and a normaliser decomposition StMod −−→∼ lim StMod . 푘퐺 푘푁퐺 (휎) [휎]∈푠푆C (퐺)

If C is S푝(퐺), A푝(퐺), or Z푝(퐺), then there is additionally a centraliser decomposition StMod −−→∼ lim StMod . 푘퐺 푘퐶퐺 (푃) 푃∈FC (퐺) The collections mentioned in the theorem are defined as follows:

(i) S푝(퐺) is the collection of non-trivial 푝-subgroups of 퐺, (ii) A푝(퐺) is the collection of non-trivial elementary abelian 푝-subgroups, (iii) B푝(퐺) is the collection of non-trivial 푝-radical subgroups, i.e. non-trivial 푝-subgroups 푃 ≤ 퐺 such that 푃 is the maximal normal 푝-subgroup in 푁퐺(푃), (iv) I푝(퐺) is the collection of all non-trivial 푝-subgroups that are the intersection of a set of Sylow 푝-subgroups, and (v) Z푝(퐺) is the subcollection of A푝(퐺) consisting of those 푉 such that 푉 is the set of elements in the centre of 퐶퐺(푉 ) whose order divides 푝, i.e. such that 푉 = Ω1푂푝푍(퐶퐺(푉 )), using standard group-theoretic notation such as found in [Asc00].

1.3. Remark. Neither B푝(퐺) nor Z푝(퐺) are closed under intersections, so the subgroup decompositions for B푝(퐺) and Z푝(퐺) in Theorem 6.4 are new and do not follow immediately from Mathew’s subgroup decomposition. For example, if 퐺 = PSL3(7), then there are rank two elementary abelian subgroups 1 ∗ ∗ 1 0 ∗ ( 1 0) and ( 1 ∗) 1 1 that are contained in both B푝(퐺) and Z푝(퐺), but whose intersection 1 0 ∗ ( 1 0) 1 80 Joshua Hunt is contained in neither collection. Acknowledgements. This work was supported by the Danish National Research Foundation through the Centre for Symmetry and Deformation (DNRF92) and the Copenhagen Centre for Geometry and Topology (DNRF151). I would like to thank Markus Land and Piotr Pstrągowski for useful conversations about the contents of this paper (in particular, Markus suggested a way to prove Theorem 4.8); my supervisor, Jesper Grodal, for suggesting the problem and his advice and encouragement; and Kaif Hilman, for catching a mistake in a previous draft as well as many other useful comments.

2. Notation, conventions, and background

Throughout the paper, 퐺 will refer to a finite group and 푝 will be a fixed prime dividing the order of 퐺. We let 푘 be a field of characteristic 푝.A collection of subgroups of 퐺 is a set of subgroups that is closed under conjugation by elements of 퐺. We let O(퐺) denote the orbit category of 퐺, whose objects are transitive left 퐺-sets and whose morphisms are 퐺-equivariant maps between them. For any collection C of subgroups of 퐺, we let OC (퐺) denote the full subcategory of O(퐺) on those 퐺-sets whose isotropy subgroups are contained in C . Note that every object of O(퐺) is isomorphic to a 퐺-set of the form 퐺/퐻, with 퐻 a subgroup of 퐺, and that 퐺/퐻 lies in OC (퐺) if and only if 퐻 ∈ C . With this identification, the morphisms in the orbit category are related to subconjugation relations between subgroups:

푔 HomO(퐺)(퐺/퐻, 퐺/퐾) ≅ { 푔 ∈ 퐺 ∶ 퐻 ≤ 퐾 }/퐾. When C is the collection of all 푝-subgroups of 퐺, including the trivial subgroup, we will use the notation O푝(퐺) instead of OC (퐺). We let FC (퐺) denote the fusion category of 퐺, whose objects are the subgroups in C and whose morphisms are homomorphisms that are induced by conjugation by an element of 퐺. We let 푠푆C (퐺) denote the orbit simplex category of 퐺, which is the poset of 퐺-conjugacy C classes of non-empty chains 휎 = (퐻0 < … < 퐻푛) of subgroups in , ordered by refinement: that is, we have [휎] ≤ [휏] if we can find representatives 휎 and 휏 for the conjugacy classes such that 휎 ⊆ 휏. The objects of 푠푆C (퐺) identify with the 퐺-conjugacy classes of non-degenerate simplices of the nerve of C . Since we deal exclusively with homotopy (co)limits, we will drop the adjective “homotopy” here: when we refer to a “colimit of 퐺-spaces” we will implicitly mean a homotopy colimit. We follow Lurie’s convention of using the prefix “∞-” instead of“(∞, 1)-”. We use Cat̂ ∞ to denote the ∞-category of large ∞-categories, which has all large limits (though we will only need the existence of small limits). Let Funcolim(C, D) denote the full subcategory of Fun(C, D) spanned by the functors that preserve small colimits. The stable module ∞-category StMod푘퐺 is defined as the localisation of the module category Mod푘퐺 at the stable equivalences, i.e. at those maps 푓 ∶ 푀 → 푁 and 푔 ∶ 푁 → 푀 such that 푓 푔 − id푁 and 푔푓 − id푀 both factor through a projective module. The homotopy category of the stable module ∞-category has been studied by representation theorists: for example, Benson–Iyengar–Krause [BIK11] use it as a way of classifying 푘퐺-modules when Mod푘퐺 has wild representation type. The objects of the homotopy category are 푘퐺-modules and the hom sets are given by

휋0 Map퐺(푀, 푁 ) ≅ Hom퐺(푀, 푁 )/(푓 ∼ 0 if 푓 factors through a projective). Decompositions of the stable module ∞-category 81

The stable module ∞-category is a presentable, stable, symmetric monoidal ∞-category. See [Car96, Section 5] for a discussion of the homotopy category and [Mat15, Definition 2.2] for a construction of StMod푘퐺 as an ∞-category. In [Mat16, Section 9.5], Mathew constructs a functor StMod(−)∶ O(퐺)op → CAlg(PrL,st) L,st whose value on 퐺/퐻 is equivalent to StMod푘퐻 . Here Pr denotes the ∞-category of presentable, stable ∞-categories and left adjoint functors between them. This functor will play a crucial role for us, so we spend the rest of this section describing its construction. For any 퐺 퐻 ≤ 퐺, we have a symmetric monoidal restriction functor res퐻 ∶ StMod푘퐺 → StMod푘퐻 , whose 퐺 right adjoint coind퐻 ∶ StMod푘퐻 → StMod푘퐺 is consequently lax symmetric monoidal. This im- 퐺 퐺 plies that coind퐻 (푘) is a commutative algebra object of StMod푘퐺, which we will denote 퐴퐻 . The 퐺 underlying module of 퐴퐻 is ∏퐺/퐻 푘 with its permutation action. In [Bal15, Theorem 1.2], Balmer proves that the homotopy category of StMod푘퐻 is equivalent 퐺 to the category of modules over 퐴퐻 internal to the homotopy category of StMod푘퐺. Proposi- tion 9.12 of [Mat16] generalises this result to the ∞-category StMod푘퐻 : 2.1. Theorem (Balmer, Mathew). There is a natural symmetric monoidal equivalence ∼ 퐺 StMod푘퐻 −−→ ModStMod푘퐺 (퐴퐻 ) ⟵ 퐺 induced by coinduction, under which the free/forget adjunction StMod푘퐺 ⟶ ModStMod푘퐺 (퐴퐻 ) ⟵ corresponds to the restriction/coinduction adjunction StMod푘퐺 ⟶ StMod푘퐻 . We have a functor O op (퐺) → CAlg(StMod푘퐺) 퐺 퐺/퐻 ↦ 퐴퐻 that on underlying modules sends a morphism 퐺/퐻 → 퐺/퐾 to the pullback map ∏퐺/퐾 푘 → O op ∏퐺/퐻 푘. This functor can be constructed by composing the analogous functor (퐺) → CAlg(Mod푘퐺) with the localisation functor Mod푘퐺 → StMod푘퐺. Theorem 2.1 implies that we obtain a functor O op StMod(−)∶ (퐺) → Cat̂ ∞ that takes 퐺/퐻 to an ∞-category equivalent to StMod푘퐻 . By using the description given in [MNN17, Construction 5.23] of the inverse to the equivalence in Theorem 2.1, one can check O that a morphism 퐺/퐻 → 퐺/퐾 in (퐺) is sent to the restriction functor StMod푘퐾 → StMod푘퐻 . 퐾 L,st Since both StMod푘퐻 and res퐻 ∶ StMod푘퐾 → StMod푘퐻 lie in CAlg(Pr ), we have constructed a functor O(퐺)op → CAlg(PrL,st) as desired.

3. The stable module ∞-category of a 퐺-space

S S Let denote the ∞-category of small spaces. Recall that the ∞-category 퐺 of small 퐺- O op S O S spaces is equivalent to Fun( (퐺) , ) and the Yoneda embedding 푦 ∶ (퐺) → 퐺 identifies with the inclusion of O(퐺) as the transitive, discrete 퐺-spaces. O op The stable module ∞-category determines a functor StMod(−)∶ (퐺) → Cat̂ ∞ that takes 퐺/퐻 to StMod푘퐻 . We can formally extend this functor to 퐺-spaces by right Kan extension 82 Joshua Hunt along the opposite of the Yoneda embedding:

O op (퐺) Cat̂ ∞ 푦 StMod(−) Sop 퐺 O Since (퐺) is small, the existence of small limits in Cat̂ ∞ guarantees the existence of the Kan extension, which we still denote by StMod(−). It sends small (homotopy) colimits of 퐺-spaces to limits in Cat̂ ∞. 3.1. Remark. We could equally well have taken StMod(−) to be a functor to CAlg(PrL,st) while carrying out the above construction, since CAlg(PrL,st) also has small limits and the compo- L,st L,st sition CAlg(Pr ) → Pr → Cat̂ ∞ preserves limits [Lur17, Corollary 3.2.2.4]. In other words, the “stable module ∞-category of a 퐺-space” is presentable, stable, and has a symmetric monoidal structure that preserves finite colimits in each variable. We will not need these facts, but mention them to point out that the extended definition of the stable module ∞-category shares many features with the standard definition.

4. Factorising Kan extensions

Our first goal is to show that StMod(−) only sees the 푆-equivariant homotopy type of a 퐺-space: that is, we have a factorisation

Sop P O op ̂ 퐺 ≃ ( (퐺)) Cat∞ 푖∗

op P(O푝(퐺)) where 푖∶ O푝(퐺) ↪ O(퐺) is the inclusion of the full subcategory of transitive 퐺-sets with 푝- group isotropy and P(A) is the ∞-category Fun(Aop, S) of presheaves on A. For simplicity of notation, we dualise and consider the following question: 4.1. Question. Let A and B be small ∞-categories, C be an ∞-category with all small colimits, and 푖 퐹 A −→ B −→ C be functors with 푖 fully faithful. When does the left Kan extension 퐹̃ of 퐹 along the Yoneda embedding 푦B admit a factorisation through P(A) as indicated in the diagram below? B

푦B 퐹 ̃ P(B) 퐹 C

푖∗ 퐹̃′ P(A) We answer this question in Theorem 4.8, providing a necessary and sufficient condition on 퐹 for such a factorisation 퐹̃′ to exist: namely, 퐹 must be equivalent to the left Kan extension of 퐹푖 along 푖. Decompositions of the stable module ∞-category 83

4.2. Notation. In this situation, there are two functors that are induced by restriction along 푖 (or its opposite) and hence could reasonably be denoted by 푖∗, namely P(B) → P(A) and Fun(B, C) → Fun(A, C). We will denote the first functor by 푖∗ and the second functor instead by 푗∗. We hope that this prevents more confusion than it causes. Both of these functors have left adjoints given by left Kan extension, which we will write as P A P B A C B C 푖! ∶ ( ) → ( ) and 푗! ∶ Fun( , ) → Fun( , ).

We will similarly use (푦B)! to denote left Kan extension along the Yoneda embedding. The ∗ adjunction 푖! ⊣ 푖 induces an adjunction Fun(P(A), C)

∗ ∗ ∗ (푖 ) ⊣ (푖!) Fun(P(B), C). 4.3. Remark. We briefly summarise the proof of Theorem 4.8, making forward reference to lemmas that we will prove later in the section. We will show that the following statements are all equivalent: (i) 퐹̃ factors through 푖∗. ∗ ∗ (ii) The natural transformation 퐹푖̃ !푖 → 퐹̃ induced by the counit of the 푖! ⊣ 푖 adjunction is an equivalence. ∗ ∗ ∗ ∗ ∗ (iii) 퐹̃ is equivalent, via the counit of the (푖 ) ⊣ (푖!) adjunction, to the composition (푖 ) ∘ ∗ (푖!) ∘ (푦퐵)! applied to 퐹. ∗ ∗ (iv) 퐹̃ is equivalent, via the counit of the 푗! ⊣ 푗 adjunction, to the composition (푦B)! ∘ 푗! ∘ 푗 applied to 퐹. ∗ ∗ (v) The natural transformation 푗!푗 퐹 → 퐹 induced by the counit of the 푗! ⊣ 푗 adjunction is an equivalence. The equivalence of (i) and (ii) is Lemma 4.4. Since 퐹̃ is the left Kan extension of 퐹 along 푦B, we see that (iii) is just a rewriting of (ii) with different notation. Lemmas 4.5 and 4.6 show that (iii) is equivalent to (iv). Checking that the natural transformation in (iv) is still induced by the counit is a straightforward but tiring diagram chase that we include in Appendix A. Finally, [Lur09, Theorem 5.1.5.6] shows that restriction along the Yoneda embedding induces an equivalence Funcolim(P(B), C) −−→∼ Fun(B, C), so (iv) is equivalent to (v). The rest of the section fills in the details omitted in Remark 4.3. We begin by showing that if 퐹̃ does factor through 푖∗, then such a factorisation is unique. 4.4. Lemma. Let 퐻 ∶ P(B) → C be a functor. Any functor 퐻 ∶ P(A) → C that satisfies 퐻 ≃ ∗ ∗ 퐻 ∘ 푖 must be given by 퐻 ≃ 퐻 ∘ 푖!. Furthermore, 퐻 factors through 푖 if and only if the natural transformation ∗ 퐻휀 ∶ 퐻푖!푖 → 퐻 ∗ P B C induced by the counit of the 푖! ⊣ 푖 adjunction is an equivalence in Fun( ( ), ). ∼ ∗ Proof. Since 푖 is fully faithful, the unit of the adjunction induces an equivalence id −−→푖 푖!. ∗ Therefore, 퐻푖! ≃ 퐻푖 푖! ≃ 퐻. The second claim is a straightforward check using naturality of ∗ ∗ □ the equivalence 퐻 ≃ 퐻 ∘ 푖 and the triangle identities for 푖! ⊣ 푖 . 84 Joshua Hunt

The condition in Lemma 4.4 for a factorisation to exist is in terms of 퐹̃, so our next goal is to rewrite this as a condition on 퐹. We will need two lemmas regarding properties of Kan extensions. 4.5. Lemma. There is a canonical equivalence ∼ 퐹푖 −−→ 퐹푖̃ !푦A, which by the universal property of left Kan extension induces a natural transformation ̃ Lan푦A (퐹푖) → 퐹푖! as functors P(A) → C. This natural transformation is an equivalence; that is, ∗ ∼ ∗ (푦A)! ∘ 푗 −−→(푖!) ∘ (푦B)! as functors Fun(B, C) → Funcolim(P(A), C).

Proof. Since 푦B is fully faithful, we have a commutative diagram

A 푖 B 퐹 C

푦A 푦B 퐹≔̃ Lan (퐹) 푖 푦B P(A) ! P(B) that gives rise to the first equivalence. ̃ The three functors 퐹, 푖!, and Lan푦A (퐹푖) all preserve small colimits, so it is enough to check that they restrict along the Yoneda embedding 푦A to equivalent functors in Fun(A, C), by [Lur09,

Theorem 5.1.5.6]. We then observe that Lan푦A (퐹푖) restricts to 퐹푖, because 푦A is also fully faithful. □

4.6. Lemma. Let 퐻 ∶ A → C be a functor. There is a natural equivalence ∗ Lan푦A (퐻) ∘ 푖 ≃ Lan푦B푖(퐻) as functors P(B) → C. That is, ∗ ∗ (푖 ) ∘ (푦A)! ≃ (푦B)! ∘ 푗! as functors Fun(A, C) → Funcolim(P(B), C).

Proof. We have a commutative diagram

A 푖 B

푦A 푦B 푖 P(A) ! P(B) that induces a commutative diagram of pullback functors:

푗∗ Fun(A, C) Fun(B, C)

∗ ∗ (4.7) (푦A) (푦B) (푖 )∗ Funcolim(P(A), C) ! Funcolim(P(B), C) Decompositions of the stable module ∞-category 85

∗ ∗ ∗ ∗ The adjunction 푖! ⊣ 푖 induces an adjunction (푖 ) ⊣ (푖!) , so all of the functors in the above diagram have left adjoints. Thus, we obtain another commutative diagram:

푗 Fun(A, C) ! Fun(B, C)

(푦A)! (푦B)! (푖∗)∗ Funcolim(P(A), C) Funcolim(P(B), C)

This is what we aimed to prove. □

Combining the above lemmas, we deduce: 4.8. Theorem. Let A and B be small ∞-categories, C be an ∞-category with all small colimits, and 푖 퐹 A −→ B −→ C be functors with 푖 fully faithful. Let 퐹̃ ∶ P(B) → C denote the left Kan extension of 퐹 along the Yoneda embedding 푦B. There is a factorisation

̃ P(B) 퐹 C

푖∗ 퐹̃′ P(A) if and only if 퐹 is the left Kan extension of its restriction to A, i.e. the natural map

퐹푖 colim(푖/푏 → A −−→ C) → 퐹(푏) is an equivalence for every 푏 ∈ B.

Proof. In Lemma 4.4 we showed that 퐹̃ factors through 푖∗ if and only if the natural transformation ∗ 퐹푖̃ !푖 → 퐹̃ ∗ induced by the counit of the 푖! ⊣ 푖 adjunction is an equivalence. By Lemma 4.5, the left-hand ∗ side of this is equivalent to Lan푦A (퐹푖) ∘ 푖 , while by Lemma 4.6, this in turn is equivalent to

Lan푦B푖(퐹푖). Since both sides of

̃ (4.9) Lan푦B푖(퐹푖) → 퐹 preserve colimits in P(B), we can check whether (4.9) is an equivalence after restricting along ∗ 푦B. Therefore, 퐹̃ factors through 푖 if and only if the natural transformation

Lan푖(퐹푖) → 퐹 is an equivalence. □ 86 Joshua Hunt

5. The stable module ∞-category is 푆-homotopy invariant

We can now return to the specific case that interests us, namely

푖 StMod(−) O op O op 푝(퐺) −→ (퐺) −−−−−−−−→ Cat̂ ∞.

′ P O op We wish to show that we have an induced functor 퐹̃ ∶ ( 푝(퐺)) → Cat̂ ∞. By the dual of Theorem 4.8, this happens if and only if the natural map

StMod op O op StMod(퐺/퐻) → lim ((푖/(퐺/퐻)) → 푝(퐺) −−−−−→ Cat̂ ∞) is an equivalence for every 퐻 ≤ 퐺. The slice category 푖/(퐺/퐻) is naturally equivalent to the 푝- orbit category O푝(퐻), with the functor O푝(퐻) → O푝(퐺) being given by 퐻/푃 ↦ 퐺/푃. Mathew proves a subgroup decomposition of StMod푘퐻 over the orbit category: 5.1. Theorem ([Mat16, Corollary 9.16]). Let C be a collection of subgroups of 퐻 that is closed under intersection and such that every elementary abelian 푝-subgroup of 퐻 is contained in a subgroup in C . There is an equivalence of symmetric monoidal ∞-categories

∼ StMod푘퐻 −−→ lim StMod푘퐾 . 퐻/퐾∈OC (퐻)op In light of Theorem 2.1, it is therefore enough to transport the above decomposition, applied C S to = 푝(퐻) ∪ {1}, up to StMod푘퐺: 5.2. Lemma. The natural map

StMod(퐺/퐻) −−→∼ lim StMod(퐺/푃) op 퐻/푃∈O푝(퐻) is an equivalence.

M 퐺 퐺 퐺 Proof. Let denote ModStMod푘퐺 (퐴퐻 ); recall from Section 2 that 퐴퐻 is coind퐻 (푘) and that StMod(퐺/퐻) is equal to M by definition. By [Lur17, Corollary 3.4.1.9], for any 퐴 ∈ CAlg(M) we have a natural equivalence

M ∼ Mod (퐴) −−→ ModStMod푘퐺 (퐴)

퐺 given by forgetting the 퐴퐻 -module structure. Therefore, we wish to prove that

퐺 M → lim ModM(퐴 ) op 푃 퐻/푃∈O푝(퐻) is an equivalence. Recall from Theorem 2.1 that coinduction induces a functor

G ∼ M coindH ∶ StMod푘퐻 −−→ Decompositions of the stable module ∞-category 87 that is an equivalence of symmetric monoidal ∞-categories. We obtain a commutative diagram

퐺 M lim ModM(퐴 ) op 푃 퐻/푃∈O푝(퐻) ∼

G 퐺 coind lim ModM(coind (퐴퐻 )) H op 퐻 푃 퐻/푃∈O푝(퐻)

G coindH StMod ∼ lim Mod (퐴퐻 ) 푘퐻 op StMod푘퐻 푃 퐻/푃∈O푝(퐻) whose bottom arrow is an equivalence by Mathew’s Theorem 5.1. □ We have therefore established: O op 5.3. Theorem. The right Kan extension of StMod(−)∶ (퐺) → Cat̂ ∞ along the opposite of O S the Yoneda embedding (퐺) → 퐺 factors through the restriction map Sop P O op P O op 퐺 ≃ ( (퐺)) → ( 푝(퐺)) . S In particular, StMod(−) sends 푆-equivalences in 퐺 to equivalences of ∞-categories. 5.4. Remark. As noted in the introduction, the only part of this argument that was non-formal was checking the descent statement in Lemma 5.2. Therefore, given a collection F of sub- O op groups of 퐺 that is closed under intersections and a functor 퐹 ∶ (퐺) → Cat̂ ∞ that has a F ̃ Sop ̂ subgroup decomposition associated with , the extended functor 퐹 ∶ 퐺 → Cat∞ inverts the weak equivalences associated with F .

6. Decompositions of the stable module ∞-category

In this section, we recall Dwyer’s construction [Dwy98, Sections 3.4–3.7] of 퐺-spaces that encode candidates for the subgroup, centraliser, and normaliser decompositions. These 퐺-spaces are (homotopy) colimits, so applying StMod(−) gives a limit of ∞-categories that receives a comparison map from StMod푘퐺; the 퐺-space encodes a decomposition precisely when this map is an equivalence. We also have 푆-homotopy equivalences between these 퐺-spaces, so can use the fact that StMod(−) inverts 푆-homotopy equivalences of 퐺-spaces to “propagate” Mathew’s subgroup decomposition to centraliser and normaliser decompositions. Let C be a collection of subgroups of 퐺. Note that we can consider a 퐺-set as a discrete 퐺-space. Centraliser: Recall from Section 2 that the fusion category FC (퐺) has objects given by the elements of C and morphisms given by group homomorphisms that are induced by F op S C conjugation in 퐺. We have a functor 훼 ∶ C (퐺) → 퐺 that takes 퐻 ∈ to the conjugacy class of the inclusion 퐻 ↪ 퐺, which is isomorphic to 퐺/퐶퐺(퐻) as a 퐺-set. The colimit of 훼 is a 퐺-space that we will denote EFC (퐺). O S O Subgroup: We have an inclusion functor 훽 ∶ C (퐺) → 퐺. We let E C (퐺) denote the colimit of 훽. Normaliser: Recall from Section 2 that the orbit simplex category 푠푆C (퐺) is the poset of 퐺- conjugacy classes of non-degenerate simplices in the nerve of C , ordered by refinement. op S We have a functor 훿 ∶ 푠푆C (퐺) → 퐺 that takes a 퐺-orbit of simplices [휎] to itself, 88 Joshua Hunt

considered as a discrete 퐺-space. This 퐺-set is isomorphic to 퐺/푁퐺(휎), where for 휎 = C (푃0 < … < 푃푛) we define 푁퐺(휎) ≔ ⋂0≤푖≤푛 푁퐺(푃푖). We let sd( ) denote the colimit of 훿. 6.1. Remark. Note that colim(훿) really is 퐺-equivalent to the subdivision of the nerve of the poset C , justifying our notation. This can be checked directly using the model for colim 훿 given by the diagonal of a bisimplicial set that is explained in [BK72, XII 5.2]. 훼 훽 훿 6.2. Remark. Dwyer calls these three spaces 푋C , 푋C , and sd 푋C , respectively. For historical reasons, EFC (퐺) is sometimes called EAC elsewhere in the literature. Since StMod(−) sends small colimits of 퐺-spaces to limits of ∞-categories, we get

StMod(EFC (퐺)) ≃ lim StMod(훼(퐻)) 퐻∈FC (퐺)

≃ lim StMod(퐺/퐶퐺(퐻)) 퐻∈FC (퐺) ≃ lim StMod . 푘퐶퐺 (퐻) 퐻∈FC (퐺) F The natural map E C (퐺) → ∗ induces a restriction functor StMod푘퐺 → limFC (퐺) StMod푘퐶 (퐻), F 퐺 so in this way E C (퐺) encodes a candidate for a centraliser decomposition for StMod푘퐺. Sim- ilarly, EOC (퐺) → ∗ induces a functor

StMod푘퐺 → lim StMod푘퐻 퐻∈OC (퐺)op corresponding to a subgroup decomposition and sd(C ) → ∗ induces a functor StMod → lim StMod 푘퐺 푘푁퐺 (휎) 휎∈푠푆C (퐺) corresponding to a normaliser decomposition. However, for a general collection C there is no reason for any of these functors from StMod푘퐺 to be an equivalence. We have comparison maps between the 퐺-spaces associated with a collection: sd(C ) ∼

EOC (퐺) C EFC (퐺) C Here the map from sd( ) is the 퐺-equivalence sending a simplex 푃0 < … < 푃푛 to 푃0; the map O F from E C (퐺) sends a point 푥 ∈ 퐺/퐻 to its stabiliser 퐺푥 ; and the map from E C (퐺) sends an inclusion 퐻 ↪ 퐺 to 퐻. Let S푝(퐺) denote the collection of all non-trivial 푝-subgroups and A푝(퐺) denote the subcollection of non-trivial elementary abelian 푝-subgroups. Recall that Mathew’s Theorem 5.1 shows that for certain collections, including C = S푝(퐺) ∪ {1} and C = A푝(퐺) ∪ {1}, we obtain a subgroup decomposition. However, to obtain a useful centraliser or normaliser decomposition, we need to remove the trivial subgroup from the collection, otherwise StMod푘퐺 itself will appear in the decomposition on the right hand side. We therefore need a minor variation of Theorem 5.1: 6.3. Lemma. Let C be a collection of subgroups of 퐺 that is closed under intersection and such that every elementary abelian 푝-subgroup of 퐺 is contained in a subgroup in C . Let C ∗ denote the non-trivial subgroups in C . There is an equivalence of symmetric monoidal ∞-categories ∼ StMod푘퐺 −−→ lim StMod푘퐻 . 퐺/퐻∈OC ∗ (퐺)op Decompositions of the stable module ∞-category 89

Proof. Since StMod(퐺/{1}) ≃ ∗, the diagram of ∞-categories that includes the trivial subgroup is a right Kan extension of the diagram that omits it, so the limits of the two diagrams agree. □ ∗ ∗ Lemma 6.3 shows we have a subgroup decomposition for C = S푝(퐺) and C = A푝(퐺). We now transfer this result to other collections and decompositions using the fact that StMod(−) inverts 푆-equivalences. We consider the following collections of 푝-subgroups: let B푝(퐺) be the collection of non-trivial 푝-radical subgroups, i.e. non-trivial 푝-subgroups 푃 ≤ 퐺 such that 푃 I is the maximal normal 푝-subgroup in 푁퐺(푃). Let the collection 푝(퐺) consist of all non-trivial subgroups that are intersections of a set of Sylow 푝-subgroups in 퐺. Finally, let Z푝(퐺) be the subcollection of A푝(퐺) consisting of those subgroups 푉 such that 푉 is the set of elements in the centre of 퐶퐺(푉 ) whose order divides 푝, i.e. such that 푉 = Ω1푂푝푍(퐶퐺(푉 )).

6.4. Theorem. Let C be one of the collections S푝(퐺), A푝(퐺), B푝(퐺), I푝(퐺), or Z푝(퐺). There is a subgroup decomposition ∼ StMod푘퐺 −−→ lim StMod푘푃 퐺/푃∈OC (퐺)op and a normaliser decomposition StMod −−→∼ lim StMod . 푘퐺 푘푁퐺 (휎) [휎]∈푠푆C (퐺)

If C is S푝(퐺), A푝(퐺), or Z푝(퐺), then there is additionally a centraliser decomposition StMod −−→∼ lim StMod . 푘퐺 푘퐶퐺 (푃) 푃∈FC (퐺)

Proof. We can restate the conclusion of Lemma 6.3 for S푝(퐺) and A푝(퐺) as saying that O O (6.5) E S푝(퐺)(퐺) → ∗ and E A푝(퐺)(퐺) → ∗ are both sent to equivalences by StMod(−). We now transport this information along 푆-homotopy equivalences. The following table is taken from [GS06, Theorem 1.1]; a solid line denotes a 퐺-homotopy equivalence, while a dashed line denotes an 푆-homotopy equivalence. The column labels represent the different collections of subgroups, where for concision we omit 푝 and 퐺 from the notation.

B I S A Z EOC (퐺)

EFC (퐺)

The subgroup decompositions arising from the maps in (6.5) correspond to the points EOS (퐺) and EOA(퐺), marked in red. The equivalences in the first row show that we have a subgroup decomposition for all of the collections in the table. More interestingly, the 푆-equivalence EOS (퐺) → S푝(퐺) induces a normaliser decomposition ∼ ∼ op S limOS (퐺) StMod(퐺/푃) StMod( 푝(퐺)) lim휎∈푠푆S (퐺) StMod(퐺/푁퐺(휎))

∼ StMod(퐺/퐺) 90 Joshua Hunt and hence (via the equivalences in the second row of the table) a normaliser decomposition for all the collections in the table. Finally, the same argument applied to the 푆-equivalence EFS (퐺) → S푝(퐺) gives centraliser decompositions for the collections S푝(퐺), A푝(퐺), and Z푝(퐺). □

6.6. Remark. We don’t have a condition analogous to Dwyer’s “ampleness” for decompositions of the stable module ∞-category. Recall that C is ample if the natural map C h퐺 → (∗)h퐺 ≃ B퐺 induces an isomorphism on mod 푝 homology, and that there are subgroup, centraliser, and normaliser decompositions associated with C if and only if C is ample. This latter statement is due to the maps sd(C )

EOC (퐺) C EFC (퐺) being h퐺-homotopy equivalences, i.e. 퐺-maps that are homotopy equivalences. Since these maps are not always 푆-equivalences, there is no analogous condition for the stable module ∞- category. The same phenomenon is discussed in [Dwy98, Remark 3.10] in the context of the behaviour of spectral sequences associated with the decompositions.

Appendix A. All relevant natural transformations are counits

In this section, we prove the assertion made in Remark 4.3 that the natural transformation in each step of the outline of Theorem 4.8 is given by the counit of some adjunction. The only step for which this is not obvious is the following: A.1. Lemma. Under the equivalence ∗ ∗ ∗ ∼ ∗ (푖 ) (푖!) (푦B)! −−→(푦B)!푗!푗 given by Lemmas 4.5 and 4.6, the natural transformation ∗ ∗ ∗ (A.2) (푖 ) (푖!) (푦B)! → (푦B)! ∗ ∗ ∗ induced by the counit of the (푖 ) ⊣ (푖!) adjunction corresponds to the natural transformation ∗ (A.3) (푦B)!푗!푗 → (푦B)! ∗ induced by the counit of the 푗! ⊣ 푗 adjunction. Proof. This amounts to a large diagram chase; the diagram is reproduced below. We first explain why the diagram proves the lemma, then explain why the diagram commutes. Every map in the diagram is an equivalence. For functors 퐹, 퐹 ′ ∈ Fun(C, D), we denote the mapping space of natural transformations from 퐹 to 퐹 ′ by Map(퐹, 퐹 ′). The natural transformation (A.2) is an element of the top-left mapping space in the diagram, and the two vertical maps on the left-hand edge are induced by the equivalences of Lemmas 4.5 and 4.6. We therefore aim to show that (A.2) is sent to (A.3) in the bottom-left corner of the region labelled ⃝3 . ∗ ∗ ∗ Since (A.2) is induced by the counit of the (푖 ) ⊣ (푖!) adjunction, it is sent to the identity natural transformation in the top-right corner of the square labelled ⃝2 . The maps along the right-hand edge of the diagram are either induced by cancelling inverse equivalences or by ∗ ∗ ∗ ∗ applying the identity 푦A (푖!) ≃ 푗 푦B of Diagram 4.7. One can check that these maps send the Decompositions of the stable module ∞-category 91 identity natural transformation in the top-right corner of ⃝2 to the identity natural transformation in the bottom-right corner of the diagram, which in turn is sent to (A.3) in the bottom-left corner of the region labelled ⃝3 . Therefore, it remains to establish that the diagram commutes. The square labelled ⃝2 commutes by definition of its left-hand edge; see Lemma 4.5. The region labelled ⃝3 also commutes by definition of its left-hand edge; see Lemma 4.6. All the other regions are easily seen to commute. □

∗ ∗ ∗ ∗ ∗ ∗ ∗ (푖 ) ⊣ (푖!) ∗ ∗ 푦A ∗ ∗ ∗ ∗ Map((푖 ) (푖!) (푦B)!, (푦B)!) Map((푖!) (푦B)!, (푖!) (푦B)!) Map(푦A (푖!) (푦B)!, 푦A (푖!) (푦B)!)

∗ ∗ ∗ ∗ Map(푗 푦B (푦B)!, 푦A (푖!) (푦B)!) ⃝1 ⃝2 ∗ ∗ ∗ Map(푗 , 푦A (푖!) (푦B)!)

∗ ∗ ∗ ∗ ∗ ∗ ∗ (푖 ) ⊣ (푖!) ∗ ∗ 푦A ∗ ∗ ∗ ∗ Map((푖 ) (푦A)!푗 , (푦B)!) Map((푦A)!푗 , (푖!) (푦B)!) Map(푦A (푦A)!푗 , 푦A (푖!) (푦B)!)

∗ (푦A)! ⊣ 푦A ⃝ ∗ ∗ ∗ 3 Map(푗 , 푦A (푖!) (푦B)!)

∗ ∗ ∗ (푦B)! ⊣ 푦B ∗ ∗ 푗! ⊣ 푗 ∗ ∗ ∗ Map((푦B)!푗!푗 , (푦B)!) Map(푗!푗 , 푦B (푦B)!) Map(푗 , 푗 푦B (푦B)!)

∗ ∗ ∗ Map(푗!푗 , id) ∗ Map(푗 , 푗 ) 푗! ⊣ 푗

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