The tt-geometry of permutation modules. Part I Stratification

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THE TT-GEOMETRY OF PERMUTATION MODULES

PART I: STRATIFICATION

PAUL BALMER AND MARTIN GALLAUER

Abstract. We consider the derived category of permutation modules over a

ﬁnite group, in positive characteristic. We stratify this tensor triangulated cat-

egory using Brauer quotients. We describe the set underlying the tt-spectrum

of compact objects, and discuss several examples.

Contents

1. Introduction 1

2. Recollections and Koszul objects 5

3. Restriction, induction and geometric ﬁxed-points 10

4. Modular ﬁxed-points functors 12

5. Conservativity via modular ﬁxed-points 17

6. The spectrum as a set 21

7. Cyclic, Klein-four and quaternion groups 26

8. Stratiﬁcation 30

References 33

1. Introduction

1.1. Convention. We place ourselves in the setting of modular representation theory

of ﬁnite groups. So unless mentioned otherwise, Gis a ﬁnite group and kis a ﬁeld

of characteristic p > 0, with ptypically dividing the order of G.

Permutation modules. Among the easiest representations to construct, permu-

tation modules are simply the k-linearizations k(X) of G-sets X. And yet they

play an important role in subjects as varied as derived equivalences [Ric96], Mackey

functors [Yos83], or equivariant homotopy theory [MNN17], to name a few. The

authors’ original interest stems from yet another connection, namely the one with

Voevodsky’s theory of motives [Voe00], speciﬁcally Artin motives. For a gentle

introduction to these ideas, we refer the reader to [BG21].

We consider a ‘small’ tensor triangulated category, the homotopy category

(1.2) K(G) := Kbperm(G;k)\

Date: October 18, 2022.

2020 Mathematics Subject Classiﬁcation. 18F99; 20C20, 20J06, 18G80.

Key words and phrases. Tensor-triangular geometry, permutation modules, stratiﬁcation.

First-named author supported by NSF grant DMS-2153758. Second-named author supported

by the Max-Planck Institute for Mathematics in Bonn. The authors thank the Hausdorﬀ Institute

for Mathematics in Bonn for its hospitality during the ﬁnal write-up of this paper.

arXiv:2210.08311v1 [math.RT] 15 Oct 2022

2 PAUL BALMER AND MARTIN GALLAUER

of bounded complexes of permutation kG-modules, idempotent-completed. (1) It

sits as the compact part K(G) = T(G)cof the ‘big’ tensor triangulated category

(1.3) T(G) := DPerm(G;k)

obtained for instance by closing K(G) under coproducts and triangles in the homo-

topy category K(Mod(kG)) of all kG-modules. We call T(G) the derived category

of permutation kG-modules. Details can be found in Recollection 2.2.

This derived category of permutation modules T(G) is amenable to techniques of

tensor-triangular geometry [Bal10], a geometric approach that brings organization

to otherwise bewildering tensor triangulated categories, in topology, algebraic ge-

ometry or representation theory. Tensor-triangular geometry has led to many new

insights, for instance in equivariant homotopy theory [BS17,BHN+19,BGH20].

Our goal in the present paper and its follow-up [BG22c] is to understand the tensor-

triangular geometry of the derived category of permutation modules, both in its

big variant T(G) and its small variant K(G). In Part III of the series [BG22d] we

extend our analysis to proﬁnite groups and to Artin motives.

Having sketched the broad context and the aims of the series, let us now turn to

the content of the present paper in more detail.

Stratiﬁcation. In colloquial terms, one of our main results says that the big de-

rived category of permutation modules is strongly controlled by its compact part:

1.4. Theorem (Theorem 8.11).The derived category of permutation modules T(G)

is stratiﬁed by Spc(K(G)) in the sense of Barthel-Heard-Sanders [BHS21a].

Let us remind the reader of BHS-stratiﬁcation. What we establish in Theo-

rem 8.11 is an inclusion-preserving bijection between the localizing ⊗-ideals of T(G)

and the subsets of the spectrum Spc(K(G)). This bijection is deﬁned via a canonical

support theory on T(G) that exists once we know that Spc(K(G)) is a noetherian

space (Proposition 8.1). Note that Theorem 1.4 cannot be obtained via ‘BIK-

stratiﬁcation’ as in Benson-Iyengar-Krause [BIK11], since the endomorphism ring

of the unit Hom•

K(G)(1,1) = kis too small. However, we shall see that [BIK11]

plays an important role in our proof, albeit indirectly. An immediate consequence

of stratiﬁcation is the Telescope Property (Corollary 8.12):

1.5. Corollary. Every smashing ⊗-ideal of T(G)is generated by its compact part.

The key question is now to understand the spectrum Spc(K(G)). For starters,

recall from [BG20, Theorem 5.13] that the innocent-looking category K(G) actually

captures much of the wilderness of modular representation theory. It admits as

Verdier quotient the derived category Db(kG) of all ﬁnitely generated kG-modules.

By Benson-Carlson-Rickard [BCR97], the spectrum of Db(kG) is the homogeneous

spectrum of the cohomology ring H•(G, k). We deduce in Proposition 2.22 that

Spc(K(G)) contains an open piece VG

(1.6) Spec•(H•(G, k)) ∼

=Spc(Db(kG)) =: VG→Spc(K(G))

that we call the cohomological open of G.

1Direct summands of ﬁnitely generated permutation modules are called p-permutation or

trivial source kG-modules and form the category denoted perm(G;k)\. It has only ﬁnitely many

indecomposable objects up to isomorphism. If Gis a p-group, all p-permutation modules are

permutation and the indecomposable ones are of the form k(G/H) for subgroups H6G.

TTG-PERM I: STRATIFICATION 3

In good logic, the closed complement of VGis the support

(1.7) Spc(K(G)) rVG= Supp(Kac(G))

of the tt-ideal Kac(G) = Ker(K(G)Db(kG)) of acyclic objects in K(G). The

problem becomes to understand this closed subset Supp(Kac(G)). To appreciate

the issue, let us say a word of closed points. Corollary 6.31 gives the complete

list: There is one closed point M(H) of Spc(K(G)) for every conjugacy class of

p-subgroups H6G. The open VGonly contains one closed point, for the trivial

subgroup H= 1. All other closed points M(H) for H6= 1 are to be found in

the complement Supp(Kac(G)). It will turn out that Spc(K(G)) is substantially

richer than the cohomological open VG, in a way that involves p-local information

about G. To understand this, we need the right notion of ﬁxed-points functor.

Modular ﬁxed-points. Let H6Gbe a subgroup. We abbreviate by

(1.8) G//H := WG(H) = NG(H)/H

the Weyl group of Hin G. If HPGis normal then of course G//H =G/H.

For every G-set X, its H-ﬁxed-points XHis canonically a (G//H)-set. We also

have a naive ﬁxed-points functor M7→ MHon kG-modules but it does not ‘lin-

earize’ ﬁxed-points of G-sets, that is, k(X)Hdiﬀers from k(XH) in general. And it

does not preserve the tensor product. We would prefer a tensor-triangular functor

(1.9) ΨH:T(G)→T(G//H)

such that ΨH(k(X)) = k(XH) for every G-set X.

A related problem was encountered long ago for the G-equivariant stable homo-

topy category SH(G), see [LMSM86]: The naive ﬁxed-points functor (a. k. a. the

‘genuine’ or ‘categorical’ ﬁxed-points functor) is not compatible with taking sus-

pension spectra, and it does not preserve the smash product. To solve both issues,

topologists invented geometric ﬁxed-points ΦH. Those functors already played an

important role in tensor-triangular geometry [BS17,BGH20,PSW22] and it would

be reasonable, if not very original, to try the same strategy for T(G). Such geomet-

ric ﬁxed-points ΦHcan indeed be deﬁned in our setting but unfortunately they do

not give us the wanted ΨHof (1.9), as we explain in Remark 3.11.

In summary, we need a third notion of ﬁxed-points functor ΨH, which is neither

the naive one (−)H, nor the ‘geometric’ one ΦHimported from topology. It turns

out (see Warning 4.1) that it can only exist in characteristic pwhen His a p-

subgroup. The good news is that this is the only restriction (see Section 4):

1.10. Proposition. For every p-subgroup H6Gthere exists a coproduct-preserving

tensor-triangular functor on the big derived category of permutation modules (1.3)

ΨH:T(G)−→ T(G//H)

such that ΨH(k(X)) ∼

=k(XH)for every G-set X. In particular, this functor pre-

serves compacts and restricts to a tt-functor ΨH:K(G)→K(G//H)on (1.2).

We call the ΨHthe modular H-ﬁxed-points functors. They already exist at

the level of additive categories perm(G;k)\→perm(G//H;k)\, where they agree

with the classical Brauer quotient, although our construction is quite diﬀerent.

See Remark 4.8. These ΨHalso recover motivic functors considered by Bachmann

in [Bac16, Corollary 5.48]. For us, modular ﬁxed-points functors are only a tool

that we want to use to prove theorems. So let us return to Spc(K(G)).

4 PAUL BALMER AND MARTIN GALLAUER

The spectrum. Each tt-functor ΨHinduces a continuous map on spectra

(1.11) ψH:= Spc(ΨH) : Spc(K(G//H)) −→ Spc(K(G)).

In particular Spc(K(G)) receives via this map ψHthe cohomological open VG//H of

the Weyl group of H:

(1.12) VG//H = Spc(Db(k(G//H))) →Spc(K(G//H)) ψH

−−→ Spc(K(G)).

Using this, we describe the set underlying Spc(K(G)) in Theorem 6.16:

1.13. Theorem. Every point of Spc(K(G)) is the image ψH(p)of a point p∈VG//H

for some p-subgroup H6G, in a unique way up to G-conjugation, i.e. we have

ψH(p) = ψH0(p0)if and only if there exists g∈Gsuch that Hg=H0and pg=p0.

In this description, the trivial subgroup H= 1 contributes the cohomological

open VG(since Ψ1= Id). Its closed complement Supp(Kac(G)), introduced in (1.7),

is covered by images of the modular ﬁxed-points maps (1.12), for Hrunning through

all non-trivial p-subgroups of G. The main ingredient in proving Theorem 1.13 is

our Conservativity Theorem 5.12 on the associated big categories: (2)

1.14. Theorem. The family of functors {T(G)ΨH

−−→ T(G//H)K Inj(k(G//H))}H,

indexed by the (conjugacy classes of) p-subgroups H6G, is conservative.

This determines the set Spc(K(G)). The topology of Spc(K(G)) is a separate

plot, involving new characters. The reader will ﬁnd them in Part II [BG22c].

Measuring progress by examples. Before the present work, we only knew the

case of cyclic group Cpof order p= 2, where Spc(K(C2)) is a 3-point space (3)

(1.15) Supp(Kac(C2)) ••

•VC2

This was the starting point of our study of real Artin-Tate motives [BG22b, Theo-

rem 3.14]. It appears independently in Dugger-Hazel-May [DHM22, Theorem 5.4].

We now have a description of Spc(K(G)) for arbitrary ﬁnite groups G. We gather

several examples in Section 7to illustrate the progress made since (1.15), and also

for later use in [BG22d]. Let us highlight the case of the quaternion group G=Q8

(Example 7.12). By Quillen, we know that the cohomological open VQ8is the same

as for its center Z(Q8) = C2, that is, the 2-point Sierpi´nski space displayed in green

on the right-hand side of (1.15), and again below:

Supp(Kac(Q8))=? •

•VQ8∼

=VC2

If intuition was solely based on (1.15) one could believe that Spc(K(G)) is just VG

with some discrete decoration for the acyclics, like the single (brown) point on the

left-hand side of (1.15). The quaternion group oﬀers a stark rebuttal.

2Recall that Krause’s homotopy category of injectives K Inj(k(G)) is a compactly generated

tensor triangulated category whose compact part identiﬁes with Db(k(G)).

3A line indicates specialization: The higher point is in the closure of the lower one.

TTG-PERM I: STRATIFICATION 5

Indeed, the spectrum Spc(K(Q8)) is the following space:

(1.16)

• • • • • •

VQ8∼

=VC2

•

Supp(Kac(Q8)) ∼

=Spc(K(C×2

2))

• • P1

··· ••• •

•

Its support of acyclics (in brown) is actually way more complicated than the co-

homological open itself: It has Krull dimension two and contains a copy of the

projective line P1

k. In fact, the map ψC2given by modular ﬁxed-points identiﬁes

the closed piece Supp(Kac(Q8)) with the whole spectrum for Q8/C2, which is a

Klein-four. We discuss the latter in Example 7.10 where we also explain the mean-

ing of P1

··· and the undulated lines in (1.16).

We hope that the outline of the paper is now clear from the above introduction

and the table of contents.

Acknowledgements. We thank Tobias Barthel, Henning Krause and Peter Symonds

for precious conversations and for their stimulating interest.

* * *

1.17. Terminology. A ‘tensor category’ is an additive category with a symmetric-

monoidal product additive in each variable. We say ‘tt-category’ for ‘tensor tri-

angulated category’ and ‘tt-ideal’ for ‘thick (triangulated) ⊗-ideal’. We say ‘big’

tt-category for a rigidly-compactly generated tt-category, as in [BF11].

We use a general notation (−)•to indicate everything related to graded rings.

For instance, Spec•(−) denotes the homogeneous spectrum.

For subgroups H, K 6G, we write H6GKto say that His G-conjugate

to a subgroup of K, that is, Hg6Kfor some g∈G. We write ∼Gfor G-

conjugation. As always Hg=g−1H g and gH=g H g−1. We write NG(H, K) for

g∈G

Hg6Kand NG(H) = NG(H, H) for the normalizer.

We write Subp(G) for the set of p-subgroups of G. It is a G-set via conjugation.

1.18. Convention. When a notation involves a subgroup Hof an ambient group G,

we drop the mention of Gif no ambiguity can occur, like with ResHfor ResG

Similarly, we sometimes drop the mention of the ﬁeld kto lighten notation.

2. Recollections and Koszul objects

2.1. Recollection. We refer to [Bal10] for elements of tensor-triangular geometry.

Recall simply that the spectrum of an essentially small tt-category Kis Spc(K) =

P(K

Pis a prime tt-ideal . For every object x∈K, its support is supp(x) :=

P∈Spc(K)

x /∈P. These form a basis of closed subsets for the topology.

2.2. Recollection. (Here kcan be a commutative ring.) Recall our reference [BG21]

for details on permutation modules. Linearizing a G-set X, we let k(X) be the

free k-module with basis Xand G-action k-linearly extending the G-action on X.

Apermutation kG-module is a kG-module isomorphic to one of the form k(X).

These modules form an additive subcategory Perm(G;k) of Mod(kG), with all kG-

linear maps. We write perm(G;k) for the full subcategory of ﬁnitely generated

permutation kG-modules and perm(G;k)\for its idempotent-completion.

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THETT-GEOMETRYOFPERMUTATIONMODULESPARTI:STRATIFICATIONPAULBALMERANDMARTINGALLAUERAbstract.Weconsiderthederivedcategoryofpermutationmodulesoveranitegroup,inpositivecharacteristic.Westratifythistensortriangulatedcat-egoryusingBrauerquotients.Wedescribethesetunderlyingthett-spectrumofcompactobjects,an...

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