Elliptic Cohomology III: Tempered lurie/papers/Elliptic-III-  · Elliptic Cohomology III:

Elliptic Cohomology III: Tempered Cohomology

April 17, 2019

Contents1 Introduction 3

2 Orientations and P-Divisible Groups 232.1 Preorientations of p-Divisible Groups . . . . . . . . . . . . . . . . . . 252.2 The p-Complete Case . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.3 Reduction to the p-Complete Case . . . . . . . . . . . . . . . . . . . 302.4 The Kpnq-Local Case . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.5 Orientations of p-Divisible Groups . . . . . . . . . . . . . . . . . . . . 342.6 P-Divisible Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392.7 Splitting of P-Divisible Groups . . . . . . . . . . . . . . . . . . . . . 432.8 Example: The Multiplicative P-Divisible Group . . . . . . . . . . . . 492.9 Example: Torsion of Elliptic Curves . . . . . . . . . . . . . . . . . . . 52

3 Orbispaces 533.1 The 8-Category of Orbispaces . . . . . . . . . . . . . . . . . . . . . . 563.2 Equivariant Homotopy Theory . . . . . . . . . . . . . . . . . . . . . . 593.3 Representable Morphisms of Orbispaces . . . . . . . . . . . . . . . . . 643.4 Formal Loop Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 693.5 Preorientations Revisited . . . . . . . . . . . . . . . . . . . . . . . . . 733.6 Example: Complex K-Theory . . . . . . . . . . . . . . . . . . . . . . 79

4 Tempered Cohomology 834.1 Equivariant K-Theory as Tempered Cohomology . . . . . . . . . . . . 884.2 Atiyah-Segal Comparison Maps . . . . . . . . . . . . . . . . . . . . . 92

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4.3 Character Isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . 1024.4 Tempered Cohomology of Eilenberg-MacLane Spaces . . . . . . . . . 1064.5 The Proof of Theorem 4.4.16 . . . . . . . . . . . . . . . . . . . . . . . 1124.6 The Tate Construction . . . . . . . . . . . . . . . . . . . . . . . . . . 1244.7 Base Change and Finiteness . . . . . . . . . . . . . . . . . . . . . . . 1334.8 Application: Character Theory for -Finite Spaces . . . . . . . . . . 1364.9 Application: The Completion Theorem . . . . . . . . . . . . . . . . . 140

5 Tempered Local Systems 1445.1 Pretempered Local Systems . . . . . . . . . . . . . . . . . . . . . . . 1485.2 The 8-Category LocSysGpXq . . . . . . . . . . . . . . . . . . . . . . 1575.3 Colimits of Tempered Local Systems . . . . . . . . . . . . . . . . . . 1615.4 Tempered Local Systems on Classifying Spaces . . . . . . . . . . . . . 1635.5 Recognition Principle for Tempered Local Systems . . . . . . . . . . . 1675.6 Extrapolation from Small Groups . . . . . . . . . . . . . . . . . . . . 1705.7 Digression: The 8-Category LocSysnulG pXq . . . . . . . . . . . . . . . 1805.8 Tensor Products of Tempered Local Systems . . . . . . . . . . . . . . 185

6 Analysis of LocSysGpXq 1896.1 Localization and Completions of Tempered Local Systems . . . . . . 1916.2 Change of Ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1986.3 The Infinitesimal Case . . . . . . . . . . . . . . . . . . . . . . . . . . 2026.4 Categorified Character Theory . . . . . . . . . . . . . . . . . . . . . . 2056.5 Isotropic Local Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 214

7 Ambidexterity for Tempered Local Systems 2207.1 Direct Images of Tempered Local Systems . . . . . . . . . . . . . . . 2247.2 The Tempered Ambidexterity Theorem . . . . . . . . . . . . . . . . . 2287.3 Projection Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . 2327.4 Transfer Maps in Tempered Cohomology . . . . . . . . . . . . . . . . 2397.5 Tempered Ambidexterity for p-Finite Spaces . . . . . . . . . . . . . . 2457.6 Induction Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2557.7 Proof of Tempered Ambidexterity . . . . . . . . . . . . . . . . . . . . 2697.8 Applications of Tempered Ambidexterity . . . . . . . . . . . . . . . . 2727.9 Dualizability of Tempered Local Systems . . . . . . . . . . . . . . . . 278

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1 IntroductionLet G be a finite group. We let ReppGq denote the complex representation ring of

G. That is, ReppGq is the abelian group generated by symbols rV s, where V rangesover the collection of all finite-dimensional complex representations of G, subject tothe relation

rV s rV 1s ` rV 2s

for every isomorphism of complex representations V V 1 V 2. It is a free abeliangroup of finite rank, equipped with a canonical basis consisting of elements rW s, whereW is an irreducible representations of G. We regard ReppGq as a commutative ring,whose multiplication is characterized by the formula rV s rW s rV bC W s.

If V is a finite-dimensional complex representation of G, we let V : G C denotethe character of V , given concretely by the formula

V pgq TrpVg V q.

The character V is an example of a class function on G: that is, it is invariant underconjugation (so V pgq V phgh1q for all g, h P G). Using the identities

VW pgq V pgq ` W pgq VbW pgq V pgqW pgq,

we see that the construction rV s V determines a ring homomorphism

ReppGq tClass functions : G Cu.

The starting point for the character theory of finite groups is the following result (seeCorollary 4.7.8):

Theorem 1.1.1. Let G be a finite group. Then the characters of the irreducible repre-sentations of G form a basis for the vector space of class functions on G. Consequently,the construction rV s V induces an isomorphism of complex vector spaces

CbZ ReppGq tClass functions : G Cu.

Theorem 1.1.1 can be reformulated using the language of equivariant complexK-theory (see [20]). Given a topological space X equipped with an action of G, we letKU0GpXq denote the (0th) G-equivariant complex K-group of X. If X is a finite G-CWcomplex, then KU0GpXq is a finitely generated abelian group, which can be realizedconcretely as the Grothendieck group of G-equivariant complex vector bundles on X.In particular, when X consists of a single point, we have a canonical isomorphismKU0Gpq ReppGq. Theorem 1.1.1 can be generalized as follows (see Corollary 4.7.7):

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Theorem 1.1.2. Let G be a finite group and let X be a finite G-CW complex. Foreach g P G, let Xg tx P X : xg xu denote the set of fixed points for the action ofG. We regard the disjoint union

gPGXg tpg, xq P GX : xg xu GX

as equipped with the right action of G given by the formula pg, xqh ph1gh, xhq. Thenthere is a canonical isomorphism

chG : CbZ KU0GpXq Hevpp

gPGXgq{G; Cq,

called the equivariant Chern character. Here

Hevpp

gPGXgq{G; Cq

nPZH2np

gPGXgq{G; Cq

denotes the product of the even cohomology groups of p

gPGXgq{G with coefficients

in the field C of complex numbers.

Example 1.1.3. In the special case where X consists of a single point, wecan identify the quotient p

gPGXgq{G appearing in Theorem 1.1.2 with the set of

conjugacy classes of elements of G (regarded as a finite set with the discrete topology),so that Hevpp

gPGXgq{G; Cq H0pp

gPGXgq{G; Cq is isomorphic to the vector space

of class functions : G C. Under this identification, the equivariant Chern characterchG : CbZ KU0GpXq Hevpp

gPGXgq{G; Cq corresponds to the isomorphism

CbZ ReppGq tClass functions : G Cu V V

of Theorem 1.1.1 (see Example 4.3.9).

Example 1.1.4. When the group G is trivial, the equivariant Chern character ofTheorem 1.1.2 specializes to the usual Chern character

ch : CbZ KU0pXq HevpX; Cq,

which is an isomorphism whenever X is a finite CW complex. In this case, it is notnecessary to work over the complex numbers: there is already a canonical isomorphismof rational vector spaces

QbZ KU0pXq HevpX; Qq,

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which induces the isomorphism ch after extending scalars along the inclusion Q C.Beware that this is not true in the equivariant case (even when X is a point): if Vis a finite-dimensional representation of G, then the character V : G C generallydoes not take values in Q.

Let EG denote a contractible space equipped with a free action of the finitegroup G. If X is any topological space equipped with a G-action, we let XhG denotethe homotopy orbit space of X by the action of G, defined as the quotient spacepX EGq{G. The projection map X EG X induces a homomorphism

: KU0GpXq KU0GpX EGq KU0pXhGq,

which we will refer to as the Atiyah-Segal comparison map. It is not far from being anisomorphism, by virtue of the following classical result (see Corollary 4.9.3):

Theorem 1.1.5 (Atiyah [1]). Let G be a finite group and let IG ReppGq be theaugmentation ideal, defined as the kernel of the ring homomorphism

ReppGq Z rV s dimCpV q.

For every finite G-CW complex X, the Atiyah-Segal comparison map

: KU0GpXq KU0pXhGq

exhibits KU0pXhGq as the IG-adic completion of KU0GpXq; here we regard KU0GpXq asa module over the representation ring ReppGq KU0Gpq.

The conclusion of Theorem 1.1.5 can be simplified by applying a further completion.Fix a prime number p. We say that an element g P G is p-singular if the order of g isa power of p, and we let Gppq G denote the subset consisting of p-singular elements.Let yKU denote the p-adic completion of the complex K-theory spectrum KU. Then,after p-adic completion, the Atiyah-Segal comparison map yields a homomorphism

p : ZpbZ KU0GpXq yKU0pXhGq

which is the projection onto a direct factor. After extending scalars to the complexnumbers, we can describe this direct factor concretely by the following variant ofTheorem 1.1.2:

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Theorem 1.1.6. Fix a prime number p and an embedding : Zp C. Then there isa canonical isomorphism of complex vector spaces

pchG : CbZpyKU0pXhGq Hevpp

gPGppqXgq{G; Cq.

Remark 1.1.7. In the situation of Theorem 1.1.6, the isomorphism pchG is related tothe equivariant Chern charac