The Chromatic Fourier Transform

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Tobias Barthel ∗Shachar Carmeli†Tomer M. Schlank‡Lior Yanovski§

Abstract

We develop a general theory of higher semiadditive Fourier transforms that includes both

the classical discrete Fourier transform for ﬁnite abelian groups at height n= 0, as well as a

certain duality for the En-(co)homology of π-ﬁnite spectra, established by Hopkins and Lurie,

at heights n≥1. We use this theory to generalize said duality in three diﬀerent directions.

First, we extend it from Z-module spectra to all (suitably ﬁnite) spectra and use it to compute

the discrepancy spectrum of En. Second, we lift it to the telescopic setting by replacing Enwith

T(n)-local higher cyclotomic extensions, from which we deduce various results on aﬃneness,

Eilenberg–Moore formulas and Galois extensions in the telescopic setting. Third, we categorify

their result into an equivalence of two symmetric monoidal ∞-categories of local systems of

K(n)-local En-modules, and relate it to (semiadditive) redshift phenomena.

The Great Wave oﬀ Kanagawa, Katsushika Hokusai.

∗Max Planck Institute for Mathematics.

†Department of Mathematics, University of Copenhagen.

‡Einstein Institute of Mathematics, Hebrew University of Jerusalem.

§Einstein Institute of Mathematics, Hebrew University of Jerusalem.

arXiv:2210.12822v1 [math.AT] 23 Oct 2022

Contents

1 Introduction 3

2 Aﬃneness and Eilenberg–Moore 12

2.1 Aﬃnefunctors ....................................... 13

2.2 Aﬃnenessforlocalsystems ................................ 19

2.3 Aﬃneness and ambidexterity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3 The Higher Fourier Transform 29

3.1 Pre-orientations....................................... 29

3.2 TheFouriertransform ................................... 32

3.3 co-Multiplicative properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4 Orientations and Orientability 41

4.1 Orientations......................................... 41

4.2 R-Cyclotomicextensions.................................. 46

4.3 Virtual orientability and aﬃneness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.4 Detectionforlocalrings .................................. 54

5 Categoriﬁcation and Redshift 57

5.1 Categoriﬁcation....................................... 57

5.2 The categorical Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.3 Orientations and categoriﬁcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

6 Orientations for Thickenings of Fp65

6.1 Fp-Orientationsandaﬃneness............................... 65

6.2 Z/pr-Orientations and higher roots of unity . . . . . . . . . . . . . . . . . . . . . . . 68

6.3 Z(p)-Orientations...................................... 72

6.4 τ≤dS(p)-Orientations and connectedness . . . . . . . . . . . . . . . . . . . . . . . . . 75

7 Chromatic Applications 88

7.1 Chromaticpreliminaries .................................. 88

7.2 Orientations of the Lubin–Tate ring spectrum . . . . . . . . . . . . . . . . . . . . . . 91

7.3 Virtual orientability of SpT(n)...............................100

References 103

1 Introduction

Background & overview

The classical m-dimensional Discrete Fourier Transform (DFT) is a linear isomorphism

Fω:Cm∼

−−! Cm,

associated to a primitive m-th root of unity ω∈C, whose characteristic property is transforming

the convolution product on the source to the pointwise product on the target. More generally, one

can associate to every commutative ring Rwith an m-th root of unity ω:Z/m !R×, a natural

transformation of R-algebras,

Fω:R[M]−! RM∗,

from the group R-algebra of an m-torsion abelian group Mto the algebra of R-valued functions

on its Pontryagin dual M∗= hom(M, Q/Z). Furthermore, Fωis an isomorphism if and only if the

image of ωis primitive in every residue ﬁeld of R. The classical case is recovered by taking R=C

and M=Z/m.

Passing from the ordinary category of abelian groups to the ∞-category of spectra, i.e., from

classical commutative algebra to stable homotopy theory, introduces new “characteristics”. The

Morava K-theory ring spectra of heights n= 0,...,∞at an (implicit) prime p,

Q=K(0) , K(1) , K(2) , . . . , K(n), . . . , K(∞) = Fp,

are in a precise sense the prime ﬁelds in the world of spectra, and can be thought of as providing an

interpolation between the classical characteristics 0and p; see [HS98]. A central tool in the study of

these intermediate characteristics is Lubin–Tate spectra. For each 0<n<∞, this is a K(n)-local

commutative algebra Enthat can be realized as the algebraic closure of the K(n)-local sphere,

and which has deep connections to the algebraic geometry of formal groups making it amenable

to computations. For example, in [HL13], Hopkins and Lurie prove the following theorem, which

resembles the discrete Fourier transform, only in higher chromatic heights:

Theorem 1.1 ([HL13, Corollary 5.3.26]).For all integers n≥1, there is a natural isomorphism of

K(n)-local commutative En-algebras

En[M]∼

−−! EΩ∞−nM∗

where Mis a connective π-ﬁnite (i.e., having only ﬁnitely many non-vanishing homotopy groups,

all of which are ﬁnite) p-local Z-module and M∗is its Pontryagin dual.

Furthermore, they deduce from this result several fundamental structural properties of local systems

of K(n)-local algebras on π-ﬁnite spaces, reproving among other things the convergence of the K(n)-

based Eilenberg–Moore spectral sequence from [Bau08].

In this paper, we develop a general theory that formalizes and substantiates the analogy between

Theorem 1.1 and the classical Fourier transform. In particular, we reinterpret both in terms of a

uniﬁed notion of a chromatic Fourier transform isomorphism for all ﬁnite chromatic heights, and

show that it shares many of the formal properties of the classical Fourier transform. We then apply

this theory to generalize Theorem 1.1 in three diﬀerent directions:

(1) We lift it to the telescopic world, by replacing Enwith certain faithful Galois extensions of

the T(n)-local sphere (Theorem A). By analogy with the K(n)-local case, we deduce several

structural results for local systems of T(n)-local algebras over π-ﬁnite spaces (Theorem B).

We also obtain an analogue of Kummer theory at heights n≥1(Theorem C).

(2) We extend it over Ento all (i.e., not just Z-module) connective π-ﬁnite p-local spectra (The-

orem D), and deduce from this the conjectured description of the discrepancy spectrum of

Ando–Hopkins–Rezk in terms of the Brown–Comenetz spectrum (Theorem E). As another

application, we construct a certain K(n)-local pro-π-ﬁnite Galois extension, which is a strong

analogue of the classical p-typical cyclotomic extension (Theorem F).

(3) We categorify it into a symmetric monoidal equivalence between ∞-categories of local systems

of K(n)-local En-modules on the underlying spaces of two dual π-ﬁnite spectra. Among other

things, this generalizes the weight space decomposition of representations of ﬁnite abelian

groups in characteristic zero (Theorem G). We also explain how this categoriﬁcation accords

with semiadditive redshift phenomena.

We shall now discuss each of these sets of results in some more detail, and outline along the way

some of the key aspects of the general theory.

Telescopic lift

Recall that the telescopic localization SpT(n)is the Bousﬁeld localization of Sp with respect to

T(n) = F(n)[v−1], where F(n)is (any) ﬁnite spectrum of type nwith a vn-self map of the form

v: ΣdF(n)!F(n). It is a classical fact that SpK(n)⊆SpT(n), and a long standing conjecture

of Ravenel, known as the telescope conjecture, states that the two localizations are in fact equal.

While proven to be true in heights n= 0,1, the telescope conjecture is widely believed to be

false for all n≥2and all primes p. In recent years, the telescopic localizations gained new

interest (independently of the status of the telescope conjecture) due to their pivotal role in several

remarkable developments, of which we mention two. First, the work [Heu21] of Heuts on unstable

chromatic homotopy theory, which generalizes Quillen’s classical rational homotopy theory to higher

chromatic heights. And second, the works [LMMT20,CMNN20], which made a major progress on

establishing the conjectural chromatic redshift philosophy pioneered by Rognes (see, e.g., [Rog14]).

The T(n)-localizations are considerably less amenable to computations than the corresponding

K(n)-localizations, largely due to the lack of a (faithful) telescopic lift of En. Nevertheless, we

show that the isomorphism of Theorem 1.1 descends from Ento a deeper base, which does admit

a faithful telescopic lift and over to which the chromatic Fourier transform lifts as well. To explain

this in more detail, we ﬁrst note that while the classical Fourier transform is not deﬁned over Q,

one does not need to go all the way up to Cor even Q. Instead, for Z/m-modules, it suﬃces to

have a primitive m-th root of unity ωm, so one can construct the Fourier transform already over

the cyclotomic ﬁeld Q(ωm), which is a ﬁnite Galois extension of Q. In the same spirit, we observe

that natural transformations as in Theorem 1.1 are in a canonical bijection with higher roots of

unity ΣnZ/pr!E×

nof En. Moreover, the natural isomorphisms are in a canonical bijection with

those higher roots of unity that are primitive in the sense of [CSY21b, Deﬁnition 4.2].

Remark 1.2. In [HL13], the isomorphism of Theorem 1.1 is constructed from a normalization of

the p-divisible group Gassociated with En, namely, an isomorphism of the top alternating power

Altn(G)with the constant p-divisible group Qp/Zp. It can be veriﬁed directly, that such data are

equivalent to compatible systems of primitive higher roots of unity of En,

ΣnQp/Zp'lim

−! ΣnZ/pr−! E×

We then proceed to show that, as in the classical case, the chromatic Fourier isomorphism exists

already over the higher cyclotomic extensions Rn,r, which are certain faithful (Z/pr)×-Galois exten-

sions of the K(n)-local sphere classifying primitive higher roots of unity in the sense of [CSY21b].

The key point now is that, by [CSY21b, Theorem A], the Rn,r -s admit faithful T(n)-local lifts

n,r, the corresponding T(n)-local higher cyclotomic extensions. Consequently, the general theory

developed in this paper, combined with the nilpotence theorem, allows us to lift the chromatic

Fourier transform to the telescopic world.

Theorem A (7.33).For every n, r ≥1, there exists a faithful (Z/pr)×-Galois extension Rf

n,r of

the T(n)-local sphere and a natural isomorphism of T(n)-local commutative Rf

n,r-algebras

Fω:Rf

n,r[M]∼

−−! (Rf

n,r)Ω∞−nM∗,

where Mis a connective π-ﬁnite Z/pr-module and M∗is its Pontryagin dual.

The natural isomorphisms of Theorem A are compatible with varying r. Thus, if we replace the

individual Rf

n,r-s with the colimit Rf

n:= lim

−! Rf

n,r, we obtain a telescopic Fourier transform for all

connective π-ﬁnite Z(p)-module (or equivalently, p-local Z-module) spectra as in Theorem 1.1. The

commutative ring spectrum Rf

ncan be viewed as the inﬁnite p-typical higher cyclotomic extension

and is a telescopic lift of Westerland’s K(n)-local commutative ring spectrum Rn(see [Wes17]).

However, in contrast with Rn, it is not known whether Rf

nis faithful. This subtle point might also

shed some new light on (the failure of) the telescope conjecture. Localizing SpT(n)with respect

to Rf

nforms an interesting intermediate localization between SpK(n)and SpT(n). In particular, if

one speculates that Rf

nis in fact K(n)-local, the telescope conjecture becomes equivalent to the

faithfulness of Rf

As in [HL13], we deduce from Theorem A several structural properties of local systems of T(n)-local

algebras over π-ﬁnite spaces.

Theorem B. Let Abe a π-ﬁnite space such that π1(A, a)is a p-group and πn+1(A, a)is of order

prime to p, for all a∈A.

(1) (7.29) For every R∈Alg(SpT(n))A, the global sections functor induces a symmetric monoidal

equivalence

ModR(SpT(n))A∼

−−! ModA∗R(SpT(n)).

(2) (7.30) For every R∈Alg(SpT(n))and spaces Band Cmapping to A, one of which is π-ﬁnite,

the canonical Eilenberg–Moore map is an isomorphism:

RB⊗RARC∼

−−! RB×AC.

(3) (7.31) Assuming Ais connected, every R∈CAlg(SpT(n))Ais ΩA-Galois, in the sense of

Rognes, over the global sections (i.e., ΩA-homotopy ﬁxed points) algebra A∗R∈CAlg(SpT(n)).

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