Background & overview

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

associated to a primitive m-th root of unity $\omega \in \mathbb {C}$ , 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 R with an m-th root of unity $\omega \colon \mathbb {Z}/m \to R^\times $ , a natural transformation of R-algebras,

$$\begin{align*}{\mathfrak{F}}_\omega \colon R[M] \longrightarrow R^{M^*}, \end{align*}$$

from the group R-algebra of an m-torsion abelian group M to the algebra of R-valued functions on its Pontryagin dual $M^* = \hom (M, \mathbb {Q}/\mathbb {Z})$ . Furthermore, ${\mathfrak {F}}_\omega $ is an isomorphism if and only if the image of $\omega $ is primitive in every residue field of R. The classical case is recovered by taking $R=\mathbb {C}$ and $M = \mathbb {Z}/m$ .

The passage from the ordinary category of abelian groups to the $\infty $ -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,\dots ,\infty $ at an (implicit) prime p,

$$\begin{align*}\mathbb{Q} = K(0) \:\: , \:\: K(1) \:\: , \:\: K(2) \:\: , \:\: \dots \:\:, \:\: K(n) \:\: , \:\: \dots \:\:, \:\: K(\infty) = \mathbb{F}_p, \end{align*}$$

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 $0$ and p; see [Reference Hopkins and SmithHS98]. A central tool in the study of these intermediate characteristics is Lubin–Tate spectra. For each $0<n<\infty $ , this is a $K(n)$ -local commutative algebra $E_n$ that 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 [Reference Hopkins and LurieHL13], Hopkins and Lurie prove the following theorem, which resembles the discrete Fourier transform only in higher chromatic heights:

Theorem 1.1 [Reference Hopkins and LurieHL13, Corollary 5.3.26].

For all integers $n\ge 1$ , there is a natural isomorphism of $K(n)$ -local commutative $E_n$ -algebras

where M is a connective $\pi $ -finite (i.e., having only finitely many nonvanishing homotopy groups, all of which are finite) p-local $\mathbb {Z}$ -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 $\pi $ -finite spaces, reproving among other things the convergence of the $K(n)$ -based Eilenberg–Moore spectral sequence from [Reference BauerBau08].

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. (1) We lift it to the telescopic world by replacing $E_n$ with 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 $\pi $ -finite spaces (Theorem B). We also obtain an analogue of Kummer theory at heights $n\ge 1$ (Theorem C).

2. (2) We extend it over $E_n$ to all (i.e., not just $\mathbb {Z}$ -module) connective $\pi $ -finite p-local spectra (Theorem 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- $\pi $ -finite Galois extension, which is a strong analogue of the classical p-typical cyclotomic extension (Theorem F).

3. (3) We categorify it into a symmetric monoidal equivalence between $\infty $ -categories of local systems of $K(n)$ -local $E_n$ -modules on the underlying spaces of two dual $\pi $ -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 $\textrm {Sp}_{T(n)}$ is the Bousfield localization of $\textrm {Sp}$ with respect to $T(n) = F(n)[v^{-1}]$ , where $F(n)$ is (any) finite spectrum of type n with a $v_n$ -self map of the form $v\colon \Sigma ^d F(n) \to F(n)$ . It is a classical fact that $\textrm {Sp}_{K(n)} \subseteq \textrm {Sp}_{T(n)}$ , 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 $n\ge 2$ and 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 [Reference HeutsHeu21] of Heuts on unstable chromatic homotopy theory, which generalizes Quillen’s classical rational homotopy theory to higher chromatic heights, and second, the works [Reference Land, Mathew, Meier and TammeLMMT20, Reference Clausen, Mathew, Naumann and NoelCMNN20], which made major progress on establishing the conjectural chromatic redshift philosophy pioneered by Rognes (see, for example, [Reference RognesRog14]).

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 $E_n$ . Nevertheless, we show that the isomorphism of Theorem 1.1 descends from $E_n$ to 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 $\mathbb {Q}$ , one does not need to go all the way up to $\mathbb {C}$ or even $\overline {\mathbb {Q}}$ . Instead, for $\mathbb {Z}/m$ -modules, it suffices to have a primitive m-th root of unity $\omega _m$ so one can construct the Fourier transform already over the cyclotomic field $\mathbb {Q}(\omega _m)$ , which is a finite Galois extension of $\mathbb {Q}$ . In the same spirit, we observe that natural transformations as in Theorem 1.1 are in a canonical bijection with higher roots of unity $\Sigma ^n \mathbb {Z}/p^r \to E_n^\times $ of $E_n$ . Moreover, the natural isomorphisms are in a canonical bijection with those higher roots of unity that are primitive in the sense of [Reference Carmeli, Schlank and YanovskiCSY21b, Definition 4.2].

Remark 1.2. In [Reference Hopkins and LurieHL13], the isomorphism of Theorem 1.1 is constructed from a normalization of the p-divisible group $\mathbb {G}$ associated with $E_n$ – namely, an isomorphism of the top alternating power $\operatorname{{Alt}}_n (\mathbb {G})$ with the constant p-divisible group $\mathbb {Q}_p/\mathbb {Z}_p$ . It can be verified directly, that such data are equivalent to compatible systems of primitive higher roots of unity of $E_n$ ,

$$\begin{align*}\Sigma^n\mathbb{Q}_p/\mathbb{Z}_p \simeq \underrightarrow{\operatorname{lim}}\, \Sigma^n\mathbb{Z}/p^r \longrightarrow E_n^\times. \end{align*}$$

We then proceed to show that, as in the classical case, the chromatic Fourier isomorphism exists already over the higher cyclotomic extensions $R_{n,r}$ , which are certain faithful $(\mathbb {Z}/p^r)^\times $ -Galois extensions of the $K(n)$ -local sphere classifying primitive higher roots of unity in the sense of [Reference Carmeli, Schlank and YanovskiCSY21b]. The key point now is that, by [Reference Carmeli, Schlank and YanovskiCSY21b, Theorem A], the extensions $R_{n,r}$ admit faithful $T(n)$ -local lifts $R^f_{n,r}$ , 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 $n,r\ge 1$ , there exists a faithful $(\mathbb {Z}/p^r)^\times $ -Galois extension $R^f_{n,r}$ of the $T(n)$ -local sphere and a natural isomorphism of $T(n)$ -local commutative $R^f_{n,r}$ -algebras

where M is a connective $\pi $ -finite $\mathbb {Z}/p^r$ -module and $M^*$ is its Pontryagin dual.

Among other things, these natural isomorphisms provide a tool for studying group ring spectra. In particular, it is used by Nikolaus, Yuan and the second-named author in [Reference Carmeli, Nikolaus and YuanCNY] to show that finitely generated abelian groups embed fully faithfully, via the spherical group ring construction, in commutative ring spectra.

The natural isomorphisms of Theorem A are compatible with varying r. Thus, if we replace the individual extensions $R^f_{n,r}$ with the colimit $R_n^f := \underrightarrow {\operatorname {lim}}\, R_{n,r}^f$ , we obtain a telescopic Fourier transform for all connective $\pi $ -finite $\mathbb {Z}_{(p)}$ -module (or equivalently, p-local $\mathbb {Z}$ -module) spectra as in Theorem 1.1. The commutative ring spectrum $R_n^f$ can be viewed as the infinite p-typical higher cyclotomic extension and is a telescopic lift of Westerland’s $K(n)$ -local commutative ring spectrum $R_n$ (see [Reference WesterlandWes17]). However, in contrast with $R_n$ , it is not known whether $R_n^f$ is faithful. This subtle point might also shed some new light on (the failure of) the telescope conjecture. Localizing $\textrm {Sp}_{T(n)}$ with respect to $R_n^f$ forms an interesting intermediate localization between $\textrm {Sp}_{K(n)}$ and $\textrm {Sp}_{T(n)}$ . In particular, if one speculates that $R_n^f$ is, in fact, $K(n)$ -local, the telescope conjecture becomes equivalent to the faithfulness of $R_n^f$ .

As in [Reference Hopkins and LurieHL13], we deduce from Theorem A several structural properties of local systems of $T(n)$ -local algebras over $\pi $ -finite spaces.

We think of the first claim as an affineness property (cf. [Reference Mathew and MeierMM15]) and study it in a general context by abstracting the $K(n)$ -local arguments of [Reference Hopkins and LurieHL13, §5.4]. In particular, in an ambidextrous setting, it both implies and is implied by (a very special case of) the second claim regarding the Eilenberg–Moore isomorphism. Roughly speaking, the affineness of A reduces the Eilenberg–Moore isomorphism to the Künneth isomorphism for the fibers of B and C over A, which in an ambidextrous setting is guaranteed by the $\pi $ -finiteness assumptions. The suitable special case of the Eilenberg–Moore isomorphism is then deduced from the chromatic Fourier transform, replacing ‘cohomology’ with ‘homology’, for which the analogous claim holds for formal reasons.

Generally, Eilenberg–Moore type isomorphisms play a vital role in relating stable and unstable homotopy theory. In particular, Gijs Heuts has been employing the second claim of Theorem B in the study of unstable telescopic chromatic homotopy theory [Reference HeutsHeu]. The third claim of Theorem B, which we learned from Dustin Clausen, that the Galois condition is automatically satisfied, is also a rather formal consequence of affineness and might have been known to other experts (in the $K(n)$ -local case), though we are not aware that it has appeared in the literature. This result is also important for the study of Galois descent in $T(n)$ -localized algebraic K-theory in an upcoming work of the last three authors.

As a further corollary of the above, we obtain an analogue of Kummer theory at higher chromatic heights $n\ge 1$ . Here, a connective $\mathbb {Z}/p^r$ -module spectrum M is called n-finite if it is $\pi $ -finite and n-truncated.

Theorem C. (7.34)

Let R be a $T(n)$ -local commutative algebra admitting a primitive higher $p^r$ -th root of unity, and let M be a connected n-finite $\mathbb {Z}/p^r$ -module spectrum. Then,

$$\begin{align*}\{\,\Omega M\text{-Galois extensions of R}\,\} \:\simeq\: \operatorname{{Map}}(\Sigma^nM^*, R^\times). \end{align*}$$

That is, we obtain a classification of abelian Galois extensions of a commutative algebra in terms of its units, in the presence of enough primitive (higher) roots of unity.Footnote ¹

Spherical orientations

Generalizing Theorem 1.1 in another direction, we show that when one works $K(n)$ -locally and over $E_n$ , the chromatic Fourier transform extends to all connective $\pi $ -finite p-local spectra (i.e., not just $\mathbb {Z}$ -modules), provided that we replace Pontryagin duality by Brown–Comenetz duality.

Theorem D (7.8).

There is a natural isomorphism of $K(n)$ -local commutative $E_n$ -algebras

where M is a connective $\pi $ -finite p-local spectrum and $I M$ is its Brown–Comenetz dual.

Theorem D generalizes Theorem 1.1, as for $\mathbb {Z}$ -module spectra the Brown–Comenetz dual identifies canonically with the Pontryagin dual. Conversely, it can be obtained from Theorem 1.1 by a bootstrap procedure. To begin with, for a $K(n)$ -local commutative algebra R, a compatible system of primitive higher $p^r$ -th roots of unity defines a map

$$\begin{align*}\Sigma^n \mathbb{Q}_p/\mathbb{Z}_p \simeq \underrightarrow{\operatorname{lim}}\, \Sigma^n \mathbb{Z}/p^r \longrightarrow R^\times. \end{align*}$$

Denoting by $I_{\mathbb {Q}_p/\mathbb {Z}_p}$ the Brown–Comenetz spectrum, we show that extensions of the chromatic Fourier transformation over R to non- $\mathbb {Z}$ -module spectra are in a natural bijection with solutions to the following extension problem:

We call such extensions spherical orientations of R and think of them as spherical analogues of (compatible systems of) primitive higher roots of unity of R. Using a devissage argument, we show that the Fourier transform associated to a spherical orientation is an isomorphism for all connective $\pi $ -finite p-local spectra. By studying the obstructions for extending higher roots of unity to spherical orientations, we construct a universal R with a spherical orientation, the spherical cyclotomic extension, and prove that it is $K(n)$ -locally faithful. In fact, we even construct it $T(n)$ -locally and show that it is faithful over the intermediate localization,

$$\begin{align*}\textrm{Sp}_{K(n)} \:\subseteq\: (\textrm{Sp}_{T(n)})_{R_n^f} \:\subseteq\: \textrm{Sp}_{T(n)}. \end{align*}$$

We further introduce a natural higher connectedness property for commutative algebras, which guarantees the canonical vanishing of said obstructions, providing a practical criterion for extending the chromatic Fourier transform to non- $\mathbb {Z}$ -module spectra for specific choices of R. As a consequence of the ‘chromatic Nullstellensatz’ of the third author with Burklund and Yuan [Reference Burklund, Schlank and YuanBSY22], this criterion is satisfied for $R = E_n$ , thus yielding Theorem D.

Remark 1.4. More specifically, the required ingredient is [Reference Burklund, Schlank and YuanBSY22, Proposition 8.14],

$$\begin{align*}\operatorname{{Map}}_{\textrm{Sp}}(C_p , \textrm{pic}(E_n)) \:\simeq\: B^{n+1} C_p. \end{align*}$$

Looping this isomorphism once yields

$$\begin{align*}\operatorname{{Map}}_{\textrm{Sp}}(C_p , E_n^\times) \:\simeq\: B^{n} C_p, \end{align*}$$

which was conjectured, and subsequently proven, by Hopkins and Lurie (see [Reference Hopkins and LurieHL13, Conjecture 5.4.14]), though we are not aware of a written account of their proof. We note that the second isomorphism suffices for establishing Theorem D for all connective $\pi $ -finite p-local spectra whose n-truncation admits a module structure over the $(n-1)$ -truncated sphere spectrum.

As a further consequence of Theorem D, we deduce that a spherical orientation of $E_n$ identifies the connective cover of $\Sigma ^nI_{\mathbb {Q}_p/\mathbb {Z}_p}$ with the universal left approximation of $E_n^\times $ by an ind- $\pi $ -finite connective p-local spectrum. We denote the latter by $\mu _{\mathbb {S}_{p}}(E_n)$ as a ‘spherical’ analogue of the spectrum of ordinary p-typical roots of unity of $E_n$ . We then reinterpret $\mu _{\mathbb {S}_{p}}(E_n)$ as the p-local part of the connective cover of the so-called discrepancy spectrum of $E_n$ , which was defined by Ando, Hopkins and Rezk to be the fiber of the localization map $E_n^\times \to L_n E_n^\times $ . To summarize, we obtain the following result, originally announced by Hopkins and Lurie:

As the discrepancy spectrum is constructed from $K(n)$ -local ingredients, it is somewhat remarkable that one can read off of it the first n stable homotopy groups of spheres, which are a ‘global’ invariant. This relation also ties the chromatic filtration with the Postnikov filtration, which are, generally speaking, two quite ‘orthogonal’ filtrations in stable homotopy theory.

To systematize the study of the various flavours of the chromatic Fourier transform, and to facilitate dévissage arguments, we introduce a general notion of an $\mathfrak {R}$ -orientation of height n for every connective p-local commutative ring spectrum $\mathfrak {R}$ . This is the type of data from which one can construct the chromatic Fourier transform for connective $\pi $ -finite $\mathfrak {R}$ -module spectra. The case $\mathfrak {R} = \mathbb {Z}/p^r$ recovers primitive height n roots of unity, while the case $\mathfrak {R} = \mathbb {S}_{(p)}$ recovers spherical orientations. Furthermore, we show that if $\mathfrak {R}$ is itself n-truncated and $\pi $ -finite, then the associated $\mathfrak {R}$ -cyclotomic extension, classifying $\mathfrak {R}$ -orientations, is $\mathfrak {R}^\times $ -Galois. Note that, in general, $\mathfrak {R}^\times $ is not a finite discrete group, but a $\pi $ -finite group in the homotopical sense. We deduce from this that the spherical cyclotomic extension is a pro- $\pi $ -finite Galois extension.

The classical p-typical cyclotomic extension $\mathbb {Q}_p(\omega _{p^\infty })$ , and the corresponding p-typical cyclotomic character classifying it

$$\begin{align*}\chi \colon \operatorname{{Gal}}(\overline{\mathbb{Q}}_p/\mathbb{Q}_p) \longrightarrow \mathbb{Z}_p^\times = \tau_{\le 0}\mathbb{S}_p^\times, \end{align*}$$

plays a fundamental role in number theory in the formulation of various arithmetic duality theorems, via the construction of Tate twists. In [Reference Carmeli, Schlank and YanovskiCSY21b], it is shown that Westerland’s ring spectrum $R_n$ , which is a $\mathbb {Z}_p^\times $ -Galois extension of $\mathbb {S}_{K(n)}$ , can be similarly viewed as a p-typical higher cyclotomic extension. As the pro-finite Galois extensions of $\mathbb {S}_{K(n)}$ are classified by the Morava stabilizer group $\mathbb {G}_n$ , associated to $R_n$ is the higher cyclotomic character

$$\begin{align*}\chi \colon \mathbb{G}_n := \operatorname{{Gal}}(E_n/\mathbb{S}_{K(n)}) \longrightarrow \mathbb{Z}_p^\times, \end{align*}$$

which is essentially the determinant map. This higher cyclotomic character plays a similarly fundamental role in chromatic homotopy theory, via the construction of the determinant sphere featuring in Gross-Hopkins duality. Theorem F, together with the theory developed in this paper, suggests that the spherical cyclotomic extension should assume a similarly pivotal role in $K(n)$ -local higher Galois theory, which deals with pro- $\pi $ -finite Galois extensions of $\mathbb {S}_{K(n)}$ . A more systematic account of this circle of ideas and their applications is a subject for a future work.

Categorification

Finally, we also extend Theorem 1.1 by way of categorification. The key feature of $\textrm {Sp}_{K(n)}$ that allows the chromatic Fourier transform to be an isomorphism is higher semiadditivity in the sense of [Reference Hopkins and LurieHL13, Definition 4.4.2]. Moreover, the chromatic height n, which appears as the ‘shift’ in the chromatic Fourier transform, can be interpreted as the semiadditive height of $\textrm {Sp}_{K(n)}$ in the sense of [Reference Carmeli, Schlank and YanovskiCSY21a]. Similarly, our telescopic lift of Theorem 1.1 relies on the higher semiadditivity of $\textrm {Sp}_{T(n)}$ , which is the maximal higher semiadditive localization of $\textrm {Sp}$ of height n (see [Reference Carmeli, Schlank and YanovskiCSY22]). We therefore construct and study the higher Fourier natural transformation in the general setting of higher semiadditive symmetric monoidal $\infty $ -categories of a given semiadditive height n (at an implicit prime p). However, it is not easy to determine, in this abstract setting, when the Fourier transform is an isomorphism, and a large portion of this paper is devoted to developing tools for answering this question.

One interesting source of examples of higher semiadditive $\infty $ -categories, outside of the stable realm, is higher category theory itself. As already observed in [Reference Hopkins and LurieHL13], the $\infty $ -category ${\textrm {Pr}}$ of presentable $\infty $ -categories is $\infty $ -semiadditive. More generally, for any presentably symmetric monoidal $\infty $ -category ${\mathscr {C}}$ , the $\infty $ -category ${{\operatorname {Mod}}}_{{\mathscr {C}}}$ of ${\mathscr {C}}$ -linear presentable $\infty $ -categories is $\infty $ -semiadditive.

Regarding roots of unity, a height n root of unity $\omega \colon \Sigma ^n {\mathbb {Z}}/m \to R^\times $ of a commutative algebra R in ${\mathscr {C}}$ deloops uniquely to a height $n+1$ root of unity

$$\begin{align*}\overline{\omega}\colon \Sigma^{n+1} \mathbb{Z}/m \longrightarrow \textrm{pic}(R) := {\operatorname{Mod}}_R^\times \end{align*}$$

of ${{\operatorname {Mod}}}_R$ , viewed as a commutative algebra in ${{\operatorname {Mod}}}_{{\mathscr {C}}}$ . Note that the ‘shift by one’ in the height is consistent with the semiadditive redshift phenomenon from [Reference Carmeli, Schlank and YanovskiCSY21a].

Namely, if ${\mathscr {C}}$ happens to be itself $\infty $ -semiadditive and R is of semiadditive height n in ${\mathscr {C}}$ , then ${{\operatorname {Mod}}}_R$ is of semiadditive height $n+1$ in ${{\operatorname {Mod}}}_{{\mathscr {C}}}$ . As the semiadditive height generalizes the chromatic height, this is strongly related to the chromatic redshift philosophy. Now, given $\omega $ as above, our theory provides a Fourier transform of commutative R-algebras in ${\mathscr {C}}$

$$\begin{align*}{\mathfrak{F}}_{\omega}\colon R[M] \longrightarrow R^{\Omega^{\infty-n}M^{*}}, \end{align*}$$

and also a ‘categorified’ Fourier transform of symmetric monoidal ${{\operatorname {Mod}}}_R$ -linear presentable $\infty $ -categories

$$\begin{align*}{\mathfrak{F}}_{\overline{\omega}}\colon {\operatorname{Mod}}_R[M] \longrightarrow {\operatorname{Mod}}_R^{\Omega^{\infty-(n+1)}M^*}. \end{align*}$$

Unpacking the definitions, the right-hand side is the $\infty $ -category of ${{\operatorname {Mod}}}_R$ -valued local systems on $\Omega ^{\infty -(n+1)}M^{*}$ with the pointwise symmetric monoidal structure, while the source is the $\infty $ -category of ${{\operatorname {Mod}}}_R$ -valued local systems on $\Omega ^\infty M$ with the Day convolution symmetric monoidal structure. Our main result regarding this situation is that ${\mathfrak {F}}_{\overline {\omega }}$ is an isomorphism if and only if ${\mathfrak {F}}_{\omega }$ is an isomorphism.

Categorification of spherical orientations is more subtle. In general, there can be an obstruction for delooping a height n spherical orientation $\Sigma ^nI_{\mathbb {Q}_p/\mathbb {Z}_p} \to R^\times $ of R to a height $n+1$ spherical orientation $\Sigma ^{n+1}I_{\mathbb {Q}_p/\mathbb {Z}_p} \to \textrm {pic}(R)$ of ${{\operatorname {Mod}}}_R$ . However, it does give an $\mathfrak {R}$ -orientation for the truncated p-local sphere spectrum ${\mathfrak {R}} = \tau _{\le n}{\mathbb {S}}_{(p)}$ . In the case of the $\infty $ -category of $K(n)$ -local $E_n$ -modules, this partial result reads as follows:

Theorem G (7.9).

For every $n\ge 0$ , there is a natural equivalence of symmetric monoidal $\infty $ -categories:

for M a connective $\pi $ -finite p-local spectrum, assuming the vanishing of the canonical map

$$\begin{align*}\pi_{n+1}\mathbb{S} \otimes \pi_0 M \longrightarrow \pi_{n+1} M. \end{align*}$$

The technical assumption on M is equivalent to the $(n+1)$ -truncation of M having a (necessarily unique) module structure over the n-truncated sphere spectrum. This happens, for example, if M admits a $\mathbb {Z}$ -module structure, as in the original statement of Theorem 1.1, in which case the equivalence exists already over $R_n$ (or even $R_n^f$ ). We conjecture, however, that this hypothesis is unnecessary – namely, that the spherical orientation of $E_n$ can be delooped to a spherical orientation of . This would happen if, for example,

$$\begin{align*}\operatorname{{Map}}(C_p, \textrm{br}(E_n)) \simeq B^{n+2}C_p, \end{align*}$$

where $\textrm {br}$ stands for the Brauer spectrum (cf. Remark 1.4).

The height $n=0$ case of Theorem G recovers a classical fact. For a finite abelian group A and $M = A^*$ , we get $\Omega ^\infty M = A^*$ and $\Omega ^{\infty - 1} IM = BA$ . We therefore obtain an equivalence between the symmetric monoidal (derived) categories of $\overline {\mathbb {Q}}$ -representations of A and of $A^*$ -graded $\overline {\mathbb {Q}}$ -vector spaces. This equivalence is provided, unsurprisingly, by decomposition into weight spaces. Assuming A is a p-group, for a general height n and $M = A^*$ , we get a similar ‘weight space decomposition’ of the symmetric monoidal $\infty $ -category of -representations of the higher group $G = B^nA$ ,

For an example of a different flavor, in height $n=1$ with $M = \Sigma A$ , we get an equivalence between -representations of A and of $A^*$ , but with different symmetric monoidal structures. A similar (non-monoidal) equivalence was considered by Treumann in [Reference TreumannTre15].

Outline of chromatic applications

The bulk of this paper is devoted to the development of the formal theory of Fourier transforms and affineness in an abstract setting. The reader primarily concerned with the applications to chromatic homotopy theory might thus be interested in a compressed outline of the path leading to these applications indicating where and how exactly this formal theory is being used. We shall hence provide such an outline for a sample of the results involving the Lubin–Tate spectrum $E_n$ , culminating in the computation of the discrepancy spectrum.

The first main result in this direction is the extension of the natural isomorphism constructed by Hopkins and Lurie,

from $\pi $ -finite $\mathbb {Z}_{(p)}$ -module spectra M to arbitrary $\pi $ -finite p-local spectra (Theorem D). As already explained, this is equivalent to solving the extension problem for orientations:

The approach we take is to break this problem up into two steps:

1. (1) Constructing some map $\overline {\omega }$ as above (i.e., an $\mathbb {S}_{(p)}$ -pre-orientation) extending $\omega $ .

2. (2) Showing that any such extension $\overline {\omega }$ is automatically an $\mathbb {S}_{(p)}$ -orientation.

The proof of (2) proceeds by a dévissage argument. Let us say that a $\pi $ -finite p-local spectrum M is oriented if the associated chromatic Fourier transform map ${\mathfrak {F}}_{\overline {\omega }}$ is an isomorphism at M. The base case consists of spectra M admitting a $\mathbb {Z}_{(p)}$ -module structure. These are shown to be oriented using the isomorphism of Hopkins and Lurie by constructing the Fourier transform sufficiently functorially to allow pushing forward (pre-)orientations along the map $\mathbb {S}_{(p)}\to \mathbb {Z}_{(p)}$ . As every $\pi $ -finite p-local spectrum M is an iterated extension of $\pi $ -finite $\mathbb {Z}_{(p)}$ -module spectra (via the Postnikov tower), it suffices to show that oriented modules are closed under extensions – namely, that given a cofiber sequence $M' \to M \to M"$ of $\pi $ -finite p-local spectra, if $M'$ and $M"$ are oriented, then so is M (see Proposition 4.11 and Proposition 4.12).

To prove the closure of oriented modules under extensions, we first show that the source and target of the Fourier transform are cofiber-preserving functors. For the source, this is obvious, and for the target, it is deduced from the Eilenberg–Moore isomorphism, which follows from affineness and is itself established by a separate dévissage argument from the orientability of $\mathbb {Z}_{(p)}$ -modules. This implies that oriented modules are closed under cofibers. As the $\infty $ -category of commutative algebras is not stable, closure under cofibers does not automatically imply closure under extensions. To get that, we show that the chromatic Fourier transform canonically promotes to a morphism of Hopf algebras (see Corollary 3.31). Loosely speaking, we exploit the fact that we are dealing with an exact sequence of group objects. Finally, as one actually deals with (co)fibers in connective spectra, there is a fringe effect at $\pi _0(M)$ . Hence, to complete the proof, one also has to use the fact that the chromatic Fourier transform intertwines Brown–Comenetz duality with Spanier–Whitehead duality (see Proposition 3.25).

The above outlined argument for (2) is quite general, as it only uses that $\mathbb {S}_{(p)} \to \mathbb {Z}_{(p)}$ is a map of local ring spectra that induces an isomorphism on residue fields. In particular, it even suffices that $\overline {\omega }$ restricts to an $\mathbb {F}_p$ -orientation (i.e., a height n primitive p-th root of unity). In contrast, the proof of (1) relies on some very special properties of $E_n$ , which we recast and abstract in the theory of categorical connectedness, as we explain next.

We say that a commutative algebra $R \in \operatorname{{CAlg}}({\mathscr {C}})$ is d-connected if the canonical map of spaces

$$\begin{align*}A \longrightarrow \operatorname{{Map}}_{ \operatorname{{CAlg}}({\mathscr{C}})}(R^A, R) \end{align*}$$

is an isomorphism for all d-finite p-spaces A. The relevance of this notion to our discussion is that it allows for inductively extending (pre-)orientations of R along the (double) tower

$$\begin{align*}\tau_{\le d-1}\mathbb{S}_{(p)} \to \dots \to \mathbb{Z}_{(p)} \to \dots \to \mathbb{Z}/p^r \to \dots \to \mathbb{F}_p \end{align*}$$

by showing that the respective obstructions vanish for formal reasons (see Proposition 6.44).

Moreover, when ${\mathscr {C}}$ is of height n, a $\tau _{\le n}\mathbb {S}_{(p)}$ -orientation of R is already the same as an $\mathbb {S}_{(p)}$ -orientation (see Remark 3.4).

The fact that $E_n$ is n-connected was conjectured and subsequently proved (though not published) by Hopkins and Lurie. Using affineness, the condition of n-connectedness can be reduced from all n-finite p-spaces A to the single special case $A = B^n C_p$ . Moreover, using the $\mathbb {Z}_{(p)}$ -linear Fourier transform of Hopkins and Lurie, this is equivalent to

$$\begin{align*}\mu_p(E_n) := \operatorname{{Map}}(C_p, E_n^\times) \simeq B^n C_p. \end{align*}$$

In this form, it was also proven by a different method in the work of the third author with Burklund and Yuan on the chromatic nullstellensatz [Reference Burklund, Schlank and YuanBSY22]. Combined with the obstruction theory for orientation extensions mentioned above, this allows one to extend an arbitrary $\mathbb {F}_p$ -orientation of $E_n$ to a $\tau _{\le n-1}\mathbb {S}_{(p)}$ -orientation. This is, however, not completely satisfactory as it is still one step away from providing a full spherical orientation for $E_n$ .

As a height n commutative algebra R is never $(n+1)$ -connected, the last extension step indeed requires an essential new ingredient. By the semiadditive redshift principle, categorification has the effect of rising the semiadditive height by 1, which in turn gives access to one additional homotopy group of $\mathbb {S}_{(p)}$ . More precisely, the symmetric monoidal $\infty $ -category of $K(n)$ -local $E_n$ -modules is a commutative algebra in ${\textrm {Pr}}$ , which is of semiadditive height $n+1$ . Thus, it can be and in fact is $(n+1)$ -connected. To establish this fact, one needs the following strengthening of the Hopkins–Lurie conjecture (also proved in [Reference Burklund, Schlank and YuanBSY22]):

In addition, we need to show that an $\mathbb {F}_p$ -orientation (of height n) of $E_n$ deloops to an $\mathbb {F}_p$ -orientation (of height $n + 1$ ) of , and conversely, the extended $\tau _{\le n}\mathbb {S}_{(p)}$ -orientation of loops back to a $\tau _{\le n}\mathbb {S}_{(p)}$ -orientation of $E_n$ . To this end, we analyze the categorical Fourier transform and derive an explicit formula (see Proposition 5.13) for its evaluation at connected modules in terms of the Fourier transform over $E_n$ . To analyze the categorical Fourier transform for arbitrary modules, we again employ the dévissage and reflection techniques discussed earlier.

We conclude this outline with an explanation of how all this relates to the discrepancy spectrum of $E_n$ . Given a $T(n)$ -local commutative ring R, its connective spectrum of units $R^\times $ need not be $T(n)$ - or even $L_n^f$ -local. In general, its p-localization fits into a cofiber sequence

$$\begin{align*}C_n^f R^\times \longrightarrow (R^\times)_{(p)} \longrightarrow L_n^f R^\times. \end{align*}$$

The $L_n^f$ -acyclic part $C_n^f R^\times $ , or some close variant thereof, is called the discrepancy spectrum of R. A result announced by Hopkins and Lurie states that the discrepancy spectrum of $E_n$ identifies with the shifted Brown–Comenetz dual $\Sigma ^n I_{\mathbb {Q}_p/\mathbb {Z}_p}$ on connective covers. This fact admits an interesting interpretation, and consequently a proof, from the perspective of our theory. First, it can be shown in general that $\tau _{\ge 0}C_n^f(R) \simeq \mu _{\mathbb {S}_{p}}(R)$ (see Theorem 7.23), where the right-hand side is the co-reflection of $R^\times $ onto the full subcategory of ind- $\pi $ -finite p-local spectra. Less formally, $\mu _{\mathbb {S}_{p}}(R)$ is a $\pi $ -finite analogue of the spectrum of p-power roots of unity of R. By adjunction, a spherical pre-orientation on R is the same as a map $\tau _{\ge 0}\Sigma ^nI_{\mathbb {Q}_p/\mathbb {Z}_p} \to \mu _{\mathbb {S}_{p}}(R)$ . Thus, given an isomorphism $\mu _{\mathbb {S}_{p}}(R) \simeq \tau _{\ge 0}\Sigma ^nI_{\mathbb {Q}_p/\mathbb {Z}_p}$ (see Corollary 6.59), there is a canonical spherical pre-orientation on R, which can be easily seen to be an orientation. Conversely, and somewhat surprisingly, we show that if R admits a spherical orientation and is n-connected, then we must have $\mu _{\mathbb {S}_{p}}(R) \simeq \tau _{\ge 0}\Sigma ^nI_{\mathbb {Q}_p/\mathbb {Z}_p}$ . Given the above, this in particular implies that $\mu _{\mathbb {S}_{p}}(E_n) \simeq \tau _{\ge 0}\Sigma ^nI_{\mathbb {Q}_p/\mathbb {Z}_p}$ , reproducing the announced result of Hopkins and Lurie.

Organization

In Section 2, we study the notion of affineness. We begin in 2.1 by introducing it in the general setting of monoidal functors. We establish equivalent characterizations for affine functors and certain closure and monoidal properties thereof. Then, in 2.2, we specialize to functors arising as pullbacks for local systems and relate the property of affineness to the behavior of Eilenberg–Moore maps and Galois extensions. We end this section, in 2.3, by studying the interaction of affineness with ambidexterity and semiadditive height, establishing among other things the mutual implications of affineness and the Eilenberg–Moore isomorphism.

In Section 3, we lay down the foundations of the abstract theory of Fourier transforms. In 3.1, we define $\mathfrak {R}$ -pre-orientations of height n for a connective p-local commutative ring spectrum $\mathfrak {R}$ and the corresponding notion of Brown–Comenetz duality for $\mathfrak {R}$ -modules. In 3.2, we proceed to construct the Fourier (not necessarily invertible) transformation associated to an $\mathfrak {R}$ -pre-orientation and discuss its functoriality and duality invariance. We conclude, in 3.3, by promoting the Fourier transform to a map of Hopf algebras and deducing an analogue of the classical translation invariance property.

In Section 4, we study $\mathfrak {R}$ -orientations, which are $\mathfrak {R}$ -pre-orientations such that the associated Fourier transform is an isomorphism for all suitably finite $\mathfrak {R}$ -modules. In 4.1, after presenting the relevant definitions and functorialities, we establish for a given $\mathfrak {R}$ -pre-orientation various closure properties of the class of oriented modules (those for which the Fourier transform is an isomorphism). Next, in 4.2, we introduce the universal $\mathfrak {R}$ -oriented algebras, the $\mathfrak {R}$ -cyclotomic extensions, and show that they are $\mathfrak {R}^\times $ -Galois under certain finiteness hypotheses. Then, in 4.3, we discuss virtual orientability, which is the property of admitting an $\mathfrak {R}$ -orientation after a faithful extension of scalars, or equivalently, that the $\mathfrak {R}$ -cyclotomic extension is faithful. We show that under the assumption of virtual $\mathfrak {R}$ -orientability, the underlying space of a suitably finite $\mathfrak {R}$ -module is affine and deduce a general form of higher Kummer theory. Finally, in 4.4, we focus on local rings $\mathfrak {R}$ and show that an $\mathfrak {R}$ -pre-orientation can be checked to be an orientation after pushforward to the residue field of $\mathfrak {R}$ . This result is essential in lifting primitive (higher) p-th roots of unity to spherical orientations.

In Section 5, we investigate the interaction of the Fourier transform with categorification. In the preliminary subsection 5.1, we review the categorification-decategorification adjunction taking, in one direction, an $\mathbb {E}_n$ -algebra to its $\mathbb {E}_{n-1}$ -monoidal category of modules, and in the other, an $\mathbb {E}_{n-1}$ -monoidal $\infty $ -category to the $\mathbb {E}_n$ -algebra of (enriched) endomorphisms of the unit. We reinterpret the property of affineness in terms of this adjunction and address certain set-theoretical issues related to ${\textrm {Pr}}$ not being itself presentable. In 5.2, we initiate the study of the ‘categorified’ Fourier transform, by describing its source and target in explicit terms and explaining how the ‘decategorified’ Fourier transform can be recovered from it. We continue, in 5.3, to show that orientations categorify to orientations and that the categorical $\mathfrak {R}$ -cyclotomic extension is the categorification of the usual $\mathfrak {R}$ -cyclotomic extension.

In Section 6, we concentrate on $\mathfrak {R}$ -(pre)-orientations for local ring spectra $\mathfrak {R}$ with residue field $\mathbb {F}_p$ , paying special attention to the following tower of rings:

$$\begin{align*}\mathbb{S}_{(p)} \to \dots \to \tau_{\le d}\mathbb{S}_{(p)} \to \dots \to \mathbb{Z}_{(p)} \to \dots \to \mathbb{Z}/p^r \to \dots \to \mathbb{F}_p. \end{align*}$$

In 6.1, we study the consequences of virtual $\mathbb {F}_p$ -orientability (at height n). In particular, we show that it implies affineness for all $\pi $ -finite spaces as in Theorem B. We also show that it implies virtual $\mathfrak {R}$ -orientability for all connective $\pi $ -finite commutative ring spectra with residue field $\mathbb {F}_p$ , such as $\mathfrak {R} = \mathbb {Z}/p^r$ . In 6.2, we relate $\mathbb {Z}/p^r$ -orientations in the stable setting to primitive higher $p^r$ -th roots of unity in the sense of [Reference Carmeli, Schlank and YanovskiCSY21b] and deduce that virtual $\mathbb {F}_p$ -orientability is detected by nil-conservative functors. We also characterize virtual $\mathbb {F}_p$ -orientability in terms of the affineness of certain spaces and the Galois condition for the higher cyclotomic extensions. We proceed, in 6.3, to study the consequences of virtual $\mathbb {Z}_{(p)}$ -orientability. First, we show that it implies virtual $\mathfrak {R}$ -orientability for all local ring spectra $\mathfrak {R}$ with residue field $\mathbb {F}_p$ , and so, in particular, that the spherical cyclotomic extension is faithful (the case $\mathfrak {R}=\mathbb {S}_{(p)}$ ). Second, we show that in the stable setting, it implies that the localization with respect to the infinite p-typical higher cyclotomic extension (or equivalently, the spherical cyclotomic extension) is a smashing localization and provide a formula for its unit. Finally, in 6.4, we study truncated spherical orientations (i.e., $\mathfrak {R}$ -orientations for $\mathfrak {R} = \tau _{\le d}\mathbb {S}_{(p)}$ ). Observing that the case $d = n$ is already equivalent to an $\mathbb {S}_{(p)}$ -orientation, we study the obstructions for lifting a $\tau _{\le d-1}\mathbb {S}_{(p)}$ -orientation to a $\tau _{\le d}\mathbb {S}_{(p)}$ -orientation for $d = 1,\dots ,n$ . To this end, we introduce the notion of d-connectedness, which ensures that the said obstructions vanish up to d, and relate this property to the connectedness of the (ordinary) roots of unity $\mu _p$ and ‘spherical’ roots of unity $\mu _{\mathbb {S}_{p}}$ spectra. The last step $d = n$ requires a categorification argument from Section 5 and, accordingly, involves the Picard spectrum. We conclude with a discussion of the spherical cyclotomic extension, showing it is pro-Galois.

In Section 7, we apply the abstract theory developed in the previous sections to chromatic homotopy theory. In the preliminary subsection 7.1, we briefly recall some material on the monochromatic $\infty $ -categories $\textrm {Sp}_{K(n)}$ and $\textrm {Sp}_{T(n)}$ , the Lubin–Tate spectra $E_n$ , and the higher cyclotomic extensions $R_{n,r}$ and $R_{n,r}^f$ . In 7.2, we study the Fourier theory over $E_n$ . First, we interpret Theorem 1.1 as the existence of a $\mathbb {Z}_{(p)}$ -orientation on $E_n$ and bootstrap it to an $\mathbb {S}_{(p)}$ -orientation, proving Theorem D. By the general theory, this readily implies Theorem F on the $K(n)$ -local spherical cyclotomic extension, and Theorem G on the categorified spherical Fourier transform. We conclude with the study of the discrepancy spectrum, proving Theorem E. Finally, in 7.3, we apply the theory of orientations and the Fourier transform to the telescopic setting. We first prove that $\textrm {Sp}_{T(n)}$ is virtually $\mathbb {Z}/p^r$ -orientable and deduce Theorem A. From the general theory, we immediately deduce all the properties of local systems of $K(n)$ -local algebras on $\pi $ -finite spaces stated in Theorem B and the higher Kummer theory as formulated in Theorem C. We conclude with a short discussion about the universal $T(n)$ -local virtually spherically oriented localization and its relation to the properties of $R_n^f$ and the telescope conjecture.

Notation and conventions

Throughout the paper, we work in the framework of $\infty $ -categories (a.k.a. quasi-categories) as developed in [Reference LurieLur09] and [Reference LurieLur]. We generally follow the terminology and notation therein. For all concepts related to semiadditivity, semiadditive height and higher cyclotomic extensions, we refer the reader to [Reference Carmeli, Schlank and YanovskiCSY21a] and [Reference Carmeli, Schlank and YanovskiCSY21b]; precise references are given in the main body of the text.

In addition, we employ the following notation:

1. (1) The underlying space of a spectrum X will be denoted by $\lfloor {X}\rfloor := \Omega ^{\infty }X$ . More generally, for an object X in a monoidal $\infty $ -category ${\mathscr {C}}$ , we write .

2. (2) A square in an $\infty $ -category is called exact if it is both a pullback and a pushout square.

3. (3) In our applications, we usually fix a prime p and work p-locally. We usually indicate the prime through a subscript; for instance, we write $\textrm {Sp}_{(p)}$ for the category of p-local spectra.

4. (4) A p-local spectrum X is said to be $\pi $ -finite if it is connective and $\bigoplus _n\pi _nX$ is a finite abelian group. We write $\textrm {Sp}_{(p)}^{\pi \text {-}\textrm {tor}}$ for the category of p-local $\pi $ -torsion spectra (Definition 6.52) (i.e., those connective spectra which can be written as filtered colimits of p-local $\pi $ -finite spectra). The p-local $\pi $ -torsion torsion part of a p-local spectrum X is denoted by $(X)_{(p)}^{\pi \text {-}\textrm {tor}}$ .