# A chromatic vanishing result for TR

Liam Keenan  Department of Mathematics, University of Minnesota, USA
<EMAIL_ADDRESS>and  Jonas McCandless  Max Planck Institute for Mathematics,
Bonn, Germany<EMAIL_ADDRESS>
###### Abstract.

In this note, we establish a vanishing result for telescopically localized TR.
More precisely, we prove that $T(k)$-local TR vanishes on connective
$L_{n}^{p,f}$-acyclic $\mathbb{E}_{1}$-rings for every $1\leq k\leq n$ and
deduce consequences for connective Morava K-theory and the Thom spectra
$y(n)$. The proof relies on the relationship between TR and the spectrum of
curves on K-theory together with fact that algebraic K-theory preserves
infinite products of additive $\infty$-categories which was recently
established by Córdova Fedeli.

##  1\. Introduction

In this note, we study the telescopic localizations of TR inspired by the work
of Land–Mathew–Meier–Tamme [24] and Mathew [28]. Our starting point is the
following result which follows from the main result of [24]: If $R$ is an
$\mathbb{E}_{1}$-ring with $L_{n}^{p,f}R\simeq 0$, then

$L_{T(k)}\operatorname{K}(R)\simeq 0$

for every $1\leq k\leq n$. For instance, if $R=\mathbb{Z}/p^{n}$ for some
integer $n\geq 1$, then $L_{T(1)}\operatorname{K}(\mathbb{Z}/p^{n})\simeq 0$.
We consider this result as an extension of Quillen’s fundamental calculation
that $\operatorname{K}(\mathbb{F}_{p})_{p}^{\wedge}\simeq H\mathbb{Z}_{p}$
which in particular yields that
$L_{T(1)}\operatorname{K}(\mathbb{F}_{p})\simeq 0$. This particular
consequence was also obtained by Bhatt–Clausen–Mathew [4] by means of a
calculation in prismatic cohomology. Additionally, the vanishing result above
for $T(k)$-local K-theory can be applied to the Morava K-theories $K(n)$ and
to the Thom spectra $y(n)$ considered by Mahowald–Ravenel–Shick in [27].

###  1.1. Results

We will be interested in similar vanishing results for $T(k)$-local
$\operatorname{TR}$111Note that $L_{T(k)}\operatorname{TR}(R)\simeq
L_{T(k)}\operatorname{TR}(R,p)$, where $\operatorname{TR}(R,p)$ denotes the
$p$-typical version of $\operatorname{TR}$. Indeed, the canonical map
$\operatorname{TR}(R)\to\operatorname{TR}(R,p)$ is a $p$-adic equivalence and
$T(n)$-localization is insensitive to $p$-completion. Therefore, we will not
distinguish between $p$-typical TR and integral TR in this note.. The
invariant $\operatorname{TR}$ plays an instrumental role in the classical
construction of topological cyclic homology in [7, 19, 6], where
$\operatorname{TC}$ is obtained as the fixedpoints of a Frobenius operator on
$\operatorname{TR}$. In §3, we briefly review the construction of
$\operatorname{TR}$ following [29] which produces $\operatorname{TR}$ together
with its Frobenius operator entirely in the Borel–equivariant formalism of
Nikolaus–Scholze [30]. Even though $\operatorname{TR}$ does not feature
prominently in the construction of $\operatorname{TC}$ given in [30],
$\operatorname{TR}$ remains an important invariant by virtue of its close
relationship to the Witt vectors and the de Rham–Witt complex [18, 19, 20,
21]. In [28], Mathew proves that $T(1)$-local $\operatorname{TR}$ is
truncating on connective $H\mathbb{Z}$-algebras which means that if $R$ is a
connective $H\mathbb{Z}$-algebra, then the canonical map of spectra

$L_{T(1)}\operatorname{TR}(R)\to L_{T(1)}\operatorname{TR}(\pi_{0}R)$

is an equivalence. This property was verified for $T(1)$-local K-theory and
$T(1)$-local $\operatorname{TC}$ in [4, 24]. Our main result is a version of
this at higher chromatic heights:

######  Theorem A.

Let $n\geq 1$. If $R$ is a connective $\mathbb{E}_{1}$-ring such that
$L_{n}^{p,f}R\simeq 0$, then

$L_{T(k)}\operatorname{TR}(R)\simeq 0$

for every $1\leq k\leq n$.

We remark that Theorem A is a consequence of the work of [24] in the case
where $R$ admits a more refined multiplicative structure; If $R$ admits an
$\mathbb{E}_{m}$-ring structure for $m\geq 2$, then the refined cyclotomic
trace $\operatorname{K}(R)\to\operatorname{TR}(R)$ is a map of
$\mathbb{E}_{1}$-rings. Consequently, the spectrum
$L_{T(k)}\operatorname{TR}(R)$ admits the structure of a
$L_{T(k)}\operatorname{K}(R)$-module and $L_{T(k)}\operatorname{K}(R)\simeq 0$
by [24, Theorem 3.8]. A similar sort of reasoning has recently been employed
with great success to study redshift phenomena for algebraic K-theory in [9,
11, 15, 31]. We deduce the following results from Theorem A:

######  Corollary B.

Let $n\geq 1$. Then $L_{T(k)}\operatorname{TR}(\mathbb{Z}/p^{n})\simeq 0$ for
every $k\geq 1$.

We stress that Corollary B is a consequence of the work of [4, 24] by the
reasoning above. For $n=1$, Corollary B can also be deduced from the work of
Mathew [28]. Since $T(1)$-local $\operatorname{TR}$ is truncating on
connective $H\mathbb{Z}$-algebra it is in particular nilinvariant by [25], so

$L_{T(1)}\operatorname{TR}(\mathbb{Z}/p^{n})\simeq
L_{T(1)}\operatorname{TR}(\mathbb{F}_{p})\simeq 0,$

where the final equivalence follows since
$\operatorname{TR}(\mathbb{F}_{p},p)\simeq H\mathbb{Z}_{p}$ by
Hesselholt–Madsen [19]. As a consquence of Theorem A we deduce a new chromatic
vanishing result for the connective Morava K-theories, which we denote by
$k(n)$. While $k(n)$ admits the structure of an $\mathbb{E}_{1}$-ring, it does
not admit the structure of an $\mathbb{E}_{2}$-ring so we cannot argue using
the refined cyclotomic trace above.

######  Corollary C.

Let $n\geq 2$. Then $L_{T(k)}\operatorname{TR}(k(n))\simeq 0$ for every $1\leq
k\leq n-1$.

Similarly, we obtain a chromatic vanishing result for the Thom spectra $y(n)$
considered in [27].

###  1.2. Methods

We end by explaining the strategy of our proof of Theorem A. They key input is
the close relationship between TR and the spectrum of curves on K-theory as
studied in [3, 5, 18, 29]. For every $\mathbb{E}_{1}$-ring $R$, the spectrum
of curves on K-theory is defined by

$\mathrm{C}(R)=\varprojlim_{i}\Omega\tilde{\operatorname{K}}(R[t]/t^{i}),$

where $\tilde{\operatorname{K}}(R[t]/t^{i})$ denotes the fiber of the map
$\operatorname{K}(R[t]/t^{i})\to\operatorname{K}(R)$ induced by the
augmentation. If we assume that $R$ is connective, then
$\operatorname{TR}(R)\simeq\mathrm{C}(R)$ by [29, Corollary 4.2.5]. This
result was preceded by Hesselholt [18] and Betley–Schlichtkrull [3] who
established the result for associative rings after profinite completion.
Combining the theorem of the weighted heart (cf. [13, 17, 16]) with the recent
result of Córdova Fedeli [12, Corollary 2.11.1] which asserts that algebraic
K-theory preserves arbitrary products of additive $\infty$-categories, we
reduce to proving that

$L_{T(k)}\operatorname{K}^{\oplus}\big{(}\prod_{i\geq
1}\operatorname{Proj}^{\omega}_{R[t]/t^{i}}\big{)}\simeq 0$

provided that $L_{n}^{p,f}R\simeq 0$, where
$\operatorname{Proj}^{\omega}_{R[t]/t^{i}}$ denotes the additive
$\infty$-category of finitely generated projective $R[t]/t^{i}$-modules and
$\operatorname{K}^{\oplus}$ denotes additive algebraic K-theory. This claim
can be verified explicitly by using [24, Proposition 3.6].

#### Acknowledgements

The authors are grateful to Akhil Mathew for discussions about and interest in
this project. The first author would also like to thank Tyler Lawson for a
number of helpful conversations. The second author was funded by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s
Excellence Strategy EXC 2044 390685587, Mathematics Münster:
Dynamics–Geometry–Structure and the Max Planck Institute for Mathematics in
Bonn while working on this project.

##  2\. Preliminaries on weight structures and K-theory

The main technical apparatus for deducing our chromatic vanishing result for
TR is the notion of a weight structure on a stable $\infty$-category in
conjunction with the closely related theorem of the weighted heart (cf. [13,
16]). This will help us reduce to studying additive algebraic K-theory of
additive $\infty$-categories.

######  Definition 2.1.

A weight structure on a stable $\infty$-category $\mathscr{C}$ consists of a
pair of full subcategories $\mathscr{C}_{[0,\infty]}$ and
$\mathscr{C}_{[-\infty,0]}$ of $\mathscr{C}$ such that the following
conditions are satisfied:

  1. (1)

The full subcategories $\mathscr{C}_{[0,\infty]}$ and
$\mathscr{C}_{[-\infty,0]}$ are closed under retracts in $\mathscr{C}$.

  2. (2)

For $X\in\mathscr{C}_{[-\infty,0]}$ and $Y\in\mathscr{C}_{[0,\infty]}$, the
mapping spectrum $\operatorname{map}_{\mathscr{C}}(X,Y)$ is connective.

  3. (3)

For every $X\in\mathscr{C}$, there is a fiber sequence

$X^{\prime}\to X\to X^{\prime\prime}$

with $X^{\prime}\in\mathscr{C}_{[-\infty,0]}$ and
$X^{\prime\prime}[-1]\in\mathscr{C}_{[0,\infty]}$.

The heart of the weight structure is the subcategory
$\mathscr{C}^{\mathrm{ht}}=\mathscr{C}_{[0,0]}$, where
$\mathscr{C}_{[a,b]}=\mathscr{C}_{[a,\infty]}\cap\mathscr{C}_{[-\infty,b]}$.
The weight structure is said to be exhaustive if every object is bounded, in
the sense that

$\mathscr{C}=\bigcup_{n\in\mathbb{Z}}\mathscr{C}_{[-n,n]}.$

A weighted $\infty$-category is a stable $\infty$-category equipped with a
weight structure.

######  Remark 2.2.

The heart of a weighted $\infty$-category is an additive $\infty$-category
([16, Lemma 3.1.2]).

We recall the following terminology which will play an important role
throughout this note. For every connective $\mathbb{E}_{1}$-ring $R$, let
$\operatorname{Proj}_{R}^{\omega}$ denote the full subcategory of the
$\infty$-category $\operatorname{LMod}_{R}^{\geq 0}$ spanned by those
connective left $R$-modules which are finitely generated and projective.
Recall that an object of $\operatorname{Proj}_{R}^{\omega}$ can be written as
a retract of a finitely generated free $R$-module (cf. [26, Proposition
7.2.2.7]). For any not necessarily connective $\mathbb{E}_{1}$-ring, let
$\operatorname{Perf}_{R}$ denote the $\infty$-category of perfect $R$-modules
defined as the smallest stable subcategory of $\operatorname{LMod}_{R}$ which
contains $R$ and is closed under retracts. The following is our main example
of interest:

######  Example 2.3.

For a connective $\mathbb{E}_{1}$-ring $R$, let $\operatorname{Perf}_{R,\geq
0}$ be the full subcategory of $\operatorname{Perf}_{R}$ spanned by those
perfect $R$-modules which are connective, and let $\operatorname{Perf}_{R,\leq
0}$ denote the full subcategory of $\operatorname{Perf}_{R}$ spanned by those
perfect $R$-modules $M$ which have projective amplitude $\leq 0$. This means
that every $R$-linear map $M\to N$ is nullhomotopic provided that $N$ is
$1$-connective. The pair $(\operatorname{Perf}_{R,\geq
0},\operatorname{Perf}_{R,\leq 0})$ defines an exhaustive weight structure on
$\operatorname{Perf}_{R}$ whose heart is equivalent to the additive
$\infty$-category $\operatorname{Proj}_{R}^{\omega}$ of finitely generated
projective $R$-modules (cf. [17, 1.38 & 1.39]); while the proofs therein are
stated for connective $\mathbb{E}_{\infty}$-rings, the same arguments work in
the $\mathbb{E}_{1}$ case.

The algebraic K-theory of a weighted $\infty$-category is often determined by
the additive algebraic K-theory of its heart by virtue of the theorem of the
weighted heart first established by Fontes [13] but we also refer the reader
to [16, Corollary 8.1.3, Remark 8.1.4]. Let $\mathscr{A}$ denote an additive
$\infty$-category regarded as a symmetric monoidal $\infty$-category with the
cocartesian symmetric monoidal structure, so that the core
$\mathscr{A}^{\simeq}$ inherits the structure of an
$\mathbb{E}_{\infty}$-monoid. Recall that the additive algebraic K-theory of
$\mathscr{A}$ is defined by

$\operatorname{K}^{\oplus}(\mathscr{A})=(\mathscr{A}^{\simeq})^{\mathrm{grp}},$

where $(\mathscr{A}^{\simeq})^{\mathrm{grp}}$ denotes the group completion of
the $\mathbb{E}_{\infty}$-monoid $\mathscr{A}^{\simeq}$. We have the following
result which will play an instrumental role below (cf. [13, Theorem 5.1] and
[16, Corollary 8.1.3]):

######  Theorem 2.4.

The canonical map of spectra

$\operatorname{K}^{\oplus}(\mathscr{C}^{\mathrm{ht}})\to\operatorname{K}(\mathscr{C})$

is an equivalence for every stable $\infty$-category $\mathscr{C}$ equipped
with an exhaustive weight structure.

##  3\. Chromatic vanishing results

The main goal of this section is to prove Theorem A from §1 and discuss
various consequences. As explained, our proof of this result relies on the
close relationship between $\operatorname{TR}$ and the spectrum of curves in
K-theory (cf. [3, 18, 29]). We will regard $\operatorname{TR}$ as a functor
$\operatorname{TR}:\mathrm{Alg}_{\mathbb{E}_{1}}^{\mathrm{cn}}\to\mathrm{Sp}$
given by

$\operatorname{TR}(R)\simeq\operatorname{map}_{\mathrm{CycSp}}(\widetilde{\operatorname{THH}}(\mathbf{S}[t]),\operatorname{THH}(R))$

following [29] and this agrees with the classical construction of
$\operatorname{TR}$ by [29, Theorem 3.3.12]. By virtue of our assumption that
$R$ is connective, there is an equivalence of spectra

$\operatorname{TR}(R)\simeq\varprojlim\Omega\tilde{\operatorname{K}}(R[t]/t^{i}),$

where $\tilde{\operatorname{K}}(R[t]/t^{i})$ denotes the fiber of the map
$\operatorname{K}(R[t]/t^{i})\to\operatorname{K}(R)$ induced by the
augmentation. In this generality, the result was obtained by the second author
in [29] preceded by Hesselholt [18] and Betley–Schlichtkrull [3] who proved
the result for associative rings after profinite completion. With this
equivalence at our disposal, we prove the following result:

######  Theorem 3.1.

Let $n\geq 1$. If $R$ is a connective $\mathbb{E}_{1}$-ring such that
$L_{n}^{p,f}R\simeq 0$, then

$L_{T(k)}\operatorname{TR}(R)\simeq 0$

for every $1\leq k\leq n$.

The limit in the definition of the spectrum of curves on K-theory above does
not commute with $T(k)$-localization. Instead, the proof of Theorem 3.1 relies
on the following result, which is proved by combining the theorem of the
weighted heart and a recent result which asserts that additive algebraic
K-theory preserves infinite products of additive $\infty$-categories, due to
Córdova Fedeli [12].

######  Proposition 3.2.

Let $R$ be a connective $\mathbb{E}_{1}$-ring which vanishes after
$L_{n}^{p,f}$-localization. If $\\{S_{i}\\}_{i\in I}$ is collection of
connective $\mathbb{E}_{1}$-rings with a map of $\mathbb{E}_{1}$-rings $R\to
S_{i}$ for every $i\in I$, then

$L_{T(k)}\big{(}\prod_{i\in I}\operatorname{K}(S_{i})\big{)}\simeq 0$

for every $1\leq k\leq n$.

###### Proof.

For $i\in I$, the stable $\infty$-category $\operatorname{Perf}_{S_{i}}$
admits an exhaustive weight structure whose heart is equivalent to the
additive $\infty$-category $\operatorname{Proj}_{S_{i}}^{\omega}$ by Example
2.3. The canonical composite

$\operatorname{K}^{\oplus}\Big{(}\prod_{i\in
I}\operatorname{Proj}_{S_{i}}^{\omega}\Big{)}\to\prod_{i\in
I}\operatorname{K}^{\oplus}(\operatorname{Proj}_{S_{i}}^{\omega})\to\prod_{i\in
I}\operatorname{K}(\operatorname{Perf}_{S_{i}})$

is an equivalence by [12, Corollary 2.11.1] and Theorem 2.4, so we have
reduced to proving that

$L_{T(k)}\operatorname{K}^{\oplus}\Big{(}\prod_{i\in
I}\operatorname{Proj}_{S_{i}}^{\omega}\Big{)}\simeq 0$

for $1\leq k\leq n$. By [24, Proposition 3.6], it suffices to prove that the
endomorphism $\mathbb{E}_{1}$-rings of

$\mathscr{A}=\prod_{i\in I}\operatorname{Proj}^{\omega}_{S_{i}}$

vanish after $L_{n}^{p,f}$-localization. If $P\in\mathscr{A}$, then the
endomorphism $\mathbb{E}_{1}$-ring of $P$ is given by

$\mathrm{End}_{\mathscr{A}}(P)\simeq\prod_{i\in
I}\operatorname{map}_{S_{i}}(P_{i},P_{i}),$

where $\operatorname{map}_{S_{i}}(P_{i},P_{i})$ denotes the mapping spectrum
in $\operatorname{LMod}_{S_{i}}$. For each $i\in I$, we may choose a positive
integer $n_{i}\geq 1$ such that $P_{i}$ is a retract of $S_{i}^{\oplus n_{i}}$
by virtue of our assumption that $P_{i}$ is a finitely generated projective
$S_{i}$-module. Consequently, we obtain a retract diagram of spectra

$\mathrm{End}_{\mathscr{A}}(P)\to\prod_{i\in I}S_{i}^{\oplus
n_{i}^{2}}\to\mathrm{End}_{\mathscr{A}}(P)$

which proves the desired statement since the middle term is a left $R$-module,
hence vanishes after $L_{n}^{p,f}$-localization by virtue of our assumption
that $R$ is $L_{n}^{p,f}$-acyclic. ∎

######  Remark 3.3.

In general, $E$-acyclic spectra are not closed under infinite products; for
each $n\geq 0$, the $n$th Postnikov truncation $\tau_{\leq n}\mathbb{S}$ is
$K(1)$-acyclic, whereas $\prod_{n\geq 0}\tau_{\leq n}\mathbb{S}$ is not, else
$L_{K(1)}\mathbb{S}\simeq 0$. The assumptions of Proposition 3.2 should be
viewed as a uniformity condition on the spectra $\operatorname{K}(S_{i})$,
forcing their product to become acyclic.

###### Proof of Theorem 3.1.

Since $R$ is a connective $\mathbb{E}_{1}$-ring, there is an equivalence of
spectra $\operatorname{TR}(R)\simeq\mathrm{C}(R)$ by [29, Corollary 4.2.5].
Thus, the spectrum $\Sigma\operatorname{TR}(R)$ is the fiber of a suitable map

$\prod_{i\geq 1}\widetilde{\operatorname{K}}(R[t]/t^{i})\to\prod_{i\geq
1}\widetilde{\operatorname{K}}(R[t]/t^{i})$

which proves the desired statement as these products vanish after
$T(k)$-localization for $1\leq k\leq n$ by virtue of Theorem 3.2. ∎

######  Remark 3.4.

As remarked above, we have used work by Córdova Fedeli [12] in a crucial way.
This result on K-theory of additive $\infty$-categories is part of a long
tradition of examining the interaction of algebraic K-theory and infinite
products of categories. One of the first results of this kind is due to
Carlsson, who showed that K-theory preserves infinite products of exact
1-categories with a cylinder functor [10]. In [23], Kasprowski–Winges proved
that K-theory preserves infinite products of additive categories. Furthermore,
Kasprowski–Winges [22] used a characterization of Grayson [14] to prove that
non-connective algebraic K-theory preserves infinite products of stable
$\infty$-categories and this was used in [8] with Bunke to prove the analogous
statement of prestable $\infty$-categories.

######  Remark 3.5.

Another attempt to prove Proposition 3.2 proceeds by invoking the recent
result of Kasprowski–Winges [22], which asserts that the canonical map of
spectra

$\operatorname{K}\big{(}\prod_{i\in
I}\operatorname{Perf}(S_{i})\big{)}\to\prod_{i\in I}\operatorname{K}(S_{i})$

is an equivalence (cf. Remark 3.5). Proceeding as in the proof of Proposition
3.2, it suffices to prove that the endomorphism $\mathbb{E}_{1}$-rings of the
product of the stable $\infty$-categories $\operatorname{Perf}(S_{i})$ vanish
after $L_{n}^{p,f}$-localization. This is closely related to the following
assertion:

  1. $(\ast)$

Let $E$ denote the endomorphism $\mathbb{E}_{1}$-ring of a finite spectrum $V$
of type $n$. If $v:\Sigma^{k}E\to E$ is the associated $v_{n}$ self-map of
$E$, then there is a canonical lift of $v$ to a map of $E$-$E$-bimodules.

By the description of the $\mathbb{E}_{1}$-center as Hochschild cohomology,
the statement $(\ast)$ is equivalent to asking for a lift of the class
$v\in\pi_{\ast}(E)$ to a class
$\tilde{v}\in\pi_{\ast}\mathscr{Z}_{\mathbb{E}_{1}}(E)$ along the
$\mathbb{E}_{1}$-map $\mathscr{Z}_{\mathbb{E}_{1}}(E)\rightarrow E$. Classes
which do lift in this way can be viewed as “homotopically central” elements of
$E$, and we remark that such lifts exist for all $\mathbb{E}_{2}$-rings, by
the universal property of the $\mathbb{E}_{1}$-center.

However, the assertion $(\ast)$ is false as we learned from Maxime Ramzi, and
we thank him for help with the following argument. If such a lift exists, then
we obtain an equivalence of $L_{K(n)}$-$L_{K(n)}$-bimodules

$\varphi:\Sigma^{k}L_{K(n)}E\to L_{K(n)}E,$

and there is an equivalence of $\mathbb{E}_{1}$-rings
$\mathrm{End}_{K(n)}(L_{K(n)}V)\simeq L_{K(n)}E$ since $V$ is a finite
spectrum. The $\infty$-category of $K(n)$-local spectra is equivalent to the
$\infty$-category $\mathrm{Mod}_{L_{K(n)}E}(\mathrm{Sp}_{K(n)})$ since
$L_{K(n)}V$ is a compact generator of $\mathrm{Sp}_{K(n)}$. As a consequence,
for every $K(n)$-local spectrum $X$, we obtain an equivalence $\Sigma^{k}X\to
X$ by base-changing along $\varphi$. This is a contradiction since the
homotopy groups of a $K(n)$-local spectrum are not periodic. We indicate an
example of this at every height $n\geq 1$. Let $k$ be a perfect field of
characteristic $p$, let $\mathbb{G}$ be a $1$-dimensional formal group of
height $n$, and let $E_{n}$ denote the associated Lubin–Tate theory which
canonically carries the structure of an $\mathbb{E}_{\infty}$-ring. For every
topological generator $g$ of $\mathbb{Z}_{p}^{\times}$, there is a map of
$\mathbb{E}_{\infty}$-rings $\psi_{g}:E_{n}\to E_{n}$, and we let $F_{n}$
denote the fiber of the map

$E_{n}\xrightarrow{1-\psi_{g}}E_{n}.$

A calculation reveals that the homotopy groups of $F_{n}$ are not periodic.
For instance, if $n=1$, then $F_{1}\simeq L_{K(1)}\mathbb{S}$ since the map
$\psi_{g}$ is induced by Adams operations on
$E_{1}\simeq\mathrm{KU}_{p}^{\wedge}$.

Finally, we explore some immediate consequences of Theorem 3.1.

######  Corollary 3.6.

Let $R$ be a connective $\mathbb{E}_{1}$-algebra over $\mathbb{Z}/p^{j}$. If
$n\geq 1$, then $L_{T(n)}\operatorname{TR}(R)\simeq 0$.

###### Proof.

Note that $L_{n}^{p,f}R$ is a module over $L_{n}^{p,f}\mathbb{Z}/p^{j}\simeq
0$, so the assertion follows from Theorem 3.1. ∎

Recall that Corollary 3.6 above also follows from [4, 24, 28] as discussed in
the introduction. We deduce some consequence for connective Morava K-theory.
Let $k(n)$ denote the connective cover of the $n$th Morava
$\operatorname{K}$-theory $K(n)$. The spectrum $k(n)$ carries the structure of
an $\mathbb{E}_{1}$-ring but not the structure of an $\mathbb{E}_{2}$-ring. We
have the following:

######  Corollary 3.7.

If $n\geq 2$, then $L_{T(k)}\operatorname{TR}(k(n))\simeq 0$ for every $1\leq
k\leq n-1$.

###### Proof.

For $n\geq 2$, the canonical map $k(n)\to\mathbb{F}_{p}$ is a
$L_{n-1}^{p,f}$-local equivalence by [24, Lemma 2.2], so the assertion follows
from Theorem 3.1. ∎

######  Remark 3.8.

There is a fiber sequence of spectra

$\operatorname{K}(\mathbb{F}_{p})\to\operatorname{K}(k(n))\to\operatorname{K}(K(n)),$

by [1, Proposition 4.4] preceded by [2]. We consider this as an analogue of
Quillen’s dévissage theorem for algebraic $\operatorname{K}$-theory of ring
spectra. One might ask whether we can establish a similar fiber sequence for
$\operatorname{TR}$. In particular, this would allow us to deduce an analogue
of Corollary 3.7 for the non-connective Morava $K$-theory.

Let $y(n)$ denote the Thom spectrum considered in [27, Section 3]. This is the
Thom spectrum associated to the map of $\mathbb{E}_{1}$-spaces

$\Omega
J_{p^{n-1}}S^{2}\hookrightarrow\Omega^{2}S^{3}\rightarrow\mathrm{BGL}_{1}(\mathbb{S}_{p}^{\wedge})$

where $J_{p^{n-1}}S^{2}$ is the $2(p^{n-1})$-skeleton of $\Omega S^{3}$, which
has a single cell in each even dimension. The map
$\Omega^{2}S^{3}\rightarrow\mathrm{BGL}_{1}(\mathbb{S}^{\wedge}_{p})$ is the
spherical fibration constructed by Mahowald (for $p=2$) and Hopkins (for $p$
odd) whose Thom spectrum is $H\mathbb{F}_{p}$. We have the following:

######  Corollary 3.9.

If $n\geq 2$, then $L_{T(k)}\operatorname{TR}(y(n))\simeq 0$ for every $1\leq
k\leq n-1$.

###### Proof.

This follows immediately by combining Theorem 3.1 with [24, Lemma 4.14]. ∎

######  Remark 3.10.

If $R$ is a connective $H\mathbb{Z}$-algebra, then the canonical map

$L_{T(1)}\operatorname{K}(R)\to L_{T(1)}\operatorname{K}(R[1/p])$

is an equivalence by [4, 24]. The analogue of this result does not hold for
$\operatorname{TC}$ as explained in [24, Remark 4.27], which in particular
means that the result also does not prolong to $\operatorname{TR}$. However,
at chromatic heights $n\geq 2$, $\operatorname{TC}$ does satisfy a version of
chromatic purity (cf. [24, Corollary 4.5]). In particular, if $A\rightarrow B$
is an $L_{n}^{p,f}$-local equivalence of $\mathbb{E}_{1}$-rings, then the
induced map

$L_{T(n)}\operatorname{TC}(\tau_{\geq
0}A)\xrightarrow{\simeq}L_{T(n)}\operatorname{TC}(\tau_{\geq 0}B).$

is an equivalence. One can wonder whether such a statement is true of
$T(n)$-local $\operatorname{TR}$, but our methods here do not seem to shed
light on this problem.

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