# Adjunction of roots, algebraic $K$-theory and chromatic redshift

Christian Ausoni, Haldun Özgür Bayındır, Tasos Moulinos

###### Abstract.

Given an $E_{1}$-ring $A$ and a class $a\in\pi_{mk}(A)$ satisfying a suitable
hypothesis, we define a map of $E_{1}$-rings $A\to A(\sqrt[m]{a})$ realizing
the adjunction of an $m$th root of $a$. We define a form of logarithmic THH
for $E_{1}$-rings, and show that root adjunction is log-THH-étale for suitably
tamely ramified extension, which provides a formula for
$\operatorname{\textup{THH}}(A(\sqrt[m]{a}))$ in terms of THH and log-THH of
$A$. If $A$ is connective, we prove that the induced map $K(A)\to
K(A(\sqrt[m]{a}))$ in algebraic $K$-theory is the inclusion of a wedge
summand. Using this, we obtain $V(1)_{*}K(ko_{p})$ for $p>3$ and also, we
deduce that if $K(A)$ exhibits chromatic redshift, so does
$K(A(\sqrt[m]{a}))$. We interpret several extensions of ring spectra as
examples of root adjunction, and use this to obtain a new proof of the fact
that Lubin-Tate spectra satisfy the redshift conjecture.

##  1\. Introduction

Let $A$ be an $E_{1}$-algebra spectrum, and let $a\in\pi_{mk}(A)$ with $m>0$
and even $k\geq 0$. In this paper, we define under a certain Hypothesis 4.4,
an $E_{1}$-algebra extension $A\to A(\sqrt[m]{a})$ realizing the adjunction of
an $m$th-root of $a$ in homotopy rings,

$\pi_{*}A\to\pi_{*}\big{(}A(\sqrt[m]{a})\big{)}\cong\pi_{*}(A)[z]/(z^{m}-a),$

and then study how the algebraic $K$-theory of $A(\sqrt[m]{a})$, or its
topological Hochschild and cyclic homology, relates to that of $A$. Hypothesis
4.4 holds for example if $A$ is an $E_{2}$-ring for which $\pi_{*}A$ is
concentrated in even degrees.

In general, the existence of a suitable root adjunction to a ring spectrum and
its effect on such invariants is an intriguing question; for example it has
been shown by Schwänzl, Vogt and Waldhausen [51, Proposition 2], precisely by
considering topological Hochschild homology, that it is not possible, in
$E_{\infty}$-ring spectra, to adjoin a fourth root of unity $i$ to the sphere
spectrum $\mathbb{S}$ (in a sense made precise in loc. cit.). Nevertheless,
Lawson [28] introduces a construction that allows, under some assumption, to
adjoin roots of a homotopy degree zero unit in $E_{\infty}$ ring spectra.

For classes in positive homotopy degrees, examples exist and have shown to be
relevant, in particular in studying redshift for algebraic $K$-theory. We have
the Adams splitting of connective complex $K$-theory $ku$ completed at an odd
prime $p$,

$ku_{p}\simeq\bigvee_{1\leq i<p-1}\Sigma^{2i}\ell_{p}\,,$

and the extension $\ell_{p}\to ku_{p}$ can be interpreted, on homotopy rings,
as the root adjunction

$\pi_{*}\ell_{p}\cong\mathbb{Z}_{p}[v_{1}]\to\mathbb{Z}_{p}[u]\cong\pi_{*}ku_{p}\,,$

were $v_{1}\mapsto u^{p-1}$, giving

$\pi_{*}ku_{p}\cong(\pi_{*}\ell_{p})[u]/(u^{p-1}-v_{1}).$

Sagave showed in [49, 4.15] that $ku_{p}$ can be constructed as an extension
of $\ell_{p}$, establishing how $\ell_{p}\to ku_{p}$ qualifies as a tamely
ramified extension in $E_{\infty}$-rings. In [4, 7], Rognes and the first
author had computed the algebraic $K$-theory of $\ell_{p}$ and $ku_{p}$ with
coefficients in a Smith-Toda complex $V(1)=\mathbb{S}/(p,v_{1})$, for $p\geq
5$. Taking $T(2)=V(1)[v_{2}^{-1}]$, one has the formula

$T(2)_{*}K(ku_{p})\cong\big{(}T(2)_{*}K(\ell_{p})\big{)}[b]/(b^{p-1}+v_{2})$

relating the two computations, hinting at a chromatic shift (or redshift) of
this tamely ramified root adjunction.

After our construction of root adjunction in $E_{1}$-rings, we offer an
investigation of how the obtained extension is reflected in algebraic
$K$-theory. In particular, we have the following spectrum-level splitting of
algebraic $K$-theory in the tamely ramified case, which applies to a wide
array of examples.

######  Theorem 1.1 (Theorem 5.8).

Assume Hypothesis 4.4 with $p\nmid m$ and $\lvert a\rvert>0$. Furthermore,
assume that $A$ is $p$-local and connective. In this situation, the map in
algebraic $K$-theory

$K(A)\to K(A(\sqrt[m]{a}))$

induced by the extension $A\to A(\sqrt[m]{a})$ is the inclusion of a wedge
summand.

This is deduced from the corresponding result for topological cyclic homology,
see Theorem 5.6. An analogous splitting result for algebraic $K$-theory in the
non-connective case is provided in Corollary 5.11.

For an integer $n>0$, we say that a spectrum $E$ is _height_ $n$ if
$L_{T(n)}E\not\simeq 0$ and $L_{T(m)}E\simeq 0$ for $m>n$, where, $T(n)$
denotes a height $n$ telescope. We say an $E_{1}$-ring $A$ of height $n$
_exhibits redshift_ if $K(A)$ is of height $n+1$. Due to [31, Purity Theorem],
$K(A)$ is of height at most $n+1$, so $A$ will exhibit redshift if
$L_{T(n+1)}K(A)\not\simeq 0$. The following result is thus an immediate
consequence of our splitting results, Theorem 5.8 and Corollary 5.11:

######  Corollary 1.2.

Assume Hypothesis 4.4 with $p\nmid m$ and $\lvert a\rvert>0$. If $A$ exhibits
redshift, then so does $A(\sqrt[m]{a})$.

A key feature of the defined root adjunction is that $A(\sqrt[m]{a})$ is
endowed with the structure of an $E_{1}$-algebra in the symmetric monoidal
category $\operatorname{Fun}(\mathbb{Z}/m,\operatorname{Sp})$ of $m$-graded
spectra, which is reflected in an Adams’ type splitting of spectra

$A(\sqrt[m]{a})\simeq\bigvee_{0\leq i<m}\Sigma^{ik}A\,.$

Such a grading on a spectrum, compatible with further additional algebraic
structures, has already proven to be very useful: let us mention the
computations of Hesselholt-Madsen [19] of the $K$-theory of truncated
polynomial algebras, and of the second and third authors [11] of the
$K$-theory of the free $E_{1}$-algebras in degree $2$ over finite fields. In
the present paper, the grading corresponding to the root adjunction, and the
induced splitting of $\operatorname{\textup{THH}}$ as developed in [2,
Appendix A], are essential ingredients in the proofs of our various splitting
results.

We use the theory of _logarithmic topological Hochschild homology_ for an in
depth study of the THH of $A(\sqrt[m]{a})$. Hesselholt and Madsen [20]
introduced logarithmic topological Hochschild homology for studying the
algebraic $K$-theory of complete discrete valuation rings in mixed
characteristic, and proved a descent property of log THH in the case of tamely
ramified extensions. Rognes [46] then initiated a study of logarithmic
structures, logarithmic André-Quillen homology and of
$\mathrm{log}\operatorname{\textup{THH}}$ in the context of $E_{\infty}$ ring
spectra. With Sagave and Schlichtkrull [47, 48], they then established the
existence of localization sequences for
$\mathrm{log}\operatorname{\textup{THH}}$ and proved that it satisfies tamely
ramified descent in the example of the extension $\ell_{p}\to ku_{p}$.

Here, we offer an alternative definition of log THH that applies to a more
general class of ring spectra. More precisely, we define log THH for a given
$E_{1}$-ring spectrum $A$ and a class $a\in\pi_{mk}(A)$ satisfying Hypothesis
4.4, associated to the pre-log structure given by the monoid generated, under
multiplication, by $a\in\pi_{*}(A)$ (see Definition 6.6); it is denoted
$\operatorname{\textup{THH}}(A\mid a)$. For instance, this applies to the
Morava $K$-theory spectrum $k(n)$ for $v_{n}\in\pi_{*}k(n)$. We prove the
following form of tamely ramified descent (see also Theorem 6.23):

######  Theorem 1.3 (Theorem 6.33).

If $A$ is $p$-local and $p\nmid m$, there is an equivalence of spectra:

$\operatorname{\textup{THH}}(A(\sqrt[m]{a}))\simeq\operatorname{\textup{THH}}(A)\vee\big{(}\bigvee_{0<i<m}\Sigma^{ik}\operatorname{\textup{THH}}(A\mid
a)\big{)}.$

We also prove the existence of a localization cofibre sequence

$\operatorname{\textup{THH}}(A)\to\operatorname{\textup{THH}}(A\mid
a)\to\Sigma\operatorname{\textup{THH}}(A/a),$

see Theorem 6.28. This is an analogue in the present setting of the
localization sequences constructed in [47], but note that our definition only
applies to the case of a pre-log structure given by a monoid on a single
generator. We would like to point out also the recent preprint of Binda,
Lundemo, Park and Østvær [10], where a version of logarithmic Hochschild
homology for simplicial commutative rings is constructed.

We now mention examples of application of the above results.

### Topological $K$-theory

We prove in Theorems 4.10 and 7.6 that, at an odd prime $p$, there are
equivalences of $E_{1}$-ring spectra

$ku_{p}\simeq\ell_{p}(\sqrt[p-1]{v_{1}})\ \ \textup{and}\ \ \
ko_{p}\simeq\ell_{p}(\sqrt[\frac{p-1}{2}]{v_{1}})\,.$

These equivalences upgrade the splitting result of Adams into a
$\mathbb{Z}/(p-1)$-graded, respectively $\mathbb{Z}/(\frac{p-1}{2})$-graded
$E_{1}$-ring structure. We prove that when $p=1$ in $\mathbb{Z}/m$, the
splitting in Theorem 1.1 can be improved to a more refined splitting of
$K(A(\sqrt[m]{a}))$. In the case of $ku_{p}\simeq\ell_{p}(\sqrt[p-1]{v_{1}})$,
this more refined splitting reads as

(1.4) $K(ku_{p})\simeq\bigvee_{0\leq i<p-1}K(ku_{p})_{i}\,.$

Here $K(ku_{p})_{0}\simeq K(\ell_{p})$, and for $p>3$, and we can compute the
$V(1)$-homotopy groups of each of the $i$-th-graded piece
$V(1)_{*}K(ku_{p})_{i}$ using first author’s computation of
$V(1)_{*}K(ku_{p})$.

Using this refined splitting, the second author makes a simplified computation
of $T(2)_{*}K(ku)$ in [8]. We also obtain complete descriptions of
$V(1)_{*}K(ko_{p})$ and $T(2)_{*}K(ko_{p})$, see Theorem 7.10.

We note that the splitting (1.4) can be considered as a version of Adams’
splitting for the cohomology theory represented by $K(ku)$, with classes
corresponding to $2$-categorical complex vector bundles, as developped in [9].

### Johnson-Wilson spectra and Morava $E$-theory

Let $n\geq 1$ be an integer, and let $E(n)$ and $E_{n}$ be the Johnson-Wilson
and Morava $E$-theory spectra. Let $\widehat{E(n)}$ be the $K(n)$-localization
of $E(n)$. These spectra have coefficient rings given as

$\pi_{*}E(n)\cong\mathbb{Z}_{p}[v_{1},...,v_{n-1},v_{n}^{\pm 1}],\hskip
21.33955pt\pi_{*}E_{n}\cong W(\mathbb{F}_{p^{n}})[\lvert
u_{1},\dots,u_{n-1}\rvert][u^{\pm 1}]$

and $\pi_{*}\widehat{E(n)}\cong\pi_{*}E(n)^{\wedge}_{I_{n}}$, where $\lvert
u_{i}\rvert=0$, $\lvert u\rvert=-2$, and $I_{n}=(p,v_{1},\dots,v_{n})$. The
Galois group $Gal=\text{Gal}(\mathbb{F}_{p^{n}}|\mathbb{F}_{p})$ acts on
$E_{n}$, and let $E_{n}^{hGal}$ be the homotopy fixed point spectrum with
coefficients $\pi_{*}E_{n}^{hGal}\cong\mathbb{Z}_{p}[\lvert
u_{1},\dots,u_{n-1}\rvert][u^{\pm 1}]$.

We prove in Theorem 9.6 that there are equivalences of $E_{1}$-rings

(1.5) $\displaystyle E_{n}$
$\displaystyle\simeq\mathbb{S}_{W(\mathbb{F}_{p^{n}})}\wedge_{\mathbb{S}_{p}}\widehat{E(n)}(\sqrt[p^{n}-1]{v_{n}}),\
\ \textup{and}$ $\displaystyle E_{n}^{hGal}$
$\displaystyle\simeq\widehat{E(n)}(\sqrt[p^{n}-1]{v_{n}})\,.$

This promotes the $E_{1}$-ring structure on Morava $E$-theories to a non-
trivial $\mathbb{Z}/(p^{n}-1)$-graded $E_{1}$-ring structure.

Using this description of $E_{n}$ and the log THH étaleness of root
adjunction, we show in Theorem 9.15 that the canonical map

$\operatorname{\textup{THH}}(\widehat{E(n)})\wedge_{\widehat{E(n)}}E_{n}\xrightarrow{\simeq_{p}}\operatorname{\textup{THH}}(E_{n}),$

is an equivalence after $p$-completion. The relationship between such
equivalences and the Galois descent question for THH are studied in [39].

Applying Corollary 5.11 to these root adjunctions of non-connective spectra,
we deduce the following result.

######  Theorem 1.6 (Theorem 9.9).

The canonical maps:

$\displaystyle K(E(n))\to$ $\displaystyle\ K(E_{n}^{hGal})$ $\displaystyle
K(\mathbb{S}_{W(\mathbb{F}_{p^{n}})}\wedge E(n))\to$ $\displaystyle\ K(E_{n})$

are inclusions of wedge summands after $T(n+1)$-localization.

### Lubin-Tate spectra

We can also apply our results to Lubin-Tate spectra that can be constructed,
in several steps, from the truncated Brown-Peterson spectra $BP\langle
n\rangle$, with coefficients $\pi_{*}BP\langle
n\rangle\cong\mathbb{Z}_{(p)}[v_{1},\dots,v_{n}]$. In more precise terms, we
consider an $E_{3}$ $MU[\sigma_{2(p^{n}-1)}]$-algebra form of $BP\langle
n\rangle$ as constructed by Hahn and Wilson [25, Remark 2.1.2]. In this
situation, we can construct $BP\langle n\rangle(\sqrt[p^{n}-1]{v_{n}})$ as an
$E_{3}$ $MU[\sigma_{2}]$-algebra (Remark 4.11).

Let be $k$ a perfect field of characteristic $p$, and let $\mathbb{S}_{W(k)}$
denote the spherical Witt vectors spectrum. We prove in Proposition 8.6 that
the $MU[\sigma_{2}]$-orientation above provides a formal group law $\Gamma$ of
height $n$ over $k$, and that there is an equivalence of $E_{3}$-rings

$L_{K(n)}(\mathbb{S}_{W(k)}\wedge BP\langle
n\rangle)(\sqrt[p^{n}-1]{v_{n}})\simeq E_{(k,\Gamma)}\,,$

where $E_{(k,\Gamma)}$ denotes the Lubin-Tate spectrum corresponding to
$\Gamma$. Due to [31, Purity Theorem], we have an equivalence

$L_{T(n+1)}K\big{(}\mathbb{S}_{W(k)}\wedge BP\langle
n\rangle(\sqrt[p-1]{v_{n}})\big{)}\simeq L_{T(n+1)}K(E_{(k,\Gamma)}).$

By [25], $BP\langle n\rangle$ satisfies the redshift conjecture; following an
argument suggested to us by Hahn, we show that $\mathbb{S}_{W(k)}\wedge
BP\langle n\rangle$ also satisfies the red-shift conjecture, c.f. Proposition
8.2. By Corollary 1.2, we deduce that $E_{(k,\Gamma)}$ satisfies the redshift
conjecture. Indeed, we deduce from this (see Theorem 8.9) a new proof of
Yuan’s result [52] that all Lubin-Tate spectra satisfy the redshift
conjecture. We also obtain the following from Corollary 5.11.

######  Theorem 1.7 (Theorem 8.8).

The induced map

$L_{T(n+1)}K(\mathbb{S}_{W(k)}\wedge BP\langle n\rangle)\to
L_{T(n+1)}K(E_{(k,\Gamma)})$

is the inclusion of a _non-trivial_ wedge summand.

We expect the above result to be relevant also for explicit computations. For
example, in [1], the authors compute
$V(2)_{*}\operatorname{\textup{TC}}(BP\langle 2\rangle)$ for $p\geq 7$ which,
provides an explicit description of $T(3)_{*}K(BP\langle 2\rangle)$. Through
the inclusion above, we deduce that $T(3)_{*}K(BP\langle 2\rangle)$ maps
isomorphically to a summand of $T(3)_{*}K(E_{(\mathbb{F}_{p},\Gamma)})$ for a
height $2$ formal group law $\Gamma$ over $\mathbb{F}_{p}$. To our knowledge,
this is the first explicit, quantitative result on the algebraic $K$-theory
groups of Lubin-Tate spectra for height larger than $1$.

Note that it is not known if the $E_{3}$ $MU$-algebra forms of $BP\langle
n\rangle$ constructed in [25] map into the Morava $E$-theories mentioned in
the preceding sub-section.

######  Remark 1.8.

In [12, Theorem G], the authors construct an $E_{\infty}$-map
$MU[\sigma_{2}]\to E_{(\overline{\mathbb{F}}_{p},\Gamma^{\prime})}$ for the
unique height $n$ formal group law $\Gamma^{\prime}$ on
$\overline{\mathbb{F}}_{p}$ solving an open question on the existence of
orientations on Lubin-Tate spectra. Our constructions provide a similar
orientation for a “smaller” form of Lubin-Tate spectra. Namely, we obtain that
$E_{(\mathbb{F}_{p},\Gamma)}$ is an $E_{3}$ $MU[\sigma_{2}]$-algebra where
$\sigma_{2}$ acts through $u^{-1}\in\pi_{*}E_{(\mathbb{F}_{p},\Gamma)}$; see
Example 8.7. Moreover, there is a grading on $E_{(\mathbb{F}_{p},\Gamma)}$
that respects this structure. To be precise, $E_{(\mathbb{F}_{p},\Gamma)}$ is
an $E_{3}$ $MU[\sigma_{2}]$-algebra in the $\infty$-category of
$\mathbb{Z}/(p^{n}-1)$-graded spectra where
$u^{-1}\in\pi_{*}E_{(\mathbb{F}_{p},\Gamma)}$ is of weight $1$.

######  Remark 1.9.

The above construction of Lubin-Tate spectra by root adjunction is used in an
essential way in the construction of a counter-example to the telescope
conjecture in forthcoming work by Burklund, Hahn, Levy and Schlank.

### Morava $K$-theory

In Section 9.3, we construct two-periodic Morava $K$-theories from Morava
$K$-theories through root adjunction. By Corollary 1.2, we deduce that two-
periodic Morava $K$-theories satisfy the redshift conjecture if the redshift
conjecture holds for Morava $K$-theories (Corollary 9.11).

For $p>3$, the $V(1)$-homotopy of $K(k(1))$ is computed by the first author
and Rognes in [5] where it is also shown that $k(1)$ satisfies the redshift
conjecture. From this, we deduce that the two-periodic first Morava $K$-theory
$ku/p$ also satisfies the redshift conjecture (Corollary 9.12). Moreover,
through the interpretation of $ku/p$ as $k(1)(\sqrt[p-1]{v_{1}})$, the second
author makes the first computation of $T(2)_{*}K\big{(}ku/p\big{)}$ in [8].

######  Outline.

We begin with a quick introduction of graded objects in Section 2. In Section
3, we construct a family of graded $E_{2}$ “polynomial” algebras and establish
their even cell decompositions. In Section 4, we provide our central
construction for root adjunctions (Construction 4.6) and prove our first
splitting result on the THH of ring spectra obtained via a root adjunction. In
Section 5 we prove Theorem 1.6. Section 6 is devoted to studying the variant
of log THH we set forth, as well as the logarithmic THH-étaleness of root
adjunctions. Section 7 contains our results on the algebraic $K$-theory of
real and complex topological $K$-theories. We apply our results to Lubin-Tate
spectra in Section 8. In Section 9, we study the THH and the algebraic
$K$-theory of Morava $E$-theories.

######  Notation 1.10.

  1. (1)

We work freely in the setting of $\infty$-categories and higher algebra from
[33, 36].

  2. (2)

For an $E_{2}$-algebra $R$ in a symmetric monoidal $\infty$-category, when we
say $T$ is an $R$-algebra (or an $E_{1}$ $R$-algebra), we mean that it is an
$E_{1}$-algebra in right $R$-modules. If we mean an $E_{1}$-algebra in left
$R$-modules, we call this a left $E_{1}$ $R$-algebra. If $R$ is $E_{\infty}$,
we do not need to denote the distinction.

  3. (3)

When we say $E_{n}$-ring, we mean an $E_{n}$-algebra in the $\infty$-category
of spectra $\operatorname{Sp}$.

######  Acknowledgements.

We would like to thank Sanath Devalapurkar for many of the ideas in Section 3.
We also thank Jeremy Hahn for showing us the proof of the redshift conjecture
for $BP\langle n\rangle$ with Witt vector coefficients. We benefited from
various conversations with Andrew Baker and Robert Burklund and we would like
to thank them as well. We are very grateful to Robert Burklund for pointing
out a mistake in the proof of the claim, in a previous version of this paper,
that the Morava $K$-theories satisfy redshift; the claim has been removed from
the present version. We would like to thank Oscar Randal-Williams for
explaining to us in depth the constructions in [17]. Finally, we would like to
thank anonymous referrees for many very useful suggestions or corrections.

The first and second authors acknowledge support from the project
ANR-16-CE40-0003 ChroK. The second author acknowledges support from the
Engineering and Physical Sciences Research Council (EPSRC) grant EP/T030771/1.
The third author acknowledges support from the grant NEDAG
ERC-2016-ADG-741501.

##  2\. Recollections on graded objects

Let $m\geq 0$ be an integer and let $\mathbb{Z}/m$ denote the discrete
$\infty$-groupoid whose objects are the elements of the set of integers modulo
$m$. For a presentably symmetric monoidal $\infty$-category $\mathcal{V}$, we
define the $\infty$-category of $m$-graded objects in $\mathcal{V}$ to be the
functor category $\operatorname{Fun}(\mathbb{Z}/m,\mathcal{V})$. For a functor
$F$ in $\operatorname{Fun}(\mathbb{Z}/m,\mathcal{V})$, we denote $F(i)$ by
$F_{i}$ for every $i\in\mathbb{Z}/m$. Since $\mathbb{Z}/m$ is discrete, we
have:

$\operatorname{Fun}(\mathbb{Z}/m,\mathcal{V})\simeq\prod_{i\in\mathbb{Z}/m}\mathcal{V}.$

For $m=0$, this is given by $\operatorname{Fun}(\mathbb{Z},\mathcal{V})$ where
$\mathbb{Z}$ is the corresponding discrete $\infty$-groupoid. In this case, we
omit $m$ and we call $\operatorname{Fun}(\mathbb{Z},\mathcal{V})$ the
$\infty$-category of graded objects in $\mathcal{V}$. We are mainly interested
in the case $\mathcal{V}=\operatorname{Sp}$. We call an object of
$\operatorname{Fun}(\mathbb{Z}/m,\operatorname{Sp})$ an $m$-graded spectrum;
for $m=0$, we drop $m$ and call it a graded spectrum.

Using the symmetric monoidal structure on $\mathbb{Z}/m$ given by addition, we
equip $\operatorname{Fun}(\mathbb{Z}/m,\mathcal{V})$ with the Day convolution
closed symmetric monoidal structure [18]. Since $\mathbb{Z}/m$ is discrete,
this boils down to the following.

$(F\otimes_{\textup{Day}}G)_{k}=\coprod_{i+j=k\textup{\ in\
}\mathbb{Z}/m}F_{i}\otimes G_{j}$

###  2.1. Algebras in graded spectra

We are interested in $E_{n}$-algebras in the $\infty$-category of $m$-graded
spectra and the algebras over these $E_{n}$-algebras.

######  Definition 2.1.

An $m$-graded $E_{n}$-ring $A$ is an $E_{n}$-algebra in
$\operatorname{Fun}(\mathbb{Z}/m,\operatorname{Sp})$. For $k<n$, an $m$-graded
$E_{k}$ $A$-algebra is an $E_{k}$ $A$-algebra in
$\operatorname{Fun}(\mathbb{Z}/m,\operatorname{Sp})$. Similarly, an $m$-graded
(left) right $A$-module is a (left) right $A$-module in
$\operatorname{Fun}(\mathbb{Z}/m,\operatorname{Sp})$.

######  Remark 2.2.

Note that the notion of an $m$-graded $E_{k}$-algebra in $\operatorname{Sp}$
is in general different than the notion of an $m$-graded object in the
$\infty$-category of $E_{k}$-algebras in $\operatorname{Sp}$.

###  2.2. Manipulations on graded objects

For a symmetric monoidal functor $\mathbb{Z}/m\to\mathbb{Z}/m^{\prime}$, there
is an induced adjunction between
$\operatorname{Fun}(\mathbb{Z}/m,\operatorname{Sp})$ and
$\operatorname{Fun}(\mathbb{Z}/m^{\prime},\operatorname{Sp})$ where the left
adjoint is symmetric monoidal and given by left Kan extension [40, Corollary
3.8]. The right adjoint is given by restriction. This provides the following
adjunctions which allow us to move between various gradings. Let $n>0$ and
$s\geq 0$ be integers.

  * •

We let $D^{n}_{sn}\dashv Q$ denote the adjunction induced by the quotient map
$\mathbb{Z}/sn\to\mathbb{Z}/n$ sending $1$ to $1$.

  * •

We often use $D^{n}_{sn}$ for $s=0$ which allows us to obtain an $n$-graded
object out of a graded object $X$ in $\operatorname{Sp}$. We let $D^{n}$
denote
$D^{n}_{0}\colon\thinspace\operatorname{Fun}(\mathbb{Z},\operatorname{Sp})\to\operatorname{Fun}(\mathbb{Z}/n,\operatorname{Sp})$
and we have

$D^{n}(X)_{i}\simeq\bigvee_{j\in\mathbb{Z}\mid j\equiv i\textup{\ mod
n}}X_{j}.$

  * •

For $n=1$, we denote $D^{1}_{s}$ by $D$. In this case,

$D\colon\thinspace\operatorname{Fun}(\mathbb{Z}/s,\operatorname{Sp})\to\operatorname{Sp}$

is given by $D(X)\simeq\vee_{j\in\mathbb{Z}/s}X_{j}$, i.e. left Kan extension
along $\mathbb{Z}/s\to 0$. For an $s$-graded (spectrum) $E_{n}$-ring $X$, we
call $D(X)$ the underlying (spectrum) $E_{n}$-ring of $X$. We often omit $D$
in our notation.

  * •

For $s\in\mathbb{Z}$, let $L_{s}\dashv R_{s}$ denote the adjunction on
$\operatorname{Fun}(\mathbb{Z},\operatorname{Sp})$ induced by the map
$\mathbb{Z}\xrightarrow{\cdot s}\mathbb{Z}$ given by multiplication by $s$.
For a graded spectrum $X$, we have

$L_{s}(X)_{si}\simeq X_{i}$

for every $i$ and $L_{s}(X)_{j}\simeq 0$ whenever $s\nmid j$.

  * •

Let $F\dashv G$ denote the adjunction induced by the trivial map
$0\to\mathbb{Z}/m$. We have $G(X)=X_{0}$. For an $m$-graded $E_{n}$-ring $A$,
$F(G(A))$ is given by $A_{0}$ in weight $0$ and it is trivial on the other
degrees. Therefore, we sometimes abuse notation and denote the $m$-graded
$E_{n}$-ring $F(G(A))$ by $A_{0}$. The counit of this adjunction provides a
map

$A_{0}\to A$

of $m$-graded $E_{n}$-rings. If $A_{i}\simeq 0$ for $i\neq 0$, then this map
is an equivalence and we say that $A$ is concentrated in weight zero. The
following lemma states that in this situation, there is an equivalence of
$E_{n}$-rings between the underlying $E_{n}$-ring of $A$ and the weight zero
piece $G(A)=A_{0}$ of $A$. Therefore, we often do not distinguish between $A$,
$G(A)=A_{0}$ and $D(A)$ in our notation when $A$ is concentrated in weight
zero.

######  Lemma 2.3.

Let $A$ be an $m$-graded $E_{n}$-ring concentrated in weight zero. There is an
equivalence of $E_{n}$-rings

$A_{0}\simeq D(A)$

where $A_{0}$ denotes $G(A)$. In particular, we have $F(D(A))\simeq A$ as
$m$-graded $E_{n}$-rings.

###### Proof.

Since $A$ is concentrated in weight zero, we have $D(A)\simeq DFG(A)$. As
$D\circ F$ Kan extends through the composite $0\to\mathbb{Z}/m\to 0$, it is
equivalent to the identity functor. We obtain that $DFG(A)\simeq G(A)\simeq
A_{0}$.

For the second statement, note that $F(D(A))\simeq F(G(A))$ due to the first
statement. Since $A$ is concentrated in weight $0$, we have $F(G(A))\simeq A$.
∎

##  3\. A family of $E_{2}$ polynomial rings in graded spectra

In this section, we introduce the construction of a family of $E_{2}$-algebras
in graded spectra. These have appeared in the work of Hahn and Wilson in [25]
and are also studied in greater depth in [14]. These will be central to our
constructions. For every $r,w\in\mathbb{Z}$, one constructs a graded
$E_{2}$-ring $\mathbb{S}[\sigma_{2r}]$ which may be thought of as a
“polynomial” algebra with a generator in homotopical degree $2r$ and grading
weight $w$. However, these are not polynomial algebras in the precise sense,
as they are only demonstrated to admit $E_{2}$ structures. By mapping these
$E_{2}$-rings into each other, we will be able to construct $E_{2}$-ring
extensions.

###  3.1. Shearing preliminaries

The main mechanism underlying this construction is that of shearing, which we
now briefly review. It has appeared in [42], and is also studied in [14]. In
what follows, $\operatorname{Gr}(\operatorname{Sp})$ denotes
$\operatorname{Fun}(\mathbb{Z},\operatorname{Sp})$, i.e. the $\infty$-category
of graded spectra.

######  Proposition 3.1.

There exists an endofunctor on graded spectra

$\operatorname{sh}:\operatorname{Gr}(\operatorname{Sp})\to\operatorname{Gr}(\operatorname{Sp})$

given by

$\operatorname{sh}(M)_{i}:=M_{i}[-2i]$

with the following properties:

  * •

$\operatorname{sh}$ is an equivalence, with inverse given by
$\operatorname{sh}^{-1}(M_{i})=M_{i}[2i]$

  * •

$\operatorname{sh}$ admits an $E_{2}$-monoidal structure, with respect to the
Day convolution product on $\operatorname{Gr}(\operatorname{Sp})$.

###### Proof.

This appears in the $\mathbb{Z}$-linear setting in [42] and is also studied in
[14]. However, for the sake of completeness, we sketch the basic ideas
underlying the construction. In [35], Lurie constructs an $E_{2}$-monoidal map
of spaces

$\phi:\mathbb{Z}\to\operatorname{Pic}(\operatorname{Sp})$

sending $n\mapsto\mathbb{S}^{-2n}$. We now define $\operatorname{sh}$ as the
functor, obtained by adjunction, from the assignment

$\mathbb{Z}\times\operatorname{Gr}(\operatorname{Sp})\to\operatorname{Sp}$

given by the composition

$\mathbb{Z}\times\operatorname{Gr}(\operatorname{Sp})\xrightarrow{(\phi,\operatorname{ev})}\operatorname{Pic}(\operatorname{Sp})\times\operatorname{Sp}\xrightarrow{\otimes}\operatorname{Sp}.$

Here, the first map sends $(n,M)\mapsto(\phi(n),M_{n})$. The fact that this
latter composition is $E_{2}$ follows from the fact that $\phi$ is itself
$E_{2}$. This further implies that $\operatorname{sh}$ is itself $E_{2}$
monoidal. To see that this is an equivalence, one displays, as in [42], an
inverse in the same way by precomposing $\phi$ with the map
$\mathbb{Z}\xrightarrow{-1\cdot}\mathbb{Z}$. ∎

######  Variant 3.2.

One can precompose the map
$\phi:\mathbb{Z}\to\operatorname{Pic}(\operatorname{Sp})$ with the map
$\cdot(k):\mathbb{Z}\to\mathbb{Z}$. We denote the composition by

$\phi^{k}:\mathbb{Z}\to\operatorname{Pic}(\operatorname{Sp})$

As in the above, we use this to define an endofunctor

$\operatorname{sh}^{k}:\operatorname{Gr}(\operatorname{Sp})\to\operatorname{Gr}(\operatorname{Sp}).$

This acquires the same formal properties as above, e.g it will be an
$E_{2}$-monoidal autoequivalence on $\operatorname{Gr}(\operatorname{Sp})$.
Furthermore, one has the description $(\operatorname{sh}^{k}M)_{i}\simeq
M_{i}[-2ki]$.

###  3.2. Sheared polynomial algebras

Recall that there exists an $E_{\infty}$ algebra
$\mathbb{S}[t]\in\operatorname{Gr}(\operatorname{Sp})$, which gives a graded
enhancement of the “flat” polynomial algebra. One can obtain this, for
example, by observing that the restriction map from filtered spectra to graded
spectra

$\operatorname{Res}:\operatorname{Fil}(\operatorname{Sp})\to\operatorname{Gr}(\operatorname{Sp})$

is lax symmetric monoidal. In more detail, this will be the restriction

$\operatorname{Fil}(\operatorname{Sp})=\operatorname{Fun}((\mathbb{Z},\leq),\operatorname{Sp})\to\operatorname{Fun}(\mathbb{Z}^{\operatorname{ds}},\operatorname{Sp})=\operatorname{Gr}(\operatorname{Sp})$

along $\mathbb{Z}\hookrightarrow(\mathbb{Z},\leq)$ so that in particular we
forget the structure maps of the filtration, cf [35]. We remind the reader
that this is different from the associated graded functor. One then sets
$\mathbb{S}[t]:=\operatorname{Res}(\mathbf{1})$, where $1$ denotes the unit of
the symmeric monoidal structure on $\operatorname{Fil}(\operatorname{Sp})$.
Thus, $\mathbb{S}[t]$ (which is given by $\mathbb{S}$ in nonpositive weights
and $0$ in positive weights) acquires the structure of an $E_{\infty}$-algebra
in graded spectra.

######  Construction 3.3.

As described in Proposition 3.1, there exists an $E_{2}$-monoidal
autoequivalence $\operatorname{sh}$. We set

$\mathbb{S}[\sigma_{2}]:=\operatorname{sh}(\mathbb{S}[t]),$

and more generally for $k>0$,

$\mathbb{S}[\sigma_{2k}]:=\operatorname{sh}^{k}(\mathbb{S}[t]);$

that is, one applies $\operatorname{sh}^{k}$ to $\mathbb{S}[t]$ to obtain a
family of $E_{2}$-algebras in graded spectra. For $k$=0, we set
$\mathbb{S}[\sigma_{0}]:=\mathbb{S}[t]$. It follows by inspection that the
underlying graded $E_{1}$-ring of $\mathbb{S}[\sigma_{2k}]$ is the free graded
$E_{1}$-ring on $\mathbb{S}^{2k}(-1)$ where $\mathbb{S}^{2k}(-1)$ is
$\mathbb{S}^{2k}$ concentrated in weight $-1$.

######  Remark 3.4.

For $w\in\mathbb{Z}$ and even $k\geq 0$, we apply the functor $L_{-w}$ which
left Kan extends along the multiplication map

$\cdot(-w):\mathbb{Z}\to\mathbb{Z}$

to obtain weight sifted variants of $\mathbb{S}[\sigma_{k}]$. We often omit
$L_{-w}$ when the weight of $\sigma_{k}$ is clear from the context but when we
wish to be explicit, we write $\mathbb{S}[\sigma_{k,w}]$ for the graded
$E_{2}$-ring $L_{-w}\mathbb{S}[\sigma_{k}]$ where $\sigma_{k,w}$ is in weight
$w$. As before, the underlying graded $E_{1}$-ring of
$\mathbb{S}[\sigma_{k,w}]$ is the free graded $E_{1}$-ring
$\operatorname{Free}_{E_{1}}(\mathbb{S}^{k}(w))$.

To adjoin roots, we often start with $\mathbb{S}[\sigma_{mk}]$ and
$\mathbb{S}[\sigma_{k}]$ with $\sigma_{mk}$ and $\sigma_{k}$ in weights $m$
and $1$ respectively where $m>0$ and $k>0$ is even.

######  Proposition 3.5.

In the situation above, there exists a map of graded $E_{2}$-rings

$\mathbb{S}[\sigma_{mk}]\to\mathbb{S}[\sigma_{k}]$

that carries $\sigma_{mk}$ to $\sigma_{k}^{m}$ in homotopy. This provides a
map of $m$-graded $E_{2}$-rings

$D^{m}(\mathbb{S}[\sigma_{mk}])\to D^{m}(\mathbb{S}[\sigma_{k}])$

where $\sigma_{k}\in\pi_{*}D^{m}(\mathbb{S}[\sigma_{k}])$ is of weight $1$ and
$D^{m}(\mathbb{S}[\sigma_{mk}])$ is concentrated in weight $0$.

Furthermore, we have $D^{m}(\mathbb{S}[\sigma_{mk}])\simeq
F(D(\mathbb{S}[\sigma_{mk}]))$ as $m$-graded $E_{2}$-rings; here $F$ Kan
extends through $0\to\mathbb{Z}/m$.

###### Proof.

We first remind the reader that there is an identification

$\mathbb{S}[\sigma_{mk,-1}]:=\operatorname{sh}^{mk}(\mathbb{S}[t])\simeq
R_{m}(\mathbb{S}[\sigma_{k,-1}])$

of graded $E_{2}$-rings. This follows from the very definition the $k$th
shearing functor $\operatorname{sh}^{k}$, in particular that it sends
$\mathbb{S}[t]$ to the negative weight part of the graded spectrum given by
the map

$\phi^{k}:\mathbb{Z}\xrightarrow{\times
k}\mathbb{Z}\to\operatorname{Pic}(\operatorname{Sp}).$

Thus we may identify $\mathbb{S}[\sigma_{mk,-1}]$ with
$R_{m}\mathbb{S}[\sigma_{k,-1}]\simeq
R_{m}\operatorname{sh}^{k}(\mathbb{S}[t])$, negative weight part of the graded
spectrum given by the map

$\phi^{mk}:\mathbb{Z}\to\operatorname{Pic}(\operatorname{Sp}).$

Therefore, the counit of the adjunction $L_{m}\dashv R_{m}$ provides a map of
graded $E_{2}$-rings

$\mathbb{S}[\sigma_{mk,-m}]\to\mathbb{S}[\sigma_{k,-1}].$

Applying the functor $L_{-1}$ to this map, we obtain the map of graded
$E_{2}$-rings $\mathbb{S}[\sigma_{mk}]\to\mathbb{S}[\sigma_{k}]$ claimed in
the proposition.

The functor $D^{m}$ gives the desired map

$D^{m}(\mathbb{S}[\sigma_{mk}])\to D^{m}(\mathbb{S}[\sigma_{k}])$

of $m$-graded $E_{2}$-algebras. Note that
$\sigma_{mk}\in\pi_{*}D^{m}(\mathbb{S}[\sigma_{mk}])$ is of weight $0$, which
ensures that $D^{m}(\mathbb{S}[\sigma_{mk}])$ is concentrated in weight $0$.

To see the last statement, we have

(3.6) $FDD^{m}(\mathbb{S}[\sigma_{mk}])\simeq D^{m}(\mathbb{S}[\sigma_{mk}])$

due to Lemma 2.3 since $D^{m}(\mathbb{S}[\sigma_{mk}])$ is conentrated in
weight $0$. Furthermore, $DD^{m}$ is Kan extension through the composite
$\mathbb{Z}\to\mathbb{Z}/m\to 0$ which is the same as the Kan extending
through $\mathbb{Z}\to 0$. Therefore, $DD^{m}(\mathbb{S}[\sigma_{mk}])\simeq
D(\mathbb{S}[\sigma_{mk}])$. This, together with (3.6) provides the desired
equivalence $D^{m}(\mathbb{S}[\sigma_{mk}])\simeq
F(D(\mathbb{S}[\sigma_{mk}]))$. ∎

We often omit the functor $D^{m}$ in our notation and denote the map of
$m$-graded $E_{2}$-rings $D^{m}(\mathbb{S}[\sigma_{mk}])\to
D^{m}(\mathbb{S}[\sigma_{k}])$ as
$\mathbb{S}[\sigma_{mk}]\to\mathbb{S}[\sigma_{k}]$.

######  Remark 3.7.

To adjoin a root to a degree $0$ class, we need the $k=0$ case of the
proposition above. In other words, we need an analogous $E_{2}$-map
$\mathbb{S}[\sigma_{mk}]\to\mathbb{S}[\sigma_{k}]$ for $k=0$. For this, we
start with the graded $E_{2}$-map
$\mathbb{S}[\sigma_{m2}]\to\mathbb{S}[\sigma_{2}]$ and apply the functor sh.
This procedure provides a graded $E_{2}$-map
$\mathbb{S}[\sigma_{{0},m}]\to\mathbb{S}[\sigma_{0,1}]$ that carries
$\sigma_{0,m}$ to $\sigma_{0,1}^{m}$ as desired.

###  3.3. Cell structures on sheared polynomial algebras

A key technical result for us will be the even cell decomposition of
$\mathbb{S}[\sigma_{k}]$ _as an $E_{2}$-algebra_. As we will see in the
remainder of this section, this is what will allow for us to define
$E_{2}$-algebra maps to a given $E_{2}$-algebra $A$, along which we will then
adjoin roots.

######  Remark 3.8.

In the second arxiv version of [25], Hahn and Wilson also construct $E_{2}$
even cell decompositions on the free $E_{1}$-algebra $\mathbb{S}[\sigma_{k}]$
using a Koszul duality argument for even $k\geq 0$. However, they removed this
result in the later versions of their paper since they found a simpler
argument for their redshift results that avoid the use of these even cell
decompositions. Since this does not appear in the published version of [25],
we give a proof of the $E_{2}$ even cell decompositions on
$\mathbb{S}[\sigma_{k}]$ that we use. We would like to note that our methods
are different than the ones used in [25, Arxiv version 2].

Before doing this, we make precise what exactly we mean by even cell
decomposition. The following notions are heavily inspired by Section 6.3 of
[17].

######  Definition 3.9.

Let $f:S\to R\in\operatorname{Alg}_{E_{2}}(\operatorname{Sp}^{\mathbb{Z}})$ be
a map of $E_{2}$-algebras in graded spectra. We say $f$ has a _filtered
cellular decomposition_ if there exists a tower in
$\operatorname{Alg}_{E_{2}}(\operatorname{Fil}(\operatorname{Fun}(\mathbb{Z},\operatorname{Sp})))$
for which

$S=\operatorname{sk}_{-1}(f)\to\operatorname{sk}_{0}(f)\to\operatorname{sk}_{1}(f)\to\cdots\to\operatorname{colim}_{i}\operatorname{sk}_{i}(f)=:\operatorname{sk}(f)\simeq
R$

such that each $\operatorname{sk}_{i}(f)$ is obtained from
$\operatorname{sk}_{i-1(f)}$ via the following pushout diagram:

${\operatorname{Free}_{E_{2}}(\bigsqcup_{\alpha\in I_{n_{i}}}\partial
D^{g_{\alpha},n_{i}}[i-1])}$${\operatorname{sk}_{i-1}(f)}$${\operatorname{Free}_{E_{2}}(\bigsqcup_{\alpha\in
I_{n_{i}}}D^{g_{\alpha},n_{i}}[i])}$${\operatorname{sk}_{i}(f).}$

The notation $X[n]$ in the above means that the object $X$ is placed in
filtering degree $n$.

In particular, in each degree $i$ of the tower we are adding cells in
increasing dimension $n_{i}$. Thus if $i\leq j$, then $n_{i}\leq n_{j}$, and
$I_{n_{i}}$ refers to the set of $n_{i}$ cells of $R$. It may be the case,
that $n_{i}=i$, but we do not require this for the sake of flexibility of the
definition, which is a point of departure from the notion in [17]. If
$f:\mathbf{1}\to R$ is the map from the unit, we call this a filtered cellular
decomposition of $R$.

######  Definition 3.10.

A map $f\colon\thinspace S\to R$ of graded $E_{2}$-rings admits a cell
decomposition if it is the colimit of a tower

$S=\operatorname{sk}_{-1}(f)\to\operatorname{sk}_{0}(f)\to\operatorname{sk}_{1}(f)\to\cdots\to\operatorname{colim}_{i}\operatorname{sk}_{i}(f)\simeq
R$

in graded $E_{2}$-rings where each stage is obtained from the previous via a
cell attachement in graded $E_{2}$-rings. In particular, if $f$ admits a
filtered cellular decomposition, taking levelwise colimits provides a cellular
decomposition of $f$.

Our first step is to establish the decomposition for $\mathbb{S}[t]$.

######  Proposition 3.11.

As an $E_{2}$-algebra in graded spectra, $\mathbb{S}[t]$ admits a (filtered)
cellular decomposition with cells in even degrees.

###### Proof.

The two key inputs for our argument are Theorem 11.21 and Theorem 13.7 of
[17]. The former result applied to the map $f:0\to I$ of non-unital
$E_{2}$-algebras in graded spectra (where $I$ is the augmentation ideal of the
map $\mathbb{S}[t]\to\mathbb{S}$) says that there exists a relative CW
decomposition

$0\to\operatorname{colim}\operatorname{sk}_{n}(f)\simeq I.$

Moreover, the proof of this fact in loc. cit. constructs a minimal cell
structure, one that has the smallest possible number of cells in a given
bidegree (the extra degree here arises since we are working in graded
spectra). In particular the colimit they construct,
$\operatorname{colim}\operatorname{sk}_{n}(f)$, will have cells precisely in
bidegree
$b^{E_{2}}_{g,d}(\mathbb{S}[t]):=\operatorname{dim}_{k}H^{E_{2}}_{g,d}(\mathbb{S}[t],\mathbb{S},k)\in\mathbb{N}\cup\\{\infty\\}$,
where $H^{\mathcal{O}}_{g,d}(R;k)$ is the $\mathcal{O}$-homology of an
$\mathcal{O}$-algebra $R$, with coefficients in the ring $k$; we will in fact
set $k=\mathbb{Z}$.

We sketch the argument given there applied to our particular case for the sake
of completeness. One proceeds by inductively constructing a factorization

$0=\operatorname{sk}_{-1}\xrightarrow{h_{0}}\cdots\xrightarrow{h_{\epsilon}}\operatorname{sk}_{\epsilon}\xrightarrow{f_{\epsilon}}I,$

in the $\infty$-category of (increasingly) filtered objects of
$\operatorname{Fun}(\mathbb{Z},\operatorname{Sp})$. Here
$h_{e}:\operatorname{sk}_{e-1}\to\operatorname{sk}_{e}$ comes with the
structure of a (filtered) CW attachment of dimension $e$, where
$\operatorname{sk}_{i}$ denotes the $i$th skeleton equipped with the skeletal
filtration leading up to that degree. Taking the colimit along $\epsilon$
gives an induced map $f_{\infty}:\operatorname{colim}sk_{\epsilon}(f)\to I$,
which is an equivalence.

For the inductive step in their argument, they show that the Hurewicz map

$\pi_{*,\epsilon}(\mathbb{S}[t],\operatorname{sk}_{\epsilon-1})\to
H_{*,\epsilon}^{E_{2}}(\mathbb{S}[t],\operatorname{sk}_{\epsilon-1};k)$

from relative homotopy to relative $E_{2}$-homology with coefficients in $k$
is surjective. Using this, one is able to choose a set of maps

$\\{E_{\alpha}:(D^{\epsilon},\partial
D^{\epsilon})\to(\mathbb{S}[t](g),\operatorname{sk}_{\epsilon-1}(g))\\},$

whose images generate
$H_{*,\epsilon}^{E_{2}}(\mathbb{S}[t],\operatorname{sk}_{\epsilon-1};k)$ as a
$k$-module. The boundary maps are then used to attach filtered cells
$(g,\epsilon)$ to $\operatorname{sk}_{\epsilon-1}$ to form
$\operatorname{sk}_{\epsilon}$ and the corresponding $E_{\alpha}$ is used to
extend $f_{\epsilon-1}$ to $f_{\epsilon}$. Putting all this together, we see
that the attachment of the cells is parameterized by the dimensions of the
$E_{2}$-algebra homology groups with coefficients in $k$. To see, in our
particular setup, that this cell decomposition is concentrated in even
degrees, it is therefore enough to verify that
$H^{E_{2}}_{g,e}(\mathbb{S}[t],k)$ vanishes whenever $e\cong 1\mod 2$. For
this we apply [17, Theorem 13.7] which states that the $k$-fold iterated bar
construction of an $E_{k}$ algebra is equivalent to the $k$-suspension of the
$E_{k}$-cotangent complex. By [35, Proposition 5.4.9], there is an equivalence
$\operatorname{Bar}^{(2)}(\mathbb{S}[t])\simeq\operatorname{gr}(\mathbb{S}[\mathbb{C}P^{n}]_{n\geq
0})$ in graded spectra, where the right hand side is the associated graded of
the filtration on spherical chains on $\mathbb{C}P^{\infty}$, with filtration
induced by the skeletal filtration on infinite projective space. Tensoring
this with $k=\mathbb{Z}$ in our particular situation, we obtain (a 2-fold
shift) of chains on $\mathbb{C}P^{\infty}$ with coefficients in $\mathbb{Z}$
which has a cell in each bidegree $(-n,2n-2)$. By taking into account units,
we conclude that $\mathbb{S}[t]$ may be constructed from $\mathbb{S}$ by
attaching the same cells. ∎

######  Remark 3.12.

We remark that we may take the levelwise colimit in the above filtered
cellular decomposition to obtain an $E_{2}$ cellular decomposition for
$\mathbb{S}\to\mathbb{S}[t]$ in the sense of Definition 3.10.

######  Corollary 3.13.

The degree zero piece of the above cellular decomposition is the free algebra
$\operatorname{Free}_{E_{2}}(\mathbb{S}(-1))$, i.e. the free $E_{2}$-algebra
with generator in degree $0$ and weight $-1$. Moreover, the map
$f_{0}:\operatorname{Free}_{E_{2}}((\mathbb{S}(-1))\to\mathbb{S}[t]$ itself
admits a cellular decomposition with even cells of positive dimensions

###### Proof.

In degree zero, we have the following pushout square in the
$\operatorname{Alg}_{E_{2}}(\operatorname{Fun}(\mathbb{Z},\operatorname{Sp}))$:

${\mathbb{S}\simeq\operatorname{Free}_{E_{2}}(\emptyset)}$${\operatorname{sk}_{-1}(\mathbb{S}[t])\simeq\mathbb{S}}$${\operatorname{Free}_{E_{2}}(\mathbb{S}(-1))\simeq\operatorname{Free}_{E_{2}}(D^{0})}$${\operatorname{sk}_{0}(\mathbb{S}[t]).}$

Since this is a pushout square we obtain an equivalence

$\operatorname{sk}_{0}(\mathbb{S}[t])\simeq\operatorname{Free}_{E_{2}}(\mathbb{S}(-1))$

Moreover, by starting in degree zero with the zero cells already attached, we
may conclude that the map

$f_{0}:\operatorname{Free}_{E_{2}}(\mathbb{S}(-1))\to\mathbb{S}[t].$

itself admits a cellular decomposition with even cells of positive dimension.

∎

By the above corollary, we have a cellular decomposition on the map of graded
$E_{2}$-rings $\operatorname{Free}_{E_{2}}(\mathbb{S}(-1))\to\mathbb{S}[t]$.
We can apply shearing to this map to obtain a map

$\operatorname{Free}_{E_{2}}(\mathbb{S}^{k}(-1))\to\mathbb{S}[\sigma_{k}].$

######  Proposition 3.14.

Let $k>0$ be even. The map
$f_{0}:\operatorname{Free}_{E_{2}}\mathbb{S}^{k}(-1)\to\mathbb{S}[\sigma_{k}]$
admits a cell decomposition with cells concentrated in even degrees. Left Kan
extending along the multiplication map
$\mathbb{Z}\xrightarrow{\times-w}\mathbb{Z}$, we conclude that
$f_{0}:\operatorname{Free}_{E_{2}}\mathbb{S}^{k}(w)\to\mathbb{S}[\sigma_{k,w}]$
admits a cell decomposition with cells concentrated in even degrees.

###### Proof.

By construction, $\mathbb{S}[t]$ may be written as a filtered colimit of a
diagram of $E_{2}$ algebras,

$\operatorname{Free}_{E_{2}}(\mathbb{S}(-1))\to\operatorname{sk}_{1}(f)\to\cdots\to\operatorname{sk}_{i-1}(f)\to\operatorname{sk}_{i}(f)\to\cdots$

where each $\operatorname{sk}_{i}(f)$ is formed as a pushout from
$\operatorname{sk}_{i-1}$ along a map
$\operatorname{Free}_{E_{2}}(\mathbb{S}^{2n+1})\to\mathbb{S}$. We may apply
$\operatorname{sh}^{k/2}$ to this diagram, and take note of the fact that this
will commute with colimits along the filtered diagram, together with the free
$E_{2}$-algebra functor. Thus we conclude with an even cell presentation for
the induced map:

$\operatorname{sh}^{k/2}(\operatorname{Free}_{E_{2}}(\mathbb{S}(-1)))\simeq\operatorname{Free}_{E_{2}}(\mathbb{S}^{k}(-1))\to\mathbb{S}[\sigma_{k}].$

By left Kan extending along the multiplication by $-w$ map on $\mathbb{Z}$
(i.e. applying $L_{-w}$), we conclude analogously for the map

$\operatorname{Free}_{E_{2}}(\mathbb{S}^{k}(w))\to\mathbb{S}[\sigma_{k,w}].$

∎

######  Proposition 3.15.

Let $A$ be a (graded) $E_{2}$-ring whose homotopy groups are concentrated in
even degrees and let $a\in\pi_{k}A$ be a weight $w$ class for some even $k\geq
0$. Then there is a (graded) $E_{2}$-ring map

$\mathbb{S}[\sigma_{k,w}]\to A$

which carries $\sigma_{k}$ to $a$.

###### Proof.

By Proposition 3.14, the map

$f:\operatorname{Free}_{E_{2}}(\mathbb{S}^{k}(w))\to\mathbb{S}[\sigma_{k,w}]$

admits an even cell decomposition. Let $a\in\pi_{k}(A)$ be as in the
hypothesis of the proposition. This induces an $E_{2}$-algebra map
$\operatorname{Free}_{E_{2}}(\mathbb{S}^{k}(w))\to A$, which we would like to
extend inductively along the above tower. In order to do this, it is enough to
note that in degree $i$, we would need to trivialize the induced map
$\operatorname{Free}_{E_{2}}(\mathbb{S}^{k+2i-1})\to A$. Using the
free/forgetful adjunction between
$\operatorname{Alg}_{E_{2}}(\operatorname{Fun}(\mathbb{Z},\operatorname{Sp}))$
and $\operatorname{Fun}(\mathbb{Z},\operatorname{Sp})$ algebras and graded
spectra, this will now follow from the fact that $\pi_{2n-1}(A)=0$ for all
$n$. ∎

######  Remark 3.16.

In Remark 3.7, we mentioned that we are going to use
$\operatorname{sh}(\mathbb{S}[\sigma_{m2,m}])$ to adjoin roots to degree $0$
classes. We remark that $\operatorname{sh}(\mathbb{S}[\sigma_{m2,m}])$ also
satisfies the lifting property in the proposition above. This follows by the
fact that the even cell decomposition for $\mathbb{S}[\sigma_{m2,m}]$ provides
an even cell decomposition for $\operatorname{sh}(\mathbb{S}[\sigma_{m2,m}])$
since $\operatorname{sh}$ is an $E_{2}$-monoidal left adjoint functor.

######  Remark 3.17.

We remark that another way to construct an $E_{2}$-algebra map
$\mathbb{S}[t]\to A$ comes from the filtration on $\mathbb{S}[t]$ given its
filtered cell decomposition. The mapping space

$\operatorname{Map}_{\operatorname{Alg}_{E_{2}}}(\mathbb{S}[t],A)$

obtains a filtration from the filtation on the source; this will have
associated graded

$\operatorname{gr}\operatorname{Map}_{\operatorname{Alg}_{E_{2}}}(\mathbb{S}[t],A)\simeq\operatorname{Map}(\operatorname{Free}_{E_{2}}(\bigoplus_{n\geq
1}\mathbb{S}^{2n},A)\simeq\operatorname{Map}(\bigoplus_{n\geq
1}\mathbb{S}^{2n},A).$

Now if $X$ has even homotopy groups, then so does the associated graded, so
that the resulting spectral sequence computing the homotopy groups of the
limit collapses. Thus, $a\in\pi_{0}(A)$ gives a class in
$x\in\pi_{0}\operatorname{Map}(\mathbb{S}^{2k},X)\subset\pi_{0}\operatorname{Map}(\bigoplus_{n\geq
1}\mathbb{S}^{2n+2nk-2},X)$, which will be an infinite cycle, and thus
corresponds to an $E_{2}$-algebra map $\mathbb{S}[t]\to A$. We remark that
this approach should allow for one to define maps $\mathbb{S}[\sigma_{k}]\to
A$ in the general case as well. We thank Oscar Randal-Williams for suggesting
this approach.

##  4\. Adjoining roots and THH

Here, we introduce our construction for adjoining roots to ring spectra and
prove our first results on the THH of ring spectra obtained through this
construction.

###  4.1. Background on algebras over $E_{n}$-algebras

Here is a quick background on some of the standard facts that we often use
from [36].

For an $E_{\infty}$-algebra $R$ in a presentably symmetric monoidal
$\infty$-category $\mathcal{C}$, the $\infty$-category of $E_{n}$ $R$-algebras
is a symmetric monoidal $\infty$-category with the pointwise tensor product
[36, Example 3.2.4.4]. Therefore, for two $E_{n}$ $R$-algebras $A$ and $B$,
$A\otimes_{R}B$ is an $E_{n}$ $R$-algebra.

In this work, we often consider algebras over an $E_{n}$-algebra $R$ and in
this case, the $\infty$-category of $E_{m}$ $R$-algebras (for $m\leq n-1$) are
not known to carry an appropriate $E_{n-1}$-monoidal structure. To work around
this problem, we use the following facts which can be extracted from [33,
Corollary 4.8.5.20].

The $\infty$-category of (left) right $R$-modules is an $E_{n-1}$-monoidal
$\infty$-category. We call an $E_{m}$-algebra in the $\infty$-category of
right $R$-modules an $E_{m}$ $R$-algebra where $m\leq n-1$.

Furthermore, for a map $f\colon\thinspace R\to S$ of $E_{n}$-algebras in
$\mathcal{C}$, one obtains an $E_{n-1}$-monoidal functor $-\otimes_{R}S$
between the respective $\infty$-categories of modules. For every $m\leq n-1$,
this induces a functor:

(4.1)
$-\otimes_{R}S\colon\thinspace\operatorname{Alg}_{E_{m}}(\operatorname{RMod}_{R})\to\operatorname{Alg}_{E_{m}}(\operatorname{RMod}_{S}).$

In particular, for an $E_{m}$ $R$-algebra $A$, $A\otimes_{R}S$ is an $E_{m}$
$S$-algebra. Furthermore, the forgetful functor induced by $f$, i.e. the right
adjoint of $-\otimes_{R}S$, is $E_{n-1}$-lax monoidal and therefore it induces
a functor:

(4.2)
$\operatorname{Alg}_{E_{m}}(\operatorname{RMod}_{S})\to\operatorname{Alg}_{E_{m}}(\operatorname{RMod}_{R}).$

The unit of this adjunction provides a map of $E_{m}$ $R$-algebras:

(4.3) $A\to A\otimes_{R}S.$

Since $S$ is the monoidal unit in $\operatorname{RMod}_{S}$, $S$ is an
$E_{n-1}$ $S$-algebra, and forgetting through (4.2), it is an $E_{n-1}$
$R$-algebra. In summary, an $E_{n}$-algebra map $R\to S$ equips $S$ with the
structure of an $E_{n-1}$ $R$-algebra.

###  4.2. A construction for adjoining roots to ring spectra

We now introduce our construction for adjoining roots to ring spectra. For
this we use the following hypothesis. Recall that we often omit the functor
$D$ and let $\mathbb{S}[\sigma_{k}]$ denote the underlying $E_{2}$-ring of the
graded $E_{2}$-ring $\mathbb{S}[\sigma_{k}]$.

######  Hypothesis 4.4 (Root adjunction hypothesis).

Given an $E_{1}$-ring $A$ with an $a\in\pi_{mk}A$, there is a chosen
$\mathbb{S}[\sigma_{mk}]$-algebra structure on $A$ for which the structure map
$\mathbb{S}[\sigma_{mk}]\to A$ carries $\sigma_{mk}$ to $a\in\pi_{mk}A$. Here,
$m>0$ and $k\geq 0$ is even. See Proposition 4.5 for the cases of interest
where this is satisfied.

The hypothesis above may not seem very natural but it is satisfied in the
following general situations.

######  Proposition 4.5.

Let $k\geq 0$ be even and $m>0$, an $E_{1}$-ring $A$ satisfies Hypothesis 4.4
for $a\in\pi_{mk}A$ if:

  1. (1)

$A$ is an $E_{2}$-ring for which $\pi_{*}A$ is concentrated in even degrees,
or

  2. (2)

$A$ is an $R$-algebra for an $E_{2}$-ring $R$ where $\pi_{*}R$ is concentrated
in even degrees and $a$ is in the image of the map $\pi_{*}R\to\pi_{*}A$.

###### Proof.

Assume that $A$ is as in (2), let $r\in\pi_{mk}R$ detect $a$ through the map
$\pi_{*}R\to\pi_{*}A$. We choose an $E_{2}$-ring map
$g\colon\thinspace\mathbb{S}[\sigma_{mk}]\to R$ that carries $\sigma_{mk}$ to
$r$, see Proposition 3.15. Forgetting through $g$, see (4.2), one obtains a
$\mathbb{S}[\sigma_{mk}]$-algebra structure on $A$. Indeed, through this
structure, $\sigma_{mk}$ acts through $a$ as desired.

If $A$ is as in (1), then $A$ is an $A$-algebra and $A$ satisfies the
assumption in (2). Therefore, $A$ satisfies Hypothesis 4.4. ∎

For instance, the Morava $K$-theory spectrum $K(n)$ and all $E_{1}$
$MU_{(p)}$-algebra forms of $BP\langle n\rangle$ satisfy Hypothesis 4.4 with
respect to their non-negative degree homotopy classes.

Notice that we are not assuming any preexisting non-trivial grading on $A$; in
fact this will allow us to view it as an $m$-graded spectrum concentrated in
weight zero. Given $a\in\pi_{mk}A$, the following construction adjoins an
$m$-root to $a$.

######  Construction 4.6.

Assume Hypothesis 4.4. We consider $\mathbb{S}[\sigma_{mk}]$ as an $m$-graded
$E_{2}$-ring and $A$ as an $m$-graded $\mathbb{S}[\sigma_{mk}]$-algebra, both
concentrated in weight $0$, using the functor $F$ from Section 2.2; we omit
$F$ in our notation.

Due to Proposition 3.5 (Remark 3.7 for $k=0$), there is a map

$\phi:\mathbb{S}[\sigma_{mk}]\to\mathbb{S}[\sigma_{k}]$

of $m$-graded $E_{2}$-rings that carries $\sigma_{mk}$ to $\sigma_{k}^{m}$ in
homotopy where $\sigma_{k}$ is of weight $1$ and $\sigma_{mk}$ is of weight
$0$. Note that we omit the functor $D^{m}$ in our notation. Considering the
corresponding extension of scalars functor
$-\wedge_{\mathbb{S}[\sigma_{mk}]}\mathbb{S}[\sigma_{k}]$ between the
$\infty$-categories of $m$-graded $\mathbb{S}[\sigma_{mk}]$-algebras and
$m$-graded $\mathbb{S}[\sigma_{k}]$-algebras, (see (4.1)), we define the
$m$-graded $E_{1}$ $\mathbb{S}[\sigma_{k}]$-algebra $A(\sqrt[m]{a})$ through:

$A(\sqrt[m]{a}):=A\wedge_{\mathbb{S}[\sigma_{mk}]}\mathbb{S}[\sigma_{k}].$

This comes equipped with a map $A\to A(\sqrt[m]{a})$ of $m$-graded $E_{1}$
$\mathbb{S}[\sigma_{mk}]$-algebras, see (4.3).

Since $\pi_{*}(\mathbb{S}[\sigma_{k}])$ is free as a
$\pi_{*}(\mathbb{S}[\sigma_{mk}])$-module, one obtains an isomorphism of
rings:

$\pi_{*}A(\sqrt[m]{a})\cong\pi_{*}(A)[z]/(z^{m}-a).$

Therefore, we say $A(\sqrt[m]{a})$ is obtained from $A$ by adjoining an
$m$-root to $a$.

When $A$ is $p$-local, observe that we have and equivalence of $m$-graded
$E_{1}$ $\mathbb{S}[\sigma_{k}]$-algebras:

$A(\sqrt[m]{a})\simeq
A\wedge_{\mathbb{S}_{(p)}[\sigma_{mk}]}\mathbb{S}_{(p)}[\sigma_{k}],$

where $\mathbb{S}_{(p)}[\sigma_{i}]$ denotes the $p$-localization of
$\mathbb{S}[\sigma_{i}]$.

It follows that the weight pieces of $A(\sqrt[m]{a})$ are given by the
following

(4.7) $A(\sqrt[m]{a})_{i}\simeq\Sigma^{ik}A$

for each $0\leq i<m$.

Note that $A(\sqrt[m]{a})$ might possibly depend on the
$\mathbb{S}[\sigma_{mk}]$-algebra structure chosen on $A$. Therefore,
everytime we apply Construction 4.6, we fix an
$\mathbb{S}[\sigma_{mk}]$-algebra structure on $A$.

######  Remark 4.8.

If $A$ is an $E_{3}$-ring with even homotopy, one may start with an
$E_{2}$-map $\mathbb{S}[\sigma_{mk}]\to A$ for a given $a\in\pi_{mk}A$ with
$k\geq 0$. By extending scalars, one obtains an $E_{2}$-ring map
$A\wedge\mathbb{S}[\sigma_{mk}]\to A$ that equips $A$ with the structure of an
$A\wedge\mathbb{S}[\sigma_{mk}]$-algebra. Through this, $A(\sqrt[m]{a})$ is
weakly equivalent as an $m$-graded $\mathbb{S}[\sigma_{k}]$-algebra to

$A\wedge_{A\wedge\mathbb{S}[\sigma_{mk}]}A\wedge\mathbb{S}[\sigma_{k}].$

In particular, $A(\sqrt[m]{a})$ admits the structure of an $m$-graded
$A\wedge\mathbb{S}[\sigma_{k}]$-algebra.

######  Remark 4.9.

In general, we do not expect the root adjunction $A\ \to A(\sqrt[m]{a})$ to
satisfy a universal property. On the other hand, if $A$ is an $E_{3}$-ring,
$A(\sqrt[m]{a})$ is an $A$-algebra and the map
$\pi_{*}A\to\pi_{*}A(\sqrt[m]{a})$ is étale, then it follows by [23, Theorem
1.10] that there is a bijection between homotopy classes of $A$-algebra maps
$A(\sqrt[m]{a})\to B$ and $\pi_{*}A$-algebra maps
$\pi_{*}A(\sqrt[m]{a})\to\pi_{*}B$ for any étale $A$-algebra $B$.

For the following, we fix an $E_{2}$-map
$\mathbb{S}_{(p)}[\sigma_{2(p-1)}]\to\ell$ carrying $\sigma_{2(p-1)}$ to
$v_{1}$.

######  Theorem 4.10.

There is an equivalence

$ku_{p}\simeq\ell_{p}(\sqrt[p-1]{v_{1}})$

of $E_{1}$ $\ell_{p}$-algebras.

###### Proof.

By Remark 4.8 above, $\ell_{p}(\sqrt[p-1]{v_{1}})$ is an $\ell_{p}$-algebra.
Let $L_{p}$ denote the non-connective $p$-completed Adams summand. The $E_{1}$
$L_{p}$-algebra

$\ell_{p}(\sqrt[p-1]{v_{1}})\wedge_{\ell_{p}}L_{p}$

is an étale $E_{1}$ $L_{p}$-algebra in the sense of Hesselholt-Pstragowski
[23] and there is an isomorphism of $\pi_{*}(L_{p})$-algebras

$\pi_{*}(\ell_{p}(\sqrt[p-1]{v_{1}})\wedge_{\ell_{p}}L_{p})\cong\pi_{*}(KU_{p}).$

It follows by [23, Theorem 1.10] that there is an equivalence of
$L_{p}$-algebras

$\ell_{p}(\sqrt[p-1]{v_{1}})\wedge_{\ell_{p}}L_{p}\simeq KU_{p}.$

Through this, $\ell_{p}(\sqrt[p-1]{v_{1}})$ serves as the connective cover of
$KU_{p}$ in $E_{1}$ $\ell_{p}$-algebras. Hence, there is an equivalence of
$E_{1}$ $\ell_{p}$-algebras $ku_{p}\simeq\ell_{p}(\sqrt[p-1]{v_{1}})$. ∎

In Theorem 9.6 we show that the Morava $E$-theory spectrum $E_{n}$ is given by
$\mathbb{S}_{W(\mathbb{F}_{p^{n}})}\wedge_{\mathbb{S}_{p}}\widehat{E(n)}(\sqrt[p^{n}-1]{v_{n}})$
as an $E_{1}$-ring where $\widehat{E(n)}$ is the $K(n)$-localized Johnson-
Wilson spectrum.

######  Remark 4.11.

In certain cases, it is possible to equip $A(\sqrt[m]{a})$ with the structure
of an $E_{n}$-algebra for $n>1$. For this, one may use the graded $E_{\infty}$
$MU$-algebra $MU[\sigma_{k}]$ [25, Construction 2.6.1] where $k>0$ is even.
Indeed, $MU[\sigma_{k}]$ is the free graded $E_{1}$ $MU$-algebra over
$\Sigma^{k}MU$. There is a map of graded $E_{\infty}$-rings

$MU[\sigma_{2(p^{n}-1)}]:=L_{p^{n}-1}R_{p^{n}-1}MU[\sigma_{2}]\to
MU[\sigma_{2}]$

and Proposition 3.15 provides maps $\mathbb{S}[\sigma_{k}]\to MU[\sigma_{k}]$
of graded $E_{2}$-rings.

It follows by [25, Remark 2.1.2] that a form of $BP\langle n\rangle$ admits
the structure of an $E_{3}$ $MU[\sigma_{2(p^{n}-1)}$]-algebra where
$\sigma_{2(p^{n}-1)}$ acts through $v_{n}\in\pi_{*}BP\langle n\rangle$. We
obtain an equivalence of $p^{n}-1$-graded $E_{1}$
$\mathbb{S}[\sigma_{2}]$-algebras

$BP\langle n\rangle(\sqrt[p^{n}-1]{v_{n}}):=BP\langle
n\rangle\wedge_{\mathbb{S}[\sigma_{2(p^{n}-1)}]}\mathbb{S}[\sigma_{2}]\simeq
BP\langle n\rangle\wedge_{MU[\sigma_{2(p^{n}-1)}]}MU[\sigma_{2}].$

This equips $BP\langle n\rangle(\sqrt[p^{n}-1]{v_{n}})$ with the structure of
a $p^{n}-1$-graded $E_{3}$ $MU[\sigma_{2}]$-algebra.

###  4.3. The weight zero piece of THH

Here, we prove our first result regarding the topological Hochschild homology
of the ring spectra obtained via root adjunction. Namely, we show that
$\operatorname{\textup{THH}}(A(\sqrt[m]{a}))$ contains
$\operatorname{\textup{THH}}(A)$ as a summand whenever $A$ is $p$-local and
$p\nmid m$.

It follows by [2, Example A.10] that for an $m$-graded $E_{1}$-ring $Y$, the
$m$-grading on $\operatorname{\textup{THH}}(Y)$ is obtained by applying the
cyclic bar construction $b_{\bullet}(Y)$ of $Y$ in the $\infty$-category of
$m$-graded spectra. In simplicial level $s$ and weight $i$, the $m$-graded
cyclic bar construction of $Y$ is given by the following.

$b_{s}(Y)_{i}\simeq\bigvee_{k_{0}+\cdots+k_{s}=i\in\mathbb{Z}/m}Y_{k_{0}}\wedge\cdots\wedge
Y_{k_{s}}$

Due to [2, Corollary A.15], one has the following equality

(4.12) $\operatorname{\textup{THH}}(D(Y))\simeq
D(\operatorname{\textup{THH}}(Y))$

where the functor $D(-)$ provides the underlying spectrum as usual; we often
omit $D$ in our notation. Furthermore, $\operatorname{\textup{THH}}(Y)$ is an
$S^{1}$-equivariant $m$-graded spectrum in a canonical way and the equivalence
above is an equivalence of $S^{1}$-equivariant spectra.

######  Construction 4.13.

Let $R$ be an $E_{2}$-ring and let $S$ be an $E_{1}$ $R$-algebra. For us this
will mean that the pair
$(R,S)\in\operatorname{RMod}^{(2)}(\operatorname{Sp})$, where

$\operatorname{RMod}^{(2)}(\mathcal{C})=\operatorname{Alg}(\operatorname{RMod}(\mathcal{C}))$

for an arbitrary symmetric monoidal $\infty$-category $\mathcal{C}$. Here,
$\operatorname{RMod}(\mathcal{C})$ is the $\infty$-category of pairs $(A,M)$
where $A$ is an $E_{1}$-algebra and
$M\in\operatorname{RMod}_{A}(\mathcal{C})$. Thus, objects in
$\operatorname{RMod}^{(2)}(\mathcal{C})$ may be identified with pairs $(A,M)$
where $A$ is an $E_{2}$-algebra, and $M$ is an $E_{1}$ A-algebra in
$\mathcal{C}$.

We remark that in general, $\operatorname{RMod}^{(2)}(\mathcal{C})$ may be
written as $\operatorname{Alg}_{\mathcal{O}}(\mathcal{C})$ where $\mathcal{O}$
is the _tensor product_ of operads

$\mathcal{O}:=\operatorname{RMod}\times E_{1}$

This tensor product of operads, studied in depth in [36, Section 2.2.5] is
symmetric and satisfies the following universal property at the level of
algebra objects:

$\operatorname{Alg}_{\mathcal{O}}(\mathcal{C})\simeq\operatorname{Alg}_{E_{1}}(\operatorname{Alg}_{\operatorname{RMod}}(\mathcal{C}))\simeq\operatorname{Alg}_{\operatorname{RMod}}(\operatorname{Alg}_{E_{1}}(\mathcal{C}))$

Hence, applying the discussion to $\mathcal{C}=\operatorname{Sp}$ and $R$ and
$S$ as above, we may view $S$ as a right $R$-module in $E_{1}$-algebras.

Since $\operatorname{\textup{THH}}$ is a symmetric monoidal functor from
$E_{1}$-rings to spectra [41, Section IV.2] we deduce that
$\operatorname{\textup{THH}}(S)$ is a right
$\operatorname{\textup{THH}}(R)$-module.

######  Proposition 4.14.

Let $F$ be an $m$-graded $E_{1}$ $E$-algebra and $E\to F^{\prime}$ be a map of
$m$-graded $E_{2}$-rings. There is a natural equivalence of $m$-graded right
$\operatorname{\textup{THH}}(F^{\prime})$-modules in $S^{1}$-equivariant
spectra:

$\operatorname{\textup{THH}}(F\wedge_{E}F^{\prime})\simeq\operatorname{\textup{THH}}(F)\wedge_{\operatorname{\textup{THH}}(E)}\operatorname{\textup{THH}}(F^{\prime}),$

whose underlying undgraded equivalence is that of right
$\operatorname{\textup{THH}}(F^{\prime})$-modules in cyclotomic spectra. If
$E$ and $F$ are concentrated in weight zero, then we have the following.

$\operatorname{\textup{THH}}(F\wedge_{E}F^{\prime})_{i}\simeq\operatorname{\textup{THH}}(F)\wedge_{\operatorname{\textup{THH}}(E)}(\operatorname{\textup{THH}}(F^{\prime})_{i})$

###### Proof.

Let us recall that the functor

$\operatorname{\textup{THH}}:\operatorname{Alg}_{\operatorname{Sp}}\to\operatorname{CycSp}$

is symmetric monoidal. Furthermore, it commutes with sifted colimits; indeed
this can be seen from the fact that it can be decomposed into a composition of
functors comprised of taking tensor products and realizations of simplicial
objects, both of which commute with sifted colimits. Thus there will be a
natural equivalence

$\displaystyle\operatorname{\textup{THH}}(F\wedge_{E}F^{\prime})$
$\displaystyle\simeq\operatorname{\textup{THH}}(||\operatorname{Bar}_{\bullet}(F,E,F^{\prime})||)$
$\displaystyle\simeq||\operatorname{Bar}_{\bullet}(\operatorname{\textup{THH}}(F)_{\bullet},\operatorname{\textup{THH}}(E)_{\bullet},\operatorname{\textup{THH}}(F^{\prime})_{\bullet})||$
$\displaystyle\simeq\operatorname{\textup{THH}}(F)\wedge_{\operatorname{\textup{THH}}(E)}\operatorname{\textup{THH}}(F^{\prime})$

This allows us to deduce that $\operatorname{\textup{THH}}$ preserves the
sifted colimit given by the double sided Bar construction; this can be
computed at the level of underlying spectra by the bilinear pairing

${}_{F}\operatorname{BMod}_{E}\times{}_{E}\operatorname{BMod}_{F^{\prime}}\to_{F}\operatorname{BMod}_{F^{\prime}}$

corresponding to the relative tensor product. Furthermore, as
$\operatorname{\textup{THH}}$ preserves the sifted colimits corresponding to
this relative tensor product, the above equivalence is compatible with right
$\operatorname{\textup{THH}}(F^{\prime})$ module structures. The analogous
claims all hold when accounting for additional gradings, by recalling that
$\operatorname{\textup{THH}}$ promotes to a sifted colimit preserving
symmetric monoidal functor from algebras in graded spectra to
$S^{1}$-equivariant objects in graded spectra. In particular, if $E$ and $F$
are concentrated in weight zero, we deduce the equivalence

$\operatorname{\textup{THH}}(F\wedge_{E}F^{\prime})_{i}\simeq\operatorname{\textup{THH}}(F)\wedge_{\operatorname{\textup{THH}}(E)}(\operatorname{\textup{THH}}(F^{\prime})_{i}).$

of graded $\operatorname{\textup{THH}}(F)$-modules. ∎

######  Remark 4.15.

The $m=1$ case of the proposition above provides the non-graded case.

One may consider $\mathbb{S}[\sigma_{k}]$ as an $E_{1}$-ring obtained by
adjoining an $m$-root to $\mathbb{S}[\sigma_{mk}]$. Proposition 4.17
identifies the weight zero piece of
$\operatorname{\textup{THH}}(\mathbb{S}_{(p)}[\sigma_{k}])$. Before
Proposition 4.17, we state and prove a well known fact.

######  Lemma 4.16.

Let $\varphi\colon\thinspace M\to N$ be a map between bounded below spectra.
Then $\varphi$ is an equivalence if and only if $H\mathbb{Z}\wedge\varphi$ is
an equivalence. If furthermore $M$ and $N$ are $p$-local, then $\varphi$ is an
equivalence if and only if $H\mathbb{Z}_{(p)}\wedge\varphi$ is an equivalence.

###### Proof.

Let $K$ be the fiber of $\varphi$ and let $i$ be the smallest $i$ such that
$\pi_{i}K\neq 0$. Due to the Tor spectral sequence of [15, Theorem IV.4.1], we
have $\pi_{i}(H\mathbb{Z}\wedge K)=\pi_{i}K$. Therefore, if $H\mathbb{Z}\wedge
K\simeq 0$ then $K\simeq 0$ and $\varphi$ is an equivalence.

If $M$ and $N$ are $p$-local, then $\varphi$ is an equivalence if and only if
$\mathbb{S}_{(p)}\wedge\varphi$ is an equivalence. It follows by the previous
result that $\varphi$ is an equivalence if and only if
$H\mathbb{Z}\wedge\mathbb{S}_{(p)}\wedge\varphi\simeq
H\mathbb{Z}_{(p)}\wedge\varphi$ is an equivalence. ∎

For the following, let $k\geq 0$ be even and let $m>1$. Furthermore, fix a map
of $m$-graded $E_{2}$-rings
$\mathbb{S}_{(p)}[\sigma_{mk}]\to\mathbb{S}_{(p)}[\sigma_{k}]$ provided by
Proposition 3.5 (Remark 3.7 for $k=0$).

######  Proposition 4.17.

In the situation above, assume that $p\nmid m$. The induced map

$\operatorname{\textup{THH}}(\mathbb{S}_{(p)}[\sigma_{mk}])_{0}\xrightarrow{\simeq}\operatorname{\textup{THH}}(\mathbb{S}_{(p)}[\sigma_{k}])_{0}$

is an equivalence of $E_{1}$-rings. Since $\mathbb{S}_{(p)}[\sigma_{mk}]$ is
concentrated in weight zero, we obtain the following chain of equivalences

$D(\operatorname{\textup{THH}}(\mathbb{S}_{(p)}[\sigma_{mk}]))\simeq\operatorname{\textup{THH}}(\mathbb{S}_{(p)}[\sigma_{mk}])_{0}\xrightarrow{\simeq}\operatorname{\textup{THH}}(\mathbb{S}_{(p)}[\sigma_{k}])_{0}$

of $E_{1}$-rings using Lemma 2.3.

###### Proof.

It suffices to prove that the map

$H\mathbb{Z}_{(p)}\wedge\operatorname{\textup{THH}}(\mathbb{S}_{(p)}[\sigma_{mk}])_{0}\to
H\mathbb{Z}_{(p)}\wedge\operatorname{\textup{THH}}(\mathbb{S}_{(p)}[\sigma_{k}])_{0}$

is an equivalence, see Lemma 4.16.

By the base change formula for THH, this is equivalent to the following map

$\operatorname{\textup{THH}}^{H\mathbb{Z}_{(p)}}(H\mathbb{Z}_{(p)}[\sigma_{mk}])\to\operatorname{\textup{THH}}^{H\mathbb{Z}_{(p)}}(H\mathbb{Z}_{(p)}[\sigma_{k}])$

being an equivalence in weight zero. Here, $H\mathbb{Z}_{(p)}[\sigma_{k}]$
denotes the free $H\mathbb{Z}_{(p)}$-algebra on $\mathbb{S}_{(p)}^{k}$ given
by $H\mathbb{Z}_{(p)}\wedge\mathbb{S}_{(p)}[\sigma_{k}]$.

The map $H\mathbb{Z}_{(p)}[\sigma_{mk}]\to H\mathbb{Z}_{(p)}[\sigma_{k}]$
induces a map

$\phi^{r}\colon\thinspace E^{r}\to{F}^{r}$

from the Bökstedt spectral sequence computing
$\operatorname{\textup{THH}}^{H\mathbb{Z}_{(p)}}(H\mathbb{Z}_{(p)}[\sigma_{mk}])$
to the Bökstedt spectral sequence computing
$\operatorname{\textup{THH}}^{H\mathbb{Z}_{(p)}}(H\mathbb{Z}_{(p)}[\sigma_{k}])$.
Since the weight grading on the THH of an $m$-graded ring spectrum comes from
a weight grading on the corresponding cyclic bar construction, the Bökstedt
spectral sequence computing THH of an $m$-graded ring spectrum admits an
$m$-grading, i.e. it splits into $m$ summands in a canonical way. Therefore,
in our situation, it is sufficient to show that $\phi^{2}$ is an isomorphism
on weight zero.

We have

(4.18)
$\phi^{2}\colon\thinspace\mathbb{Z}_{(p)}[\sigma_{mk}]\otimes\Lambda_{\mathbb{Z}_{(p)}}(d(\sigma_{mk}))\to\mathbb{Z}_{(p)}[\sigma_{k}]\otimes\Lambda_{\mathbb{Z}_{(p)}}(d(\sigma_{k}))$

where $d$ denotes the Connes operator. The degrees of the classes above are
given by the following.

$\text{deg}(\sigma_{mk})=(0,mk)\ \ \ \text{deg}(d(\sigma_{mk}))=(1,mk)$
$\text{deg}(\sigma_{k})=(0,k)\textup{\ \ \ \ }\text{deg}(d(\sigma_{k}))=(1,k)$

Furthermore, $\sigma_{mk}$ and $d(\sigma_{mk})$ are in weight $0$ and
$\sigma_{k}$ and $d(\sigma_{k})$ are in weight $1$. In particular, all of
$E^{2}$ is weight zero and the weight zero piece of $F^{2}$ is the
$\mathbb{Z}_{(p)}$-module generated by the classes $\sigma_{k}^{im}$ and
$\sigma_{k}^{(i+1)m-1}d(\sigma_{k})$ over $i\geq 0$.

Since $\phi^{2}(\sigma_{mk})=\sigma_{k}^{m}$, we obtain that

$\phi^{2}(d(\sigma_{mk}))=d(\phi^{2}(\sigma_{mk}))=d(\sigma_{k}^{m})=m\sigma_{k}^{m-1}d(\sigma_{k}).$

Therefore, we have

$\phi^{2}(\sigma_{mk}^{i})=\sigma_{k}^{im}\textup{\ \ \ and\ \
}\phi^{2}(\sigma_{mk}^{i}d(\sigma_{mk}))=m\sigma_{k}^{(i+1)m-1}d(\sigma_{k}).$

Since $p\nmid m$, we have that $m$ is a unit. Using this, one observes that
$\phi^{2}$ is an isomorphism after restricting and corestricting to weight
zero as desired. ∎

In the situation of Hypothesis 4.4, $A\to A(\sqrt[m]{a})$ is a map of
$m$-graded $E_{1}$-rings and $A$ is concentrated in weight zero. Therefore,
there is a map

(4.19)
$\operatorname{\textup{THH}}(A)\to\operatorname{\textup{THH}}(A(\sqrt[m]{a}))_{0}$

where $A$ above denotes the underlying $E_{1}$-ring of $A$.

######  Theorem 4.20.

Assume Hypothesis 4.4 and that $A$ is $p$-local and $p\nmid m$. Then (4.19) is
an equivalence.

###### Proof.

Recall that $A(\sqrt[m]{a})$ is given by

$A\wedge_{\mathbb{S}_{(p)}[\sigma_{mk}]}\mathbb{S}_{(p)}[\sigma_{k}]$

where $A$ and $\mathbb{S}_{(p)}[\sigma_{mk}]$ are concentrated in weight zero.
Due to Proposition 4.14, we have

$\operatorname{\textup{THH}}(A(\sqrt[m]{a}))_{0}\simeq\operatorname{\textup{THH}}(A)\wedge_{\operatorname{\textup{THH}}(\mathbb{S}_{(p)}[\sigma_{mk}])}(\operatorname{\textup{THH}}(\mathbb{S}_{(p)}[\sigma_{k}])_{0})$

and it follows by Proposition 4.17 that the map

$\operatorname{\textup{THH}}(\mathbb{S}_{(p)}[\sigma_{mk}])\xrightarrow{\simeq}\operatorname{\textup{THH}}(\mathbb{S}_{(p)}[\sigma_{k}])_{0}$

is an equivalence. This identifies
$\operatorname{\textup{THH}}(A(\sqrt[m]{a}))_{0}$ with
$\operatorname{\textup{THH}}(A)$ as desired. ∎

##  5\. Adjoining roots and algebraic $K$-theory

We now prove Theorem 1.1 from the introduction. For the rest of this section,
assume Hypothesis 4.4. We established that $A(\sqrt[m]{a})$ is an $m$-graded
ring spectrum and therefore $\operatorname{\textup{THH}}(A(\sqrt[m]{a}))$ is
an $S^{1}$-equivariant $m$-graded spectrum, see (4.12). One might define
$\operatorname{\textup{TC}^{-}}(A(\sqrt[m]{a}))$ as an $m$-graded spectrum
given by:

$\operatorname{\textup{TC}^{-}}(A(\sqrt[m]{a}))_{i}\simeq\operatorname{\textup{THH}}(A(\sqrt[m]{a}))_{i}^{hS^{1}}.$

Since $m$ is finite, the underlying spectrum of an $m$-graded spectrum,
provided by the functor $D$, is given by a finite coproduct which is
equivalent to the corresponding finite product. In particular, $D$ commutes
with all limits and colimits. Because of this, we have

$D(\operatorname{\textup{TC}^{-}}(A(\sqrt[m]{a})))\simeq\operatorname{\textup{TC}^{-}}(D(A(\sqrt[m]{a})))$

and therefore, we often omit $D$ in our notation.

Similarly, $\operatorname{\textup{TP}}(A(\sqrt[m]{a}))$ and
$(\operatorname{\textup{THH}}(A(\sqrt[m]{a}))^{tC_{p}})^{hS^{1}}$ admit the
structure of $m$-graded spectra and these constructions commute with the
functor $D$ as above. Combining this, with Theorem 4.20, we obtain the
following result.

######  Theorem 5.1.

Assume Hypothesis 4.4, that $A$ is $p$-local, and that $p\nmid m$. The maps

$\displaystyle\operatorname{\textup{TC}^{-}}(A)\xrightarrow{\simeq}\operatorname{\textup{TC}^{-}}(A(\sqrt[m]{a}))_{0}$
$\displaystyle\operatorname{\textup{TP}}(A)\xrightarrow{\simeq}\operatorname{\textup{TP}}(A(\sqrt[m]{a}))_{0}$
$\displaystyle(\operatorname{\textup{THH}}(A)^{tC_{p}})^{hS^{1}}\xrightarrow{\simeq}((\operatorname{\textup{THH}}(A(\sqrt[m]{a}))^{tC_{p}})^{hS^{1}})_{0}$

induced by $A\to A(\sqrt[m]{a})$ are all equivalences. ∎

When $A$ is connective and $p$-local, $A(\sqrt[m]{a})$ is also connective and
$p$-local. By [41, Corollary 1.5] (and the discussion afterwards), the
topological cyclic homology of $A(\sqrt[m]{a})$ is defined via the following
fiber sequence.

(5.2)
$\operatorname{\textup{TC}}(A(\sqrt[m]{a}))\to\operatorname{\textup{THH}}(A(\sqrt[m]{a}))^{hS^{1}}\xrightarrow{\varphi_{p}^{hS^{1}}-can}(\operatorname{\textup{THH}}(A(\sqrt[m]{a}))^{tC_{p}})^{hS^{1}}$

As mentioned above, the middle term and the third term above admit canonical
splittings into $m$-cofactors. Furthermore, $can$ respects this splitting
since it only depends on the $S^{1}$-equivariant structure of
$\operatorname{\textup{THH}}(A(\sqrt[m]{a}))$.

However, $\operatorname{\textup{TC}}(A(\sqrt[m]{a}))$ do not necessarily split
into $m$-cofactors. This is due to the fact that the Frobenius map does not
necessarily respect the grading. Indeed, the Frobenius is given by maps

$\varphi_{p}\colon\thinspace\operatorname{\textup{THH}}(A(\sqrt[m]{a}))_{i}\to\operatorname{\textup{THH}}(A(\sqrt[m]{a}))^{tC_{p}}_{ip},$

see [2, Corollary A.9]. On the other hand, we obtain the following splitting
of $\operatorname{\textup{TC}}(A(\sqrt[m]{a}))$.

######  Construction 5.3.

Assume Hypothesis 4.4 and that $A$ is connective and $p$-local where $p\nmid
m$. In this situation, $p$ is a non-zero divisor in $\mathbb{Z}/m$. Therefore,
the Frobenius map on $\operatorname{\textup{THH}}(A(\sqrt[m]{a}))$ carries
pieces of non-zero weight to non-zero weight pieces. Moreover, $\varphi_{p}$
carries weight zero to weight zero. Therefore, the map $\varphi_{p}-can$
splits as a coproduct of their restriction to weight zero and their
restriction to non-zero weight. In particular, the fiber sequence in (5.2)
admits a splitting as follows.

$\operatorname{\textup{TC}}(A(\sqrt[m]{a}))_{0}\vee\operatorname{\textup{TC}}(A(\sqrt[m]{a}))_{1}\to\operatorname{\textup{THH}}(A(\sqrt[m]{a}))_{0}^{hS^{1}}\vee\operatorname{\textup{THH}}(A(\sqrt[m]{a}))_{>0}^{hS^{1}}\\\
\xrightarrow{(\varphi_{p})_{0}-can_{0}\vee(\varphi_{p})_{>0}-can_{>0}}(\operatorname{\textup{THH}}(A(\sqrt[m]{a}))_{0}^{tC_{p}})^{hS^{1}}\vee(\operatorname{\textup{THH}}(A(\sqrt[m]{a}))^{tC_{p}})_{>0}^{hS^{1}}$

Here, $(-)_{>0}$ denotes restriction to weight not equal to $0$. We have

(5.4)
$\operatorname{\textup{TC}}(A(\sqrt[m]{a}))\simeq\operatorname{\textup{TC}}(A(\sqrt[m]{a}))_{0}\vee\operatorname{\textup{TC}}(A(\sqrt[m]{a}))_{1}$

where $\operatorname{\textup{TC}}(A(\sqrt[m]{a}))_{0}$ denotes the fiber of
the map $(\varphi_{p})_{0}-can_{0}$ and
$\operatorname{\textup{TC}}(A(\sqrt[m]{a}))_{1}$ denotes the fiber of the map
$(\varphi_{p})_{>0}-can_{>0}$.

######  Remark 5.5.

There are interesting cases where one obtains further splittings of the
topological cyclic homology spectrum
$\operatorname{\textup{TC}}(A(\sqrt[m]{a}))$. For instance, if $p=1$ in
$\mathbb{Z}/m$, then and one obtains that
$\operatorname{\textup{TC}}(A(\sqrt[m]{a}))$ splits into $m$-summands. This
happens to be the case when $m=p-1$ or when $p$ is odd and $m=2$. We exploit
this in Construction 7.3 to obtain a splitting of
$\operatorname{\textup{TC}}(ku_{p})$ into $p-1$ summands. Moreover, if
$m=p^{n}-1$, then one obtains an underlying $p-1$-grading of
$\operatorname{\textup{THH}}(A(\sqrt[m]{a}))$ by Kan extending through
$\mathbb{Z}/(p^{n}-1)\to\mathbb{Z}/(p-1)$. This provides a $p-1$-grading for
$\operatorname{\textup{TC}}(A(\sqrt[m]{a}))$.

######  Theorem 5.6.

Assume Hypothesis 4.4 with $p\nmid m$ and that $A$ is $p$-local and
connective. Under the equivalence (5.4), the canonical map

$\operatorname{\textup{TC}}(A)\to\operatorname{\textup{TC}}(A(\sqrt[m]{a}))$

is equivalent to the inclusion of the first wedge summand.

###### Proof.

Since $A$ is concentrated in weight zero, the map
$\operatorname{\textup{TC}}(A)\to\operatorname{\textup{TC}}(A(\sqrt[m]{a}))$
factors through the map

(5.7)
$\operatorname{\textup{TC}}(A)\to\operatorname{\textup{TC}}(A(\sqrt[m]{a}))_{0}$

induced by the canonical map
$\operatorname{\textup{THH}}(A)\to\operatorname{\textup{THH}}(A(\sqrt[m]{a}))_{0}$.
The map
$\operatorname{\textup{THH}}(A)\to\operatorname{\textup{THH}}(A(\sqrt[m]{a}))_{0}$
of cyclotomic spectra is an equivalence due to Theorem 4.20. Considering the
construction of $\operatorname{\textup{TC}}(A(\sqrt[m]{a}))_{0}$, one observes
that this equivalence induces an equivalence between the fiber sequences
defining $\operatorname{\textup{TC}}(A)$ and
$\operatorname{\textup{TC}}(A(\sqrt[m]{a}))_{0}$. In other words, (5.7) is an
equivalence as desired. ∎

Finally, we obtain the desired splitting for $K(A(\sqrt[m]{a}))$.

######  Theorem 5.8 (Theorem 1.1).

Assume Hypothesis 4.4 with $p\nmid m$ and $k>0$. Furthermore, assume that $A$
is $p$-local and connective. In this situation, the following map

$K(A)\to K(A(\sqrt[m]{a}))$

is the inclusion of a wedge summand.

###### Proof.

Since $\lvert a\rvert=mk$ and since $k>0$, we have

(5.9) $\pi_{0}A(\sqrt[m]{a})=\pi_{0}A.$

We start by constructing a map of $m$-graded $E_{1}$-algebras

(5.10) $A(\sqrt[m]{a})\to H\pi_{0}A$

that induces an isomorphism on $\pi_{0}$ where $H\pi_{0}A$ is concentrated in
weight $0$. Weight $0$ Postnikov truncation [25, Lemma B.0.6] provides a map
of graded $E_{2}$-rings $\mathbb{S}[\sigma_{k}]\to\mathbb{S}$ that we consider
as a map of $m$-graded $E_{2}$-rings by left Kan extending through
$\mathbb{Z}\to\mathbb{Z}/m$.

This provides a map of $m$-graded $E_{1}$-rings

$A(\sqrt[m]{a})\simeq
A\wedge_{\mathbb{S}[\sigma_{mk}]}\mathbb{S}[\sigma_{k}]\to
A\wedge_{\mathbb{S}[\sigma_{mk}]}\mathbb{S}$

(see (4.3)) where the right hand side is concentrated in weight $0$.
Postcomposing with the ordinary Postnikov truncation, we obtain (5.10).

Due to the Dundas-Goodwillie-McCarthy theorem [13], there is a pullback square

${K(A(\sqrt[m]{a}))}$${\operatorname{\textup{TC}}(A(\sqrt[m]{a}))\simeq\operatorname{\textup{TC}}(A)\vee\operatorname{\textup{TC}}(A(\sqrt[m]{a}))_{1}}$${K(H\pi_{0}A)}$${\operatorname{\textup{TC}}(H\pi_{0}A)}$

provided by the map (5.10). The equivalence on the upper right corner follows
by Construction 5.3 and Theorem 5.6.

The map $A(\sqrt[m]{a})\to H\pi_{0}A$ induces a map of $m$-graded spectra

$f\colon\thinspace\operatorname{\textup{THH}}(A(\sqrt[m]{a}))\to\operatorname{\textup{THH}}(H\pi_{0}A).$

Since $H\pi_{0}A$ is concentrated in weight zero,
$\operatorname{\textup{THH}}(H\pi_{0}A)$ is also concentrated in weight zero.
Therefore, the map $f$ is trivial on
$\operatorname{\textup{THH}}(A(\sqrt[m]{a}))_{>0}$. This shows that the right
vertical map above induces the trivial map on
$\operatorname{\textup{TC}}(A(\sqrt[m]{a}))_{1}$. Using this, we obtain that
the pullback square above splits as a coproduct of the pullback squares

${K(A)}$${\operatorname{\textup{TC}}(A)}$${K(H\pi_{0}A)}$${\operatorname{\textup{TC}}(H\pi_{0}A)}$

and

${\operatorname{\textup{TC}}(A(\sqrt[m]{a}))_{1}}$${\operatorname{\textup{TC}}(A(\sqrt[m]{a}))_{1}}$${*}$${*.}$$\scriptstyle{\simeq}$

This shows that

$K(A(\sqrt[m]{a}))\simeq
K(A)\vee\operatorname{\textup{TC}}(A(\sqrt[m]{a}))_{1}$

as desired.

∎

Using the Purity theorem for algebraic $K$-theory, we obtain the following for
non-connective $A$.

######  Corollary 5.11.

Assume Hypothesis 4.4 with $p\nmid m$ and $k>0$. If $A$ is $p$-local, then the
map

$L_{T(i)}K(A)\to L_{T(i)}K(A(\sqrt[m]{a}))$

is the inclusion of a wedge summand for every $i\geq 2$. In particular, if $A$
is of height larger than $0$ and $A$ satisfies the redshift conjecture, then
$A(\sqrt[m]{a})$ also satisfies the redshift conjecture.

###### Proof.

Let $cA$ denote the connective cover of $A$ in
$\mathbb{S}[\sigma_{mk}]$-algebras. We consider the following commuting
diagram of $m$-graded $E_{1}$-rings.

${cA}$${A}$${(cA)(\sqrt[m]{a})}$${A(\sqrt[m]{a})}$

Every spectrum with bounded above homotopy is $T(i)$-locally trivial for every
$i\geq 1$. Taking fibers, one obtains that the horizontal arrows above are
$T(i)$-equivalence for every $i\geq 1$, see (4.7).

It follows by [31, Purity Theorem] that the horizontal maps above induce
$T(i)$-equivalences in algebraic $K$-theory for every $i\geq 2$. The result
follows by applying Theorem 5.8 to the left vertical map. ∎

##  6\. A variant of logarithmic THH

Here, we introduce our definition of logarithmic THH and identify
$\operatorname{\textup{THH}}(A(\sqrt[m]{a}))$ using
$\operatorname{\textup{THH}}(A)$ and logarithmic THH of $A$ whenever $A$ is
$p$-local and $p\nmid m$. Through our definition, logarithmic THH admits a
canonical structure of a cyclotomic spectrum; in upcoming work, Devalapurkar
and the third author develop a very general notion of logarithmic structures
for $E_{2}$-algebras and a corresponding theory of log THH which subsumes the
definition we use here. This will in particular recover the variant due to
Rognes, which is defined by way of the replete bar construction, cf. [46, 47]

Our definition of log THH starts with a definition of the log THH of the free
algebra $\mathbb{S}[\sigma_{k}]$ where $k\geq 0$ is even as before. We
consider $\sigma_{k}$ to be in weight 1.

For a graded $E_{n}$-ring spectrum $E$, we denote the _weight connective
cover_ of $E$ by $E_{\geq 0}$. Indeed, the weight connective cover is obtained
by restricting and then left Kan extending through the inclusion
$\mathbb{N}\to\mathbb{Z}$. The counit of this adjunction provides a map
$E_{\geq 0}\to E$ of graded $E_{n}$-algebras.

######  Construction 6.1.

Analogous to Construction 3.3, let $\mathbb{S}[\sigma_{k}^{{\pm
1}}]:=\operatorname{sh}^{k}(\mathbb{S}[t^{\pm 1}])$. The graded
$E_{\infty}$-map $\mathbb{S}[t]\to\mathbb{S}[t^{\pm 1}]$ provides a graded
$E_{2}$-map $\mathbb{S}[\sigma_{k}]\to\mathbb{S}[\sigma_{k}^{\pm 1}]$.
Furthermore, by the definition of the shearing functor,
$\mathbb{S}[\sigma_{k}^{\pm 1}]$ is indeed given by $\phi^{k}$ of Variant 3.2;
in particular, $\mathbb{S}[\sigma_{mk}^{\pm 1}]$ is the restriction of
$\mathbb{S}[\sigma_{k}^{\pm 1}]$ along $\mathbb{Z}\xrightarrow{\cdot
m}\mathbb{Z}$. Applying the adjunction $L_{m}\dashv R_{m}$ induced by $\cdot
m$ to the map $\mathbb{S}[\sigma_{mk}]\to\mathbb{S}[\sigma_{k}]$, one observes
that $\mathbb{S}[\sigma_{mk}]$ is also the restriction of
$\mathbb{S}[\sigma_{k}]$ along $\cdot m$. Therefore, the counit of
$L_{m}\dashv R_{m}$ provides a commutative diagram of graded $E_{2}$-rings:

${\mathbb{S}[\sigma_{mk}]}$${\mathbb{S}[\sigma_{mk}^{\pm
1}]}$${\mathbb{S}[\sigma_{k}]}$${\mathbb{S}[\sigma_{k}^{\pm 1}].}$

######  Remark 6.2.

One may also take weight connective covers in the $\infty$-category of graded
$S^{1}$-equivariant spectra by using the left Kan extension/restriction
adjunction induced by $\mathbb{N}^{ds}\to\mathbb{Z}$. This provides a map

$\operatorname{\textup{THH}}(\mathbb{S}[\sigma_{k}])\to\operatorname{\textup{THH}}(\mathbb{S}[\sigma_{k}^{\pm
1}])_{\geq 0}$

of graded $E_{1}$-algebras in $S^{1}$-equivariant spectra factoring the map
$\operatorname{\textup{THH}}(\mathbb{S}[\sigma_{k}])\to\operatorname{\textup{THH}}(\mathbb{S}[\sigma_{k}^{\pm
1}])$.

The following is analogous to the description of the replete bar construction
of commutative $\mathcal{J}$-space monoids generated by a single element; c.f.
[46, Proposition 3.21], [50, Section 8.5] and [47, Sections 6 and 7].

######  Definition 6.3.

Let $k\geq 0$ be even. The logarithmic THH of $\mathbb{S}[\sigma_{k}]$ with
respect to $\sigma_{k}\in\pi_{k}\mathbb{S}[\sigma_{k}]$ is the weight
connective cover of the topological Hochschild homology of
$\mathbb{S}[{\sigma_{k}}^{{\pm 1}}]$. In other words, it is the
$S^{1}$-equivariant $E_{1}$-algebra:

$\operatorname{\textup{THH}}(\mathbb{S}[\sigma_{k}]\mid\sigma_{k}):=\operatorname{\textup{THH}}(\mathbb{S}[\sigma_{k}^{\pm
1}])_{\geq 0}.$

Similarly, the $p$-local counterpart is defined as follows.

$\operatorname{\textup{THH}}(\mathbb{S}_{(p)}[\sigma_{k}]\mid\sigma_{k}):=\operatorname{\textup{THH}}(\mathbb{S}_{(p)}[\sigma_{k}^{\pm
1}])_{\geq 0}$

The following example provides a justification for this definition of
logarithmic THH by showing that its $H\mathbb{Z}$-homology provides what
should be the logarithmic Hochschild homology of the free algebra
$\mathbb{Z}[\sigma_{k}]$, c.f. [27, Example 10.3].

######  Example 6.4.

Considering $H\mathbb{Z}$ as a graded $E_{\infty}$-algebra concentrated in
weight $0$, we deduce that

$H\mathbb{Z}\wedge(\operatorname{\textup{THH}}(\mathbb{S}[\sigma_{k}^{\pm
1}])_{\geq
0})\simeq(H\mathbb{Z}\wedge\operatorname{\textup{THH}}(\mathbb{S}[\sigma_{k}^{\pm
1}]))_{\geq 0}.$

Therefore,
$H\mathbb{Z}_{*}\operatorname{\textup{THH}}(\mathbb{S}[\sigma_{k}]\mid\sigma_{k})$
is given by the weight connective cover of

(6.5)
$\operatorname{\textup{THH}}^{H\mathbb{Z}}_{*}(H\mathbb{Z}[\sigma_{k}^{\pm
1}])\cong\mathbb{Z}[\sigma_{k}^{\pm 1}]\otimes\Lambda(d\sigma_{k})$

where $d\sigma_{k}$ is of weight $1$ and degree $k+1$ and $\sigma_{k}$ is of
weight $1$ and degree $k$. The isomorphism above follows by the usual Bökstedt
spectral sequence considerations applied together with the HKR theorem. Taking
the weight connective cover of (6.5), we obtain:

$H\mathbb{Z}_{*}\operatorname{\textup{THH}}(\mathbb{S}[\sigma_{k}]\mid\sigma_{k})\cong\mathbb{Z}[\sigma_{k}]\otimes\Lambda(\textup{dlog}\sigma_{k})$

where $\text{dlog}\sigma_{k}$ is of weight $0$ and homotopical degree $1$ and
it corresponds to $(d\sigma_{k})/\sigma_{k}$. Furthermore, the map

$H\mathbb{Z}_{*}\operatorname{\textup{THH}}(\mathbb{S}[\sigma_{k}])\to
H\mathbb{Z}_{*}\operatorname{\textup{THH}}(\mathbb{S}[\sigma_{k}]\mid\sigma_{k})$

carries $d\sigma_{k}$ to $d\sigma_{k}=\sigma_{k}\text{dlog}\sigma_{k}$.

Recall from Construction 4.13 that when $A$ is a
$\mathbb{S}[\sigma_{k}]$-algebra, $\operatorname{\textup{THH}}(A)$ admits the
structure of a right
$\operatorname{\textup{THH}}(\mathbb{S}[\sigma_{k}])$-module. We use this
structure in the following definition. Recall that Proposition 4.5 provides
various cases of interest where the assumptions on $A$ in the following
definition are satisfied.

######  Definition 6.6.

Let $A$ be an $E_{1}$ $\mathbb{S}[\sigma_{k}]$-algebra and assume that the
unit map $\mathbb{S}[\sigma_{k}]\to A$ carries
$\sigma_{k}\in\pi_{k}\mathbb{S}[\sigma_{k}]$ to $a\in\pi_{k}A$ with even
$k\geq 0$. We define the logarithmic THH of $A$ relative to $a$ as the
following $S^{1}$-equivariant spectrum.

$\operatorname{\textup{THH}}(A\mid
a):=\operatorname{\textup{THH}}(A)\wedge_{\operatorname{\textup{THH}}(\mathbb{S}[\sigma_{k}])}\operatorname{\textup{THH}}(\mathbb{S}[\sigma_{k}]\mid\sigma_{k})$

If $A$ is assumed to be $p$-local, we use the following equivalent definition

$\operatorname{\textup{THH}}(A\mid
a):=\operatorname{\textup{THH}}(A)\wedge_{\operatorname{\textup{THH}}(\mathbb{S}_{(p)}[\sigma_{k}])}\operatorname{\textup{THH}}(\mathbb{S}_{(p)}[\sigma_{k}]\mid\sigma_{k}).$

The definition of logarithmic THH we provide above is analogous to the
definitions used in [46, 50, 47].

######  Remark 6.7.

We remark that
$\operatorname{\textup{THH}}(\mathbb{S}[\sigma_{k}]\mid\sigma_{k})$ should be
a cyclotomic spectrum as the Frobenius maps of THH multiply the weight by $p$
and this should provide $\operatorname{\textup{THH}}(A\mid a)$ above with the
structure of a cyclotomic spectrum. However, since we don’t explicitly need
this for our application, displaying this will take us too far afield, and so,
we leave the details to the future work of Devalapurkar and the third author.

Since the definition of logarithmic THH is given by the extension of scalars
functor:

$-\wedge_{\operatorname{\textup{THH}}(\mathbb{S}[\sigma_{k}])}\operatorname{\textup{THH}}(\mathbb{S}[\sigma_{k}]\mid\sigma_{k})\colon\thinspace\operatorname{RMod}_{\operatorname{\textup{THH}}(\mathbb{S}[\sigma_{k}])}\to\operatorname{RMod}_{\operatorname{\textup{THH}}(\mathbb{S}[\sigma_{k}]\mid\sigma_{k})},$

corresponding to the $E_{1}$-algebra map
$\operatorname{\textup{THH}}(\mathbb{S}[\sigma_{k}])\to\operatorname{\textup{THH}}(\mathbb{S}[\sigma_{k}]\mid\sigma_{k})$,
we deduce that $\operatorname{\textup{THH}}(A\mid a)$ is equipped with the
structure of a right
$\operatorname{\textup{THH}}(\mathbb{S}[\sigma_{k}]\mid\sigma_{k})$-module.
Furthermore, the unit of the adjunction given by the extension of scalars
functor above and the corresponding forgetful functor provides a map

(6.8) $\operatorname{\textup{THH}}(A)\to\operatorname{\textup{THH}}(A\mid a)$

of right $\operatorname{\textup{THH}}(\mathbb{S}[\sigma_{k}])$-modules, see
(4.3).

######  Remark 6.9.

Using $MU[\sigma_{k}]$ mentioned in Remark 4.11, it is possible to equip
logarithmic THH with the structure of an $E_{n}$-algebra for $n>0$ in
favorable cases. For instance, for the $E_{3}$
$MU[\sigma_{2(p^{n}-1)}]$-algebra form of $BP\langle n\rangle$ constructed in
[25], $\operatorname{\textup{THH}}(BP\langle n\rangle\mid v_{n})$ admits the
structure of an $E_{1}$-ring. Indeed, using the map of $E_{2}$-rings
$\operatorname{\textup{THH}}(MU[\sigma_{2(p^{n}-1)}])\to\operatorname{\textup{THH}}(BP\langle
n\rangle)$, we obtain an $E_{1}$-ring:

$\operatorname{\textup{THH}}(BP\langle
n\rangle)\wedge_{\operatorname{\textup{THH}}(MU[\sigma_{2(p^{n}-1)}])}\operatorname{\textup{THH}}(MU[\sigma_{2(p^{n}-1)}^{\pm
1}])_{\geq 0},$

equivalent to $\operatorname{\textup{THH}}(BP\langle n\rangle\mid v_{n})$.
This equivalence follows by the following chain of equivalences

$\begin{split}\operatorname{\textup{THH}}&(BP\langle
n\rangle)\wedge_{\operatorname{\textup{THH}}(MU[\sigma_{2(p^{n}-1)}])}\operatorname{\textup{THH}}(MU[\sigma_{2(p^{n}-1)}^{\pm
1}])_{\geq 0}\\\ &\simeq\operatorname{\textup{THH}}(BP\langle
n\rangle)\wedge_{\operatorname{\textup{THH}}(MU)\wedge\operatorname{\textup{THH}}(\mathbb{S}[\sigma_{2(p^{n}-1)}])}\operatorname{\textup{THH}}(MU)\wedge\operatorname{\textup{THH}}(\mathbb{S}[\sigma_{2(p^{n}-1)}^{\pm
1}])_{\geq 0}\\\ &\simeq\operatorname{\textup{THH}}(BP\langle
n\rangle)\wedge_{\operatorname{\textup{THH}}(\mathbb{S}[\sigma_{2(p^{n}-1)}])}\operatorname{\textup{THH}}(\mathbb{S}[\sigma_{2(p^{n}-1)}])\end{split}$

obtained from the equivalence of $E_{2}$ $MU$-algebras
$MU[\sigma_{2(p^{n}-1)}]\simeq MU\wedge\mathbb{S}[\sigma_{2(p^{n}-1)}]$
mentioned in Remark 4.11.

Furthermore, Hahn and Yuan [26, 1.11 and 1.12] show that there is an
$E_{\infty}$-map $MU[\sigma_{2}]\to ku_{p}$ for $p=2$ and claim that their
methods provide such a map for odd primes too. In this situation,
$\operatorname{\textup{THH}}(ku_{p}\mid u_{2})$ is equipped with the structure
of an $E_{\infty}$-ring (by arguing as above) where $u_{2}$ denotes the Bott
element. Note that the logarithmic THH of $ku_{p}$ relative to $u_{2}$ is also
constructed as an $E_{\infty}$-ring in [48].

######  Remark 6.10.

In work in progress, S. Devalapurkar and the second author show in a general
context, that for every $E_{2}$-ring with even homotopy, logarithmic THH, as
in our definition, may be equipped with a canonical $E_{1}$-algebra structure
in cyclotomic spectra.

To study the log THH of $E_{1}$-rings obtained via root adjunctions, we use
the following constructions.

######  Construction 6.11.

Using Construction 6.1 and the weight connective cover adjunction mentioned in
Remark 6.2, we obtain the following commuting diagram of graded
$E_{1}$-algebras.

(6.12)
${\mathbb{S}[\sigma_{mk}]}$${\operatorname{\textup{THH}}(\mathbb{S}[\sigma_{mk}])}$${\operatorname{\textup{THH}}(\mathbb{S}[\sigma_{mk}]\mid\sigma_{mk})}$${\mathbb{S}[\sigma_{k}]}$${\operatorname{\textup{THH}}(\mathbb{S}[\sigma_{k}])}$${\operatorname{\textup{THH}}(\mathbb{S}[\sigma_{k}]\mid\sigma_{k})}$

######  Construction 6.13.

Assume Hypothesis 4.4. By Proposition 4.14, there is an equivalence:

$\operatorname{\textup{THH}}(A(\sqrt[m]{a}))\simeq\operatorname{\textup{THH}}(A)\wedge_{\operatorname{\textup{THH}}(\mathbb{S}[\sigma_{mk}])}\operatorname{\textup{THH}}(\mathbb{S}[\sigma_{k}]).$

This equips $\operatorname{\textup{THH}}(A(\sqrt[m]{a}))$ with the structure
of a right $\operatorname{\textup{THH}}(\mathbb{S}[\sigma_{k}])$-module in
$m$-graded spectra. Considering the map

$\operatorname{\textup{THH}}(\mathbb{S}[\sigma_{k}])\to\operatorname{\textup{THH}}(\mathbb{S}[\sigma_{k}]\mid\mathbb{S}[\sigma_{k}])$

as a map of $m$-graded $E_{1}$-ring spectra, Definition 6.6 may be employed at
the level of $m$-graded spectra. This shows that
$\operatorname{\textup{THH}}(A(\sqrt[m]{a})\mid\sqrt[m]{a})$ admits a
canonical structure of a right
$\operatorname{\textup{THH}}(\mathbb{S}[\sigma_{k}]\mid\sigma_{k})$-module in
$m$-graded spectra. Furthermore, the map
$\operatorname{\textup{THH}}(A(\sqrt[m]{a}))\to\operatorname{\textup{THH}}(A(\sqrt[m]{a})\mid\sqrt[m]{a})$
is a map of $m$-graded
$\operatorname{\textup{THH}}(\mathbb{S}[\sigma_{k}])$-modules.

###  6.1. Logarithmic THH-étale root adjunctions

Here, our goal is to show that when $A$ is $p$-local and $p\nmid m$, root
adjunction is logarithmic THH-étale. In other words, we show that there is an
equivalence of $m$-graded spectra:

$\operatorname{\textup{THH}}(A\mid
a)\wedge_{\mathbb{S}_{(p)}[\sigma_{mk}]}\mathbb{S}_{(p)}[\sigma_{k}]\simeq\operatorname{\textup{THH}}(A(\sqrt[m]{a})\mid\sqrt[m]{a}).$

######  Remark 6.14.

A notion of logarithmic THH-étalenes is already defined in [48]. In the
language of Rognes, Sagave and Schlichtkrull [48], logarithmic THH-étaleness
of $A\to A(\sqrt[m]{a})$ would be expressed by an equivalence:

(6.15) $\operatorname{\textup{THH}}(A(\sqrt[m]{a})\mid\sqrt[m]{a})\simeq
A(\sqrt[m]{a})\wedge_{A}\operatorname{\textup{THH}}(A\mid a).$

Since we only assume $A$ to be $E_{1}$, $\operatorname{\textup{THH}}(A\mid a)$
may not admit an $A$-module structure and therefore, the right hand side above
may not be defined in our generality. On the other hand, if one starts with an
$E_{3}$-algebra $A$ with even homotopy, the logarithmic THH of $A$ may be
given an $A$-module structure and we obtain that $A\to A(\sqrt[m]{a})$ is
logarithmic THH-étale in the sense of (6.15) whenever $A$ is $p$-local and
$p\nmid m$.

######  Proposition 6.16.

For $k\geq 0$, the spectra
$\operatorname{\textup{THH}}(\mathbb{S}[\sigma_{k}]\mid\sigma_{k})$ and
$\operatorname{\textup{THH}}(\mathbb{S}_{(p)}[\sigma_{k}]\mid\sigma_{k})$ are
connective in homotopy.

###### Proof.

This follows by the fact that the weight connective part of the cyclic bar
construction on $\mathbb{S}[\sigma_{k}^{\pm 1}]$
($\mathbb{S}_{(p)}[\sigma_{k}^{\pm 1}]$) is connective in homotopy in each
simplicial degree. ∎

We start with proving a logarithmic THH étaleness result for the $p$-localized
free $E_{1}$-algebra $\mathbb{S}_{(p)}[\sigma_{mk}]$.

######  Proposition 6.17.

Let $k\geq 0$ be even and let $m>0$ with $p\nmid m$. In this situation, there
is an equivalence of left
$\operatorname{\textup{THH}}(\mathbb{S}_{(p)}[\sigma_{mk}]\mid\sigma_{mk})$-modules
in $m$-graded spectra:

$\operatorname{\textup{THH}}(\mathbb{S}_{(p)}[\sigma_{mk}]\mid\sigma_{mk})\wedge_{\mathbb{S}_{(p)}[\sigma_{mk}]}\mathbb{S}_{(p)}[\sigma_{k}]\simeq\operatorname{\textup{THH}}(\mathbb{S}_{(p)}[\sigma_{k}]\mid\sigma_{k})$

where the left
$\operatorname{\textup{THH}}(\mathbb{S}_{(p)}[\sigma_{mk}]\mid\sigma_{mk})$-module
structure on the right hand is provided by Construction 6.11.

###### Proof.

We start by constructing the desired map. First, there is a composite map of
$m$-graded $E_{1}$-ring spectra,

(6.18)
$\mathbb{S}_{(p)}[\sigma_{k}]\to\operatorname{\textup{THH}}(\mathbb{S}_{(p)}[\sigma_{k}])\to\operatorname{\textup{THH}}(\mathbb{S}_{(p)}[\sigma_{k}]\mid\sigma_{k})$

which is in particular a map of left $\mathbb{S}_{(p)}[\sigma_{mk}]$-modules
in $m$-graded spectra by forgetting structure trough the $m$-graded
$E_{1}$-ring map
$\mathbb{S}_{(p)}[\sigma_{mk}]\to\mathbb{S}_{(p)}[\sigma_{k}]$. Using the
extension of scalars functor induced by the map

(6.19)
$\mathbb{S}_{(p)}[\sigma_{mk}]\to\operatorname{\textup{THH}}(\mathbb{S}_{(p)}[\sigma_{mk}]\mid\sigma_{mk})$

of $m$-graded $E_{1}$-algebras, we obtain the desired map:

$f\colon\thinspace\operatorname{\textup{THH}}(\mathbb{S}_{(p)}[\sigma_{mk}]\mid\sigma_{mk})\wedge_{\mathbb{S}_{(p)}[\sigma_{mk}]}\mathbb{S}_{(p)}[\sigma_{k}]\to\operatorname{\textup{THH}}(\mathbb{S}_{(p)}[\sigma_{k}]\mid\sigma_{k}),$

of left
$\operatorname{\textup{THH}}(\mathbb{S}_{(p)}[\sigma_{mk}]\mid\sigma_{mk})$-modules
in $m$-graded spectra from (6.18). Here, we used the fact that the left
$\mathbb{S}_{(p)}[\sigma_{mk}]$-module structure on
$\operatorname{\textup{THH}}(\mathbb{S}_{(p)}[\sigma_{k}]\mid\sigma_{k})$ used
in (6.18) is compatible with the one obtained by forgetting the canonical left
$\operatorname{\textup{THH}}(\mathbb{S}_{(p)}[\sigma_{mk}]\mid\sigma_{mk})$-module
structure on
$\operatorname{\textup{THH}}(\mathbb{S}_{(p)}[\sigma_{k}]\mid\sigma_{k})$
through (6.19); this follows by the $p$-local version of Diagram (6.12).

What remains is to show that $f$ is an equivalence. Since $f$ is a map between
$p$-local connective spectra (Proposition 6.16), it is sufficient to show that
$H\mathbb{Z}_{(p)}\wedge f$ is an equivalence, see Lemma 4.16.

By inspection on the two sided bar construction defining relative smash
products, one obtains that

$\displaystyle
H\mathbb{Z}_{(p)}\wedge\operatorname{\textup{THH}}(\mathbb{S}_{(p)}[\sigma_{mk}]\mid\sigma_{mk})$
$\displaystyle\wedge_{\mathbb{S}_{(p)}[\sigma_{mk}]}\mathbb{S}_{(p)}[\sigma_{k}]\simeq$
$\displaystyle(H\mathbb{Z}_{(p)}\wedge\operatorname{\textup{THH}}(\mathbb{S}_{(p)}[\sigma_{mk}]\mid\sigma_{mk}))\wedge_{H\mathbb{Z}_{(p)}[\sigma_{mk}]}H\mathbb{Z}_{(p)}[\sigma_{k}].$

Using the base change formula for THH, we obtain that $H\mathbb{Z}_{(p)}\wedge
f$ is given by the canonical map

$H\mathbb{Z}_{(p)}\wedge
f\colon\thinspace\operatorname{\textup{THH}}^{H\mathbb{Z}_{(p)}}(H\mathbb{Z}_{(p)}[\sigma_{mk}^{\pm
1}])_{\geq
0}\wedge_{H\mathbb{Z}_{(p)}[\sigma_{mk}]}H\mathbb{Z}_{(p)}[\sigma_{k}]\to\operatorname{\textup{THH}}^{H\mathbb{Z}_{(p)}}(H\mathbb{Z}_{(p)}[\sigma_{k}^{\pm
1}])_{\geq 0}.$

To prove that $H\mathbb{Z}_{(p)}\wedge f$ is an equivalence, we argue as in
the proof of Proposition 4.17. The map of Bökstedt spectral sequences
computing the map

(6.20)
$\operatorname{\textup{THH}}^{H\mathbb{Z}_{(p)}}_{*}(H\mathbb{Z}_{(p)}[\sigma_{mk}^{\pm
1}])\to\operatorname{\textup{THH}}^{H\mathbb{Z}_{(p)}}_{*}(H\mathbb{Z}_{(p)}[\sigma_{k}^{\pm
1}])$

is given on the second page, due to the HKR theorem, by the ring map

(6.21) $\phi\colon\thinspace\mathbb{Z}_{(p)}[\sigma_{mk}^{\pm
1}]\otimes\Lambda_{\mathbb{Z}_{(p)}}(d(\sigma_{mk}))\to\mathbb{Z}_{(p)}[\sigma_{k}^{\pm
1}]\otimes\Lambda_{\mathbb{Z}_{(p)}}(d(\sigma_{k}))$

satisfying

$\phi(\sigma_{mk})=\sigma_{k}^{m}\textup{ \ and \
}\phi(d(\sigma_{mk}))=m\sigma_{k}^{m-1}d(\sigma_{k})$

where $m$ is a unit as $p\nmid m$. In particular,

(6.22)
$\phi(\sigma_{mk}^{-1}d(\sigma_{mk}))=\sigma_{k}^{-m}m\sigma_{k}^{m-1}d(\sigma_{k})=m\sigma_{k}^{-1}d(\sigma_{k}).$

Here, $\sigma_{mk}$ and $\sigma_{k}$ are in degrees $(0,mk)$ and $(0,k)$
respectively and $d(\sigma_{mk})$ and $d(\sigma_{k})$ are in degrees $(1,mk)$
and $(1,k)$ respectively. Furthermore, $\sigma_{mk}$ and $d(\sigma_{mk})$ are
of weight $m$ and $\sigma_{k}$ and $d(\sigma_{k})$ are of weight $1$. In
particular, both Bökstedt spectral sequences degenerate on the second page and
the map $\phi$ provides the map (6.20).

Taking connective covers in weight direction and identifying
$\sigma_{mk}^{-1}d(\sigma_{mk})$ as $\text{dlog}\sigma_{mk}$ and
$\sigma_{k}^{-1}d(\sigma_{k})$ as $\text{dlog}\sigma_{k}$, we obtain that the
map

$\operatorname{\textup{THH}}^{H\mathbb{Z}_{(p)}}_{*}(H\mathbb{Z}_{(p)}[\sigma_{mk}^{\pm
1}])_{\geq
0}\to\operatorname{\textup{THH}}^{H\mathbb{Z}_{(p)}}_{*}(H\mathbb{Z}_{(p)}[\sigma_{k}^{\pm
1}])_{\geq 0}$

is given by a map

$\mathbb{Z}_{(p)}[\sigma_{mk}]\otimes\Lambda_{\mathbb{Z}_{(p)}}(\textup{dlog}\sigma_{mk})\to\mathbb{Z}_{(p)}[\sigma_{k}]\otimes\Lambda_{\mathbb{Z}_{(p)}}(\textup{dlog}\sigma_{k})$

that carries $\sigma_{mk}$ to $\sigma_{k}^{m}$ and $\text{dlog}\sigma_{mk}$ to
$\text{dlog}\sigma_{k}$ up to a unit due to (6.22) as $p\nmid m$.

Upon extending scalars with respect to the map
$\mathbb{Z}_{(p)}[\sigma_{mk}]\to\mathbb{Z}_{(p)}[\sigma_{k}]$, this map
becomes an isomorphism. In other words, $\pi_{*}(H\mathbb{Z}_{(p)}\wedge f)$
is an isomorphism and therefore, $f$ is an equivalence.

∎

The following provides the logarithmic THH-étaleness of root adjunction in
ring spectra.

######  Theorem 6.23.

Assume Hypothesis 4.4 with $p\nmid m$ and that $A$ is $p$-local. In this
situation, there is an equivalence of $m$-graded spectra

$\operatorname{\textup{THH}}(A(\sqrt[m]{a})\mid\sqrt[m]{a})\simeq\operatorname{\textup{THH}}(A\mid
a)\wedge_{\mathbb{S}_{(p)}[\sigma_{mk}]}\mathbb{S}_{(p)}[\sigma_{k}].$

In other words, as an $m$-graded spectrum,
$\operatorname{\textup{THH}}(A(\sqrt[m]{a})\mid\sqrt[m]{a})$ is given by

$\operatorname{\textup{THH}}(A(\sqrt[m]{a})\mid\sqrt[m]{a})_{i}\simeq\Sigma^{ik}\operatorname{\textup{THH}}(A\mid
a)$

for every $0\leq i<m$.

###### Proof.

We have the following chain of equivalences

$\begin{split}&\operatorname{\textup{THH}}(A(\sqrt[m]{a})\mid\sqrt[m]{a})\simeq\operatorname{\textup{THH}}(A(\sqrt[m]{a}))\wedge_{\operatorname{\textup{THH}}(\mathbb{S}_{(p)}[\sigma_{k}])}\operatorname{\textup{THH}}(\mathbb{S}_{(p)}[\sigma_{k}]\mid\sigma_{k})\\\
&\simeq\big{(}\operatorname{\textup{THH}}(A)\wedge_{\operatorname{\textup{THH}}(\mathbb{S}_{(p)}[\sigma_{mk}])}\operatorname{\textup{THH}}(\mathbb{S}_{(p)}[\sigma_{k}])\big{)}\wedge_{\operatorname{\textup{THH}}(\mathbb{S}_{(p)}[\sigma_{k}])}\operatorname{\textup{THH}}(\mathbb{S}_{(p)}[\sigma_{k}]\mid\sigma_{k})\\\
&\simeq\operatorname{\textup{THH}}(A)\wedge_{\operatorname{\textup{THH}}(\mathbb{S}_{(p)}[\sigma_{mk}])}\operatorname{\textup{THH}}(\mathbb{S}_{(p)}[\sigma_{k}]\mid\sigma_{k})\\\
&\simeq\operatorname{\textup{THH}}(A)\wedge_{\operatorname{\textup{THH}}(\mathbb{S}_{(p)}[\sigma_{mk}])}\operatorname{\textup{THH}}(\mathbb{S}_{(p)}[\sigma_{mk}]\mid\sigma_{mk})\wedge_{\operatorname{\textup{THH}}(\mathbb{S}_{(p)}[\sigma_{mk}]\mid\sigma_{mk})}\operatorname{\textup{THH}}(\mathbb{S}_{(p)}[\sigma_{k}]\mid\sigma_{k})\\\
&\simeq\operatorname{\textup{THH}}(A\mid
a)\wedge_{\operatorname{\textup{THH}}(\mathbb{S}_{(p)}[\sigma_{mk}]\mid\sigma_{mk})}\operatorname{\textup{THH}}(\mathbb{S}_{(p)}[\sigma_{k}]\mid\sigma_{k})\\\
&\simeq\operatorname{\textup{THH}}(A\mid
a)\wedge_{\operatorname{\textup{THH}}(\mathbb{S}_{(p)}[\sigma_{mk}]\mid\sigma_{mk})}\big{(}\operatorname{\textup{THH}}(\mathbb{S}_{(p)}[\sigma_{mk}]\mid\sigma_{mk})\wedge_{\mathbb{S}_{(p)}[\sigma_{mk}]}\mathbb{S}_{(p)}[\sigma_{k}]\big{)}\\\
&\simeq\operatorname{\textup{THH}}(A\mid
a)\wedge_{\mathbb{S}_{(p)}[\sigma_{mk}]}\mathbb{S}_{(p)}[\sigma_{k}]\\\
\end{split}$

The first and the fifth equivalences follow by the definition of logarithmic
THH, the second equivalence follows by our definition of root adjunction and
Proposition 4.14 and the sixth equivalence follows by Proposition 6.17. ∎

######  Remark 6.24.

In [48], the authors show that $\ell\to ku_{(p)}$ is logarithmic THH-étale.
This compares to our result above since we show that
$ku_{p}\simeq\ell_{p}(\sqrt[p-1]{v_{1}})$ in Theorem 4.10.

###  6.2. Relating THH and logarithmic THH

The goal of this section is to show that there is a fiber sequence

$\operatorname{\textup{THH}}(A)\to\operatorname{\textup{THH}}(A\mid
a)\to\Sigma\operatorname{\textup{THH}}(A/a)$

under our usual assumptions. The $E_{1}$-ring $A/a$ above is described in the
following construction which is analogous to [47, Lemmas 6.14 and 6.15].

######  Construction 6.25.

Let $A$ be an $\mathbb{S}[\sigma_{k}]$-algebra where $\sigma_{k}$ acts through
$a\in\pi_{k}A$ where $k\geq 0$ is even. The weight $0$ Postnikov section of
$\mathbb{S}[\sigma_{k}]$ provides a map $\mathbb{S}[\sigma_{k}]\to\mathbb{S}$
of $E_{2}$-rings [25, B.0.6]. Considering the extension of scalars functor
$-\wedge_{\mathbb{S}[\sigma_{k}]}\mathbb{S}$ from the $\infty$-category of
$E_{1}$ $\mathbb{S}[\sigma_{k}]$-algebras to the $\infty$-category of $E_{1}$
$\mathbb{S}$-algebras, one equips

$A/a:=A\wedge_{\mathbb{S}[\sigma_{k}]}\mathbb{S}$

with the structure of an $E_{1}$-ring spectrum. Since $\mathbb{S}$ is the
cofiber of the map
$\mathbb{S}[\sigma_{k}]\xrightarrow{\cdot\sigma_{k}}\mathbb{S}[\sigma_{k}]$,
$A/a$ is indeed the cofiber of the map $A\xrightarrow{\cdot a}A$.

Considering $\mathbb{S}[\sigma_{k}]$ as a graded $E_{2}$-ring, we have

$\operatorname{\textup{THH}}(\mathbb{S}[\sigma_{k}])_{0}\simeq\mathbb{S}.$

This can be observed by inspection on the cyclic bar construction on
$\mathbb{S}[\sigma_{k}]$ or by computing the $H\mathbb{Z}$-homology of the
left hand side above. This is used in the statement of the following
proposition.

######  Remark 6.26.

The following proposition is analogous to [47, Proposition 6.11]. We remark
that unlike in loc cit., we do not take $S^{1}$-equivariance into account,
which leads to a simpler proof.

######  Proposition 6.27.

The cofiber of the map

$\operatorname{\textup{THH}}(\mathbb{S}[\sigma_{k}])\to\operatorname{\textup{THH}}(\mathbb{S}[\sigma_{k}]\mid\sigma_{k})$

is given by $\Sigma\mathbb{S}$ concentrated in weight $0$ as a left
$\operatorname{\textup{THH}}(\mathbb{S}[\sigma_{k}])$-module in graded
spectra. Here, the left
$\operatorname{\textup{THH}}(\mathbb{S}[\sigma_{k}])$-module structure on
$\mathbb{S}$ is given by the weight-Postnikov truncation map of graded
$E_{1}$-rings

$\operatorname{\textup{THH}}(\mathbb{S}[\sigma_{k}])\to\operatorname{\textup{THH}}(\mathbb{S}[\sigma_{k}])_{0}\simeq\mathbb{S}.$

###### Proof.

Let $M$ be the cofiber of the map $f$ below in left
$\operatorname{\textup{THH}}(\mathbb{S}[\sigma_{k}])$-modules in graded
spectra.

$\operatorname{\textup{THH}}(\mathbb{S}[\sigma_{k}])\xrightarrow{f}\operatorname{\textup{THH}}(\mathbb{S}[\sigma_{k}],\mid\sigma_{k})\to
M$

We start by computing $H\mathbb{Z}_{*}M$. The map

$H\mathbb{Z}_{*}f\colon\thinspace\operatorname{\textup{THH}}^{H\mathbb{Z}}_{*}(H\mathbb{Z}[\sigma_{k}])\to\operatorname{\textup{THH}}^{H\mathbb{Z}}_{*}(H\mathbb{Z}[\sigma_{k}^{\pm
1}])_{\geq 0}$

is given by the ring map

$\mathbb{Z}[\sigma_{k}]\otimes\Lambda(d(\sigma_{k}))\to\mathbb{Z}[\sigma_{k}]\otimes\Lambda(\textup{dlog}\sigma_{k})$

that carries $\sigma_{k}$ to $\sigma_{k}$ and $d(\sigma_{k})$ to
$\sigma_{k}\text{dlog}\sigma_{k}$; this follows by the Böksedt spectral
sequences in (4.18) and (6.21). This map is injective and the only class that
is not in the image is $\text{dlog}\sigma_{k}$. We obtain,

$H\mathbb{Z}\wedge M\simeq\Sigma H\mathbb{Z}$

where the right hand side is concentrated in weight $0$. Due to Proposition
6.16, $f$ is a map between connective spectra. In particular, $M$ is
connective and we obtain an equivalence of spectra

$M\simeq\Sigma\mathbb{S}.$

We need to improve this to an equivalence of left
$\operatorname{\textup{THH}}(\mathbb{S}[\sigma_{k}])$-modules in graded
spectra. Since $M$ is a left
$\operatorname{\textup{THH}}(\mathbb{S}[\sigma_{k}])$-module in graded
spectra, there is a map

$\Sigma\operatorname{\textup{THH}}(\mathbb{S}[\sigma_{k}])\to M$

of left $\operatorname{\textup{THH}}(\mathbb{S}[\sigma_{k}])$-modules in
graded spectra carrying $1$ to $1$ in homotopy. Taking weight $0$ Postnikov
sections [25, B.0.6], we obtain an equivalence of left
$\operatorname{\textup{THH}}(\mathbb{S}[\sigma_{k}])$-modules in graded
spectra

$\Sigma\operatorname{\textup{THH}}(\mathbb{S}[\sigma_{k}])_{0}\xrightarrow{\simeq}M.$

This map is an equivalence because it carries $1$ to $1$ in homotopy by
construction and since both sides are equivalent as spectra to
$\Sigma\mathbb{S}$.

∎

We are ready to provide the cofiber sequence relating THH to logarithmic THH.

######  Theorem 6.28.

Let $A$ be an $\mathbb{S}[\sigma_{k}]$-algebra where $\sigma_{k}$ acts through
$a\in\pi_{k}A$ with even $k\geq 0$. In this situation, there is a cofiber
sequence of spectra:

$\operatorname{\textup{THH}}(A)\to\operatorname{\textup{THH}}(A\mid
a)\to\Sigma\operatorname{\textup{THH}}(A/a).$

The corresponding cofiber sequence for
$\operatorname{\textup{THH}}(A(\sqrt[m]{a})\mid\sqrt[m]{a})$ is a cofiber
sequence of $m$-graded spectra.

###### Proof.

Proposition 6.27 provides the following cofiber sequence of left
$\operatorname{\textup{THH}}(\mathbb{S}[\sigma_{k}])$-modules in graded
spectra.

$\operatorname{\textup{THH}}(\mathbb{S}[\sigma_{k}])\to\operatorname{\textup{THH}}(\mathbb{S}[\sigma_{k}]\mid\sigma_{k})\to\Sigma\mathbb{S}$

Applying the functor
$\operatorname{\textup{THH}}(A)\wedge_{\operatorname{\textup{THH}}(\mathbb{S}[\sigma_{k}])}-$
to this cofiber sequence, we obtain the following cofiber sequence

(6.29) $\operatorname{\textup{THH}}(A)\to\operatorname{\textup{THH}}(A\mid
a)\to\operatorname{\textup{THH}}(A)\wedge_{\operatorname{\textup{THH}}(\mathbb{S}[\sigma_{k}])}\Sigma\mathbb{S}.$

What is left is to identify the cofiber above as
$\operatorname{\textup{THH}}(A/a)$. We have

(6.30)
$\begin{split}\operatorname{\textup{THH}}(A)\wedge_{\operatorname{\textup{THH}}(\mathbb{S}[\sigma_{k}])}\Sigma\mathbb{S}&\simeq\Sigma\operatorname{\textup{THH}}(A)\wedge_{\operatorname{\textup{THH}}(\mathbb{S}[\sigma_{k}])}\mathbb{S}.\\\
\end{split}$

Here, $\mathbb{S}$ on the right hand side denotes the de-suspension of
$\Sigma\operatorname{\textup{THH}}(\mathbb{S}[\sigma_{k}])_{0}$ as a left
$\operatorname{\textup{THH}}(\mathbb{S}[\sigma_{k}])$-module in graded
spectra, see Proposition 6.27. This is
$\operatorname{\textup{THH}}(\mathbb{S}[\sigma_{k}])_{0}$ which admits the
structure of a graded $E_{1}$-ring spectrum equipped with a map
$\operatorname{\textup{THH}}(\mathbb{S}[\sigma_{k}])\to\operatorname{\textup{THH}}(\mathbb{S}[\sigma_{k}])_{0}$
of graded $E_{1}$-ring spectra given by the relevant weight $0$ Postnikov
section map. Indeed, due to the universal property of Postnikov sections, this
weight $0$ Postnikov section map factors the map of graded $E_{1}$-rings
$\operatorname{\textup{THH}}(\mathbb{S}[\sigma_{k}])\to\operatorname{\textup{THH}}(\mathbb{S})$
induced by the weight $0$ Postnikov section map
$\mathbb{S}[\sigma_{k}]\to\mathbb{S}$; i.e. we have a factorization of this
map of graded $E_{1}$-algebras as

$\operatorname{\textup{THH}}(\mathbb{S}[\sigma_{k}])\to\operatorname{\textup{THH}}(\mathbb{S}[\sigma_{k}])_{0}\xrightarrow{\simeq}\operatorname{\textup{THH}}(\mathbb{S}).$

The second map above is an equivalence as its domain and codomain are
equivalent to $\mathbb{S}$ as spectra and it carries the unit to the unit by
construction. In particular, we can replace $\mathbb{S}$ on the right hand
side of (6.30) with $\operatorname{\textup{THH}}(\mathbb{S})$. This provides
the first equivalence below.

(6.31)
$\begin{split}\Sigma\operatorname{\textup{THH}}(A)\wedge_{\operatorname{\textup{THH}}(\mathbb{S}[\sigma_{k}])}\mathbb{S}&\simeq\Sigma\operatorname{\textup{THH}}(A)\wedge_{\operatorname{\textup{THH}}(\mathbb{S}[\sigma_{k}])}\operatorname{\textup{THH}}(\mathbb{S})\\\
&\simeq\Sigma\operatorname{\textup{THH}}(A\wedge_{\mathbb{S}[\sigma_{k}]}\mathbb{S})\\\
&\simeq\Sigma\operatorname{\textup{THH}}(A/a)\end{split}$

The second equivalence follows by Proposition 4.14 and the third equivalence
follows by our description of the $E_{1}$-algebra $A/a$ in Construction 6.25.
Equations (6.30) and (6.31) identify the cofiber in (6.29) as
$\operatorname{\textup{THH}}(A/a)$ providing the cofiber sequence claimed in
the theorem.

The statement regarding the cofiber sequence in $m$-graded spectra for
$A(\sqrt[m]{a})$ follows by utilizing same arguments. ∎

######  Remark 6.32.

The above localization sequence is of fundamental importance in the theory of
log THH. A proof of the above localization sequence for general $E_{2}$ log
structures, using more general methods, will be supplied in [14].

###  6.3. THH after root adjunction

Here, we identify $\operatorname{\textup{THH}}(A(\sqrt[m]{a}))$ in terms of
$\operatorname{\textup{THH}}(A)$ and $\operatorname{\textup{THH}}(A\mid a)$.

######  Theorem 6.33 (Theorem 1.3).

Assume Hypothesis 4.4 with $p\nmid m$ and that $A$ is $p$-local. In this
situation, the $m$-graded spectrum
$\operatorname{\textup{THH}}(A(\sqrt[m]{a}))$ is given by

$\operatorname{\textup{THH}}(A(\sqrt[m]{a}))_{0}\simeq\operatorname{\textup{THH}}(A)$

and

$\operatorname{\textup{THH}}(A(\sqrt[m]{a}))_{i}\simeq\Sigma^{ik}\operatorname{\textup{THH}}(A\mid
a)\textup{ \ for \ }0<i<m.$

In particular, there is an equivalence of spectra:

$\operatorname{\textup{THH}}(A(\sqrt[m]{a}))\simeq\operatorname{\textup{THH}}(A)\vee\big{(}\bigvee_{0<i<m}\Sigma^{ik}\operatorname{\textup{THH}}(A\mid
a)\big{)}.$

###### Proof.

The identification of $\operatorname{\textup{THH}}(A(\sqrt[m]{a}))_{0}$ is
provided by Proposition 4.20. Therefore, it is sufficient to provide the
identification of $\operatorname{\textup{THH}}(A(\sqrt[m]{a}))_{i}$ for $i\neq
0$.

Due to Theorem 6.23,

(6.34)
$\operatorname{\textup{THH}}(A(\sqrt[m]{a})\mid\sqrt[m]{a})_{i}\simeq\Sigma^{ik}\operatorname{\textup{THH}}(A\mid
a).$

Therefore, it is sufficient to show that

(6.35)
$\operatorname{\textup{THH}}(A(\sqrt[m]{a}))_{i}\simeq\operatorname{\textup{THH}}(A(\sqrt[m]{a})\mid\sqrt[m]{a})_{i}$

whenever $i\neq 0$. This follows once we show that the cofiber of the the map

(6.36)
$\operatorname{\textup{THH}}(A(\sqrt[m]{a}))\to\operatorname{\textup{THH}}(A(\sqrt[m]{a})\mid\sqrt[m]{a})$

of $m$-graded spectra is concentrated in weight $0$. Due to Theorem 6.28, the
cofiber of this map is given by

$\operatorname{\textup{THH}}(A(\sqrt[m]{a})/\sqrt[m]{a})$

where $A(\sqrt[m]{a})/\sqrt[m]{a}$ is defined to be
$A(\sqrt[m]{a})\wedge_{\mathbb{S}[\sigma_{k}]}\mathbb{S}$. Therefore, we have

$A(\sqrt[m]{a})/\sqrt[m]{a}:=A(\sqrt[m]{a})\wedge_{\mathbb{S}[\sigma_{k}]}\mathbb{S}\simeq
A\wedge_{\mathbb{S}[\sigma_{mk}]}\mathbb{S}[\sigma_{k}]\wedge_{\mathbb{S}[\sigma_{k}]}\mathbb{S}\simeq
A\wedge_{\mathbb{S}[\sigma_{mk}]}\mathbb{S}.$

Since $A$, $\mathbb{S}[\sigma_{mk}]$ and $\mathbb{S}$ are concentrated in
weight $0$, we obtain that $A(\sqrt[m]{a})/\sqrt[m]{a}$ and therefore
$\operatorname{\textup{THH}}(A(\sqrt[m]{a})/\sqrt[m]{a})$ are also
concentrated in weight $0$. This proves that the cofiber of (6.36) is
concentrated in weight $0$ which proves (6.35) and this, together with (6.34)
proves the theorem. ∎

##  7\. Algebraic $K$-theory of complex and real topological $K$-theories

Here, we start by showing that $K(ku_{p})$ splits into $p-1$ non-trivial
summands. Afterwards, we show that $ku_{p}$ may be constructed from $ko_{p}$
via root adjunction. We use this to obtain an explicit description of the
$V(1)$-homotopy of $K(ko_{p})$ from the first authors computation of
$V(1)_{*}K(ku_{p})$ [7].

###  7.1. Adams’ splitting result for $2$-vector bundles

Recall that $\pi_{*}ku_{p}\cong\mathbb{Z}_{p}[u]$ and
$\pi_{*}\ell_{p}\cong\mathbb{Z}_{p}[v_{1}]$ where $\lvert u\rvert=2$ and
$\lvert v_{1}\rvert=2p-2$. The $E_{\infty}$-map $\ell_{p}\to ku_{p}$ carries
$v_{1}$ to $u^{p-1}$ in homotopy. For the rest of this section, we fix a map
$\mathbb{S}[\sigma_{2(p-1)}]\to\ell_{p}$ of $E_{2}$-algebras carrying
$\sigma_{2(p-1)}$ to $v_{1}$ and perform root adjunction using this map.
Recall from Theorem 4.10 that there is an equivalence of $E_{1}$
$\ell_{p}$-algebras

$\ell_{p}(\sqrt[p-1]{v_{1}})\simeq ku_{p}.$

This equips $ku_{p}$ with the structure of a $p-1$-graded $E_{1}$
$\ell_{p}$-algebra which further equips $\operatorname{\textup{THH}}(ku_{p})$
with the structure of a $p-1$-graded $S^{1}$-equivariant spectrum.

Let $p>3$ be a prime and let $V(1)$ denote the type-$2$ finite spectrum used
in [6]; $V(1)$ is a homotopy ring spectrum.

There is another grading on $V(1)_{*}\operatorname{\textup{THH}}(ku_{p})$ that
the first author calls the $\delta$-grading [6]. The group
$\Delta:=\mathbb{Z}/(p-1)$ acts on the $E_{\infty}$-ring $ku_{p}$ through
Adams operations. Let $\delta\in\Delta$ be a chosen generator and let
$\alpha\in\mathbb{F}_{p}^{\times}$ satisfy
$\pi_{*}(\mathbb{S}/p\wedge\delta)(u)=\alpha u$ where

$\pi_{*}(\mathbb{S}/p\wedge\delta)\colon\thinspace\pi_{*}(\mathbb{S}/p\wedge
ku_{p})\to\pi_{*}(\mathbb{S}/p\wedge ku_{p})\cong\mathbb{F}_{p}[u].$

We say $u^{i}$ has $\delta$-weight $i$ as
$\pi_{*}(\mathbb{S}/p\wedge\delta)(u^{i})=\alpha^{i}u^{i}$. Similarly, one
says $x\in V(1)_{*}\operatorname{\textup{THH}}(ku_{p})$ has $\delta$-weight
$i$ if the self map of $V(1)_{*}\operatorname{\textup{THH}}(ku_{p})$ induced
by $\delta$ carries $x$ to $\alpha^{i}x$. One defines $\delta$-weight in a
similar way on other invariants of $ku_{p}$ [6, Definition 8.2].

######  Proposition 7.1.

The group $V(1)_{*}\operatorname{\textup{THH}}(ku_{p})_{i}$ is given by the
classes of $\delta$-weight $i$ in
$V(1)_{*}\operatorname{\textup{THH}}(ku_{p})$.

###### Proof.

Since $H\mathbb{F}_{p}\wedge ku_{p}$ is a $p-1$ graded $E_{1}$
$H\mathbb{F}_{p}$-algebra, there is a $p-1$-grading on
$\textup{HH}^{\mathbb{F}_{p}}_{*}(H{\mathbb{F}_{p}}_{*}ku_{p})$. By inspection
on the Hochschild complex, one observes that the $\delta$-weight grading on
$\textup{HH}^{\mathbb{F}_{p}}_{*}(H{\mathbb{F}_{p}}_{*}ku_{p})$ agrees with
the weight grading. In particular, the $\delta$-weight grading and the weight
grading agree on the second page of the Bökstedt spectral sequence computing
$\operatorname{\textup{HH}^{\mathbb{F}_{p}}}(H\mathbb{F}_{p}\wedge ku_{p})$.
Due to [6, Section 9], this shows that the $\delta$-weight grading and the
weight grading agree on
$\textup{HH}_{*}^{\mathbb{F}_{p}}(H\mathbb{F}_{p}\wedge ku_{p})$. Furthermore,
there is a basis of $\textup{HH}_{*}^{\mathbb{F}_{p}}(H\mathbb{F}_{p}\wedge
ku_{p})$ as an $\mathbb{F}_{p}$-module where $\delta$-weight is defined for
each basis element. Therefore, the $H\mathbb{F}_{p}$-module
$\operatorname{\textup{HH}^{H\mathbb{F}_{p}}}(H\mathbb{F}_{p}\wedge ku_{p})$
splits as a coproduct of suspensions of $H\mathbb{F}_{p}$ in a way that the
map
$\operatorname{\textup{HH}^{H\mathbb{F}_{p}}}(H\mathbb{F}_{p}\wedge\delta)$ is
given by the respective multiplication map corresponding to the
$\delta$-weight on each cofactor. Using this, one observes that the
$\delta$-weight and the weight grading agree on
$H_{*}(V(1)\wedge\operatorname{\textup{THH}}(ku_{p});\mathbb{F}_{p})$.

The Hurewicz map

$V(1)_{*}\operatorname{\textup{THH}}(ku_{p})\to
H_{*}(V(1)\wedge\operatorname{\textup{THH}}(ku_{p});\mathbb{F}_{p})$

is injective and this map preserves both gradings. From this, we deduce that
the weight grading and the $\delta$-weight grading agree on
$V(1)_{*}\operatorname{\textup{THH}}(ku_{p})$. ∎

In general, THH of $m$-graded ring spectra may not result in an $m$-graded
cyclotomic spectrum as the Frobenius map do not preserve the grading; it
multiplies the grading by $p$. On the other hand, for $ku_{p}$,
$\operatorname{\textup{THH}}(ku_{p})$ is $p-1$-graded and $p=1$ in
$\mathbb{Z}/(p-1)$. In particular, the Frobenius map preserves the grading and
one obtains that $\operatorname{\textup{THH}}(ku)$ is a $p-1$-graded
cyclotomic spectrum.

######  Proposition 7.2.

The $S^{1}$-equivariant structure on $\operatorname{\textup{THH}}(ku_{p})_{i}$
lifts to a cyclotomic structure for which there is an equivalence

$\operatorname{\textup{THH}}(ku)\simeq\prod_{i\in\mathbb{Z}/(p-1)}\operatorname{\textup{THH}}(ku)_{i}$

of cyclotomic spectra.

###### Proof.

The monoid $\mathbb{Z}/(p-1)$ satisfies the conditions in [2, Appendix A]
needed endow $\operatorname{\textup{THH}}(ku)$ with an $L_{p}$ twisted
cyclotomic structure. However, since $p\cong 1\mod p-1$, this ends up being
the identity functor on $\mathbb{Z}/(p-1)$-graded spectra. Thus one obtains a
sequence of $S^{1}$-equivariant maps

$\operatorname{\textup{THH}}(ku)_{i}\to\operatorname{\textup{THH}}(ku)^{tC_{p}}_{i}$

for each $i\in\mathbb{Z}/(p-1)$, which is precisely the relevant additional
piece of structure needed to view this as a cyclotomic object. ∎

######  Construction 7.3.

Here, we construct a splitting of $K(ku_{p})$ using Proposition 7.2. Since the
product mentioned in Proposition 7.2 is a finite product, it is at the same
time a coproduct. In particular, it commutes with all limits and colimits.
Therefore, the fiber sequence defining $\operatorname{\textup{TC}}(ku_{p})$
splits into a product of fiber sequences

$\operatorname{\textup{TC}}(ku_{p})_{i}\to\operatorname{\textup{THH}}(ku_{p})_{i}^{hS^{1}}\xrightarrow{(\varphi_{p})_{i}-can_{i}}(\operatorname{\textup{THH}}(ku_{p})_{i}^{tC_{p}})^{hS^{1}}.$

Hence, there is a splitting of $\operatorname{\textup{TC}}(ku_{p})$:

$\operatorname{\textup{TC}}(ku_{p})\simeq\prod_{i\in\mathbb{Z}/(p-1)}\operatorname{\textup{TC}}(ku_{p})_{i}$

where
$\operatorname{\textup{TC}}(ku_{p})_{i}:=\operatorname{\textup{TC}}(\operatorname{\textup{THH}}(ku_{p})_{i})$.
Arguing as in the proof of Theorem 5.8, one obtains a map $ku_{p}\to
H\mathbb{Z}_{p}$ of $p-1$-graded $E_{1}$-rings where $H\mathbb{Z}_{p}$ is
concentrated in weight $0$. Therefore, the induced map
$\operatorname{\textup{THH}}(ku_{p})\to\operatorname{\textup{THH}}(\mathbb{Z}_{p})$
of $p-1$-graded spectra is trivial in non-zero weight. By inspection on the
product splitting of the fiber sequence defining
$\operatorname{\textup{TC}}(ku_{p})$, we consider
$\operatorname{\textup{TC}}(ku_{p})\to\operatorname{\textup{TC}}(\mathbb{Z}_{p})$
as a map of $p-1$ graded spectra where
$\operatorname{\textup{TC}}(\mathbb{Z}_{p})$ is concentrated in weight $0$.
Again, as in the proof of Theorem 5.8, this splits the pull-back square (from
Dundas-Goodwillie-McCarthy theorem) relating
$\operatorname{\textup{TC}}(ku_{p})$ to $K(ku_{p})$ resulting in a splitting
of $K(ku_{p})$ that we denote by

$K(ku_{p})\simeq\bigvee_{i\in\mathbb{Z}/(p-1)}K(ku_{p})_{i}.$

Here, $K(ku_{p})_{0}\simeq K(\ell_{p})$ due to Theorem 5.8.

To understand the resulting splitting of $K(ku_{p})$, we identify the
$V(1)$-homotopy of each weight piece. The computation of $V(1)_{*}K(ku_{p})$
is due to the first author [7, Theorem 8.1] and these groups are given below.

(7.4)
$\begin{split}V(1)_{*}K(ku_{p})\cong&\mathbb{F}_{p}[b]\otimes\Lambda(\lambda_{1},a_{1})\oplus\mathbb{F}_{p}[b]\otimes\mathbb{F}_{p}\\{\partial\lambda_{1},\partial
b,\partial a_{1},\partial\lambda_{1}a_{1}\\}\\\
&\oplus\mathbb{F}_{p}[b]\otimes\Lambda(a_{1})\otimes\mathbb{F}_{p}\\{t^{d}\lambda_{1}\mid
0<d<p\\}\\\
&\oplus\mathbb{F}_{p}[b]\otimes\Lambda(\lambda_{1})\otimes\mathbb{F}_{p}\\{\sigma_{n},\lambda_{2}t^{p^{2}-p}\mid
1\leq n\leq p-2\\}\\\ &\oplus\mathbb{F}_{p}\\{s\\}\end{split}$

Here, $\lvert b\rvert=2p+2$, $\lvert\partial\rvert=-1$,
$\lvert\lambda_{1}\rvert=2p-1$, $\lvert a_{1}\rvert=2p+3$,
$\lvert\sigma_{n}\rvert=2n+1$, $\lvert t\rvert=-2$,
$\lvert\lambda_{2}\rvert=2p^{2}-1$ and $\lvert s\rvert=2p-3$. We assign
weights to these classes in a way that turns $V(1)_{*}K(ku_{p})$ into a
$p-1$-graded abelian group. The weights of $\sigma_{n}$, $b$, $a_{1}$,
$\partial$, $\lambda_{1}$, $t$, $\lambda_{2}$ and $s$ are given by $n$, $1$,
$1$, $0$, $0$, $0$, $0$ and $0$ respectively. Classes denoted by tensor
products or products above have the canonical degrees and weights.
Furthermore, the isomorphism above is that of $\mathbb{F}_{p}[b]$-modules and
$b^{p-1}=-v_{2}$.

######  Theorem 7.5.

For the equivalence of spectra

$K(ku_{p})\simeq\bigvee_{i\in\mathbb{Z}/(p-1)}K(ku_{p})_{i}$

provided by Construction 7.3, there is an equivalence:

$K(ku_{p})_{0}\simeq K(\ell_{p})$

and there are isomorphisms

$V(1)_{*}(K(ku_{p})_{i})\cong\big{(}V(1)_{*}K(ku_{p})\big{)}_{i}$

for each $i\in\mathbb{Z}/(p-1)$ where the right hand side denotes the weight
$i$ piece of the $p-1$-grading on $V(1)_{*}K(ku_{p})$ described above.

###### Proof.

The identification of $K(ku_{p})_{0}$ is given in Construction 7.3. This
provides the identification of $V(1)_{*}K(ku_{p})_{0}$ as
$\big{(}V(1)_{*}K(ku_{p})\big{)}_{0}$ since this is precisely the image of the
map

$V(1)_{*}K(\ell_{p})\to V(1)_{*}K(ku_{p}),$

see [6, Theorem 10.2]. The identification of $V(1)_{*}(K(ku_{p})_{i})$ for
$i\neq 0$ follows by noting from Proposition 7.1 that it is sufficient to keep
track of the contribution of $\delta$-weight $i$ classes in
$V(1)_{*}\operatorname{\textup{THH}}(ku_{p})$ to
$V(1)_{*}\operatorname{\textup{TC}}(ku_{p})$. This follows by inspection on
[7, Section 7] and [7, Section 5]. ∎

###  7.2. Algebraic $K$-theory of real $K$-theory

Let $p>3$. Using Theorem 5.8, the splitting of $K(ku_{p})$ discussed above and
our root adjunction formalism, we obtain a straightforward computation of
$V(1)_{*}K(ko_{p})$ from our knowledge of $V(1)_{*}K(ku_{p})$ from [7]. Here,
$ko_{p}$ denotes the connective cover of the $p$-completed real topological
$K$-theory spectrum $KO_{p}$. We have
$\pi_{*}KO_{p}\cong\mathbb{Z}_{p}[\alpha^{\pm 1}]$ with
$\lvert\alpha\rvert=4$.

There is a subgroup of $C_{2}$ of $\Delta\cong\mathbb{Z}/(p-1)$ such that
$KO_{p}\simeq KU_{p}^{hC_{2}}$. Through this, the induced map $KO_{p}\to
KU_{p}$ carries $\alpha$ to $u^{2}$ up to a unit that we are going to omit.
Since $L\simeq(KU_{p})^{h\Delta}$, we obtain a sequence of $E_{\infty}$-maps

$L_{p}\to KO_{p}\to KU_{p}$

where the first map carries $v_{1}$ to $\alpha^{\frac{p-1}{2}}$ in homotopy.

######  Theorem 7.6.

For $p>3$, there is an equivalence

$ko_{p}\simeq\ell_{p}(\sqrt[\frac{p-1}{2}]{v_{1}})$

of $E_{1}$ $\ell_{p}$-algebras.

###### Proof.

This follows as in the proof of Theorem 4.10 by noting that
$p\nmid\frac{p-1}{2}$. ∎

Furthermore, $ku_{p}$ may also be obtained from $ko_{p}$ via root adjunction;
for this root adjunction, we use the $\mathbb{S}_{p}[\sigma_{4}]$-algebra
structure on $ko_{p}$ provided by Theorem 7.6. To identify the resulting
$2$-graded $E_{1}$-ring structure on $ku_{p}$, we use the symmetric monoidal
functor

$D^{\prime}\colon\thinspace\operatorname{Fun}(\mathbb{Z}/(p-1),\operatorname{Sp})\to\operatorname{Fun}(\mathbb{Z}/2,\operatorname{Sp})$

given by left Kan extension through the canonical map
$\mathbb{Z}/(p-1)\to\mathbb{Z}/2$.

######  Proposition 7.7.

For $p>3$, there is an equivalence

$ko_{p}(\sqrt[2]{\alpha})\simeq D^{\prime}(ku_{p})$

of $2$-graded $E_{1}$-algebras where $D^{\prime}$ is defined above and the
$p-1$-grading on $ku_{p}$ is given by Theorem 4.10.

###### Proof.

Due to Theorem 7.6, $ko_{p}$ is an $\mathbb{S}[\sigma_{4}]$-algebra given by

$\ell_{p}\wedge_{\mathbb{S}[\sigma_{2(p-1)}]}\mathbb{S}[\sigma_{4}].$

To adjoin a root to $ko_{p}$ using this structure, we use the sequence of maps

$\mathbb{S}[\sigma_{2(p-1)}]\to\mathbb{S}[\sigma_{4}]\to
D^{\prime}(\mathbb{S}[\sigma_{2}])$

of $2$-graded $E_{2}$-ring spectra where $\mathbb{S}[\sigma_{2(p-1)}]$ and
$\mathbb{S}[\sigma_{4}]$ are concentrated in weight $0$ and
$\mathbb{S}[\sigma_{2}]$ above is given its canonical $p-1$-grading so that
$\sigma_{2}$ in $D^{\prime}(\mathbb{S}[\sigma_{2}])$ lies in weight $1$.

We obtain the following equivalences of $2$-graded $E_{1}$-rings.

(7.8)
$ko_{p}(\sqrt[2]{\alpha})\simeq\ell_{p}\wedge_{\mathbb{S}[\sigma_{2(p-1)}]}\mathbb{S}[\sigma_{4}]\wedge_{\mathbb{S}[\sigma_{4}]}D^{\prime}(\mathbb{S}[\sigma_{2}])\simeq\ell_{p}\wedge_{\mathbb{S}[\sigma_{2(p-1)}]}D^{\prime}(\mathbb{S}[\sigma_{2}])$

The functor $D^{\prime}$ is a left adjoint and it is symmetric monoidal.
Therefore, it commutes with the two sided bar construction defining relative
smash products. This provides the second equivalence in the following
equivalences of $2$-graded $E_{1}$-algebras.

(7.9) $D^{\prime}(ku_{p})\simeq
D^{\prime}(\ell_{p}\wedge_{\mathbb{S}[\sigma_{2(p-1)}]}\mathbb{S}[\sigma_{2}])\simeq
D^{\prime}(\ell_{p})\wedge_{D^{\prime}(\mathbb{S}[\sigma_{2(p-1)}])}D^{\prime}(\mathbb{S}[\sigma_{2}])$

The first equivalence above follows by Theorem 4.10 and the relative smash
product in the middle is taken in $p-1$-graded spectra. Since $\ell_{p}$ and
$\mathbb{S}[\sigma_{2(p-1)}]$ are concentrated in weight $0$, the right hand