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.

2.2 Manipulations on graded objects

For a symmetric monoidal functor $\mathbb{Z}/m\to \mathbb{Z}/m'$ (i.e. a group homomorphism), there is an induced adjunction between the $\infty$ -categories $\operatorname{Fun}(\mathbb{Z}/m,\operatorname{Sp})$ and $\operatorname{Fun}(\mathbb{Z}/m',\operatorname{Sp})$ where the left adjoint is symmetric monoidal and given by left Kan extension [Reference NikolausNik16, Corollary 3.8]. The right adjoint is given by restriction. This provides the following adjunctions, which allow us to move between various gradings. Let $n\gt 0$ 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 $D^n_{0} \colon \operatorname{Fun}(\mathbb{Z},\operatorname{Sp}) \to \operatorname{Fun}(\mathbb{Z}/n,\operatorname{Sp})$ and we have

  \[D^n(X)_i \simeq \bigvee_{j\in \mathbb{Z} \mid j \equiv i\ \operatorname{mod}\,n}X_j.\]

• – For $n = 1$ , we denote $D^1_s$ by D. In this case,

  \[D \colon \operatorname{Fun}(\mathbb{Z}/s,\operatorname{Sp}) \to \operatorname{Sp}\]
  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 $s \nmid j$ ; furthermore, $R_s(X)_j \simeq X_{sj}$ for every 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 0. 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-0 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 0.

Lemma 2.3. Let A be an m-graded $E_n$ -ring concentrated in weight 0. 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.

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 $F(G(A))\simeq A$ .

3.1 Shearing preliminaries

The main mechanism underlying this construction is that of shearing, which we now briefly review. It has appeared in [Reference RaksitRak20], and is also studied by Devalpurkar and Moulinos, Logarithmic structures for $E_2$ -algebras, in progress, 2022). 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:

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

• – 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 [Reference RaksitRak20] and also appears in [Reference Devalapurkar, Hahn, Lawson, Senger and WilsonDHL⁺23]. For the sake of completeness, we sketch the basic ideas underlying the construction. In [Reference LurieLur15], 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 [Reference RaksitRak20], 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. [Reference LurieLur15]. 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 symmetric 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 $k\gt 0$ ,

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

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 $m\gt 0$ and $k\gt 0$ 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 of the kth 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

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

Here, $\mathbb{S}[\sigma_{mk,-m}]$ is the free graded $E_1$ -algebra on a class $\sigma_{mk,-m}$ of degree mk and weight $-m$ and by inspection on the adjunction $L_m \dashv R_m$ , this map carries $\sigma_{mk,-m}$ to $\sigma_{k,-1}^m$ . 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

(3.1) \begin{equation} F DD^m(\mathbb{S}[\sigma_{mk}]) \simeq D^m(\mathbb{S}[\sigma_{mk}])\end{equation}

due to Lemma 2.3, since $D^m(\mathbb{S}[\sigma_{mk}])$ is concentrated in weight 0. Furthermore, $D D^m$ is the 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 (3.1), 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 $\mathbb{S}[\sigma_{mk}] \to \mathbb{S}[\sigma_k]$ .

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 show in the remainder of this section, this is what will allow us to define $E_2$ -algebra maps to a given $E_2$ -algebra A, along which we will then adjoin roots.

Before doing this, we make precise what exactly we mean by even cell decomposition. The following notions are heavily inspired by § 6.3 of [Reference Galatius, Kupers and Randal-WilliamsGKR25].

Definition 3.8. 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 $\operatorname{Alg}_{E_2}(\operatorname{Fil}(\operatorname{Gr}(\operatorname{Sp})))$ 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.

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 [Reference Galatius, Kupers and Randal-WilliamsGKR25]. If $f: \mathbf{1} \to R$ is the map from the unit, we call this a filtered cellular decomposition of R.

Definition 3.9. A map $f\colon S \to R$ 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, then taking levelwise colimits provides a cellular decomposition of f.

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

Proof. The two key inputs for our argument are Theorems 11.21 and 13.7 of [Reference Galatius, Kupers and Randal-WilliamsGKR25]. 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 the same work 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 ith 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 $e \cong1\operatorname{mod} 2$ . For this, we apply [Reference Galatius, Kupers and Randal-WilliamsGKR25, 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 [Reference LurieLur15, 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 grading 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.

We begin with the particular filtered cellular decomposition on $\mathbb{S}[t]$ given in the proof above and apply levelwise colimits to obtain a cellular decomposition for $\mathbb{S} \to \mathbb{S}[t]$ in the sense of Definition 3.9. In the following corollary, we use this particular cellular decomposition.

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 0 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

$$\operatorname{Free}_{E_2}(\mathbb{S}^{k}(-1)) \to \mathbb{S}[\sigma_k].$$

Proof. By Corollary 3.11, $\mathbb{S}[t]$ 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 the $-w$ 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 3.13. 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 $k\geq0$ . 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 3.12, 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 $\operatorname{Alg}_{E_2}(\operatorname{Gr}(\operatorname{Sp}))$ and $\operatorname{Gr}(\operatorname{Sp})$ algebras and graded spectra, this will now follow from the fact that $\pi_{2n-1}(A)= 0$ for all n.

Remark 3.15. 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 filtration on the source; this will have associated grading

$$\operatorname{gr}\operatorname{Map}_{\operatorname{Alg}_{E_2}}(\mathbb{S}[t], A )\simeq {\operatorname{Map}}\bigg({\operatorname{Free}_{E_2}}\bigg(\bigoplus_{n \geq 1} \mathbb{S}^{2n }, A\bigg)\bigg) \simeq {\operatorname{Map}}\bigg(\bigoplus_{n \geq 1} \mathbb{S}^{2n }, A\bigg).$$

Now, if X has even homotopy groups, then so does the associated grading, so 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.

4.1 Background on algebras over $E_n$ -algebras

Here is a quick background on some of the standard facts that we often use from [Reference LurieLur17].

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 [Reference LurieLur17, 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 [Reference LurieLur17, 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 $f\colon R \to S$ 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) \begin{equation}- \otimes_{R} S \colon \operatorname{Alg}_{E_{m}}(\operatorname{RMod}_{R}) \to \operatorname{Alg}_{E_{m}}(\operatorname{RMod}_{S}).\end{equation}

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) \begin{equation} \operatorname{Alg}_{E_{m}}(\operatorname{RMod}_{S}) \to \operatorname{Alg}_{E_{m}}(\operatorname{RMod}_{R}). \end{equation}

The unit of this adjunction provides a map of $E_m$ R-algebras

(4.3) \begin{equation} A \to A \otimes_R S. \end{equation}

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 $\mathbb{S}[\sigma_k]$ .

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

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 $\pi_* R \to \pi_* A$ .

If A is as in part (1), then A is an A-algebra and A satisfies the assumption in part (2). Therefore, A satisfies Hypothesis 4.1.

For instance, the Morava K-theory spectrum K(n) and all $E_1$ $MU_{(p)}$ -algebra forms of $BP \langle n \rangle$ satisfy Hypothesis 4.1 with respect to their non-negative degree homotopy classes.

Note that we are not assuming any pre-existing non-trivial grading on A; in fact, this will allow us to view it as an m-graded spectrum concentrated in weight 0. Given $a \in \pi_{mk} A$ , the following construction adjoins an m-root to a.

Owing to Proposition 3.5 (Remark 3.6 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 an 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:

(4.4) \begin{equation}A(\!\sqrt[m]{a})_i \simeq \Sigma^{ik}A\end{equation}

for each $0\leq i\lt m$ .

Note that $A(\!\sqrt[m]{a})$ might possibly depend on the $\mathbb{S}[\sigma_{mk}]$ -algebra structure chosen on A. Therefore, every time we apply Construction 4.3, we fix an $\mathbb{S}[\sigma_{mk}]$ -algebra structure on A.

Remark 4.4. If A is an $E_3$ -ring with even homotopy, we 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, we obtain 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.

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 4.6. There is an equivalence

\[ku_p\simeq \ell_p(\!\sqrt[p-1]{v_1})\]

of $E_1$ $\ell_p$ -algebras.

Proof. By Remark 4.4 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 and Pstragowski [Reference Hesselholt and PstragowskiHP25], 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 [Reference Hesselholt and PstragowskiHP25, 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 9.3, 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 4.7. In certain cases, it is possible to equip $A(\!\sqrt[m]{a})$ with the structure of an $E_n$ -algebra for $n\gt 1$ . For this, one may use the graded $E_\infty$ MU-algebra $MU[\sigma_k]$ [Reference Hahn and WilsonHW22, Construction 2.6.1], where $k\gt 0$ is even. Indeed, $MU[\sigma_k]$ is the free graded $E_1$ MU-algebra over $\Sigma^kMU$ . 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 3.13 provides maps $\mathbb{S}[\sigma_k] \to MU[\sigma_k]$ of graded $E_2$ -rings.

It follows by [Reference Hahn and WilsonHW22, 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-0 piece of THH

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

It follows by [Reference Antieau, Mathew, Morrow and NikolausAMM⁺22, Example A.10] that for an m-graded $E_1$ -ring Y, the m-grading on $\operatorname{THH}(Y)$ 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:

\[b_s(Y)_i \simeq \bigvee_{k_0+ \cdots +k_s =i \in \mathbb{Z}/m}Y_{k_0} \wedge \cdots \wedge Y_{k_s}.\]

Owing to [Reference Antieau, Mathew, Morrow and NikolausAMM⁺22, Corollary A.15], we have the following equality:

(4.5) \begin{equation} \operatorname{THH}(D(Y)) \simeq D(\operatorname{THH}(Y)),\end{equation}

where the functor $D(-)$ provides the underlying spectrum as usual; we often omit D in our notation. Furthermore, $\operatorname{THH}(Y)$ 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 4.8. Let R be an $E_2$ -ring and S 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 [Reference LurieLur17, 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 $\operatorname{THH}$ is a symmetric monoidal functor from $E_1$ -rings to spectra [Reference Nikolaus and ScholzeNS18, § IV.2], we deduce that $\operatorname{THH}(S)$ is a right $\operatorname{THH}(R)$ -module.

Proposition 4.9. Let F be an m-graded $E_1$ E-algebra and $E\to F'$ a map of m-graded $E_2$ -rings. There is a natural equivalence of m-graded right $\operatorname{THH}(F')$ -modules in $S^1$ -equivariant spectra,

\[\operatorname{THH}(F\wedge_E F') \simeq \operatorname{THH}(F) \wedge_{\operatorname{THH}(E)} \operatorname{THH}(F'),\]

whose underlying ungraded equivalence is that of right $\operatorname{THH}(F')$ -modules in cyclotomic spectra. If E and F are concentrated in weight 0, then we have the following:

\[\operatorname{THH}(F\wedge_E F')_i \simeq \operatorname{THH}(F) \wedge_{\operatorname{THH}(E)} (\operatorname{THH}(F')_i).\]

Proof. Let us recall that the functor

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

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

\begin{align*} \operatorname{THH}(F \wedge_{E} F') &\simeq \operatorname{THH}( ||\operatorname{Bar}_\bullet(F,E,F')||) \\ &\simeq ||\operatorname{Bar}_{\bullet}(\operatorname{THH}(F)_\bullet, \operatorname{THH}(E)_\bullet, \operatorname{THH}(F')_\bullet)|| \\ &\simeq \operatorname{THH}(F) \wedge_{\operatorname{THH}(E)}\operatorname{THH}(F').\end{align*}

This allows us to deduce that $\operatorname{THH}$ 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

$${}_{F}\operatorname{BMod}_E \times {}_{E}\operatorname{BMod}_{F'} \to _{F} \operatorname{BMod}_{F'}$$

corresponding to the relative tensor product. Furthermore, as $\operatorname{THH}$ preserves the sifted colimits corresponding to this relative tensor product, the above equivalence is compatible with right $\operatorname{THH}(F')$ module structures. The analogous claims all hold when accounting for additional gradings, by recalling that $\operatorname{THH}$ 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 0, we deduce the equivalence

$$\operatorname{THH}(F\wedge_E F')_i \simeq \operatorname{THH}(F) \wedge_{\operatorname{THH}(E)} (\operatorname{THH}(F')_i)$$

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

One may consider $\mathbb{S}[\sigma_k]$ as an $E_1$ -ring obtained by adjoining an m-root to $\mathbb{S}[\sigma_{mk}]$ . Proposition 4.12 identifies the weight-0 piece of $\operatorname{THH}(\mathbb{S}_{(p)}[\sigma_k])$ . Before Proposition 4.12, we state and prove a well-known fact.

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 $k\geq0$ be even and let $m\gt 1$ . 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 3.6 for $k=0$ ).

Proposition 4.12. In the situation above, assume that $p \nmid m$ . The induced map

\[ \operatorname{THH}(\mathbb{S}_{(p)}[\sigma_{mk}])_0 \xrightarrow{\simeq} \operatorname{THH}(\mathbb{S}_{(p)}[\sigma_k])_0\]

is an equivalence of $E_1$ -rings. Since $\mathbb{S}_{(p)}[\sigma_{mk}]$ is concentrated in weight 0, we obtain the following chain of equivalences:

\[D(\operatorname{THH}(\mathbb{S}_{(p)}[\sigma_{mk}])) \simeq \operatorname{THH}(\mathbb{S}_{(p)}[\sigma_{mk}])_0 \xrightarrow{\simeq} \operatorname{THH}(\mathbb{S}_{(p)}[\sigma_k])_0\]

of $E_1$ -rings using Lemma 2.3.

Proof. It suffices to prove that the map

\[ H\mathbb{Z}_{(p)} \wedge \operatorname{THH}(\mathbb{S}_{(p)}[\sigma_{mk}])_0 \to H\mathbb{Z}_{(p)} \wedge \operatorname{THH}(\mathbb{S}_{(p)}[\sigma_k])_0\]

is an equivalence; see Lemma 4.11.

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

\begin{equation*} \operatorname{THH}^{H\mathbb{Z}_{(p)}}(H\mathbb{Z}_{(p)}[\sigma_{mk}]) \to \operatorname{THH}^{H\mathbb{Z}_{(p)}}(H\mathbb{Z}_{(p)}[\sigma_{k}]),\end{equation*}

being an equivalence in weight 0. 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

\[\phi^r \colon E^r \to {F}^r\]

from the Bökstedt spectral sequence computing $\operatorname{THH}^{H\mathbb{Z}_{(p)}}(H\mathbb{Z}_{(p)}[\sigma_{mk}])$ to the Bökstedt spectral sequence computing $\operatorname{THH}^{H\mathbb{Z}_{(p)}}(H\mathbb{Z}_{(p)}[\sigma_{k}])$ . 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 0.

We have

(4.6) \begin{equation} \phi^2 \colon \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)),\end{equation}

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

\begin{align*}\operatorname{deg}(\sigma_{mk}) &= (0,mk), \quad \operatorname{deg}(d(\sigma_{mk})) = (1,mk),\\\operatorname{deg}(\sigma_k) &= (0,k), \quad \operatorname{deg}(d(\sigma_k)) = (1,k).\end{align*}

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 0 and the weight-0 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

\[\phi^2(\sigma_{mk}^i) = \sigma_k^{im}\quad\mathrm{and}\quad \phi^2(\sigma_{mk}^id(\sigma_{mk})) = m \sigma_k^{(i+1)m-1} d(\sigma_k).\]

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 0 as desired.

In the situation of Hypothesis 4.1, $A \to A(\!\sqrt[m]{a})$ is a map of m-graded $E_1$ -rings and A is concentrated in weight 0. Therefore, there is a map

(4.7) \begin{equation} \operatorname{THH}(A) \to \operatorname{THH}(A(\!\sqrt[m]{a}))_0,\end{equation}

where A above denotes the underlying $E_1$ -ring of A. This map is $S^1$ -equivariant since it is the restriction of the $S^1$ -equivariant m-graded map $\operatorname{THH}(A) \to \operatorname{THH}(A(\!\sqrt[m]{a}))$ along $0 \to \mathbb{Z}/m$ as $\operatorname{THH}(A)$ is concentrated in weight 0.

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 0. Owing to Proposition 4.9, we have

\[\operatorname{THH}(A(\!\sqrt[m]{a}))_0 \simeq \operatorname{THH}(A) \wedge_{\operatorname{THH}(\mathbb{S}_{(p)}[\sigma_{mk}])} (\operatorname{THH}(\mathbb{S}_{(p)}[\sigma_k])_0)\]

and it follows by Proposition 4.12 that the map

\[\operatorname{THH}(\mathbb{S}_{(p)}[\sigma_{mk}])\xrightarrow{\simeq} \operatorname{THH}(\mathbb{S}_{(p)}[\sigma_k])_0\]

is an equivalence. This identifies $\operatorname{THH}(A(\!\sqrt[m]{a}))_0$ with $\operatorname{THH}(A)$ as desired.

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 log-THH-étale. In other words, we show that there is an equivalence of m-graded spectra

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

Remark 6.11. A notion of log-THH-étalenes is already defined in [Reference Rognes, Sagave and SchlichtkrullRSS18]. In the language of Rognes, Sagave and Schlichtkrull [Reference Rognes, Sagave and SchlichtkrullRSS18], log-THH-étaleness of $A\to A(\!\sqrt[m]{a})$ would be expressed by an equivalence

(6.4) \begin{equation}\operatorname{THH}(A(\!\sqrt[m]{a}) \mid \sqrt[m]{a}) \simeq A(\!\sqrt[m]{a}) \wedge_A \operatorname{THH}(A \mid a).\end{equation}

Since we only assume A to be $E_1$ , $\operatorname{THH}(A\mid a)$ 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 log THH of A may be given an A-module structure and we obtain that $A\to A(\!\sqrt[m]{a})$ is log-THH-étale in the sense of (6.4) whenever A is p-local and $p \nmid m$ .

We start by proving a log-THH-étaleness result for the p-localized free $E_1$ -algebra $\mathbb{S}_{(p)}[\sigma_{mk}]$ .

Proposition 6.13. Let $k\geq 0$ be even and let $m\gt 0$ with $p \nmid m$ . In this situation, there is an equivalence of left $\operatorname{THH}(\mathbb{S}_{(p)}[\sigma_{mk}] \mid \sigma_{mk})$ -modules in m-graded spectra:

\[ \operatorname{THH}(\mathbb{S}_{(p)}[\sigma_{mk}] \mid \sigma_{mk}) \wedge_{\mathbb{S}_{(p)}[\sigma_{mk}]} \mathbb{S}_{(p)}[\sigma_k] \simeq \operatorname{THH}(\mathbb{S}_{(p)}[\sigma_k] \mid \sigma_k), \]

where the left $\operatorname{THH}(\mathbb{S}_{(p)}[\sigma_{mk}] \mid \sigma_{mk})$ -module structure on the right-hand side is provided by Construction 6.9.

Proof. We start by constructing the desired map. First, there is a composite map of m-graded $E_1$ -ring spectra

(6.5) \begin{equation}\mathbb{S}_{(p)}[\sigma_k] \to \operatorname{THH}(\mathbb{S}_{(p)}[\sigma_k]) \to \operatorname{THH}(\mathbb{S}_{(p)}[\sigma_k] \mid \sigma_k),\end{equation}

which is, in particular, a map of left $\mathbb{S}_{(p)}[\sigma_{mk}]$ -modules in m-graded spectra by forgetting structure through 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

(6.6) \begin{equation}\mathbb{S}_{(p)}[\sigma_{mk}] \to \operatorname{THH}(\mathbb{S}_{(p)}[\sigma_{mk}]\mid \sigma_{mk})\end{equation}

of m-graded $E_1$ -algebras, we obtain the desired map:

\begin{equation*} f \colon \operatorname{THH}(\mathbb{S}_{(p)}[\sigma_{mk}] \mid \sigma_{mk}) \wedge_{\mathbb{S}_{(p)}[\sigma_{mk}]} \mathbb{S}_{(p)}[\sigma_k] \to \operatorname{THH}(\mathbb{S}_{(p)}[\sigma_k] \mid \sigma_k)\end{equation*}

of left $\operatorname{THH}(\mathbb{S}_{(p)}[\sigma_{mk}] \mid \sigma_{mk})$ -modules in m-graded spectra from (6.5). Here, we used the fact that the left $\mathbb{S}_{(p)}[\sigma_{mk}]$ -module structure on $\operatorname{THH}(\mathbb{S}_{(p)}[\sigma_k] \mid \sigma_{k})$ used in (6.5) is compatible with the one obtained by forgetting the canonical left $\operatorname{THH}(\mathbb{S}_{(p)}[\sigma_{mk}] \mid \sigma_{mk})$ -module structure on $\operatorname{THH}(\mathbb{S}_{(p)}[\sigma_k] \mid \sigma_{k})$ through (6.6); this follows by the p-local version of (6.3).

What remains is to show that f is an equivalence. Since f is a map between p-local connective spectra (Proposition 6.12), it is sufficient to show that $H\mathbb{Z}_{(p)} \wedge f$ is an equivalence; see Lemma 4.11.

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

\begin{align*}& H\mathbb{Z}_{(p)} \wedge \operatorname{THH}(\mathbb{S}_{(p)}[\sigma_{mk}] \mid \sigma_{mk}) \wedge_{\mathbb{S}_{(p)}[\sigma_{mk}]} \mathbb{S}_{(p)}[\sigma_k] \\&\quad \simeq (H\mathbb{Z}_{(p)} \wedge \operatorname{THH}(\mathbb{S}_{(p)}[\sigma_{mk}] \mid \sigma_{mk}) ) \wedge_{H\mathbb{Z}_{(p)}[\sigma_{mk}]} H\mathbb{Z}_{(p)}[\sigma_{k}].\end{align*}

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

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

To prove that $H\mathbb{Z}_{(p)} \wedge f$ is an equivalence, we argue as in the proof of Proposition 4.12. The map of Bökstedt spectral sequences computing the map

(6.7) \begin{equation} \operatorname{THH}^{H\mathbb{Z}_{(p)}}_*(H\mathbb{Z}_{(p)}[\sigma_{mk}^{\pm 1}]) \to \operatorname{THH}^{H\mathbb{Z}_{(p)}}_*(H\mathbb{Z}_{(p)}[\sigma_k^{\pm 1}])\end{equation}

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

(6.8) \begin{equation}\phi \colon \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))\end{equation}

satisfying

\[\phi(\sigma_{mk}) = \sigma_k^m \quad\mathrm{and}\quad\phi(d(\sigma_{mk})) = m \sigma_k^{m-1}d(\sigma_k),\]

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

(6.9) \begin{equation} \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).\end{equation}

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 (6.7).

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

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

is given by a map

\[\mathbb{Z}_{(p)}[\sigma_{mk}] \otimes \Lambda_{\mathbb{Z}_{(p)}}(d\log \sigma_{mk}) \to \mathbb{Z}_{(p)}[\sigma_{k}] \otimes \Lambda_{\mathbb{Z}_{(p)}}(d\log \sigma_{k})\]

that carries $\sigma_{mk}$ to $\sigma_{k}^m$ and $d\log \sigma_{mk}$ to $d\log \sigma_k$ up to a unit due to (6.9) 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 theorem provides the log-THH-étaleness of root adjunction in ring spectra.

Theorem 6.14. Assume Hypothesis 4.1 with $p\nmid m$ and that A is p-local. In this situation, there is an equivalence of m-graded spectra

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

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

\[\operatorname{THH}(A(\!\sqrt[m]{a}) \mid \sqrt[m]{a})_i \simeq \Sigma^{ik} \operatorname{THH}(A\mid a)\]

for every $0\leq i\lt m$ .

Proof. We have the following chain of equivalences:

\begin{equation*} \begin{split} &\operatorname{THH}(A(\!\sqrt[m]{a}) \mid \sqrt[m]{a}) \simeq \operatorname{THH}(A(\!\sqrt[m]{a})) \wedge_{\operatorname{THH}(\mathbb{S}_{(p)}[\sigma_k])} \operatorname{THH}(\mathbb{S}_{(p)}[\sigma_k] \mid \sigma_k)\\ &\quad\simeq (\operatorname{THH}(A) \wedge_{\operatorname{THH}(\mathbb{S}_{(p)}[\sigma_{mk}])}\operatorname{THH}(\mathbb{S}_{(p)}[\sigma_k]) ) \wedge_{\operatorname{THH}(\mathbb{S}_{(p)}[\sigma_k])} \operatorname{THH}(\mathbb{S}_{(p)}[\sigma_k] \mid \sigma_k)\\ &\quad\simeq \operatorname{THH}(A) \wedge_{\operatorname{THH}(\mathbb{S}_{(p)}[\sigma_{mk}])} \operatorname{THH}(\mathbb{S}_{(p)}[\sigma_{mk}] \mid \sigma_{mk}) \wedge_{\operatorname{THH}(\mathbb{S}_{(p)}[\sigma_{mk}] \mid \sigma_{mk})} \operatorname{THH}(\mathbb{S}_{(p)}[\sigma_k] \mid \sigma_k)\\ &\quad\simeq \operatorname{THH}(A \mid a) \wedge_{\operatorname{THH}(\mathbb{S}_{(p)}[\sigma_{mk}] \mid \sigma_{mk})} \operatorname{THH}(\mathbb{S}_{(p)}[\sigma_k] \mid \sigma_k)\\ &\quad\simeq \operatorname{THH}(A \mid a) \wedge_{\operatorname{THH}(\mathbb{S}_{(p)}[\sigma_{mk}]\mid \sigma_{mk})} ( \operatorname{THH}(\mathbb{S}_{(p)}[\sigma_{mk}] \mid \sigma_{mk}) \wedge_{\mathbb{S}_{(p)}[\sigma_{mk}]} \mathbb{S}_{(p)}[\sigma_k] )\\ &\quad\simeq \operatorname{THH}(A\mid a) \wedge_{\mathbb{S}_{(p)}[\sigma_{mk}]} \mathbb{S}_{(p)}[\sigma_k].\\ \end{split}\end{equation*}

The first and fourth equivalences follow by the definition of log THH, the second equivalence follows by our definition of root adjunction and Proposition 4.9 and the fifth equivalence follows by Proposition 6.13.

6.2 Relating THH and logarithmic THH

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

\[\operatorname{THH}(A) \to \operatorname{THH}(A \mid a) \to \Sigma \operatorname{THH}(A/a)\]

under our usual assumptions. The $E_1$ -ring $A/a$ above is described in the following construction, which is analogous to [Reference Rognes, Sagave and SchlichtkrullRSS15, Lemmas 6.14 and 6.15].

Construction 6.16. Let A be an $\mathbb{S}[\sigma_k]$ -algebra where $\sigma_k$ acts through $a \in \pi_kA$ , 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 [Reference Hahn and WilsonHW22, 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 $\Sigma^{k}\mathbb{S}[\sigma_k] \xrightarrow{\cdot \sigma_k } \mathbb{S}[\sigma_k]$ , $A/a$ is indeed the cofiber of the map $\Sigma^{k} A \xrightarrow{\cdot a} A$ .

Considering $\mathbb{S}[\sigma_k]$ as a graded $E_2$ -ring, we have

\[\operatorname{THH}(\mathbb{S}[\sigma_k])_0 \simeq \mathbb{S}.\]

This can be observed by inspection of 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.

Proposition 6.18. The cofiber of the map

\[\operatorname{THH}(\mathbb{S}[\sigma_k]) \to \operatorname{THH}(\mathbb{S}[\sigma_k] \mid \sigma_k)\]

is given by $\Sigma \mathbb{S}$ concentrated in weight 0 as a left $\operatorname{THH}(\mathbb{S}[\sigma_k])$ -module in graded spectra. Here, the left $\operatorname{THH}(\mathbb{S}[\sigma_k])$ -module structure on $\mathbb{S}$ is given by the weight-Postnikov truncation map of graded $E_1$ -rings

\[\operatorname{THH}(\mathbb{S}[\sigma_k]) \to \operatorname{THH}(\mathbb{S}[\sigma_k])_{0} \simeq \mathbb{S}.\]

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

\[ \operatorname{THH}(\mathbb{S}[\sigma_k]) \xrightarrow{f} \operatorname{THH}(\mathbb{S}[\sigma_k], \mid \sigma_k) \to M.\]

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

\[H\mathbb{Z}_*f \colon \operatorname{THH}^{H\mathbb{Z}}_*(H\mathbb{Z}[\sigma_k]) \to \operatorname{THH}^{H\mathbb{Z}}_*(H\mathbb{Z}[\sigma_k^{\pm 1}])_{\geq0}\]

is given by the ring map

\[\mathbb{Z}[\sigma_k] \otimes \Lambda(d(\sigma_k)) \to\mathbb{Z}[\sigma_k] \otimes \Lambda(d\log \sigma_k)\]

that carries $\sigma_k$ to $\sigma_k$ and $d(\sigma_k)$ to $\sigma_k d\log \sigma_k$ ; this follows by the Böksedt spectral sequences in (4.6) and (6.8). This map is injective and the only class that is not in the image is $d\log \sigma_k$ . We obtain an equivalence of graded $H \mathbb{Z}$ -modules

(6.10) \begin{equation} H\mathbb{Z} \wedge M \simeq \Sigma H\mathbb{Z},\end{equation}

where the right-hand side is concentrated in weight 0.

Since M is connective in homotopy (by Proposition 6.12), we deduce that it is concentrated in weight 0 by the equivalence above. Again, since M is connective, the Tor spectral sequence shows that the lowest-degree non-trivial homotopy group of $H\mathbb{Z} \wedge M$ is given by the lowest-degree non-trivial homotopy group of M. By this, and (6.10), there is a map of graded spectra $\Sigma \mathbb{S} \to M$ (with $\Sigma \mathbb{S}$ placed in weight 0), which when smashed with $H \mathbb{Z}$ gives the degree-1 class in $H \mathbb{Z}$ homology. This map

\[ \Sigma \mathbb{S} \xrightarrow{\simeq} M \]

is an equivalence (by Lemma 4.11) because one gets an equivalence after smashing with $H \mathbb{Z}$ .

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

\[\Sigma \operatorname{THH}(\mathbb{S}[\sigma_k]) \to M\]

of left $\operatorname{THH}(\mathbb{S}[\sigma_k])$ -modules in graded spectra carrying 1 to 1 in homotopy. Taking weight-0 Postnikov sections [Reference Hahn and WilsonHW22, B.0.6], we obtain an equivalence of left $\operatorname{THH}(\mathbb{S}[\sigma_k])$ -modules in graded spectra

\[ \Sigma \operatorname{THH}(\mathbb{S}[\sigma_k])_0 \xrightarrow{\simeq } M.\]

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 log THH.

Theorem 6.19. Let A be an $\mathbb{S}[\sigma_k]$ -algebra where $\sigma_k$ acts through $a \in \pi_kA$ with even $k \geq 0$ . In this situation, there is a cofiber sequence of spectra

\[\operatorname{THH}(A) \to \operatorname{THH}(A \mid a) \to \Sigma \operatorname{THH}(A/a).\]

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

Proof. Proposition 6.18 provides the following cofiber sequence of left $\operatorname{THH}(\mathbb{S}[\sigma_k])$ -modules in graded spectra:

\[\operatorname{THH}(\mathbb{S}[\sigma_k]) \to \operatorname{THH}(\mathbb{S}[\sigma_k] \mid \sigma_k) \to \Sigma \mathbb{S}.\]

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

(6.11) \begin{equation} \operatorname{THH}(A) \to \operatorname{THH}(A \mid a) \to \operatorname{THH}(A) \wedge_{\operatorname{THH}(\mathbb{S}[\sigma_k])} \Sigma \mathbb{S}.\end{equation}