Since the map $\GG_a^\sharp \cong W[F] \to W$ sends $x\in \GG_a^\sharp$ to the Witt vector whose ghost coordinates are $(x, 0, 0, \cdots)$, the claim <ref> follows from the observation that $p^n\in W[F](\Z_p)$ is sent to zero in $W[F](\Z/p^n)$.

As pointed out by Sasha Petrov, the preceding calculation also determines the $\GG_m^\sharp$-action on $\spec(\Z/p^n)^{\slashed{D}}$ as follows. The above discussion says that the isomorphism $\GG_a^\sharp \xar{\sim} \spec(\Z/p^n)^{\slashed{D}}$ sends $x\mapsto x + V(p^{n-2})$. Under this isomorphism, the action of $g\in \GG_m^\sharp$ on $x+V(p^{n-2}) \in \spec(\Z/p^n)^{\slashed{D}}$ is given by

$$g(x + V(p^{n-2})) = gx + gV(p^{n-2}) = gx + V(F(g) p^{n-2});$$

but $F(g) = 1$ since $\GG_m^\sharp = W^\times[F]$, so that this can be identified with $gx + V(p^{n-2})$. In other words, the isomorphism $\GG_a^\sharp \xar{\sim} \spec(\Z/p^n)^{\slashed{D}}$ is equivariant for the scaling action of $\GG_m^\sharp$ on $\GG_a^\sharp$.

One can get a formula which is more “accurate” than <ref> via the following (see also <cit.>, where part of this statement is attributed to Gabber)[Our understanding is that this result is quite well-known; some form is heavily used in [23].].

Let $y$ denote the element of $W(\Z_p)$ associated to the ghost coordinates $(1-p^{p-1}, 1-p^{p^2-1}, \cdots)$. Then $[p] + V(y) = p$. Moreover, $y = Fx$ for some $x\in W(\Z_p)$ if and only if $p>2$; in this case, $x\in W(\Z_p)^\times$ (and hence $y\in W(\Z_p)^\times$). If $p=2$, then $y[2^m]$ is in the image of $F$ for any $m\geq 2$.

Let us assume $p$ is odd for simplicity. Then <ref> implies that $p-[p]\in W(\Z_p)$ is a unit multiple of $V(1)$, since $p-[p] = V(y) = xV(1)$ and $x\in W(\Z_p)^\times$.[Analogously, $[2](2 - [2]) = [2] V(y) = V(y[4])\in W(\Z_p)$ is divisible by $V(1)$ for $p=2$.] It follows from <cit.> that if $X = \spf(R)$ is a bounded $p$-adic formal scheme, then the diffracted Hodge complex $X^{\slashed{D}}$ is given on $p$-nilpotent rings $S$ by $X^{\slashed{D}}(S) = X(W(R)/(p-[p]))$.

Applying $F$ to the identity $[p]+V(y)=p$, we see that $[p^2] = p(1-y)$. In particular, the element $a\in W(\Z_p)$ of <cit.> can be identified with $1-y$.

Using <ref>, we can give an “alternative” formula for a preimage of $p^n$ under $VF$. Indeed, we have $p = [p] + V(y)$ for some $y\in W(\Z_p)$, so that $p^n = [p^n] + \sum_{i=0}^{n-1} \binom{n}{i} [p^i] V(y)^{n-i}$ in $W(\Z_p)$. Because $V(a)b = V(aFb)$ and $FV = p$, we have $V(a)^n = V(p^{n-1} a^n)$ by an easy induction on $n$. Moreover, $[p^i] V(a) = V(aF[p^i]) = V([p^{pi}]a)$. Since $[p^n] = 0\in W(\Z/p^n)$ (and hence in $W(R)$), we have

$$p^n = \sum_{i=0}^{n-1} \binom{n}{i} [p^i] V(y)^{n-i} = \sum_{i=0}^{n-1} \binom{n}{i} V(p^{n-i-1} y^{n-i} F[p^i]) \in W(\Z/p^n).$$

Assume $p>2$, so that <ref> implies that $y = Fx$ for some $x\in W(\Z_p)$. The multiplicativity of $F$ now lets us conclude that

$$p^n = VF\left(\sum_{i=0}^{n-1} \binom{n}{i} p^{n-i-1} x^{n-i} [p^i]\right) \in W(\Z/p^n),$$

so that $p^n\in W(R)$ is in the image of $VF$, as desired.

One can check that

$$\sum_{i=0}^{n-1} \binom{n}{i} p^{n-i-1} y^{n-i} [p^{pi}] = p^{n-1} \in W(\Z/p^n).$$

This is essentially an elaboration on the proof of <ref>. Indeed, applying $w_j$, we have

\begin{align*}
w_j\left(\sum_{i=0}^{n-1} \binom{n}{i} p^{n-i-1} y^{n-i} [p^{pi}]\right) & = \frac{1}{p} \sum_{i=0}^{n-1} \binom{n}{i} (p-p^{p^{j+1}})^{n-i} p^{p^{j+1}i} \\
%& = \frac{1}{p}((p-p^{p^{j+1}} + p^{p^{j+1}})^n - p^{p^{j+1}n})
& = p^{n-1} - p^{p^{j+1}n-1}.
\end{align*}

It therefore suffices to show that the Witt vector $a\in W(\Z_p)$ with coordinates $w_j(a) = p^{p^{j+1}n-1}$ vanishes in $W(\Z/p^n)$, which follows from a direct calculation.

Let us end with a proof of <ref>; the explicit formulas below are unnecessary for any conceptual development, but we included it since the computation was rather fun.

First, it is easy to see that $y$ is well-defined.

Let us now check that $p = [p] + V(y)$. If $w(x) = (w_0(x), w_1(x), \cdots)$ are the ghost coordinates of $x\in W(R)$, then $w_{n+1}(Vx) = pw_n(x)$. It follows that $w_n(Vy) = p-p^{p^n}$. Since $w([p]) = (p, p^p, p^{p^2}, \cdots)$, we have

$$w([p]+Vy) = w([p]) + w(Vy) = (p, p, \cdots) = w(p),$$

so that $p = [p] + V(y)$, as claimed.

To prove the claim about $y$ being in the image of $F$, recall that if $x\in W(R)$, then the ghost coordinates of $Fx$ are given by $w_n(Fx) = w_{n+1}(x)$. In particular, $y = Fx$ for some $x\in W(\Z_p)$ if and only if we can solve

$$1-p^{p^n-1} = x_0^{p^n} + px_1^{p^{n-1}} + \cdots + p^n x_n$$

for some $x_0,\cdots, x_n\in \Z_p$ and all $n\geq 1$. This is impossible for $p=2$. Indeed, first note that we need $x_0^2 + 2x_1 = 1-p^{p-1} = -1$, so that $x_0^2 \equiv 1\pmod{2}$ (and hence $x_0 \equiv 1\pmod{2}$). Write $x_0 = 1 + 2s$, so that $x_0^2 + 2x_1 = 1 + 4s(1+s) + 2x_1$. In order for this to equal $1-p^{p-1}=-1$, we need $4s(1+s) + 2x_1 = -2$, i.e., $x_1\equiv 1\pmod{2}$. This implies that $x_0^4 \equiv 1\pmod{8}$ and $x_1^2 \equiv 1\pmod{4}$ (so $2x_1^2 \equiv 2\pmod{8}$). Since $1 - p^{p^2-1} = -7 = x_0^4 + 2x_1^2 + 4x_2$, we can reduce modulo $8$ to find that $1 \equiv 1 + 2 + 4x_2\pmod{8}$. But then $x_2$ would solve $4x_2 \equiv -2\pmod{8}$, which is impossible.

Now assume $p>2$. Since $x_0^p + px_1 = 1-p^{p-1}$, we have $x_0^p \equiv 1\pmod{p}$; this implies that $x_0^{p^n} \equiv 1\pmod{p^{n+1}}$. Writing $x_0^{p^n} = 1 - p^{n+1} s_n$ for some $s_n\in \Z_p$, we have $x_1 = ps_1-p^{p-2}$. Since $p>2$, we see that $x_1 = p(s_1 - p^{p-3})\in p\Z_p$. We claim that $x_n$ exists and is an element of $p\Z_p$ for $n\geq 1$. We established the base case $n=1$ above, so assume that $x_1, \cdots, x_{n-1} \in p\Z_p$, and let $x_i = pt_i$. We then have

\begin{align*}
p^n x_n & = 1 - p^{p^n-1} - (x_0^{p^n} + px_1^{p^{n-1}} + \cdots + p^{n-1} x_{n-1}^p)\\
%& = 1 - p^{p^n-1} - (1 - p^{n+1} s_n + p p^{p^{n-1}} t_1^{p^{n-1}} + \cdots + p^{n-1} p^p t_{n-1}^p)\\
& = p^{n+1} s_n - p^{p^n-1} - p^{p^{n-1}+1} t_1^{p^{n-1}} - \cdots - p^{p+n-1} t_{n-1}^p,
\end{align*}

so that

$$x_n = ps_n - p^{p^n-1-n} - p^{p^{n-1}+1-n} t_1^{p^{n-1}} - \cdots - p^{p-1} t_{n-1}^p.$$

This is clearly divisible by $p$ since $p>2$ (so that $p^n-1-n \geq 1$ for $n\geq 1$). Therefore, $x_n$ exists and lives in $p\Z_p$, as desired. (Note that if $p=2$ and $n=1$, then $p^n-1-n = 0$, so $x_1\not\in 2\Z_2$.) If one prefers an explicit formula, the above argument shows that once one writes $x_0 = 1-ps_0$, then $x_j = pt_j$ for $j\geq 1$ can be defined inductively by

$$t_n =
%- p^{p^n-2-n} - p^{p^{n-1}-n} t_1^{p^{n-1}} - \cdots - p^{p-2} t_{n-1}^p + \sum_{i=1}^{p^n} \frac{(-1)^{i+1}}{p^{n+1-i}} \binom{p^n}{i} s_0^i
\sum_{i=1}^{p^n} \frac{(-1)^{i+1}}{p^{n+1-i}} \binom{p^n}{i} s_0^i - p^{p^n-2-n} - \sum_{k=1}^{n-1} p^{p^k-k-1} t_{n-k}^{p^k}.

The first term is $s_n$; note that $\frac{1}{p^{n+1-i}} \binom{p^n}{i}\in \Z$. Since $x_0 \equiv 1\pmod{p}$ and $x_i\equiv 0\pmod{p}$ for $i\geq 1$, it is easy to see that all the ghost components of $x$ lie in $1+p\Z_p\subseteq \Z_p^\times$; this implies that $x\in W(\Z_p)$ is invertible, as claimed.

Let us now assume that $p=2$, and show that $y[2^m]$ is in the image of $F$ for any $m\geq 2$. To see this, observe that the ghost components of $y[2^m]$ are given by

$$w_n(y[2^m]) = w_n(y) w_n([2^m]) = 2^{m2^n}(1-2^{2^{n+1}-1}).$$

We therefore need to solve

$$2^{m2^{n-1}}(1-2^{2^n-1}) = x_0^{2^n} + 2x_1^{2^{n-1}} + \cdots + 2^n x_n$$

for some $x_0,\cdots, x_n\in \Z_2$ and all $n\geq 1$. When $n=1$, we have $x_0^2 + 2x_1 = -2^m$, so that $x_0^2 \equiv 0\pmod{2}$ since $m>0$. It follows that $x_0 = 2t_0$ for some $t_0\in \Z_2$.
We now claim that $x_n$ exists for $n\geq 0$ and lives in $2\Z_2$. We established the base case $n=0$ above, so assume $x_0,x_1, \cdots, x_{n-1}\in 2\Z_2$, and write $x_i = 2t_i$. Then

\begin{align*}
2^n x_n & = 2^{m2^{n-1}}(1-2^{2^n-1}) - (x_0^{2^n} + 2x_1^{2^{n-1}} + \cdots + 2^{n-1} x_{n-1}^2)\\
%& = 2^{m2^{n-1}}(1-2^{2^n-1}) - (2^{2^n} t_0^{2^n} + 2\cdot 2^{2^{n-1}} t_1^{2^{n-1}} + \cdots + 2^{n-1} \cdot 2^2 t_{n-1}^2)\\
& = 2^{m2^{n-1}}(1-2^{2^n-1}) - (2^{2^n} t_0^{2^n} + 2^{2^{n-1}+1} t_1^{2^{n-1}} + \cdots + 2^{n+1} t_{n-1}^2),
\end{align*}

so that

$$x_n = 2^{m2^{n-1}-n}(1-2^{2^n-1}) - (2^{2^n-n} t_0^{2^n} + 2^{2^{n-1}+1-n} t_1^{2^{n-1}} + \cdots + 2 t_{n-1}^2).$$

Because $m\geq 2$ and $2^j-j\geq 1$ for every $j\geq 0$, we see that $x_n\in 2\Z_2$, as desired. (Of course, the key case is $m=2$; when $m=1$ and $n=1$, the term $2^{m2^{n-1}-n}(1-2^{2^n-1}) = -1\not\in 2\Z_2$.) If one prefers an explicit formula, note that the above argument shows that once one writes $x_0 = 2t_0$, then $x_j = 2t_j$ can be defined inductively by

$$t_n =
%2^{m2^{n-1}-n-1}(1-2^{2^n-1}) - 2^{2^n-n-1} t_0^{2^n} - 2^{2^{n-1}-n} t_1^{2^{n-1}} - \cdots - t_{n-1}^2
2^{m2^{n-1}-n-1}(1-2^{2^n-1}) - \sum_{i=1}^n 2^{2^i - i - 1} t_{n-i}^{2^i}.

Note that $x$ is not invertible in $W(\Z_2)$; instead, since $x_j\in 2\Z_2$, the $n$th ghost component $w_n(x)\in 2^{n+1} \Z_2$.

§ CARTIER DUALS OF $W[F^N]$ AND $W^\TIMES[F^N]$

This section was inspired by the results proved above, but it does not play an essential role in the body of this article.
<ref> below can be viewed as an algebraic way to bookkeep the structure possessed by the topological Sen operators; and, as we hope to show in future work, it sits as an intermediary between the topological and algebraic Sen operators of <ref> and <ref> (see <ref>). We begin with the following (presumably well-known) result. I am (again) grateful to Sasha Petrov for a relevant discussion on it.

There is an isomorphism of group schemes over $\Z_{(p)}$ between $W[F^n] := \ker(F^n: W \to W)$ and the Cartier dual of the completion of $W_n = W/V^n$ at the origin.

Let us model $W$ by the $p$-typical big Witt vectors. Given $f(t)\in W$, let $a_0,a_1,a_{2},\cdots$ denote the ghost components of $f$, so that $td\log(f(t)) = \sum_{m\geq 0} a_{m} t^{p^m}$. Then $f(t)\in W[F^n]$ if and only if $a_{m} = 0$ for $m\geq n$.

Let us first prove the claim of the proposition when $n=1$. Then, $d\log(f(t))$ is a constant, and $f(0) = 1$; we claim that this is equivalent to the condition that $f$ defines a homomorphism $\hat{\GG}_a \to \GG_m$, i.e., that $f(x+y) = f(x) f(y)$. To check this, first suppose that $f(x+y) = f(x) f(y)$. Then $\partial_x f(x+y) = f(y) f'(x)$, so that

$$\frac{\partial_x f(x+y)}{f(x+y)} = \frac{f'(x)}{f(x)} = d\log(f(x))$$

is independent of $y$. Taking $x = 0$, we see that $d\log(f(x))$ is constant, as desired. The reverse direction (that $d\log(f(x))$ being constant and $f(0) = 1$ implies that $f(x+y) = f(x) f(y)$) is similar.

In the general case, note that since the Frobenius on $W$ shifts the ghost components by $F: (a_0, a_1, a_{2}, \cdots) \mapsto (a_1, a_{2}, a_{3}, \cdots)$, the Frobenius $F$ applied to $f$ satisfies:

$$d\log(F^j(f)(t)) = \sum_{m=0}^{n-j} a_{{m+j}} t^{p^m},$$

so that there is an equality of power series

$$F^j(f)(t) = \exp\left(\sum_{m=0}^{n-j} \frac{a_{{m+j}}}{p^m} t^{p^m}\right).$$

Note that this is a slight variant of the the Artin-Hasse exponential.
Define a map $g: \hat{W}_n \to \GG_m$ on Witt components $(x_0,\cdots,x_{n-1})$ (not ghost components!) as follows:

\begin{align*}
g(x_0,\cdots,x_{n-1}) & = \prod_{j=0}^{n-1} F^j(f)(x_j) = \exp\left(\sum_{j=0}^{n-1} \sum_{m=0}^{n-j} \frac{a_{{m+j}}}{p^m} x_j^{p^m}\right) \\
& = \exp\left(\sum_{m=0}^{n-1} \frac{a_{m}}{p^m} \left(\sum_{j=0}^{m} p^j x_j^{p^{m-j}}\right) \right).
\end{align*}

The coefficient of $\tfrac{a_{m}}{p^m}$ is precisely the $m$th Witt polynomial, so that the function $g$ is indeed additive on $\hat{W}_n$. Moreover, the assignment $f\mapsto g$ indeed gives an isomorphism $W[F^n] \xar{\sim} \Hom(\hat{W}_n, \GG_m)$, as one can check inductively using the case $n=1$ and the fact that it induces an isomorphism over $\QQ$.

One does not need $p$-typicality for the above statement to hold. Namely, if $\WW$ denotes the big Witt ring scheme, it is a classical fact that the Cartier dual of $\WW$ over $\Z$ is canonically identified with $\hat{\WW}$. As in the $p$-typical case above, the pairing $\WW \times \hat{\WW} \to \GG_m$ sends

$$(a,b) \mapsto \exp\left(\sum_{n\geq 1} \frac{w_n(a) w_n(b)}{n}\right).$$

One only needs to check that this expression is in fact defined over $\Z$. To see this, first observe that if the Witt components of $b$ are $(b_1, b_2, \cdots)$, we have

$$\exp\left(\sum_{n\geq 1} \frac{w_n(a) w_n(b)}{n}\right) = \prod_{j\geq 1} \exp\left(\sum_{n\geq 1} \frac{w_{nj}(a) b_j^n}{n}\right).$$

Note that $w_{nj}(a) = w_n(F_j a)$, so that if $(F_j a)_d$ denote the Witt components of $F_j a$, we have

$$\exp\left(\sum_{n\geq 1} \frac{w_n(a) w_n(b)}{n}\right) = \prod_{j,d\geq 1} (1 - (F_j a)_d b_j^d),$$

giving the desired integral representation.
In fact, the last step can be generalized via the following rephrasing of the Dwork lemma:

Let $R$ be a torsionfree ring equipped with ring maps $\phi_p: R \to R$ for each prime $p$ such that $\phi_p(r) \equiv r^p\pmod{p}$ for all $r\in R$. Let $(x_n)_{n\geq 1}$ be a sequence of elements such that $x_n \equiv \phi_p(x_{n/p})\pmod{p^{v_p(n)}}$ for each prime $p$ and every $n\in p\Z_{\geq 0}$. Then $f(t) := \exp\left(\sum_{n\geq 1} \frac{x_n t^n}{n}\right)$ lies in $1 + tR\pw{t}\subseteq 1 + t(R \otimes \QQ)\pw{t}$.

Let $g(t) = 1-t \in R\pw{t}$, so that there is an identity

$$g(t) = \exp\left(\log(1-t)\right) = \exp\left(-\sum_{n\geq 1} \frac{t^n}{n}\right).$$

Because $f(0) = 1$, we can write $f(t) = \prod_{j\geq 1} (1-r_j t^j) = \prod_{j\geq 1} g(r_j t^j)$ for unique $r_j\in R\otimes \QQ$. Since $g(t)$ is integral, it is sufficient to show that the elements $r_j$ are also integral. Applying $d\log$, we find that

$$\sum_{n\geq 1} \frac{x_n t^n}{n} = d\log(f)(t) = \sum_{j\geq 1} d\log(g)(r_j t^j) = -\sum_{j,m\geq 1} \frac{r_j^m t^{jm}}{m}.$$

It follows that $x_n = -\sum_{j|n} j r_j^{n/j}$. One can now argue in exactly the same way as the usual Dwork lemma (i.e., by induction on $r_j\in R$ for $j|n$ with $j\neq n$) to argue that each $r_j$ is integral.

Write the underlying scheme of $W_n$ as $\prod_{i=0}^{n-1} \GG_a$ (where the $i$th copy of $\GG_a$ has coordinate $\Phi_i$).
There is a fully faithful functor $\QCoh(BW[F^n]) \hookrightarrow \QCoh(W_n)$ whose essential image consists of those $p$-complete $M\in \QCoh(W_n)$ such that $\Phi_i$ acts locally nilpotently on $\H^\ast(M/p)$ for each $0\leq i \leq n-1$. Furthermore, this functor is symmetric monoidal for the convolution tensor product on $\QCoh(W_n)$.

If $\cf\in \QCoh(BW[F^n])$ is sent to $M\in \QCoh(W_n)$ under this functor, one obtains a cube[Recall that $[n]$ denotes the set $\{0,\cdots,n\}$.] $\Phi_\bull: 2^{[n-1]} \to \Mod_{\Z_p}$ whose vertices are all $M$ and such that the edge from the subset $\{i_1, \cdots, i_{j-1}\}$ to $\{i_1, \cdots, i_j\}$ is given by the operator $\Phi_j: M \to M$. Then, the global sections $\Gamma(BW[F^n]; \cf)$ can be identified with the total fiber of the cube $\Phi_\bull$.

More generally, the argument of <ref> shows that there is an isomorphism of group schemes over $\Z_{(p)}$ between $W_m[F^n] := \ker(F^n: W_m \to W_m)$ and the Cartier dual of $W_n[F^m]$.
One can give a simpler proof of this fact over a perfect field $k$ of characteristic $p>0$ using the theory of Dieudonné modules: the Dieudonné module of $W_m[F^n]$ over $k$ is $W(k)[F,V]/(F^n, V^m)$, while the Dieudonné module of $W_n[F^m]$ over $k$ is $W(k)[F,V]/(F^m, V^n)$.

The argument of <ref> also shows the following (which is already discussed in <cit.>):

Let $\hat{\GG}_\lambda$ be the degeneration of $\hat{\GG}_m$ to $\hat{\GG}_a$ given by $\spf \Z_{(p)}[t, \lambda, \frac{1}{1+t\lambda}]^\wedge_t$ with group law $x + y + \lambda xy$. Then the $\Z_{(p)}[\lambda]$-linear Cartier dual of $\hat{\GG}_\lambda$ is isomorphic to the group scheme $\bD(\hat{\GG}_\lambda) = \spec \Z_{(p)}[\lambda, z, \frac{\prod_{j=0}^{n-1} (z-j\lambda)}{n!}]$ over $\AA^1_\lambda = \spec \Z_{(p)}[\lambda]$ with coproduct $z\mapsto z\otimes 1 + 1\otimes z$.

A homomorphism $f: \hat{\GG}_\lambda \to \GG_m \times \AA^1_\lambda$ is an element of $\Z_{(p)}[t, \lambda, \frac{1}{1+t\lambda}]^\wedge_t$ such that

$$f(x+y+\lambda xy) = f(x) f(y).$$

This condition implies that

$$(1+\lambda y) f'(x+y+\lambda xy) = f(y) f'(x),$$

so that dividing both sides by $f(x+y+\lambda xy)$, we have

$$(1+\lambda y) \cdot d\log(f)(x+y+\lambda xy) = d\log(f)(x).$$

Taking $x = 0$, we see that $d\log(f)(y)$ is a constant multiple of $\frac{1}{1+\lambda y}$ (where the constant is given by $\frac{f'(0)}{f(0)}$), and hence

$$f(y) = (1+\lambda y)^{z/\lambda} = \sum_{n\geq 0} y^n \frac{\prod_{j=0}^{n-1} (z-j\lambda)}{n!}$$

for some fixed $z$. This gives the desired claim, similarly to <ref>. Note that $f(y) = \exp(z\log_F(y))$, where $\log_F$ is the logarithm of the formal group law $x + y + \lambda xy$ over $\AA^1_\lambda$.

Observe that $\bD(\hat{\GG}_\lambda)$ is isomorphic to the subgroup $(W\times \AA^1_\lambda)[F+[-\lambda]^{p-1}]$ of $W \times \AA^1_\lambda$ cut out by $\{x | Fx = [-\lambda]^{p-1} x\}$; see <cit.> and <cit.>. The key point is that if $f(x)\in W$ and $xd\log(f(x)) = \sum_{m\geq 1} a_{m} x^{m}$, then $f(t)\in (W\times \AA^1_\lambda)[F+[-\lambda]^{p-1}]$ if and only if

$$a_{p^{n+1}} = ((-\lambda)^{p-1})^{p^n} a_{p^n} = (-\lambda)^{p^{n+1} - p^n} a_{p^n}.$$

To check this, note that

$$xd\log(f)(x) = \frac{zx}{1+\lambda x} = \sum_{n\geq 0} (-\lambda)^n z x^{n+1},$$

so that $a_m = (-\lambda)^{m-1} z$, and $a_m = (-\lambda)^{m-n} a_n$ if $m\geq n$.

A similar argument shows that if $\GG$ denotes the group scheme $\spec \Z/p^N[\lambda]\pdb{x}$ with group law $x + y + \lambda xy$ (so that when $\lambda = 0$, we get $\GG_a^\sharp$), then the $\Z/p^N[\lambda]$-linear Cartier dual of $\GG$ is isomorphic to the completion of $\spec \Z/p^N[\lambda, z]$ at the locus $\prod_{j=0}^{p-1} (z-j\lambda) = z(z^{p-1} - \lambda^{p-1})$ (see, e.g., <cit.>). It follows that the $\infty$-category of $\GG$-representations is equivalent to the $\infty$-category of $\Z/p^N$-modules $M$ equipped with an operator $z: M \to M$ such that $z(z^{p-1} - \lambda^{p-1})$ acts locally nilpotently on $\H^\ast(M \otimes_{\Z/p^N} \FF_p)$.

In <cit.>, Bhatt and Lurie show that the following is a Cartesian square of group schemes over $\Z/p^k$:

\begin{equation}\label{gm-sharp-log-square}
\xymatrix{
\GG_m^\sharp \ar[r]^-{\log} \ar[d] & \GG_a^\sharp \ar[d]^-{x\mapsto \exp(px)} \\
\GG_m \ar[r]_-{x\mapsto x^p} & \GG_m^{(1)}.
\end{equation}

We will generalize this below in <ref>.
In [43], we prove another generalization of this square, albeit in a different direction: $\GG_a^\sharp$ is replaced by the Cartier dual of a formal group $\hat{\GG}$, and $\GG_m^\sharp$ is replaced by an appropriate $\hat{\GG}$-analogue of the divided power completion.

Let $k\geq 0$. There is an isomorphism of group schemes over $\Z/p^k$ between the Cartier dual of $W^\times[F^n] := \ker(F^n: W^\times \to W^\times)$ and the completion of $W_n$ at its $\FF_p$-rational points $W_n(\FF_p) \cong \Z/p^n$.

Following <cit.>, it suffices to prove the following analogue of <cit.>: there is a Cartesian diagram of flat group schemes over $\Z/p^k$ given by

\begin{equation}\label{log-square}
\xymatrix{
W^\times[F^n] \ar[r]^-{\log} \ar[d] & W[F^n] \ar[d]^-{x\mapsto \exp(p^n x)} \\
\GG_m \ar[r]_-{x\mapsto x^{p^n}} & \GG_m^{(n)}.
\end{equation}

Here, the left vertical map $W^\times[F^n] \to \GG_m$ is the composite

$$W^\times[F^n] \to W^\times \to W^\times/V \cong \GG_m.$$

Indeed, taking the Cartier dual of <ref> and using <ref>, we obtain a pushout diagram of formal group schemes

p^n \Z \ar[r] \ar[d] & \Z \ar[d] \\
\hat{W}_n \ar[r] & \bD(W^\times[F^n]).

This implies that $\bD(W^\times[F^n])$ is the completion of $W_n$ at its $\FF_p$-rational points $W_n(\FF_p) \cong \Z/p^n$, as desired.

The proof that the square <ref> is Cartesian is in fact a consequence of <cit.>. As in <cit.>, since all group schemes involved are flat over $\Z/p^k$, it suffices to prove that the diagram is Cartesian after base-changing to $\FF_p$ (i.e., assume that $k=1$). We begin by noting that there is an isomorphism $V: W^\times \xar{\sim} \{x\in W | x_0 = 0\}$ of $p$-adic formal schemes sending $x\mapsto Vx$. This gives an isomorphism $W[F^n] \times \mu_{p^n} \cong W^\times[F^n]$ of group schemes over $\FF_p$, sending $(x, a) \mapsto [a] + Vx$. It therefore suffices to show that the composite

$$W[F^n] \xar{x\mapsto 1 + Vx} W^\times[F^n] \xar{\log} W[F^n]$$

is an isomorphism. But there is a commutative diagram

W[F^{n-1}] \ar[r]_-{x\mapsto 1 + Vx} \ar[d] & W^\times[F^{n-1}] \ar[r]^-{\log} \ar[d] & W[F^{n-1}] \ar[d] \\
W[F^{n}] \ar[r]^-{x\mapsto 1 + Vx} \ar[d] & W^\times[F^{n}] \ar[r]^-{\log} \ar[d] & W[F^{n}] \ar[d] \\
\GG_a^\sharp \ar[r]^-{x\mapsto 1 + Vx} & \GG_m^\sharp \ar[r]^-{\log} & \GG_a^\sharp,

where the columns exhibit the third term as the quotient of the first two. By induction on $n$ (with the base case being provided by <cit.>), we may conclude that the middle horizontal composite is an isomorphism.

One could have alternatively/equivalently proved <ref> by observing that the square <ref> for $n-1$ maps to <ref> for $n$; all components of this map of squares are the canonical ones, except on the bottom-right $\GG_m$ (where it is given by the $p$th power map $\GG_m \to \GG_m^{(1)}$). Diagramatically:

\begin{equation}
\begin{tikzcd}[row sep=2.5em]
W^\times[F^{n-1}] \arrow[rr,"\log"] \arrow[dr,swap] \arrow[dd,swap] &&
W[F^{n-1}] \arrow[dd,swap, "x\mapsto \exp(p^{n-1} x)" near start] \arrow[dr] \\
& W^\times[F^n] \arrow[rr,crossing over,swap,"\log" near start] &&
W[F^n] \arrow[dd,"x\mapsto \exp(p^n x)"] \\
\GG_m \arrow[rr,"x\mapsto x^{p^{n-1}}" near end] \arrow[dr,swap,"\id"] && \GG_m^{(n-1)} \arrow[dr,"x\mapsto x^p"] \\
& \GG_m \arrow[rr,"x\mapsto x^{p^n}"] \arrow[uu,<-,crossing over] && \GG_m^{(n)}.
\end{tikzcd}
\end{equation}

Taking Cartier duals, we obtain a map of pushout squares:

\begin{equation}
\begin{tikzcd}[row sep=2.5em]
p^n \Z \arrow[rr] \arrow[dr,swap] \arrow[dd,swap] &&
\hat{W}_n \arrow[dd,swap] \arrow[dr] \\
& \Z \arrow[rr,crossing over] &&
\bD(W^\times[F^n]) \arrow[dd] \\
p^{n-1} \Z \arrow[rr] \arrow[dr,swap] && \hat{W}_{n-1} \arrow[dr] \\
& \Z \arrow[rr] \arrow[uu,<-,crossing over] && \bD(W^\times[F^{n-1}]).
\end{tikzcd}
\end{equation}

Again, <ref> follows by induction on $n$, using <cit.> for the base case.

Write the underlying scheme of $W_n$ as $\prod_{i=0}^{n-1} \GG_a$ (where the $i$th copy of $\GG_a$ has coordinate $\Psi_i$).
There is a fully faithful functor $\QCoh(BW^\times[F^n]) \hookrightarrow \QCoh(W_n)$ whose essential image consists of those $p$-complete $M\in \QCoh(W_n)$ such that $\Psi_i^p-\Psi_i$ acts locally nilpotently on $\H^\ast(M/p)$ for each $0\leq i \leq n-1$. Furthermore, this functor is symmetric monoidal for the convolution tensor product on $\QCoh(W_n)$.

If $\cf\in \QCoh(BW^\times[F^n])$ is sent to $M\in \QCoh(W_n)$ under this functor, one obtains a cube $\Psi_\bull: 2^{[n-1]} \to \Mod_{\Z_p}$ whose vertices are all $M$ and such that the edge from the subset $\{i_1, \cdots, i_{j-1}\}$ to $\{i_1, \cdots, i_j\}$ is given by the operator $\Psi_j: M \to M$. Then, the global sections $\Gamma(BW^\times[F^n]; \cf)$ can be identified with the total fiber of the cube $\Psi_\bull$.

If $\cf, \cg\in \QCoh(BW^\times[F^2])$ correspond to tuples $(M, \Psi_0^M, \Psi_1^M)$ and $(M', \Psi_0^{M'}, \Psi_1^{M'})$, then the global sections $\Gamma(BW^\times[F^2]; \cf)$ can be identified with the total fiber of the square

M \ar[r]^-{\Psi_0^M} \ar[d]_-{\Psi_1^M} & M \ar[d]^-{\Psi_1^M} \\
M \ar[r]_-{\Psi_0^M} & M.

Moreover, $\cf\otimes \cg$ corresponds to the module $M \otimes M'$, where

\begin{align*}
\Psi_0^{M \otimes M'} & = \Psi_0^{M}\otimes 1 + 1 \otimes \Psi_0^{M'},\\
\Psi_1^{M \otimes M'} & = \Psi_1^{M'} \otimes 1 + 1\otimes \Psi_1^{M'} - \frac{1}{p}\sum_{i=1}^{p-1} \binom{p}{i} (\Psi_0^M)^i \otimes (\Psi_0^{M'})^{p-i}.
\end{align*}

More generally, if $\cf, \cg\in \QCoh(BW^\times[F^n])$ correspond to tuples $(M, \Psi_0^M, \cdots, \Psi_{n-1}^M)$ and $(M', \Psi_0^{M'}, \cdots, \Psi_{n-1}^{M'})$, let us write $\Psi := (\Psi_0, \Psi_0, \cdots)$. Let $w_j(\Psi) = \sum_{i=0}^j p^i \Psi_{i}^{p^{j-i}}$ denote the corresponding Witt polynomial; then

\begin{equation}\label{wj-tensor}
w_j(\Psi^{M \otimes M'}) = w_j(\Psi^M) \otimes 1 + 1\otimes w_j(\Psi^{M'}).
\end{equation}

Define a homomorphism $W^\times[F^n] \to \GG_m$ via the composite

$$W^\times[F^n] \to W^\times \to (W/V)^\times \cong \GG_m.$$

Let $\co\{1\}$ denote the line bundle over $BW^\times[F^n]$ determined by the resulting map $BW^\times[F^n] \to B\GG_m$. Under the functor of <ref>, the total space of the line bundle $\co\{1\}$ corresponds to the $p$-completion of $\Z_p[x^{\pm 1}]$ with the action of $\Psi_{j}$ determined by the following requirement on Witt polynomials:

\begin{equation}\label{witt-poly-psi}
w(\Psi) = (w_0(\Psi), w_1(\Psi), w_2(\Psi), \cdots) = (x\partial_x, x\partial_x, x\partial_x, \cdots).
\end{equation}

The map $BW^\times[F^n] \to B\GG_m$ determines the stack $\GG_m/W^\times[F^n]$ over $BW^\times[F^n]$, so that the corresponding object in $\QCoh(W_n)$ under the functor of <ref> has underlying module given by $\co_{\GG_m} = \Z[x^{\pm 1}]$.
It is not too hard to show from the definition of the map $BW^\times[F^n] \to B\GG_m$ that under the functor of <ref>, the line bundle $\co\{1\}$ over $BW^\times[F^n]$ corresponds to the $p$-complete module $\Z_p$ (with generator $x$) where $\Psi_0$ acts on $x$ by $1$, and $\Psi_j$ acts on $x$ by zero for $j\geq 1$. The action of $w_j(\Psi)$ on $\co\{m\} = \Z_p \cdot x^m$ then follows from <ref>.

For instance, it follows from <ref> that

\begin{align*}
\Psi_0 & = x\partial_x, \\
\Psi_1 & = \frac{x\partial_x}{p}\left(1 - (x\partial_x)^{p-1}\right),\\
\Psi_2
%& = \frac{1}{p^2}\left(x\partial_x - (x\partial_x)^{p^2} - p \left(\frac{x\partial_x - (x\partial_x)^p}{p}\right)^p\right)
%& = \frac{1}{p^2}\left(x\partial_x - (x\partial_x)^{p^2} - \frac{1}{p^{p-1}} \sum_{j=0}^p (-1)^{j} \binom{p}{j} (x\partial_x)^{pj + p-j} \right)
& = \frac{x\partial_x}{p^2}\left(1 - (x\partial_x)^{p^2-1} - \frac{1}{p^{p-1}} \sum_{j=0}^p (-1)^{j} \binom{p}{j} (x\partial_x)^{(p-1)(j+1)} \right).
\end{align*}

Using <ref>, a similar calculation can be used to describe the $\GG_a$-bundle $\GG_a/W[F^n]$ over $BW[F^n]$; and, in particular, the $\infty$-category $\QCoh((\GG_a/W[F^n])/\GG_m)$. Let us summarize this calculation as follows. Recall that $\GG_a/W[F] \cong \GG_a/\GG_a^\sharp$ is isomorphic to $\GG_a^\dR$, so that if $\cA_1 := \Z_p\{x,\partial_x\}/([\partial_x,x] = 1)$ is the Weyl algebra, then $\QCoh(\GG_a/W[F]) \simeq \LMod_{\cA_1}^{\partial_x\text{-nilp}}$. Similarly, if we write $\frac{1}{m!} \partial_x^m = \partial_x^{[m]}$ (so that $\partial_x^{[m]}(x^k) = \binom{k}{m} x^{k-m}$), define $\cA_1^{[n]}$ via

$$\cA_1^{[n]} = \Z_p\left\{x, \partial_x, \cdots, \partial_x^{[p^{n-1}]} \right\}/([\partial_x^{[p^j]}, x] = \partial_x^{[p^j-1]}).$$

Note that $\partial_x^{[p^j-1]}$ is a $p$-adic unit multiple of $\prod_{k=0}^{j-1} (\partial_x^{[p^k]})^{p-1}$. Then, the action of $W[F^n]$ on $\GG_a$ implies that there is an equivalence

$$\QCoh(\GG_a/W[F^n]) \simeq \LMod_{\cA_1^{[n]}}^{\partial_x^{[p^j]}\text{-}\mathrm{nilp}};$$

this can be extended to an equivalence between $\QCoh((\GG_a/W[F^n])/\GG_m)$ and the $\infty$-category of graded $\cA_1^{[n]}$-modules such that $\partial_x^{[p^j]}$ acts nilpotently for $0\leq j \leq n-1$, where $x\in \cA_1^{[n]}$ has weight $1$ and $\partial_x^{[p^j]}\in \cA_1^{[n]}$ has weight $-p^j$. Algebras of divided power differential operators such as $\cA_1^{[n]}$ were initially studied by Berthelot in [14].

Note that $\GG_a/W[F^n]$ is not a ring stack. Indeed, the map $W[F^n] \to \GG_a$ is not a quasi-ideal: the $W$-module structure on $W[F^n]$ does not factor through $W \twoheadrightarrow W_1 = \GG_a$ (if $F^n(x) = 0$, then $xV(y) = V(F(x)y)$ need not vanish). However, it does factor through $W \twoheadrightarrow W_n$ (if $F^n(x) = 0$, then $xV^n(y) = V^n(F^n(x)y) = 0$); indeed, $W_n/W[F^n] \cong W/p^n$ admits the structure of a ring stack.

The proof of <ref> showed that there is an isomorphism

$$W^\times[F^n] \cong W[F^n] \times \mu_{p^n}$$

over $\FF_p$. Let $\fr{X}$ be a smooth $p$-adic formal scheme over $\Z_p$, and let $X = \fr{X} \otimes_{\Z_p} \FF_p$. Suppose that the $\GG_m^\sharp$-action on $\diffr_{\fr{X}} \otimes_{\Z_p} \FF_p = F_{X,\ast} \Omega^\bull_{X/\FF_p}$ refines to a $W^\times[F^n]$-action. (For instance, let $(\diffr_{\fr{X}})_0$ denote the weight $0$ piece of the $\Z/p$-grading on $\diffr_{\fr{X}}$ inherited from the $\GG_m^\sharp$-action. The datum of a refinement to a $W^\times[F^2]$-action leads to an operator on $(\diffr_{\fr{X}})_0$ which acts on $\gr^{pi}_\conj \diffr_{\fr{X}}$ by multiplication by $-i$.)
In this case, the $\Z/p$-grading on $F_{X,\ast} \Omega^\bull_{X/\FF_p}$ from <cit.> would refine to a $\Z/p^n$-grading; this would imply a refinement of the Deligne-Illusie theorem [40], stating that $\tau_{\geq -p^n+1} F_{X,\ast} \Omega^\bull_{X/\FF_p}$ would be decomposable.

The work of [65] suggests
that the base-change along $\BP{n-1}_\ast \to \FF_p$ (even along $\BP{n-1}_\ast \to \Z_p$) of the stack constructed from the associated graded of the motivic filtration [53] on $\THH(\BP{n-1})^{t\Cp}$ (resp. $\THH(\BP{n-1})$) is isomorphic to the stack $(\GG_m/W^\times[F^n])/\GG_m \cong BW^\times[F^n]$ (resp. $(\GG_a/W[F^n])/\GG_m \cong (F_\ast W/pF^{n-1})/\GG_m$). We are currently investigating this and its consequences with Jeremy Hahn and Arpon Raksit. In particular, this suggests that if a $\Z_p$-scheme “lifts to $\BP{n-1}$”, the $\GG_m^\sharp$-action on $\diffr_{\fr{X}}$ refines to a $W^\times[F^n]$-action. From this perspective, the operators $\Psi_j$ from above are closely related to the topological Sen operators $\Theta_j$ from the body of this article: roughly, $\Theta_j$ can be understood as $w_{j-1}(\Psi)$.

Given <ref>, one is therefore naturally led to the following question: if $X$ is a smooth and proper $\FF_p$-scheme which “lifts to $\BP{n-1}$” and $\dim(X)<p^n$, does the Hodge-de Rham spectral sequence for $X$ degenerate at the $E_1$-page? This question need not make sense, since $\BP{n-1}$ is generally not an $\Eoo$-ring [62, 101].
However, since $\BP{n-1}$ admits the structure of an $\E{3}$-ring, one can nevertheless ask whether such a degeneration statement holds noncommutatively if $\QCoh(X)$ admits a lift to a left $\BP{n-1}$-linear $\infty$-category.
This line of thinking was motivation for the following result (see [35]): if $\QCoh(X)$ lifts to a left $\BP{n-1}$-linear $\infty$-category, and $\dim(X)<p^n$, then the Tate spectral sequence for $\HP(X/\FF_p)$ degenerates at the $E_2$-page.

[1]
G. Angelini-Knoll, D. Culver, and E. Honing.
Topological Hochschild homology of truncated Brown-Peterson spectra
<https://arxiv.org/abs/2106.06785>, 2021.

[2]
V. Angeltveit, M. Hill, and T. Lawson.
Topological Hochschild homology of $\ell$ and $ko$.
Amer. J. Math., 132(2):297–330, 2010.

[3]
V. Angeltveit and J. Lind.
Uniqueness of $BP \langle n\rangle$.
J. Homotopy Relat. Struct., 12(1):17–30, 2017.

[4]
J. Anschütz and A.-C. Le Bras.
The $p$-completed cyclotomic trace in degree 2.
Ann. K-Theory, 5(3):539–580, 2020.

[5]
S. Amelotte.
A homotopy decomposition of the fibre of the squaring map on
Homology Homotopy Appl., 20(1):141–154, 2018.

[6]
B. Antieau, A. Mathew, and T. Nikolaus.
On the Blumberg–Mandell Künneth theorem for $\mathrm{TP}$.
Selecta Mathematica, 24(5):4555–4576, 2018.

[7]
M. Ando, J. Morava, and H. Sadofsky.
Completions of $\Z/(p)$-Tate cohomology of periodic spectra.
Geom. Topol., 2:145–174, 1998.

[8]
F. Adams and S. Priddy.
Uniqueness of $B{\rm SO}$.
Math. Proc. Cambridge Philos. Soc., 80(3):475–509, 1976.

[9]
G. Angelini-Knoll and J. D. Quigley.
The Segal Conjecture for topological Hochschild homology of the
Ravenel spectra.
<https://arxiv.org/abs/1705.03343>, 2017.

[10]
G. Angelini-Knoll and J. D. Quigley.
Chromatic Complexity of the Algebraic K-theory of $y(n)$.
<https://arxiv.org/abs/1908.09164>, 2019.

[11]
V. Angeltveit and J. Rognes.
Hopf algebra structure on topological Hochschild homology.
Algebr. Geom. Topol., 5:1223–1290, 2005.

[12]
A. Blumberg, R. Cohen, and C. Schlichtkrull.
Topological Hochschild homology of Thom spectra and the free
loop space.
Geom. Topol., 14(2):1165–1242, 2010.

[13]
J. Beardsley.
Relative Thom spectra via operadic Kan extensions.
Algebr. Geom. Topol., 17(2):1151–1162, 2017.

[14]
P. Berthelot.
$\mathcal{D}$-modules arithmétiques. I. Opérateurs
différentiels de niveau fini.
Ann. Sci. École Norm. Sup. (4), 29(2):185–272, 1996.

[15]
B. Bhatt.
$p$-adic derived de Rham cohomology.
<https://arxiv.org/abs/1204.6560>, 2012.

[16]
B. Bhatt.
Prismatic F-gauges.
Lecture notes available at
<https://www.math.ias.edu/ bhatt/teaching/mat549f22/lectures.pdf>, 2022.

[17]
J. Beardsley and T. Lawson.
Skeleta and categories of algebras.
<https://arxiv.org/abs/2110.09595>, 2021.

[18]
B. Bhatt and J. Lurie.
Absolute prismatic cohomology.
<https://arxiv.org/abs/2201.06120>, 2022.

[19]
B. Bhatt and J. Lurie.
The prismatization of $p$-adic formal schemes.
<https://arxiv.org/abs/2201.06124>, 2022.

[20]
M. Bokstedt and I. Madsen.
Topological cyclic homology of the integers.
Number 226, pages 7–8, 57–143. 1994.
$K$-theory (Strasbourg, 1992).

[21]
B. Bhatt and A. Mathew.
Syntomic complexes and $p$-adic étale Tate twists.
<https://arxiv.org/abs/2202.04818>, 2022.

[22]
R. Bruner, P. May, J. McClure, and M. Steinberger.
$H_\infty $ ring spectra and their applications, volume 1176
of Lecture Notes in Mathematics.
Springer-Verlag, Berlin, 1986.

[23]
B. Bhatt, M. Morrow, and P. Scholze.
Integral $p$-adic Hodge theory.
Publ. Math. Inst. Hautes Études Sci., 128:219–397, 2018.

[24]
B. Bhatt, M. Morrow, and P. Scholze.
Topological Hochschild homology and integral $p$-adic Hodge
Publ. Math. Inst. Hautes Études Sci., 129:199–310, 2019.

[25]
P. Berthelot and A. Ogus.
Notes on crystalline cohomology.
Princeton University Press, Princeton, N.J.; University of Tokyo
Press, Tokyo, 1978.

[26]
R. Bruner and J. Rognes.
Differentials in the homological homotopy fixed point spectral
Algebr. Geom. Topol., 5:653–690, 2005.

[27]
B. Bhatt and P. Scholze.
Prisms and Prismatic Cohomology.
<https://arxiv.org/abs/1905.08229>, 2019.

[28]
M. Bökstedt.
The topological Hochschild homology of $\Z$ and of $\Z/p\Z$.
Unpublished preprint from Bielefeld; available online, 1985.

[29]
M. Crabb and S. Mitchell.
The loops on ${\U}(n)/{\O}(n)$ and ${\U}(2n)/{\Sp}(n)$.
Math. Proc. Cambridge Philos. Soc., 104(1):95–103, 1988.

[30]
F. Cohen, J. Moore, and J. Neisendorfer.
The double suspension and exponents of the homotopy groups of
Ann. of Math. (2), 110(3):549–565, 1979.

[31]
F. Cohen, J. Moore, and J. Neisendorfer.
Torsion in homotopy groups.
Ann. of Math. (2), 109(1):121–168, 1979.

[32]
S. Devalapurkar.
The Ando-Hopkins-Rezk orientation is surjective.
<https://arxiv.org/abs/1911.10534>, 2019.

[33]
S. Devalapurkar.
Hodge theory for elliptic curves and the Hopf element $\nu$.
Bull. Lond. Math. Soc.; available at
<https://doi.org/10.1112/blms.12759>, 2022.

[34]
S. Devalapurkar.
Higher chromatic Thom spectra via unstable homotopy theory.
<https://arxiv.org/abs/2004.08951>; to appear in Algebr.
Geom. Topol., 2023.

[35]
S. Devalapurkar.
Lifting to truncated Brown-Peterson spectra and Hodge-de Rham
degeneration in characteristic $p>0$.
<https://arxiv.org/abs/2303.00739>, 2023.

[36]
C. Douglas, J. Francis, A. Henriques, and M. Hill.
Topological Modular Forms, volume 201 of Mathematical
Surveys and Monographs.
American Mathematical Society, 2014.

[37]
B. Dundas, T. Goodwillie, and R. McCarthy.
The local structure of algebraic K-theory, volume 18 of
Algebra and Applications.
Springer-Verlag London, Ltd., London, 2013.

[38]
S. Devalapurkar, J. Hahn, T. Lawson, A. Senger, and D. Wilson.
Examples of disk algebras.
<https://arxiv.org/abs/2302.11702>, 2023.

[39]
E. Devinatz, M. Hopkins, and J. Smith.
Nilpotence and stable homotopy theory. I.
Ann. of Math. (2), 128(2):207–241, 1988.

[40]
P. Deligne and L. Illusie.
Relèvements modulo $p^2$ et décomposition du complexe de de
Invent. Math., 89(2):247–270, 1987.

[41]
D. Davis, D. Johnson, J. Klippenstein, M. Mahowald, and S. Wegmann.
The spectrum $(P\wedge\mathrm{BP}\langle 2\rangle)_{-\infty}$.
Trans. Amer. Math. Soc., 296(1):95–110, 1986.

[42]
D. Davis and M. Mahowald.
$v_{1}$- and $v_{2}$-periodicity in stable homotopy theory.
Amer. J. Math., 103(4):615–659, 1981.

[43]
S. Devalapurkar and M. Misterka.
Generalized $n$-series and de Rham complexes.
Forthcoming, 2023.

[44]
E. Dotto, K. Moi, I. Patchkoria, and S. Reeh.
Real topological Hochschild homology.
<https://arxiv.org/abs/1711.10226>, 2017.

[45]
S. Devalapurkar and A. Raksit.
Forthcoming, 2023.

[46]
V. Drinfeld.
A 1-dimensional formal group over the prismatization of $\spf
\Z_p$.
<http://arxiv.org/abs/2107.11466>, 2021.

[47]
V. Drinfeld.
<https://arxiv.org/abs/2005.04746>, 2022.

[48]
B. Gheorghe, D. Isaksen, A. Krause, and N. Ricka.
$\mathbb{C}$-motivic modular forms.
<https://arxiv.org/abs/1810.11050>, 2018.

[49]
J. Hahn, A. Horev, I. Klang, D. Wilson, and F. Zou.
Equivariant nonabelian Poincaré duality and equivariant
factorization homology of Thom spectra.
<https://arxiv.org/abs/2006.13348>, 2020.

[50]
M. Hopkins and M. Mahowald.
The structure of 24 dimensional manifolds having normal bundles which
lift to $B{\rm O}[8]$.
In Recent progress in homotopy theory (Baltimore, MD,
2000), volume 293 of Contemp. Math., pages 89–110. Amer. Math. Soc.,
Providence, RI, 2002.

[51]
M. Hopkins.
Stable decompositions of certain loop spaces.
ProQuest LLC, Ann Arbor, MI, 1984.
Thesis (Ph.D.)–Northwestern University.

[52]
R. Horiuchi.
The non-nil-invariance of periodic topological cyclic homology.
Homology Homotopy Appl., 22(2):173–179, 2020.

[53]
J. Hahn, A. Raksit, and D. Wilson.
A motivic filtration on the topological cyclic homology of
commutative ring spectra.
<https://arxiv.org/abs/2206.11208>, 2022.

[54]
J. Hahn and D. Wilson.
Redshift and multiplication for truncated Brown-Peterson spectra.
<https://arxiv.org/abs/2012.00864>, 2020.

[55]
L. Illusie.
New perspectives on de Rham cohomology, after Bhatt-Lurie, Drinfeld,
et al.
<https://www.imo.universite-paris-saclay.fr/ illusie/DI-slides-new.pdf>,

[56]
D. Kaledin.
Non-commutative Hodge-to-de Rham degeneration via the method of
Pure Appl. Math. Q., 4(3, Special Issue: In honor of Fedor
Bogomolov. Part 2):785–875, 2008.

[57]
D. Kaledin.
Spectral sequences for cyclic homology.
In Algebra, geometry, and physics in the 21st century, volume
324 of Progr. Math., pages 99–129. Birkhäuser/Springer, Cham,

[58]
N. Kitchloo.
$\H\Z/p^k$ as a Thom spectrum and topological Hochschild homology.
Proc. Amer. Math. Soc., 148:3647–3651, 2020.

[59]
S. Kelly and M. Morrow.
$K$-theory of valuation rings.
Compos. Math., 157(6):1121–1142, 2021.

[60]
S. Kobayashi and K. Nomizu.
Foundations of differential geometry. Vol. I.
Wiley Classics Library. John Wiley & Sons, Inc., New York, 1996.
Reprint of the 1963 original, A Wiley-Interscience Publication.

[61]
A. Krause and T. Nikolaus.
Bökstedt periodicity and quotients of DVRs.
<https://arxiv.org/abs/1907.03477>, 2019.

[62]
T. Lawson.
Secondary power operations and the Brown-Peterson spectrum at
the prime 2.
Ann. of Math. (2), 188(2):513–576, 2018.

[63]
T. Lawson.
$E_n$-ring spectra and Dyer–Lashof operations.
<http://www-users.math.umn.edu/ tlawson/papers/dyerlashof.pdf>,

[64]
T. Lawson.
Unwinding the relative Tate diagonal.
J. Topol., 14(2):674–699, 2021.

[65]
D. Lee.
Integral topological Hochschild homology of connective complex
<https://arxiv.org/abs/2206.02411>, 2022.

[66]
A. Lindenstrauss and I. Madsen.
Topological Hochschild homology of number rings.
Trans. Amer. Math. Soc., 352(5):2179–2204, 2000.

[67]
M. Land, A. Mathew, L. Meier, and G. Tamme.
Purity in chromatically localized algebraic k-theory.
<https://arxiv.org/abs/2001.10425>, 2020.

[68]
T. Lawson and N. Naumann.
Strictly commutative realizations of diagrams over the Steenrod
algebra and topological modular forms at the prime 2.
Int. Math. Res. Not. IMRN, 10:2773–2813, 2014.

[69]
J. Lurie.
Rotation invariance in algebraic K-theory.
<https://www.math.ias.edu/ lurie/papers/Waldhaus.pdf>, 2015.

[70]
J. Lurie.
Elliptic Cohomology II: Orientations.
<http://www.math.harvard.edu/ lurie/papers/Elliptic-II.pdf>,

[71]
B. Loo and A. Verjovsky.
On quotients of Hopf fibrations.
Rend. Istit. Mat. Univ. Trieste, 26(1-2):103–108 (1995),

[72]
R. Liu and G. Wang.
Topological Cyclic Homology of Local Fields.
<https://arxiv.org/abs/2012.15014>, 2020.

[73]
M. Mahowald.
Ring spectra which are Thom complexes.
Duke Math. J., 46(3):549–559, 1979.

[74]
M. Mahowald.
$b{\rm o}$-resolutions.
Pacific J. Math., 92(2):365–383, 1981.

[75]
M. Mahowald.
The primary $v_{2}$-periodic family.
Math. Z., 177(3):381–393, 1981.

[76]
M. Mahowald.
The image of $J$ in the $EHP$ sequence.
Ann. of Math. (2), 116(1):65–112, 1982.

[77]
Z. Mao.
Perfectoid rings as Thom spectra.
<https://arxiv.org/abs/2003.08697>, 2020.

[78]
A. Mathew.
Kaledin's degeneration theorem and topological Hochschild homology.
Geom. Topol., 24(6):2675–2708, 2020.

[79]
A. Mathew.
On $K(1)$-local TR.
Compos. Math., 157(5):1079–1119, 2021.

[80]
A. Mathew.
Some recent advances in topological Hochschild homology.
Bull. Lond. Math. Soc., 54(1):1–44, 2022.

[81]
P. May.
A general algebraic approach to Steenrod operations.
In The Steenrod Algebra and its Applications (Proc.
Conf. to Celebrate N. E. Steenrod's Sixtieth Birthday,
Battelle Memorial Inst., Columbus, Ohio, 1970), Lecture Notes in
Mathematics, Vol. 168, pages 153–231. Springer, Berlin, 1970.

[82]
H. Miller.
Comodules, sheaves, and the exact functor theorem.
In Homotopy theory: tools and applications, volume 729 of
Contemp. Math., pages 205–220. Amer. Math. Soc., [Providence], RI,

[83]
M. Mahowald and C. Rezk.
Brown-Comenetz duality and the Adams spectral sequence.
Amer. J. Math., 121(6):1153–1177, 1999.

[84]
M. Mahowald, D. Ravenel, and P. Shick.
The triple loop space approach to the telescope conjecture.
In Homotopy methods in algebraic topology (Boulder, CO,
1999), volume 271 of Contemp. Math., pages 217–284. Amer. Math. Soc.,
Providence, RI, 2001.

[85]
T. Moulinos, M. Robalo, and B. Toën.
A Universal HKR Theorem.
<https://arxiv.org/abs/1906.00118v2>, 2019.

[86]
J. McClure and R. Staffeldt.
On the topological Hochschild homology of $b{\rm u}$. I.
Amer. J. Math., 115(1):1–45, 1993.

[87]
M. Mahowald and A. Unell.
Bott periodicity at the prime $2$ in the unstable homotopy of
<http://www.mit.edu/ sanathd/mahowald-unell-bott.pdf>, 1977.

[88]
J. Neisendorfer.
$3$-primary exponents.
Math. Proc. Cambridge Philos. Soc., 90(1):63–83, 1981.

[89]
T. Nikolaus and P. Scholze.
On topological cyclic homology.
Acta Math., 221(2):203–409, 2018.

[90]
E. Peterson.
Formal Geometry and Bordism Operations.
<https://github.com/ecpeterson/FormalGeomNotes>, October 2017.

[91]
A. Petrov.
Non-decomposability of the de Rham complex and non-semisimplicity of
the Sen operator.
<https://arxiv.org/abs/2302.11389>, 2023.

[92]
P. Pstragowski.
Synthetic spectra and the cellular motivic category.
<https://arxiv.org/abs/1803.01804>, 2018.

[93]
A. Petrov and V. Vologodsky.
On the periodic topological cyclic homology of DG categories in
characteristic $p$.
<https://arxiv.org/abs/1912.03246>, 2019.

[94]
J. Quigley and J. Shah.
On the equivalence of two theories of real cyclotomic spectra.
<https://arxiv.org/abs/2112.07462>, 2021.

[95]
A. Raksit.
$q$-de Rham cohomology and complex K-theory.
Talk at UChicago; poster at
<http://math.uchicago.edu/ algtop/posters/20201201.pdf>, 2020.

[96]
D. Ravenel.
Localization with respect to certain periodic homology theories.
Amer. J. Math., 106(2):351–414, 1984.

[97]
D. Ravenel.
Complex cobordism and stable homotopy groups of spheres.
Academic Press, 1986.

[98]
N. Riggenbach.
The $S^1$ Assembly Map on $K$-Theory and Topological Cyclic
ProQuest LLC, Ann Arbor, MI, 2021.
Thesis (Ph.D.) – Indiana University.

[99]
R. Schultz.
Compact fiberings of homogeneous spaces. I.
Compositio Math., 43(2):181–215, 1981.

[100]
P. Scholze.
Canonical $q$-deformations in arithmetic geometry.
Ann. Fac. Sci. Toulouse Math. (6), 26(5):1163–1192, 2017.

[101]
A. Senger.
The Brown-Peterson spectrum is not $\mathbb{E}_{2(p^2+2)}$ at odd
<https://arxiv.org/abs/1710.09822>, 2017.

[102]
H. Toda.
Composition methods in homotopy groups of spheres.
Annals of Mathematics Studies, No. 49. Princeton University Press,
Princeton, N.J., 1962.

[103]
S. Tsalidis.
On the topological cyclic homology of the integers.
Amer. J. Math., 119(1):103–125, 1997.

[104]
D. Yan.
On the Thom spectra over $\Omega({\SU}(n)/{\SO}(n))$ and
Mahowald's $X_k$ spectra.
Proc. Amer. Math. Soc., 116(2):567–573, 1992.