Proof. For (i), the fact that $E$ is nilinvariant follows from [Reference Land and TammeLT19, Theorem B]. For (ii), since $E$ is nilinvariant, we may assume $C$ is an $\mathbb {F}_p$-algebra, so that $E(C)$ is by the theory of noncommutative motives [Reference Blumberg, Gepner and TabuadaBGT13] a $K(\mathbb {F}_p; \mathbb {Z}_p) = H\mathbb {Z}_p$-module (the last identification by [Reference QuillenQui72]); since $E$ is $K(1)$-local we get $E(C) = 0$.

For (iii), the kernel of $A \to A'$ is annihilated by a power of $p$, so by (ii) (and excision) we can assume that $A \subset A'$. Let $n \gg 0$, so $p^n A' \subset A$. Then the diagram

is a Milnor square of rings. Applying the localizing invariant $E$ and using excision and (ii) again, we conclude (iii).

Theorem 2.3 ($h$-descent for truncating $K(1)$-local invariants)

Let $E$ be a $K(1)$-local localizing invariant on $B$-linear $\infty$-categories which is truncating. Then $E$ satisfies $h$-descent on qcqs $B$-schemes.

Proof. By the results of [Reference Land and TammeLT19, Appendix A], $E$ satisfies $cdh$-descent, and in particular satisfies excision for abstract blow-up squares. The results of [Reference Clausen, Mathew, Naumann and NoelCMNN20] imply that $E$ satisfies finite locally free descent. Since $E$ also satisfies Nisnevich descent (as does any localizing invariant [Reference Thomason and TrobaughTT90]), we obtain that $E$ satisfies fppf descent thanks to [Sta20, Tag 05WN]. By [Reference Bhatt and ScholzeBS17, Theorem 2.9], $h$-descent (for any sheaf) is implied by fppf descent and excision for abstract blow-up squares. Combining these facts, we conclude.

Proof. Replacing $S$ with $S \times R/p$, we may assume without loss of generality that $R \to S$ is an $h$-cover. Then, by Theorem 2.3, we have

(4)\begin{equation} E(R) \simeq \mathrm{Tot}( E(C(R \to S)^\bullet)) = \mathrm{Tot} \Big(E(S) \rightrightarrows E(S \otimes_R S ) \begin{smallmatrix} \to \\ \to \\\to \end{smallmatrix} \ldots \Big). \end{equation}

Since the group $G$ acts on $S$, we have a natural map of cosimplicial rings from the Čech nerve $C(R \to S)^\bullet$ to the standard resolution $S \rightrightarrows \prod _G S \begin{smallmatrix} \to \\ \to \\\to \end{smallmatrix}$ for $G$ acting on $S$ (which calculates $S^{hG}$). This map of cosimplicial rings is an isogeny in each degree; for example, in degree $1$, $S \otimes _R S \to \prod _G S$ is an isogeny because it is a map of finitely presented $R$-modules (see [Sta20, Tag 0564]) which induces an isomorphism after inverting $p$ thanks to the Galois hypothesis. Therefore, by Proposition 2.2, we find that the map induces an equivalence after applying $E$, and we find from (4) that

\[ E(R) \xrightarrow{\sim} \mathrm{Tot} \bigg( E(S) \rightrightarrows E \biggl( \prod_G S \biggr) \begin{smallmatrix} \to \\ \to \\\to \end{smallmatrix} \ldots \bigg) = E(S)^{hG}, \]

which is the desired claim. Finally, to see that the $G$-action on $E(S)$ is nilpotent, we use that $E(S)$ is a module $G$-equivariantly over $L_{K(1)} K(S) \xrightarrow {\sim } L_{K(1)} K(S[1/p])$ (via (2)), and the $G$-action on $L_{K(1)} K(S[1/p])$ is nilpotent by [Reference Clausen, Mathew, Naumann and NoelCMNN20, Theorem 5.6] (see also [Reference Clausen and MathewCM19, Lemma 4.20]).

Theorem 3.1 The construction $L_{K(1)} \mathrm {TR}(-)$ is truncating on connective $E_1$-$B$-algebras. More generally, for any set $Q$,Footnote ² the construction $L_{K(1)} (\prod _Q \mathrm {TR}(-))$ is truncating on connective $E_1$-$B$-algebras.

Proof. Compare [Reference Mathew, Naumann and NoelMNN17, § 7.4]. In fact, via the projection formula, $M^{hC_{p^n}} \simeq (M \otimes _{ku} ku^{S^1/C_{p^n}})^{hS^1}$. Here $ku^{S^1/C_{p^n}}$ denotes the $ku$-valued function spectrum of $S^1/C_{p^n}$ with the corresponding $S^1$-action, and the tensor product is taken in $\mathrm {Fun}(BS^1, \mod (ku))$. Let $V_n$ denote the one-dimensional complex representation of $S^1$ where $z \in S^1$ acts by multiplication by $z^{p^n}$, and let $S(V_n)$ denote the unit circle in $V_n$ as an $S^1$-space, so $S(V_n) \simeq S^1/C_{p^n}$. The Spanier–Whitehead dual of the Euler sequence in $\mathrm {Fun}(BS^1, {\rm Sp} )$, $S(V_n)_+ \to S^0 \to S^{V_n}$ and the complex orientability of $ku$ together show that $ku^{S^1/C_{p^n}} \in \mathrm {Fun}(BS^1, \mod (ku))$ belongs to the thick subcategory generated by the unit. Therefore, applying the right adjoint $(-)^{hS^1} \colon \mathrm {Fun}(BS^1, \mod (ku)) \to \mod (ku^{BS^1})$, we find that the natural map

\[ M^{hS^1} \otimes_{ku^{BS^1}} (ku^{S^1/C_{p^n}})^{hS^1} \to (M \otimes_{ku} ku^{S^1/C_{p^n}})^{hS^1} = M^{hC_{p^n}} \]

is an equivalence, whence the result.

Proof. Let $R = A \otimes ku/p$, so $R$ is an associative $ku$-algebra spectrum with $S^1$-action. Consider the $S^1$-homotopy fixed points $R^{hS^1}$, which is a $ku^{BS^1}$-algebra. Since $p = 0$ in $\pi _0 R$, we have by Proposition 3.2 and (6),

\[ R^{hC_{p^n}} = R^{hS^1} \otimes_{ku^{BS^1}} ku^{BC_{p^n}} = R^{hS^1}/ [p^n](x) = R^{hS^1}/ \big(\beta^{p^n-1}x^{p^n}\big). \]

Our assumption is that $\beta ^r =0$ in $\pi _*(R)$, which means that we can write $\beta ^r = x v \in \pi _*(R^{hS^1})$ for some $v$ since $R = R^{hS^1}/x$. It follows that in $\pi _*(R^{hC_{p^n}})$, we have $\beta ^{p^n -1 + rp^n} = \beta ^{p^n-1} (xv)^{p^n} = 0$.

Proof. To prove (i), using the transfer and restricting to a $p$-Sylow subgroup, we may reduce to the case where $i$ is a power of $p$, say $i = p^n$. Then the claim follows from Proposition 3.3, since $\beta$ is nilpotent in $ku/p \otimes \mathrm {THH}(B)$ since $B$ is $K(1)$-acyclic. It follows that we can find a $\kappa _1$ such that $\beta ^{\kappa _1 i} = 0$ in $(ku/p \otimes \mathrm {THH}(B))^{hC_i}$ for all $i > 0$.

For (ii), consider a $\mathrm {THH}(B) \otimes ku$-module $N$ in $\mathrm {Fun}(BS^1, {\rm Sp} )$ which is the induction of a $\mathrm {THH}(B) \otimes ku$-module $N'$ in $\mathrm {Fun}(BC_{i}, {\rm Sp} )$. It follows that $N_{hS^1} /p = (N' /p)_{hC_i}$ is a module over $(ku/p \otimes \mathrm {THH}(B))^{hC_i}$, writing homotopy orbits as a module over homotopy fixed points; this is therefore annihilated by $\beta ^{\kappa _1 i}$ by the previous part of the result. For any $t$, let $W_t$ be the one-dimensional complex representation of $S^1$ where $z$ acts by multiplication by $z^t$, so the unit circle $S(W_t)_+ = (S^1/C_t)_+$ as $S^1$-spaces. We have $N_{hC_t} = (N \otimes S(W_t)_+)_{hS^1}$ by the projection formula for induction and restriction along $C_t \subset S^1$. Then the Euler sequence $S(W_t)_+ \to S^0 \to S^{W_t}$ (as in the proof of Proposition 3.2) and the complex orientability of $ku$ yield a fiber sequence $N_{h C_t} \to N_{hS^1} \to \Sigma ^2 N_{hS^1}$. Therefore, $N_{hC_t} /p$ is annihilated by $\beta ^{2 \kappa _1 i}$; taking $\kappa = 2\kappa _1$ we conclude.

Proof. For simplicity of notation, we write $\mathrm {TR}_Q(-) = \prod _Q \mathrm {TR}(-)$. The construction $\mathrm {TR}_Q(-)$ is exact and commutes with geometric realizations on $\mathrm {CycSp}_{\geq 0}$; therefore, it commutes with tensoring with $ku$. Without loss of generality, we can therefore assume that $X$ is a $\mathrm {THH}(B) \otimes ku$-module in graded cyclotomic spectra. Here we regard $ku$ as a trivial cyclotomic spectrum, that is, via the image of the unique symmetric monoidal functor ${\rm Sp} \to \mathrm {CycSp}$.

We consider the descending filtration $\{ \mathrm {Fil}^{\geq n } X = X_{\geq p^{n}}\}_{n \geq 0}$ on the cyclotomic spectrum $X$; note that the associated graded terms have trivialized Frobenii for degree reasons. This yields a filtration on the spectrum $\mathrm {TR}_Q(X)$, with $\mathrm {Fil}^{\geq n} \mathrm {TR}_Q(X) = \mathrm {TR}_Q(X_{\geq p^n})$. Since $\mathrm {TR}(-)$ preserves connectivity, our assumptions imply that $\mathrm {Fil}^{\geq n} \mathrm {TR}_Q(X)$ is $(2p + 1) \kappa p^n$-connective.

We have by (7) that $\mathrm {gr}^n \mathrm {TR}_Q(X) = \bigoplus _{p^{n} \leq i < p^{n+1}} \prod _Q\prod _{t \geq 0} (X_i)_{hC_{p^t}}$, since the cyclotomic Frobenius is trivial on $\mathrm {gr}^n X$ for grading reasons. Since $X_i$ is induced from $C_i \subset S^1$, it follows from Proposition 3.4 that the $ku$-module $\mathrm {gr}^n \mathrm {TR}_Q(X)/p$ is annihilated by $\beta ^{\kappa p^{n+1}}$.

Now $|\beta | = 2$, and $2\sum _{i = 0}^{n-1} \kappa p^{i+1} < (2p+1)\kappa p^{n}$ with the difference tending to $\infty$ as $n \to \infty$. Applying Lemma 3.7 below, we conclude that inverting $\beta$ annihilates $\mathrm {TR}_Q(X) /p$, whence $\mathrm {TR}_Q(X)$ is $K(1)$-acyclic as desired.

Proof. Let $y \in \pi _s(Y)$. For each $n > 0$, the class $x^{f(0) + f(1) + \cdots + f(n-1)} y \in \pi _*(Y)$ naturally lifts to $\pi _{s + t(f(0) + \cdots + f(n-1))}( \mathrm {Fil}^{\geq n} Y)$. But for $n \gg 0$, the connectivity of $\mathrm {Fil}^{\geq n} Y$ forces this last group to vanish. Therefore, the image of $y$ in $\pi _s(Y[1/x])$ vanishes as desired.

Theorem 3.8 ($\mathrm {THH}$ of a free associative algebra)

Let $M$ be a spectrum, and let $T(M) = \bigoplus _{n \geq 0}M^{\otimes n}$ be the free $E_1$-algebra spectrum generated by $M$. Then there is a natural equivalence in $\mathrm {Fun}(BS^1, {\rm Sp} )$,

(8)\begin{equation} \mathrm{THH}( T(M)) \simeq \bigoplus_{n \geq 0} \mathrm{Ind}_{C_n}^{S^1} (M^{{\otimes} n}) , \end{equation}

where we use the natural $C_n$-action on $M^{\otimes n}$ by permuting the factors.

Proof. The results of [Reference Ayala, Francis and TanakaAFT17, Reference Ayala and FrancisAF15] (applied to the framed manifold $S^1$) imply that $\mathrm {THH}( T(M)) \simeq \int _{S^1} T(M) = \bigoplus _{n \geq 0} (\mathrm {Conf}_n(S^1)_+ \otimes M^{\otimes n})_{h \Sigma _n}$, for $\mathrm {Conf}_n(S^1)$ the configuration space of $n$ ordered points on the circle. One checks now (see [Reference Crabb and JamesCJ98, Example II.14.4]) that $\mathrm {Conf}_n(S^1)$, as a space with $S^1 \times \Sigma _n$-action, is homotopy equivalent to $(S^1 \times \Sigma _n)/C_n$ (with $C_n$ embedded diagonally), whence the claim.

Proof. We can consider the tensor algebra $T(M \oplus N)$ as a graded $E_1$-ring spectrum where $M$ is placed in degree $0$ and $N$ is placed in degree $1$. In this case, if we collect the terms in (8), we find that the $i$th graded piece of $\mathrm {THH}( T(M \oplus N))$ is the component which is $i$-homogeneous. Explicitly, for any subset $I \subset \langle n\rangle = \{1, 2, \ldots , n\}$, we write $(M, N)^{\otimes (\langle n \rangle \setminus I, I)}$ for the ordered tensor product of $n$ factors, where the $j$th factor is $M$ if $j \notin I$ and $N$ if $j \in I$. Expanding (8) gives

(9)\begin{equation} \mathrm{THH}( T(M \oplus N))_i = \bigoplus_{n \geq 0} \mathrm{Ind}_{C_n}^{S^1} \bigg( \bigoplus_{I \subset \langle n \rangle, |I| = i} (M, N)^{{\otimes} (\langle n \rangle\setminus I, I)} \bigg). \end{equation}

Here the $C_n$-action on the parenthesized term in (9) permutes the various summands. Note in particular that the stabilizer of $I \subset \langle n\rangle$ in the $n$th summand is a subgroup of a cyclic group $C_i \subset C_n$ since $|I| = i$. In particular, it follows that the $i$th graded piece of $\mathrm {THH}( T(M \oplus N))$ is induced from $C_i \subset S^1$. Furthermore, since $N$ is at least $\kappa (2p+1)$-connective, it follows that the $i$th graded piece of $\mathrm {THH}( T( M \oplus N))$ is at least $\kappa (2p+1)$-connective. By Proposition 3.6, it follows that the positively graded part of $\mathrm {THH}(T(M \oplus N)) \otimes \mathrm {THH}(B)$ has vanishing $K(1)$-local $\prod _Q \mathrm {TR}(-)$ for any set $Q$, whence the result.

Proof. Simplicially resolving $N$ by free $(R, R)$-bimodules (since $\prod _Q \mathrm {TR}(-)$ commutes with geometric realizations), we may assume that $N$ is free on generators (possibly infinitely many) in degrees $\kappa (2p+1)$ or higher. Simplicially resolving $R$ by free $B$-algebras, we may assume that $R$ is free as well, on some classes in degrees at least $0$. In this case, the result follows from Proposition 3.9.

Proof. We show by descending inductionFootnote ³ that for any $i \geq 0$ and for any connective $B$-algebra $R$, the map $R \to \tau _{\leq i} R$ induces an equivalence on $E$; taking $i = 0$ gives the theorem. For $i \geq k$, we already know the claim by assumption. Suppose we know the claim for $i+1$; to prove the claim for $i$, we need to see that $\tau _{\leq i+1} R \to \tau _{\leq i} R$ induces an equivalence on $E$. Now $\tau _{\leq i+1} R \to \tau _{\leq i} R$ is a square-zero extension, that is, we have the following pullback diagram of $B$-algebras.

Since $E$ is a localizing invariant, the main result of Land and Tamme [Reference Land and TammeLT19] yields a $B$-algebra $\tilde {R}$ with underlying spectrum $H(\pi _0 R) \otimes _{\tau _{\leq i+1} R } \tau _{\leq i} R$ fitting into a commutative diagram of $B$-algebras

which is carried to a pullback by $E$. But the map $H(\pi _0 R) \to \tilde {R}$ is an equivalence in degrees up to $i+1$ and therefore induces an equivalence on $E$ by the inductive hypothesis. Therefore, $E( \tau _{\leq i+1} R) \to E( \tau _{\leq i} R)$ is an equivalence, whence the inductive step and the result.

Proof of Theorem 3.1 As before, we write $\mathrm {TR}_Q(-) = \prod _Q \mathrm {TR}(-)$ for a set $Q$. For any connective $B$-algebra $R$, we claim that $R \to \tau _{\leq \kappa (2p+1)} R$ induces an equivalence on $L_{K(1)} \mathrm {TR}_Q(-)$. Indeed, this follows from Proposition 3.10 because we can simplicially resolve $\tau _{\leq \kappa (2p+1)} R$ using free $R$-algebras over free $(R, R)$-bimodules on classes in degrees $\kappa (2p+1)$ or higher and since $\mathrm {TR}_Q(-)$ commutes with geometric realizations. The result now follows from Lemma 3.11.

Proof. The result for $L_{K(1)} \mathrm {TC}(-)$ follows from Theorem 3.1 by taking Frobenius fixed points. The result for $L_{K(1)} K(-)$ is a formal consequence since the fiber of the trace $K(-) \to \mathrm {TC}(-)$ is truncating by the Dundas–Goodwillie–McCarthy theorem [Reference Dundas, Goodwillie and McCarthyDGM13].

Proof. Take $B = H\mathbb {Z}$ in Theorem 3.1, so $L_{K(1)} \mathrm {TC}(-)$ is truncating and therefore nilinvariant on connective $H\mathbb {Z}$-algebras. Then the result follows because the Dundas–Goodwillie–McCarthy theorem and comparison with $\mathbb {F}_p$ yield $L_{K(1)} K(\mathbb {Z}/p^n) = L_{K(1)} \mathrm {TC}(\mathbb {Z}/p^n)$, but the above shows that this equals $L_{K(1)} \mathrm {TC}(\mathbb {F}_p) = 0$.

Theorem 4.1 (Hesselholt–Madsen [Reference Hesselholt and MadsenHM03, Reference Hesselholt and MadsenHM04])

Let $K$ be a complete, discretely valued field of characteristic $0$ with ring of integers $\mathcal {O}_K \subset K$ and perfect residue field $k$ of characteristic $p > 2$. Let $R$ be a smooth $\mathcal {O}_K$-algebra of relative dimension $d$. Then the map

\[ \mathrm{TR}(R; \mathbb{F}_p) \to L_{K(1)} \mathrm{TR}(R; \mathbb{F}_p) \]

is $d$-truncated.

Proof. Without loss of generality, we can assume that $\mu _p \subset K$, since otherwise $\mathrm {TR}(R; \mathbb {F}_p)$ is a retract of $\mathrm {TR}(R[\zeta _p]; \mathbb {F}_p)$ via the transfer. In this case, the result follows from [Reference Hesselholt and MadsenHM04, Theorem E]. Indeed, [Reference Hesselholt and MadsenHM04, Theorem E] gives a calculation of the cofiber $\mathrm {TR}(R|R_K; \mathbb {F}_p)$ of the transfer map $\mathrm {TR}(R \otimes _{\mathcal {O}_K} k; \mathbb {F}_p) \to \mathrm {TR}(R; \mathbb {F}_p)$. Since $\mathrm {TR}(R \otimes _{\mathcal {O}_K} k; \mathbb {F}_p)$ is $K(1)$-acyclic and $d$-truncated in view of the identification [Reference HesselholtHes96] with the de Rham–Witt complex of $R \otimes _{\mathcal {O}_K} k$, it suffices to verify the (stronger) claim that $\mathrm {TR}(R|R_K; \mathbb {F}_p) \to L_{K(1)}\mathrm {TR}(R|R_K; \mathbb {F}_p)$ is $(d-2)$-truncated. Equivalently (see Lemma 5.8 below), it suffices to show that the cofiber of the Bott element on $\mathrm {TR}(R|R_K; \mathbb {F}_p)$ is $(d+1)$-truncated. In fact, this follows from the calculation in [Reference HesselholtHes96], once we know that the absolute de Rham–Witt complex $W \Omega ^{\ast }_{(R, M_R)}$ is $p$-divisible in degrees $d+2$ or higher. This in turn follows from the case where $R = \mathcal {O}_K$ (see [Reference Hesselholt and MadsenHM03, Corollary 3.2.7]; the case of polynomial rings via the functor $P$ (see [Reference Hesselholt and MadsenHM04, Lemma 7.1.4] and its proof); and finally, étale base-change [Reference Hesselholt and MadsenHM04, Lemma 7.1.1].

Proof. The homotopy fiber of the map in question is, by Quillen's dévissage theorem, the mod $p$ $K$-theory of the abelian category of finitely generated $R/p$-modules, that is, the mod $p$ $G$-theory of $R/p$, $G(R/p; \mathbb {F}_p)$. So it suffices to show that $G(R/p; \mathbb {F}_p)$ is concentrated in degrees up to $d$. Using the filtration by codimension of support and dévissage (see [Reference QuillenQui73, Theorem 5.4]), we find that $G(R/p; \mathbb {F}_p)$ has a filtration whose associated graded terms are direct sums of the $K( \kappa (x); \mathbb {F}_p)$ for $x \in \mathrm {Spec}(R/p)$; it therefore suffices to show that these terms are $d$-truncated. But now the Geisser–Levine theorem [Reference Geisser and LevineGL00] implies that for any field $E$ of characteristic $p$, there is an embedding $K_*(E; \mathbb {F}_p) \hookrightarrow \Omega ^{\ast }_{E/\mathbb {F}_p}$. This implies that $K(\kappa (x); \mathbb {F}_p)$ is $\dim _{\kappa (x)} \Omega ^1_{\kappa (x)/\mathbb {F}_p}$-truncated. But $\dim _{\kappa (x)} \Omega ^1_{\kappa (x)/\mathbb {F}_p} = \log _p [\kappa (x): \kappa (x)^p] \leq d$ by the theory of $p$-bases [Reference MatsumuraMat86, Theorem 26.5]. Combining these facts, the result follows.

Proof sketch Using Nisnevich descent [Reference Thomason and TrobaughTT90], we reduce to the case where $R$ is henselian local, and even a field of characteristic not equal to $p$ by Gabber–Suslin rigidity [Reference GabberGab92]; note that we do not have to worry about the distinction between connective and nonconnective $K$-theory by regularity. Then the result follows from the Beilinson–Lichtenbaum conjecture (proved by Voevodsky and Rost; see [Reference Haesemeyer and WeibelHW19]) describing the associated graded terms of the motivic filtration on $K(R; \mathbb {F}_p)$ (see, for example, [Reference Clausen and MathewCM19, § 6.2] for an account).

Proof. By standard continuity arguments, it suffices to show that for any affine open $U = \mathrm {Spec}( R[1/f]) \subset \mathrm {Spec}(R[1/p])$ and any constructible $p$-torsion sheaf $\mathcal {F}$ on $U$, we have $H^n( U, \mathcal {F}) = 0$ for $n > d +1$. Denote by $j\colon U \hookrightarrow \mathrm {Spec}(R)$ the open inclusion, and denote by $i_0\colon \mathrm {Spec}(R/I) \hookrightarrow \mathrm {Spec}(R)$ the closed embedding. Using that $\mathrm {Spec}(R/I)$ has $p$-cohomological dimension at most $1$ and the affine analog of proper base change [Reference GabberGab94, Reference HuberHub93] applied to the henselian ideal $I \subset R$, we see that it suffices to show that $i_0^* R^n j_* \mathcal {F} = 0$ for $n > d$.

Working stalkwise on $R$ and using the compatibility of étale cohomology with filtered colimits, we can now reduce to the case where $R$ is an excellent, strictly henselian normal local domain with residue field $k$ (and $I \subset R$ is contained in the maximal ideal), since excellence and normality are preserved by strict henselization (see [Reference GrecoGre76] for the former). The statement then becomes that $H^n( U, \mathcal {F}) = 0$ for $n > d$, which follows from the Gabber–Orgogozo bound [Reference Gabber and OrgogozoGO08, Theorem 6.1] (noting that the $p$-dimension of the residue field $k$ is $\log _p [k:k^p]$ since $k$ is separably closed).

Proof. We have that $R_0[[x]]$ is generated by $p$ elements over the subring generated by its $p$th powers and by $R_0$. For every residue field $\kappa$ of $R_0$, it follows that $R_0[[x]] \otimes _{R_0} \kappa$ is generated by $p$ elements over the subring generated by $p$th powers together with $\kappa$. In particular, it is generated by $p^{d+1}$ elements as a module over its $p$th powers. It follows that the same holds for any residue field of $R_0[[x]] \otimes _{R_0} \kappa$ and, varying $\kappa$, we obtain the conclusion for every residue field of $R_0$.

Theorem 4.8 Let $R$ be an excellent, $p$-torsion-free regular noetherian ring. Suppose $R/p$ is finitely generated as a module over its subring of $p$th powers. Suppose, furthermore, that for all $\mathfrak {p} \in \mathrm {Spec}(R)$ containing $(p)$, we have $\dim R_{\mathfrak {p}} + \log _p [ \kappa (\mathfrak {p}):\kappa (\mathfrak {p})^p] \leq d$ for some $d \geq 0$. Then the map $\mathrm {TR}(R; \mathbb {F}_p) \to L_{K(1)} \mathrm {TR}(R; \mathbb {F}_p)$ is $(d-1)$-truncated.

Proof. Without loss of generality, we can assume that $R$ is $p$-henselian (since henselization preserves excellence; see [Reference GrecoGre76]), so $\dim R \leq d$. By Construction 4.2, it suffices to show that $C(R)/p \to L_{K(1)} C(R)/p$ is $(d-1)$-truncated. Now $C(R)$ is the desuspension of the fiber of the map $\varprojlim _n K( R[x]/x^n) \to K(R)$. With $p$-adic coefficients, the fact that $R/p$ is finitely generated as a module over its $p$th powers allows us to pass to the limit [Reference Clausen, Mathew and MorrowCMM21, Theorem F] and we obtain

\[ C(R; \mathbb{F}_p) = \Omega K(R[[x]], (x); \mathbb{F}_p). \]

Since $K(R)$ is a retract of $K(R[[x]])$, it suffices to show that the fiber of $K(R[[x]]; \mathbb {F}_p) \to L_{K(1)}K(R[[x]]; \mathbb {F}_p)$ is $d$-truncated. We will verify this by comparing both sides with the intermediate term $K( R[[x]][1/p]; \mathbb {F}_p)$.

First, for every characteristic $p$ residue field $\kappa$ of $R$, corresponding to a prime ideal $\mathfrak {p} \subset R$ containing $(p)$, we have $[\kappa : \kappa ^p] \leq p^{d-1}$ by our assumption, since $\dim R_{\mathfrak {p}} \geq 1$. Now $R[[x]]$ is also a $p$-torsion-free regular ring of dimension $\dim (R) + 1$. For every characteristic $p$ residue field $\kappa '$ of $R[[x]]$, we have $[\kappa ': \kappa '^p] \leq p^{d}$ by Lemma 4.7. By Proposition 4.3, it follows that $K(R[[x]]; \mathbb {F}_p) \to K(R[[x]][1/p]; \mathbb {F}_p)$ is $d$-truncated.

Second, we apply Proposition 4.5 to the ring $R[[x]]$ and the ideal $I = (p, x)$. The power series ring $R[[x]]$ remains excellent (and regular) since $R$ is excellent, thanks to [Reference Kurano and ShimomotoKS16]. For any prime ideal $\mathfrak {p} \in \mathrm {Spec}( R[[x]])$ containing $I$, we let $\mathfrak {p}_0 = \mathfrak {p} \cap R \subset R$, so that $R[[x]]_{\mathfrak {p}}/(x) = R_{\mathfrak {p}_0}$. Then $\dim ( R[[x]]_{\mathfrak {p}}) = \dim ( R_{\mathfrak {p}_0})+ 1$ and $\kappa ( \mathfrak {p}) = \kappa (\mathfrak {p}_0)$. Thus, Proposition 4.5 applies to $R[[x]]$ (with $d$ replaced by $d+1$) and we find that the characteristic $0$ residue fields of $R[[x]]$ have $p$-cohomological dimension at most $d + 2$. Therefore, by Proposition 4.4, $K( R[[x]][1/p]; \mathbb {F}_p) \to L_{K(1)} K( R[[x]][1/p])$ is $(d-1)$-truncated.

Combining the above, we find that the composite map $K(R[[x]]; \mathbb {F}_p) \to L_{K(1)} K(R[[x]]; \mathbb {F}_p) \simeq L_{K(1)} K( R[[x]][1/p]; \mathbb {F}_p)$ (the last identification by (2)) is $d$-truncated, whence the result.

Proof. Since $X$ is $K(1)$-acyclic, it follows that $V\colon \mathrm {TR}( X)_{hC_p} \to \mathrm {TR}(X)$ is $K(1)$-locally an equivalence. The $K(1)$-localization $L_{K(1)} \mathrm {TR}(X)$ acquires the structure of a topological Cartier module as well by $K(1)$-localizing $F, V$ and using the comparison map $L_{K(1)} ( \mathrm {TR}(X)^{hC_p}) \to (L_{K(1)} \mathrm {TR}(X))^{hC_p}$. The composite map $(L_{K(1)} \mathrm {TR}(X))_{hC_p} \to (L_{K(1)} \mathrm {TR}(X))^{hC_p}$ is an equivalence after $p$-completion since Tate constructions vanish in the $K(1)$-local category. Since we saw that $V$ is an equivalence on $L_{K(1)} \mathrm {TR}(X)$ after $p$-completion, it follows that the Frobenius on $L_{K(1)} \mathrm {TR}(X)$ induces an equivalence

\[ L_{K(1)} \mathrm{TR}( X) \xrightarrow{\sim} (L_{K(1)} \mathrm{TR}(X))^{hC_p}. \]

For a topological Cartier module $Y$, we consider the fiber of $F = F_Y\colon Y \to Y^{hC_p}$, which we denote $\mathrm {fib}(F)$. As we saw above, $\mathrm {fib}(F)$ is contractible for the topological Cartier module $L_{K(1)} \mathrm {TR}(X)$. Moreover, $\mathrm {fib}(F)$ is $d$-truncated for the topological Cartier module $\mathrm {fib}( \mathrm {TR}(X) \to L_{K(1)} \mathrm {TR}(X))$ because this topological Cartier module is itself $d$-truncated. In particular, we find that $\mathrm {fib}(F\colon \mathrm {TR}(X) \to \mathrm {TR}(X)^{hC_p})$ is $d$-truncated. Taking the cofiber on both the domain and codomain of the Verschiebung, we find that $\mathrm {fib}(F\colon \mathrm {TR}(X) \to \mathrm {TR}(X)^{hC_p}) = \mathrm {fib}(\varphi \colon X \to X^{tC_p})$ which is therefore $d$-truncated as desired.

Proof. The homotopy groups $\pi _*(M), \pi _*(M')$ form graded modules over the ring $\pi _*( \mathbb {F}_p[S^1]) = \mathbb {F}_p[\epsilon ]/\epsilon ^2, \ |\epsilon | = 1$. By assumption, $\pi _*(M')$ is a free graded $\mathbb {F}_p[\epsilon ]/\epsilon ^2$-module and $\pi _*(M)$ is the submodule of those elements in degree $r$ or higher. Choosing an $\mathbb {F}_p$-subspace $V$ of $\pi _r(M) = \pi _r(M')$ which is complementary to $\epsilon \pi _{r-1}(M') \subset \pi _r(M')$, we can modify $M$ to form $\tilde {M}$ with $\pi _r(\tilde {M}) = V$; it is now easy to see that $\pi _*(\tilde {M})$ is a free graded $\mathbb {F}_p[\epsilon ]/\epsilon ^2$-module, so we can conclude.

Proof. It follows that the cyclotomic spectrum $Y = X/(p, \beta ) = X \otimes _{ku} H \mathbb {F}_p \in \mathrm {CycSp}_{\geq 0}$ has the property that $\varphi \colon Y \to Y^{tC_p}$ is $(d+4)$-truncated. It follows that the comparison map $Y^{C_p} \to Y^{hC_p}$ is $(d+4)$-truncated as well, via the fiber square

Now, for each $n \geq 1$, we have the following pullback diagram [Reference Nikolaus and ScholzeNS18, Lemma II.4.5].

(11)

The bottom horizontal map is $Y^{C_{p^{n-1}}} \to Y^{hC_{p^{n-1}}} \xrightarrow {\varphi ^{hC_{p^{n-1}}}} (Y^{tC_p})^{hC_{p^{n-1}}}$, and the last term is identified with $Y^{tC_{p^n}}$ using the Tate orbit lemma; see [Reference Nikolaus and ScholzeNS18, Lemma II.4.1].

Now $Y^{C_p} \to Y^{hC_p}$ is an equivalence in degrees at least $d+ 6$, hence $Y^{C_{p^{n-1}}} \to Y^{hC_{p^{n-1}}}$ is an equivalence in degrees at least $d+ 6$ by [Reference Nikolaus and ScholzeNS18, Corollary II.4.9] (a generalization of results of Tsalidis [Reference TsalidisTsa98] and Bökstedt, Bruner, Lunø, Nielsen, and Rognes [Reference Bökstedt, Bruner, Lunøe-Nielsen and RognesBBLR14]). Since $\varphi \colon Y \to Y^{tC_p}$ is an equivalence in degrees at least $d+6$, it follows that the bottom horizontal map in (11) is an equivalence in degrees at least $d+ 6$. Note also that $Y^{tC_p}$ is a module over $\mathbb {F}_p^{tC_p}$ in $\mathrm {Fun}(BS^1, \mod (H\mathbb {F}_p))$; since the latter has induced $S^1$-action, it follows that the former does too.

By Lemma 5.6, we can find an at least $(d+6)$-connected $Y'$ with an $S^1$-equivariant map $Y' \to Y$ which induces an equivalence in degrees at least $d + 7$ and such that $Y'$ is induced as an object of $\mathrm {Fun}(BS^1, \mod (H \mathbb {F}_p))$. It follows that the map $Y'^{hC_{p^n}} \to Y^{hC_{p^n}}$ is an equivalence in degrees at least $d + 7$. But the commutative square

now shows that any $\alpha \in \pi _r( Y^{hC_{p^n}})$ for $r \geq d + 7$ has vanishing image in $\pi _r (Y^{tC_{p^n}})$. Using the commutative square (11) again (since the horizontal maps are equivalences in degrees at least $d+6$), it follows that the restriction map $Y^{C_{p^n}} \to Y^{C_{p^{n-1}}}$ induces the zero map in degrees at least $d + 7$. Consequently, taking the inverse limit over $R$, we find that $\mathrm {TR}(Y) \in {\rm Sp} _{\leq d + 6}$, whence the fiber of $\mathrm {TR}(X)/p \to L_{K(1)} \mathrm {TR}(X)/p$ belongs to ${\rm Sp} _{\leq d + 3}$ by Lemma 5.8.

Proof. Given a $ku$-module $N$, it suffices to show that the fiber of $N \to N[1/\beta ]$ (which after $p$-completion is $L_{K(1)} N$) is concentrated in degrees at most $d$ if and only if $N/\beta$ is concentrated in degrees at most $d+3$; then the result will follow by taking $N = M/p$. To prove the claim, first replace $N$ by $\mathrm {fib}( N \to N[1/\beta ])$; thus we may assume that $N$ is actually $\beta$-power torsion.

Then the claim follows from the observation that a $\beta$-power torsion $ku$-module $N$ is concentrated in degrees at most $d$ if and only if $N/\beta$ is concentrated in degrees at most $d+3$. The ‘only if’ direction is evident, so we verify the ‘if’ direction. In fact, if there exists a nonzero $x \in \pi _i(N)$ for $i > d$, then by multiplying by a power of $\beta$ we may assume $\beta x = 0$, whence there exists a nonzero class in $\pi _{i+3}(N/\beta )$ from the long exact sequence.

Proof. Let $X_i, i \in \mathcal {C}$ be a diagram of spectra. Suppose that for each $i \in \mathcal {C}$, the map $X_i \to L_{K(1)} X_i$ is $m$-truncated for some $m$. Then the map $\varprojlim _i X_i \to L_{K(1)} ( \varprojlim _i X_i)$ is $m$-truncated, and $L_{K(1)} ( \varprojlim _i X_i) \to \varprojlim _i L_{K(1)} X_i$ is an equivalence. This follows because $L_{K(1)}(-)$ annihilates bounded-above spectra. Applying the above observation, and using flat descent of $\mathrm {TR}(-; \mathbb {Z}_p)$ (which follows from [Reference Bhatt, Morrow and ScholzeBMS19, § 3]) and $\mathrm {THH}(-; \mathbb {Z}_p), \mathrm {THH}(-; \mathbb {Z}_p)^{tC_p}$, we reduce to the case where $R$ is a $\widehat {\mathbb {Z}_p[\zeta _{p^\infty }]}$-algebra (e.g. using André's lemma; see [Reference Bhatt and ScholzeBS19, Theorem 7.12]), so $\mathrm {THH}(R; \mathbb {Z}_p)$ is a $ku_{\hat {p}}$-module in $\mathrm {CycSp}_{\geq 0}$ (Example 5.5). With this reduction in mind, the first claim implies the second in view of Proposition 5.7 (applied to the cyclotomic spectrum $\mathrm {THH}(R; \mathbb {F}_p)$).

Thus, we prove the first claim (i.e. the Segal conjecture for $\mathrm {THH}(R; \mathbb {F}_p)$); this result appears as [Reference Bhatt, Morrow and ScholzeBMS19, Corollary 9.12] in the case where $R_0$ is the ring of integers in a complete, algebraically closed nonarchimedean field of mixed characteristic, and in [Reference Bhatt, Morrow and ScholzeBMS19, § 6] when $R = R_0$.

Let $(A, I)$ be the perfect prism associated to the perfectoid ring $R_0$, and let $\tilde {\xi } \in I$ be a generator. We use the prismatic cohomology ${{\mathbb {\Delta }}}_{R/A}$ of [Reference Bhatt and ScholzeBS19] and its Nygaard completion $\hat {{{\mathbb {\Delta }}}}_{R/A}$. For a quasisyntomic $R_0$-algebra $S$, there are [Reference Bhatt, Morrow and ScholzeBMS19] complete, exhaustive descending $\mathbb {Z}$-indexed filtrations on $\mathrm {TC}^-(S; \mathbb {Z}_p), \mathrm {TP}(S; \mathbb {Z}_p)$ with

\[ \mathrm{gr}^i \mathrm{TC}^-( S; \mathbb{Z}_p) \simeq \mathcal{N}^{{\geq} i} \hat{{{\mathbb{\Delta}}}}_{S/A}[2i], \quad \mathrm{gr}^i \mathrm{TP}(S; \mathbb{Z}_p) \simeq \hat{{{\mathbb{\Delta}}}}_{S/A}[2i], \]

and the map $\varphi \colon \mathrm {TC}^-(S; \mathbb {Z}_p) \to \mathrm {TP}(S; \mathbb {Z}_p)$ on graded pieces is given by the prismatic divided Frobenius

(12)\begin{equation} \varphi/\tilde{\xi}^i \colon \mathcal{N}^{{\geq} i} \hat{{{\mathbb{\Delta}}}}_{S/A}[2i] \to \hat{{{\mathbb{\Delta}}}}_{S/A}[2i]; \end{equation}

see [Reference Bhatt and ScholzeBS19, § 13] for the comparison between prismatic cohomology and the objects of [Reference Bhatt, Morrow and ScholzeBMS19]. Here we have trivialized the Breuil–Kisin twists involved since we are over the base perfectoid ring $R_0$, and we can compute the Nygaard-completed absolute prismatic cohomology as the Nygaard completed relative prismatic cohomology over $R_0$.

Now the map $\varphi \colon \mathrm {TC}^-(S; \mathbb {Z}_p) \to \mathrm {TP}(S; \mathbb {Z}_p)$ arises by taking $S^1$-invariants on the cyclotomic Frobenius $\varphi \colon \mathrm {THH}(S; \mathbb {Z}_p) \to \mathrm {THH}(S; \mathbb {Z}_p)^{tC_p}$. Both sides of this map are filtered (again, as in [Reference Bhatt, Morrow and ScholzeBMS19]) and the associated graded pieces are

\[ \mathrm{gr}^i \mathrm{THH}(S; \mathbb{Z}_p) \simeq \mathcal{N}^{i} \hat{{{\mathbb{\Delta}}}}_{S/A}[2i], \quad \mathrm{gr}^i \mathrm{THH}(S; \mathbb{Z}_p)^{tC_p} = (\hat{{{\mathbb{\Delta}}}}_{S/A})/\tilde{\xi} [2i], \]

and the cyclotomic Frobenius is induced from (12). But since $R$ is formally smooth over $R_0$, the Nygaard filtration is complete and $\varphi /\tilde {\xi }^i$ induces an equivalence $\mathcal {N}^i {{\mathbb {\Delta }}}_{R/A} \xrightarrow {\sim } \tau ^{\leq i} ( {{\mathbb {\Delta }}}_{R/A}/ \tilde {\xi })$ (see [Reference Bhatt and ScholzeBS19, § 12.4]), where the right-hand side is the $i$th stage of the conjugate filtration on ${{\mathbb {\Delta }}}_{R/A}/\tilde {\xi }$. Using the Hodge–Tate comparison [Reference Bhatt and ScholzeBS19, Theorem 4.10], it follows that the cohomology groups of ${{\mathbb {\Delta }}}_{R/A}/\tilde {\xi }$ are $p$-torsion-free. It follows that

\[ \mathrm{gr}^i \varphi \colon \mathrm{gr}^i \mathrm{THH}(R; \mathbb{F}_p) \to \mathrm{gr}^i \mathrm{THH}(R; \mathbb{F}_p)^{tC_p} \]

exhibits the domain as the $i$-connective cover of the codomain, and therefore has $(i -2)$-truncated homotopy fiber for all $i$. The codomain is $(2i-d)$-connective by the Hodge–Tate comparison, since $R/R_0$ is formally smooth of relative dimension $d$; therefore, $\mathrm {gr}^i \varphi$ is an equivalence for $i \geq d$. Therefore, the fiber of $\varphi \colon \mathrm {THH}(R; \mathbb {F}_p) \to \mathrm {THH}(R; \mathbb {F}_p)^{tC_p}$ has a complete filtration such that $\mathrm {gr}^i$ is $(i-2)$-truncated for all $i$ and contractible for $i \geq d$. This proves that $\varphi \colon \mathrm {THH}(R; \mathbb {F}_p) \to \mathrm {THH}(R; \mathbb {F}_p)^{tC_p}$ is $(d-3)$-truncated, whence the result.