# Perfectoid rings as Thom spectra

Zhouhang Mao

(Date: May 30, 2021)

###### Abstract.

The Hopkins-Mahowald theorem realizes the Eilenberg-Maclane spectra
$H\mathbb{F}_{p}$ as Thom spectra for all primes $p\in\mathbb{N}_{>0}$. In
this article, we record a known proof of a generalization of Hopkins-Mahowald
theorem, realizing $Hk$ as Thom spectra for perfect rings $k$, and we provide
a further generalization by realizing $HR$ as Thom spectra for perfectoid
rings $R$. We also discuss even further generalizations to prisms $(A,I)$ and
indicate how to adapt our proofs to Breuil-Kisin case.

Note: This document has been written using GNU TEXmacs; see
https://www.texmacs.org.

###### Contents

  1. 1 Introduction
  2. 2 Recollection of spherical Witt vectors
  3. 3 Perfect rings being Thom spectra
  4. 4 Recollection of perfectoid rings
    1. 4.1 Basic definitions and properties
    2. 4.2 Universal properties of Fontaine’s map (and a spherical analogue)
  5. 5 Proof of the main theorem
    1. 5.1 Finiteness and completeness of $Mf_{R,\xi}$ and $HR$ as $W^{+}(R^{\flat})$-modules
    2. 5.2 $t_{R,\xi}$ is an equivalence after the base change along $W^{+}(R^{\flat})\rightarrow W^{+}(\kappa)$
    3. 5.3 $t_{R,\xi}$ is an equivalence after the base change along $W^{+}(R^{\flat})\rightarrow HR^{\flat}$
    4. 5.4 Conclude: $t_{R,\xi}$ is an equivalence
    5. 5.5 An intermezzo: Identifying $\operatorname{THH}\left(-/W^{+}(k)\right)$ with $\operatorname{THH}\left(-\right)$ after $p$-completion
  6. 6 Analogues
    1. 6.1 Preparations
    2. 6.2 The Breuil-Kisin case
    3. 6.3 Complete regular local rings
  7. 7 Characterizing Thom spectra as quotients of free $\mathbb{E}_{2}$-algebras
  8. A Recollection of Higher Algebra
    1. A.1 Finiteness properties of rings and modules
    2. A.2 Nilpotent, local and complete modules

##  1\. Introduction

In this article, since most of our results are $p$-typical, we fix a prime
$p\in\mathbb{N}_{>0}$. We first describe the classical Hopkins-Mahowald
theorem. We know that $\mathbb{F}_{p}\cong\mathbb{Z}_{p}/p$, that is to say,
$\mathbb{F}_{p}$ is the free $\mathbb{Z}_{p}$-algebra in which $p=0$. For some
reasons, we need to extend this kind of results to a category of “less linear”
algebras in which the addition is not commutative or even associative on the
nose, but only up to coherent homotopy. To be more precise, we need to
understand whether the (Eilenberg-Maclane) ring spectrum $H\mathbb{F}_{p}$ is
still the free object in the category of $\mathbb{S}_{p}^{\wedge}$-algebras
satisfying certain associativity and commutativity with $p=0$? The classical
Hopkins-Mahowald theorem answers this affirmatively: they are the free object
in the category of $\mathbb{E}_{2}$-$\mathbb{S}_{p}^{\wedge}$-algebras with
$p=0$. There are two ways to describe “free $\mathbb{E}_{2}$-algebras with
$p=0$”. In this article, we will mainly adopt the description via Thom
spectra. We will go to another, more direct and natural but technically more
burdened description in Section 7. We start with formal definitions of Thom
spectra with informal illustrations and refer to [AB19] for further
discussions.

######  Definition 1.1.

Given a ring spectrum $R$, we define the $\infty$-category
$\operatorname{BGL}_{1}(R)$ to be the full subcategory of
$\operatorname{LMod}_{R}^{\simeq}$ spanned by left $R$-module spectra
equivalent to $R$, where we denote by $\mathcal{C}^{\simeq}$ the maximal
groupoid associated to an $\infty$-category $\mathcal{C}$.

######  Remark 1.2.

The $\infty$-category $\operatorname{BGL}_{1}(R)$ is in fact an
$\infty$-groupoid, and if we further suppose that $R$ is an
$\mathbb{E}_{n+1}$-ring spectrum, then $\operatorname{BGL}_{1}(R)$ inherits an
$\mathbb{E}_{n}$-monoidal structure from $\operatorname{LMod}_{R}$.

We admit the following result, which could be understand as an analogue of the
fact that $\pi_{1}(BG)=G$ for any discrete group $G$:

######  Proposition 1.3.

$\pi_{1}(\operatorname{BGL}_{1}(R))=\operatorname{GL}_{1}(\pi_{0}R)$ for any
ring spectra $R$. Concretely, an invertible element $a\in\pi_{0}R$ corresponds
to a multiplication map $m_{a}:R\rightarrow R$ in $\operatorname{BGL}_{1}(R)$.

######  Remark 1.4.

In fact, $\operatorname{BGL}_{1}(R)$ is a delooping of the group of invertible
elements in $R$.

Now we recall the definition of Thom spectra:

######  Definition 1.5.

Given a ring spectrum $R$, a space $X$ and a map
$f:X\rightarrow\operatorname{BGL}_{1}(R)$, the Thom spectrum $Mf$ associated
to $f$ is the colimit of the composition

$X\rightarrow\operatorname{BGL}_{1}(R)\rightarrow\operatorname{LMod}_{R}$

We note that by definition of colimits, we can understand the colimit as a
kind of “free objects satisfying several equations”. We will choose a special
space $X$ to encode the $\mathbb{E}_{2}$-commutativity (understood as a
generalized version of classical associativity, a collection of equations) and
a map $f:X\rightarrow\operatorname{BGL}_{1}(R)$ to encode the “equation”
$p=0$.

######  Remark 1.6.

As a special case of [Lur09, Proposition 4.1.2.6], any homotopy equivalence of
Kan complexes is cofinal, therefore the formation of the colimit does not
depend on the choice of models of the space $X$.

######  Remark 1.7.

In this article, we only consider the case that $R$ is a connective
$\mathbb{E}_{\infty}$-ring spectrum. As a consequence, we can replace
$\operatorname{LMod}_{R}$ by $\operatorname{Mod}_{R}$ and the Thom spectrum
$Mf$ is connective.

######  Remark 1.8.

In Definition 1.5, if $X$ is endowed with an $\mathbb{E}_{n}$-algebra
structure, and $f$ is assumed to be $\mathbb{E}_{n}$-monoidal, then the Thom
spectrum $Mf$ naturally inherits an $\mathbb{E}_{n}$-$R$-algebra structure. In
this case, we will call $Mf$ the $\mathbb{E}_{n}$-Thom spectrum associated to
$f$.

In the classical Hopkins-Mahowald theorem, we will choose $X=\Omega^{2}S^{3}$,
the free $\mathbb{E}_{2}$-group in the $\infty$-category $\mathcal{S}$ of
spaces.

######  Remark 1.9.

As a special case,
$\pi_{1}(\operatorname{BGL}_{1}(\mathbb{S}_{p}^{\wedge}))=\operatorname{GL}_{1}(\mathbb{Z}_{p})=\\{a\in\mathbb{Z}_{p}|a\operatorname{mod}p\neq
0\\}$. The invertible element $1-pu$ in $\mathbb{Z}_{p}$ gives rise to a map
$S^{1}\rightarrow\operatorname{BGL}_{1}(\mathbb{S}_{p}^{\wedge})$ where
$u\in\operatorname{GL}_{1}(\mathbb{Z}_{p})$ is an invertible element in
$\mathbb{Z}_{p}$. Since the $p$-adic sphere spectrum $\mathbb{S}_{p}^{\wedge}$
is an $\mathbb{E}_{\infty}$-ring spectrum, by Remark 1.2 this map extends to a
double loop map
$\Omega^{2}S^{3}\simeq\Omega^{2}\Sigma^{2}S^{1}\rightarrow\operatorname{BGL}_{1}(\mathbb{S}_{p}^{\wedge})$,
which we denote by $f_{\mathbb{F}_{p},pu}$.

We note that the choice of $1-pu$ essentially imposes an equation $1-pu=1$.
This could be seen by the fact that taking the colimit along
$f_{\mathbb{F}_{p},pu}$ is essentially taking the homotopy orbits of the
$\Omega^{2}S^{3}$-action, which is somehow “multiplying by” $1-pu$.

######  Remark 1.10.

In the first drafts of this article, we simply took $u=1.$ Later, we realized
that it might be easier to introduce $u$ to fix a gap in commutative algebra
for technical reasons.

Now we formulate the classical Hopkins-Mahowald theorem (cf. [AB19, Theorem
5.1], where $u=1$, but the proof works for the general case. See also [KN,
Theorem A.1]):

######  Theorem 1.11 (Hopkins-Mahowald).

The Eilenberg-Maclane spectrum $H\mathbb{F}_{p}$ is the $\mathbb{E}_{2}$-Thom
spectrum associated to the map
$f_{\mathbb{F}_{p},pu}:\Omega^{2}S^{3}\rightarrow\operatorname{BGL}_{1}(\mathbb{S}_{p}^{\wedge})$.

This arouses a natural question: what other discrete rings are Thom spectra in
a similar fashion? The first guess will come from the observation that
$\mathbb{Z}_{p}\cong W(\mathbb{F}_{p})$, so it would be natural to ask whether
we have similar results for perfect $\mathbb{F}_{p}$-algebras?

In this article, our main purpose is to show that this is the case for
perfectoid rings (which is inspired by computational results of topological
Hochschild homology of perfectoid rings in [BMS19]), and consequently, for
perfect $\mathbb{F}_{p}$-algebras. In order to do so, we need the concept of
spherical Witt vectors $W^{+}(k)$ for perfect $\mathbb{F}_{p}$-algebras $k$,
which we will recall in section 2. For the moment, we will take advantage of
the fact that $\pi_{0}(W^{+}(k))=W(k)$ where $W(k)$ is the ring of (classical)
Witt vectors. One example is that
$W^{+}(\mathbb{F}_{p})\simeq\mathbb{S}_{p}^{\wedge}$.

######  Remark 1.12.

Given a perfectoid ring $R$, denote by $\xi$ a generator of the kernel of
Fontaine’s pro-infinitesimal thickening $\theta:W(R^{\flat})\rightarrow R$,
which we will review in section 4. As in Remark 1.9, the invertible element in
$W(R^{\flat})$,
$1-\xi\in\operatorname{GL}_{1}(W(R^{\flat}))=\pi_{1}(\operatorname{BGL}_{1}(W^{+}(R^{\flat})))$
gives rise to a map $S^{1}\rightarrow\operatorname{BGL}_{1}(W^{+}(R^{\flat}))$
which extends to a double loop map
$f_{R,\xi}:\Omega^{2}S^{3}\rightarrow\operatorname{BGL}_{1}(W^{+}(R^{\flat}))$.

######  Theorem 1.13 (Main Theorem).

The Eilenberg-Maclane spectrum $HR$ is the $\mathbb{E}_{2}$-Thom spectrum
associated to the map $f_{R,\xi}$ for any perfectoid ring $R$.

Fontaine’s pro-infinitesimal thickening $\theta$ is in fact surjective. Note
that $R\cong W(R^{\flat})/\xi$, and our result is amount to say that the ring
spectrum $HR$ is a free $\mathbb{E}_{2}$-$W^{+}(R^{\flat})$-algebra with
$\xi=0$.

######  Remark 1.14.

When $R$ is a perfect $\mathbb{F}_{p}$-algebra, we can take $\xi=pu$ where
$u\in\operatorname{GL}_{1}(R)$ is an invertible element in $R$, and we note
that $R^{\flat}=R$. Especially, when $R=\mathbb{F}_{p}$, $f_{R,pu}$ coincides
with $f_{\mathbb{F}_{p},pu}$, hence Theorem 1.13 generalizes Theorem 1.11.

######  Remark 1.15.

The composite map $W^{+}(R^{\flat})\xrightarrow{\tau_{\leq
0}}HW(R^{\flat})\xrightarrow{H\theta}HR$ should be understood as a spherical
analogue of Fontaine’s map $\theta:W(R^{\flat})\rightarrow R$. We will
establish a universal property, Proposition 4.18, similar to Fontaine’s,
Proposition 4.16, which might be of independent interest.

The motivation to realize $H\mathbb{F}_{p}$ as a free $\mathbb{E}_{2}$-algebra
with $p=0$ is that it describes a direct “generation-relation” like
description with respect to the (p-completed) sphere spectrum
$\mathbb{S}_{p}^{\wedge}$. Similarly, realization of $HR$ as a free
$\mathbb{E}_{2}$-$W^{+}(R)$-algebra with $\xi=0$ enables us to relate $HR$
more directly to the ring $W^{+}(R^{\flat})$ of spherical Witt vectors, which
allows us to deduce “topological” results about these rings. For example, as a
consequence, we can compute the topological Hochschild homology
$\operatorname{THH}(HR)$ (of a perfectoid ring $R$) as an
$\mathbb{E}_{1}$-ring spectrum and deduce Bökstedt’s periodicity. By [KN,
Proposition 4.7], as in the proof of Theorem 4.1 there, we have

######  Proposition 1.16.

The (relative) topological Hochschild homology
$\operatorname{THH}(HR/W^{+}(R^{\flat}))\simeq HR\otimes\Omega S^{3}$ as
$\mathbb{E}_{1}$-$W^{+}(R^{\flat})$-algebras for any perfectoid ring $R$.

The proof is somehow technical, but essentially it is similar to the classical
computation of the Hochschild homology $\operatorname{HH}(R/W(R^{\flat}))$,
via resolving $R$ by $W(R^{\flat})$-CDGAs. We refer to first paragraphs of the
proof of [HN19, Theorem 1.3.2] for this classical case. As a consequence of
Proposition 1.16, we have (see subsection 5.5):

######  Proposition 1.17.

The (absolute) topological Hochschild homology
$\operatorname{THH}(HR)_{p}^{\wedge}\simeq HR\otimes\Omega S^{3}$ as
$\mathbb{E}_{1}$-ring spectra.

By known results on the homology of $\Omega S^{3}$ (a classical reference is
[Bot82]), we deduce Bökstedt’s periodicity for perfectoid rings (cf. [BMS19,
Theorem 6.1]).

######  Corollary 1.18 (Bökstedt’s periodicity).

$\pi_{\ast}(\operatorname{THH}(HR)_{p}^{\wedge})\cong R[u]$ where $u$ is any
generator of $\pi_{2}(\operatorname{THH}(HR)_{p}^{\wedge})$ as a
$\pi_{0}(\operatorname{THH}(HR)_{p}^{\wedge})$-module.

In fact, our question was motivated by Bökstedt’s periodicity for perfectoid
rings: we wanted to understand why Bökstedt’s periodicity holds.

Further generalizations of Theorem 1.13 to prisms, the concept introduced in
[BS19], seem plausible. However, we are only capable to reach another special
case of prisms motivated by Breuil-Kisin cohomology, parallel to the
perfectoid case, proposed by Matthew Morrow:

######  Theorem 1.19.

Let $A$ be complete discrete valuation ring of mixed characteristic with
residue field $k$ being perfect of characteristic $p$. Then the Eilenberg-
Maclane spectrum $HA$ is the $\mathbb{E}_{2}$-Thom spectrum associated to a
map $f_{E}:\Omega^{2}S^{3}\rightarrow\operatorname{BGL}_{1}(W^{+}(k)[[u]])$.

Inspired by [KN19, Section 9], we will also provide a version of Hopkins-
Mahowald theorem for complete regular local rings:

######  Theorem 1.20.

Let $(A,\mathfrak{m})$ be a complete regular local ring of mixed
characteristic with residue field $k$ being perfect of characteristic $p$. Let
$(a_{1},\ldots,a_{n})\subseteq\mathfrak{m}$ be a regular sequence which
generates the maximal ideal $\mathfrak{m}$. Then the Eilenberg-Maclane
spectrum $HA$ is the $\mathbb{E}_{2}$-Thom spectrum associated to a map
$f_{A}:\Omega^{2}S^{3}\rightarrow\operatorname{BGL}_{1}(W^{+}(k)[[u_{1},\ldots,u_{n}]])$.

In this article, we will first review spherical Witt vectors. We then record a
known proof of perfect rings being Thom spectrum, the special case of Theorem
1.13 for perfect rings, which we learn from Sanath Devalapurkar, but the proof
is also well-known to experts such as Achim Krause and Thomas Nikolaus, see
[KN19]. This result is needed in the proof of the general case of Theorem
1.13. Then we start with recalling the definition and some basic properties of
perfectoid rings, and prove Theorem 1.13. As far as we know, although this is
known to several experts, the proof is not found in the literature. We will
finally discuss further generalizations to prisms in Section 6, and especially
Hopkins-Mahowald theorem for Breuil-Kisin cases, which seems also to be known
by experts (see [KN19, Remark 3.4]). We take an opportunity to write down
those proofs. The author thanks Matthew Morrow for various suggestions during
the construction of this article.

######  Warning 1.21.

For spectra $M,N$, we will denote the smash product of $M,N$ by $M\otimes N$.
Let $R$ be an $\mathbb{E}_{1}$-ring (spectrum), $M$ a right $R$-module
(spectrum) and $N$ a left $R$-module (spectrum), we will denote the relative
tensor product by $M\otimes_{R}N$. In order to avoid possible ambiguities, for
discrete rings $A$, right $A$-modules $P$ and left $A$-modules $Q$, we will
denote the ordinary (algebraic) tensor product by
$\operatorname{Tor}_{0}^{A}(P,Q)$ (instead of $P\otimes_{A}Q$). It is
important that in general the Eilenberg-Maclane spectrum
$H\operatorname{Tor}_{0}^{A}(P,Q)$ do not coincide with the relative tensor
product $HP\otimes_{HA}HQ$ of spectra. Rather, the relative tensor
$HP\otimes_{HA}HQ$ coincides with the Eilenberg-Maclane spectrum
$H(P\otimes_{A}^{\mathbb{L}}Q)$ of the derived tensor product. Since the
concept of the derived tensor product does not play a great role in this
article, we will not use the notation $\otimes_{A}^{\mathbb{L}}$, and we will
uniformly preserve the notation $\otimes$ for smash products and relative
tensor products of spectra.

######  Notation 1.22.

In this article, we mainly adopt notations in [Lur17], [Lur18a] and [Lur18b].
In particular, we will let $\operatorname{LMod}_{R}$ denote the
$\infty$-category of an $\mathbb{E}_{1}$-ring $R$, let
$\operatorname{Mod}_{R}$ denote the symmetric monoidal $\infty$-category of an
$\mathbb{E}_{\infty}$-ring $R$ and let
$\operatorname{Alg}_{R}^{\mathbb{E}_{n}}$ denote the $\infty$-category of
$\mathbb{E}_{n}$-$R$-algebras for an $\mathbb{E}_{\infty}$-ring $R$ and a
positive integer $n\in\mathbb{N}_{>0}$. In particular, we will denote
$\operatorname{Alg}_{R}^{\mathbb{E}_{\infty}}$ by $\operatorname{CAlg}_{R}$,
referred to as the $\infty$-category of commutative $R$-algebras. On the other
hand, we will denote $\operatorname{Mod}_{R}^{\heartsuit}$ the
$\infty$-category of discrete $R$-modules, and
$\operatorname{CAlg}_{R}^{\heartsuit}$ the $\infty$-category of discrete
commutative $R$-algebras.

##  2\. Recollection of spherical Witt vectors

In this section, we will review the definition and some basic properties of
spherical Witt vectors. We quote some definitions and propositions from
[Lur18a, Section 5.2].

######  Definition 2.1 ([Lur18a, Definition 5.2.1]).

Let $A$ be a connective $\mathbb{E}_{\infty}$-ring, let $I\subseteq\pi_{0}A$
be a finitely generated ideal, and set $A_{0}=\pi_{0}(A)/I$. Suppose that we
are given a commutative diagram of connective $\mathbb{E}_{\infty}$-rings

$\begin{array}[]{ccc}A&\xrightarrow{f}&B\\\
{\mbox{\rotatebox[origin={c}]{-90.0}{$\longrightarrow$}}}&\sigma&{\mbox{\rotatebox[origin={c}]{-90.0}{$\longrightarrow$}}}\\\
HA_{0}&\xrightarrow{f_{0}}&HB_{0}\end{array}$

where $B_{0}$ is a discrete commutative ring. We will say that $\sigma$
exhibits $f$ as an $A$-thickening of $f_{0}$ if the following conditions are
satisfied:

  1. a)

The $\mathbb{E}_{\infty}$-ring $B$ is $I$-complete as an $A$-module;

  2. b)

The diagram $\sigma$ induces an isomorphism of commutative rings
$\pi_{0}(B)/I\pi_{0}(B)\rightarrow B_{0}$;

  3. c)

Let $R$ be any connective $\mathbb{E}_{\infty}$-algebra over $A$ which is
$I$-complete. Then the canonical map

$\operatorname{Map}_{\operatorname{CAlg}_{A}}(B,R)\rightarrow\operatorname{Hom}_{\operatorname{CAlg}_{A_{0}}^{\heartsuit}}(B_{0},\pi_{0}(R)/I\pi_{0}(R))$

is a homotopy equivalence. In particular, the mapping space
$\operatorname{Map}_{\operatorname{CAlg}_{A}}(B,R)$ is discrete up to homotopy
equivalence, that is, each connected component is contractible.

######  Remark 2.2 (Uniqueness, [Lur18a, Remark 5.2.2]).

Let $A$ be a connective $\mathbb{E}_{\infty}$-ring, let $I\subseteq\pi_{0}A$
be a finitely generated ideal, and set $A_{0}=\pi_{0}(A)/I$. Suppose that we
are given a homomorphism of commutative rings $f_{0}:A_{0}\rightarrow B_{0}$.
It follows immediately from the definition that if there exists a diagram
$\sigma$:

$\begin{array}[]{ccc}A&\xrightarrow{f}&B\\\
{\mbox{\rotatebox[origin={c}]{-90.0}{$\longrightarrow$}}}&\sigma&{\mbox{\rotatebox[origin={c}]{-90.0}{$\longrightarrow$}}}\\\
HA_{0}&\xrightarrow{f_{0}}&HB_{0}\end{array}$

which exhibits $f$ as an $A$-thickening of $f_{0}$, then the morphism $f$ (and
the diagram $\sigma$) is uniquely determined up to equivalence.

######  Remark 2.3 ([Lur18a, Remark 5.2.4]).

Suppose that we are given a commutative diagram of commutative
$\mathbb{E}_{\infty}$-rings

$\begin{array}[]{ccc}A&\xrightarrow{f}&B\\\
{\mbox{\rotatebox[origin={c}]{-90.0}{$\longrightarrow$}}}&&{\mbox{\rotatebox[origin={c}]{-90.0}{$\longrightarrow$}}}\\\
A^{\prime}&\xrightarrow{f^{\prime}}&B^{\prime}\\\
{\mbox{\rotatebox[origin={c}]{-90.0}{$\longrightarrow$}}}&&{\mbox{\rotatebox[origin={c}]{-90.0}{$\longrightarrow$}}}\\\
HA_{0}&\xrightarrow{f_{0}}&HB_{0}\end{array}$

Assume that $A_{0},B_{0}$ are discrete rings and the left vertical maps induce
surjective ring morphisms $\pi_{0}A\rightarrow\pi_{0}A^{\prime}\rightarrow
A_{0}$ whose composition has kernel $I\subseteq\pi_{0}A$. Suppose that the
outer rectangle exhibits $f$ as an $A$-thickening of $f_{0}$ and that the
upper square exhibits $B^{\prime}$ as an $I$-completion of
$B\otimes_{A}A^{\prime}$. Then the lower square exhibits $f^{\prime}$ as an
$A^{\prime}$-thickening of $f_{0}$.

######  Theorem 2.4 ([Lur18a, Theorem 5.2.5]).

Let $A$ be a connective $\mathbb{E}_{\infty}$-ring, let $I\subseteq\pi_{0}A$
be a finitely generated ideal, and set $A_{0}=\pi_{0}(A)/I$. Suppose that
$A_{0}$ is an $\mathbb{F}_{p}$-algebra such that $HA_{0}$ is almost perfect as
an $A$-module and that the Frobenius map $\varphi_{A_{0}}:A_{0}\rightarrow
A_{0}$ is flat. Let $f:A_{0}\rightarrow B_{0}$ be a morphism of commutative
$\mathbb{F}_{p}$-algebras which is relatively perfect, then there exists a
diagram

$\begin{array}[]{ccc}A&\xrightarrow{f}&B\\\
{\mbox{\rotatebox[origin={c}]{-90.0}{$\longrightarrow$}}}&\sigma&{\mbox{\rotatebox[origin={c}]{-90.0}{$\longrightarrow$}}}\\\
HA_{0}&\xrightarrow{f_{0}}&HB_{0}\end{array}$

which exhibits $f$ as an $A$-thickening of $f_{0}$. Moreover, $\sigma$ is a
pushout square.

######  Example 2.5 (Classical Witt vectors, [Lur18a, Example 5.2.6]).

In the statement of Theorem 2.4 take $A=H\mathbb{Z}_{p}$ and
$I=p\mathbb{Z}_{p}$. Then $A_{0}=\pi_{0}(A)/I$ is the finite field
$\mathbb{F}_{p}$ and a map $f_{0}:A_{0}\rightarrow B_{0}$ of discrete rings is
relative perfect if and only if $B_{0}$ is a perfect $\mathbb{F}_{p}$-algebra.
If this condition is satisfied, then Theorem 2.4 allows us to lift $B_{0}$ to
an $\mathbb{E}_{\infty}$-$H\mathbb{Z}_{p}$-algebra which is complete with
respect to the ideal $p\mathbb{Z}_{p}$ and for which the quotient
$\pi_{0}(B)/p\pi_{0}(B)$ is isomorphic to $B_{0}$. This
$\mathbb{Z}_{p}$-algebra is in fact the Eilenberg-Maclane spectrum of the ring
of Witt vectors $W(B_{0})$. See also [Ser79, Section II.5, Proposition 10] for
a classical description of this universal property.

######  Example 2.6 (Spherical Witt vectors, [Lur18a, Example 5.2.7]).

In the statement of Theorem 2.4 take $A=\mathbb{S}_{p}^{\wedge}$ and $I=(p)$.
Then $A_{0}=\pi_{0}(A)/I$ is the finite field $\mathbb{F}_{p}$ and a morphism
$f_{0}:A_{0}\rightarrow B_{0}$ is relative perfect if and only if $B_{0}$ is a
perfect $\mathbb{F}_{p}$-algebra. If this condition is satisfied, Theorem 2.4
allows us to lift $B_{0}$ to an
$\mathbb{E}_{\infty}$-$\mathbb{S}_{p}^{\wedge}$-algebra which is complete with
respect to the ideal $(p)$ and the tensor product
$H\mathbb{F}_{p}\otimes_{\mathbb{S}_{p}^{\wedge}}B\simeq\pi_{0}(B)/p\pi_{0}(B)$
is isomorphic to $B_{0}$. This is the $\mathbb{E}_{\infty}$-ring
$W^{+}(B_{0})$ of “spherical” Witt vectors.

######  Proposition 2.7.

$\pi_{0}(W^{+}(k))$ is isomorphic to $W(k)$, the ring of Witt vectors, and
$HW(k)\simeq W^{+}(k)\otimes_{\mathbb{S}_{p}^{\wedge}}H\mathbb{Z}_{p}$ for any
perfect $\mathbb{F}_{p}$-algebra $k$.

###### Proof.

First, we have a commutative diagram

Figure 1.

where the outer square is given by Theorem 2.4. The right vertical map
$W^{+}(k)\rightarrow Hk$ factors through the pushout
$W^{+}(k)\otimes_{\mathbb{S}_{p}^{\wedge}}H\mathbb{Z}_{p}$ in the category of
$\mathbb{E}_{\infty}$-rings. Note that $\mathbb{S}_{p}^{\wedge}$ is a coherent
ring as in Definition A.10, and $H\mathbb{Z}_{p}\simeq
H\pi_{0}(\mathbb{S}_{p}^{\wedge})$ is an almost perfect
$\mathbb{S}_{p}^{\wedge}$-module by Corollary A.12, which implies that
$W^{+}(k)\otimes_{\mathbb{S}_{p}^{\wedge}}H\mathbb{Z}_{p}$ is an almost
perfect $W^{+}(k)$-module by Proposition A.8. By Definition 2.1, $W^{+}(k)$ is
a $p$-complete $\mathbb{E}_{\infty}$-$\mathbb{S}_{p}^{\wedge}$-algebra,
therefore by Proposition A.27, the spectrum
$W^{+}(k)\otimes_{\mathbb{S}_{p}^{\wedge}}H\mathbb{Z}_{p}$ is $p$-complete.
Now we take $A=\mathbb{S}_{p}^{\wedge}$, $A^{\prime}=H\mathbb{Z}_{p}$,
$A_{0}=H\mathbb{F}_{p}$, $B=W^{+}(k)$,
$B^{\prime}=W^{+}(k)\otimes_{\mathbb{S}_{p}^{\wedge}}H\mathbb{Z}_{p}$ and
$B_{0}=Hk$ in Remark 2.3, we deduce that the lower square

$\begin{array}[]{ccc}H\mathbb{Z}_{p}&\longrightarrow&W^{+}(k)\otimes_{\mathbb{S}_{p}^{\wedge}}H\mathbb{Z}_{p}\\\
{\mbox{\rotatebox[origin={c}]{-90.0}{$\longrightarrow$}}}&&{\mbox{\rotatebox[origin={c}]{-90.0}{$\longrightarrow$}}}\\\
H\mathbb{F}_{p}&\longrightarrow&Hk\end{array}$

constitutes a commutative diagram of thickening as in Definition 2.1. Then it
follows from Remark 2.2 and Example 2.5 that
$W^{+}(k)\otimes_{\mathbb{S}_{p}^{\wedge}}H\mathbb{Z}_{p}$ is equivalent to
$HW(k)$ as $\mathbb{E}_{\infty}$-$H\mathbb{Z}_{p}$-algebras, which implies
that
$W(k)\cong\pi_{0}(HW(k))\cong\operatorname{Tor}_{0}^{\pi_{0}(\mathbb{S}_{p}^{\wedge})}(\pi_{0}(W^{+}(k)),\pi_{0}(H\mathbb{Z}_{p}))\cong\operatorname{Tor}_{0}^{\mathbb{Z}_{p}}(\pi_{0}(W^{+}(k)),\mathbb{Z}_{p})\cong\pi_{0}(W^{+}(k))$.
∎

######  Proposition 2.8 (Recognition of Thickenings, [Lur18a, Proposition
5.2.9]).

Let $A$ be a connective $\mathbb{E}_{\infty}$-ring, let $I\subseteq\pi_{0}A$
be a finitely generated ideal, and set $A_{0}=\pi_{0}(A)/I$. Suppose that
$A_{0}$ is an $\mathbb{F}_{p}$-algebra which is almost perfect as an
$A$-module and that the Frobenius map $\varphi_{A_{0}}:A_{0}\rightarrow A_{0}$
is flat. Suppose we are given a commutative diagram of connective
$\mathbb{E}_{\infty}$-rings $\sigma$:

$\begin{array}[]{ccc}A&\xrightarrow{f}&B\\\
{\mbox{\rotatebox[origin={c}]{-90.0}{$\longrightarrow$}}}&\sigma&{\mbox{\rotatebox[origin={c}]{-90.0}{$\longrightarrow$}}}\\\
A_{0}&\xrightarrow{f_{0}}&B_{0}\end{array}$

where $f_{0}$ is a relative perfect morphism of commutative
$\mathbb{F}_{p}$-algebras. Then $\sigma$ exhibits $f$ as an $A$-thickening of
$f_{0}$ if and only if the following conditions are satisfied:

  1. i.

The $\mathbb{E}_{\infty}$-ring $B$ is $I$-complete as an $A$-module;

  2. ii.

The diagram $\sigma$ is a pushout square.

##  3\. Perfect rings being Thom spectra

We first admit a (superficially) slightly stronger Hopkins-Mahowald’s theorem
for sake of convenience. Given a perfect $\mathbb{F}_{p}$-algebra $k$ and an
invertible element $u\in\operatorname{GL}_{1}(W(k))$, as a special case of
Remark 1.12, we have a map
$f_{k,pu}:\Omega^{2}S^{3}\rightarrow\operatorname{BGL}_{1}(W^{+}(k))$.

######  Theorem 3.1 (Hopkins-Mahowald for $k$).

The Eilenberg-Maclane spectrum $Hk$ is the $\mathbb{E}_{2}$-Thom spectrum
associated to the map $f_{k,pu}$.

For technical reasons, we start with the special case that
$u\in\operatorname{GL}_{1}(\mathbb{Z}_{p})\subseteq\operatorname{GL}_{1}(W(k))$.
In this case, it is a direct consequence of that for $\mathbb{F}_{p}.$

######  Lemma 3.2.

Theorem 3.1 is true when
$u\in\operatorname{GL}_{1}(\mathbb{Z}_{p})\subseteq\operatorname{GL}_{1}(W(k))$.

###### Proof.

We note that the image of the multiplication map
$m_{1-pu}:\mathbb{S}_{p}^{\wedge}\rightarrow\mathbb{S}_{p}^{\wedge}$ given by
$1-pu\in\pi_{0}(\mathbb{S}_{p}^{\wedge})\cong\mathbb{Z}_{p}$ under the
canonical (symmetric monoidal) functor
$W^{+}(k)\otimes_{\mathbb{S}_{p}^{\wedge}}-:\operatorname{Mod}_{\mathbb{S}_{p}^{\wedge}}\rightarrow\operatorname{Mod}_{W^{+}(k)}$
is still a multiplication map $m_{1-pu}:W^{+}(k)\rightarrow W^{+}(k)$ given by
$1-pu\in\pi_{0}(W^{+}(k))\cong W(k)$, and therefore the map $f_{k,pu}$
coincides with the composition map

$\Omega^{2}S^{3}\mathop{\longrightarrow}\limits^{f_{\mathbb{F}_{p},pu}}\operatorname{BGL}_{1}(\mathbb{S}_{p}^{\wedge})\xrightarrow{W^{+}(k)\otimes_{\mathbb{S}_{p}^{\wedge}}-}\operatorname{BGL}_{1}(W^{+}(k))$

Since $Mf_{\mathbb{F}_{p},pu}\simeq H\mathbb{F}_{p}$ as $\mathbb{E}_{2}$-ring
spectra by Theorem 1.11,

$Mf_{k,pu}\simeq
W^{+}(k)\otimes_{\mathbb{S}_{p}^{\wedge}}Mf_{\mathbb{F}_{p},pu}\simeq
W^{+}(k)\otimes_{\mathbb{S}_{p}^{\wedge}}H\mathbb{F}_{p}\simeq Hk$

as $\mathbb{E}_{2}$-ring spectra, where the first equivalence follows from the
fact that the functor $W^{+}(k)\otimes_{\mathbb{S}_{p}^{\wedge}}-$ is a left
adjoint therefore commutes with colimits and the last equivalence is given by
the last claim in Theorem 2.4. ∎

In order to prove Theorem 3.1, it suffices to show that $Mf_{k,pu}\simeq
Mf_{k,p}$ holds for all $u\in\operatorname{GL}_{1}(W(k))$, therefore
$Mf_{k,pu}\simeq Mf_{k,p}\simeq Hk$ by Lemma 3.2. We will base the proof on a
universal property of Thom spectra which we will not use elsewhere, and the
author looks forward to an alternative proof which does not depend on this
universal property.

######  Lemma 3.3 (Proposition 4.9 in [AB19] along with the discussions after
Lemma 4.6).

The $\mathbb{E}_{2}$-Thom spectrum $Mf_{k,pu}$ satisfies the following
universal property: For all $\mathbb{E}_{2}$-$W^{+}(k)$-algebras $A$, the
mapping space
$\operatorname{Map}_{\operatorname{Alg}_{W^{+}(k)}^{\mathbb{E}_{2}}}(Mf_{k,pu},A)$
could be naturally identified with the space of null-homotopies of the
composite map $W^{+}(k)\xrightarrow{m_{pu}}W^{+}(k)\xrightarrow{\eta}A$ in the
category of $W^{+}(k)$-modules where $\eta:W^{+}(k)\rightarrow A$ is the
canonical map given by the $\mathbb{E}_{2}$-$W^{+}(k)$-algebra structure, and
$m_{pu}:W^{+}(k)\rightarrow W^{+}(k)$ is the multiplication map given by
$pu\in W(k)=\pi_{0}(W^{+}(k))$.

Proof of Theorem 3.1 Note that the multiplication map
$m_{u}:W^{+}(k)\rightarrow W^{+}(k)$ is an equivalence of $W^{+}(k)$-modules
since $u\in W(k)=\pi_{0}(W^{+}(k))$ is invertible. Hence by Lemma 3.3, the map
$m_{u}$ induces an equivalence of spaces
$\operatorname{Map}_{\operatorname{Alg}_{W^{+}(k)}^{\mathbb{E}_{2}}}(Mf_{k,p},A)\rightarrow\operatorname{Map}_{\operatorname{Alg}_{W^{+}(k)}^{\mathbb{E}_{2}}}(Mf_{k,pu},A)$
which is natural in $A$. By the Yoneda lemma, we deduce that $Mf_{k,pu}\simeq
Mf_{k,p}$ as $\mathbb{E}_{2}$-$W^{+}(k)$-algebras. $\Box$

##  4\. Recollection of perfectoid rings

In this section, we will review basic definitions and properties of perfectoid
rings.

###  4.1. Basic definitions and properties

######  Definition 4.1.

Let $A$ be a ring and $I\subseteq A$ be an ideal. Then the ring $A$ is called
$I$-adically complete if the canonical map from $A$ to the (inverse) limit of
the tower

$\cdots\rightarrow A/I^{n}\rightarrow\cdots\rightarrow A/I^{2}\rightarrow A/I$

is an isomorphism. The ring $A$ is called $I$-adically separated if the
intersection $\bigcap_{n=0}^{\infty}I^{n}=0$.

######  Warning 4.2.

In the literature, sometimes authors call a ring $A$ is $I$-adically complete
when the canonical map $A\rightarrow\lim_{n\in(\mathbb{N},>)}(A/I^{n})$ is
only supposed to be surjective, and our $I$-adic completeness is equivalent to
their $I$-adic completeness plus $I$-adic separateness.

######  Definition 4.3.

Let $A$ be an $\mathbb{F}_{p}$-algebra. The direct limit perfection
$A_{\operatorname{perf}}$ of $A$ is the direct limit of the telescope
$A\xrightarrow{\varphi}A\xrightarrow{\varphi}A\xrightarrow{\varphi}\cdots$.

######  Definition 4.4.

An $\mathbb{F}_{p}$-algebra $A$ is called semiperfect if the Frobenius map
$\varphi:A\rightarrow A$ is surjective.

######  Remark 4.5.

For a semiperfect $\mathbb{F}_{p}$-algebra $A$, the direct limit perfection
$A_{\operatorname{perf}}$ coincides with $A_{\operatorname{red}}=A/\sqrt{0}$,
by checking that $A_{\operatorname{red}}$ satisfies the universal property of
$A_{\operatorname{perf}}$.

######  Remark 4.6.

The canonical map $R\rightarrow R_{\operatorname{perf}}$ is initial among all
$\mathbb{F}_{p}$-algebra morphisms $R\rightarrow S$ such that $S$ is a perfect
$\mathbb{F}_{p}$-algebra. This follows directly from the universal property of
direct limits in the definition of direct limit perfections.

######  Definition 4.7.

Let $R$ be a commutative ring which is $p$-adically complete. The tilt of $R$,
denoted by $R^{\flat}$, is a perfect $\mathbb{F}_{p}$-algebra defined by the
limit of the tower

$\cdots\xrightarrow{\varphi}R/p\xrightarrow{\varphi}R/p\xrightarrow{\varphi}R/p$

where $\varphi:R/p\rightarrow R/p$ is the Frobenius map. In particular, if $R$
is an $\mathbb{F}_{p}$-algebra, then $R^{\flat}$ is the inverse limit
perfection of $R$, and if furthermore $R$ is semiperfect, then the canonical
map $R^{\flat}\rightarrow R$ is a surjection.

We need the following classical proposition to define the Fontaine’s pro-
infinitesimal thickening map. We omit the proof which is routine. One can find
a proof in, say, [HN19, Section 1.3].

######  Proposition 4.8.

Let $R$ be a $p$-adically complete commutative ring. Then there exists a
multiplicative map (that is to say, a morphism of multiplicative monoids)
$R^{\flat}\xrightarrow{(-)^{\sharp}}R$ that sends
$a=(x_{n})_{n\in\mathbb{N}}\in R^{\flat}$ to
$a^{\sharp}:=\lim_{n\rightarrow\infty}y_{n}^{p^{n}}$ where
$(x_{n})_{n\in\mathbb{N}}$ satisfies $\varphi(x_{n+1})=x_{n}$ for all
$n\in\mathbb{N}$, and $(y_{n})_{n\in\mathbb{N}}\in R^{\mathbb{N}}$ is any
sequence such that for each $n\in\mathbb{N}$, $y_{n}$ is a lift of $x_{n}\in
R/p$ in $R$. We note that $a^{\sharp}$ does not depend on choice of
$(y_{n})_{n\in\mathbb{N}}$.

######  Definition 4.9.

Fontaine’s map $\theta:W(R^{\flat})\rightarrow R$ is given by
$\theta\left(\sum_{i=0}^{\infty}[a_{i}]p^{i}\right)=\sum_{i=0}^{\infty}a_{i}^{\sharp}p^{i}$,
where $[-]:R^{\flat}\rightarrow W(R^{\flat})$ is the Teichmüller
representative.

######  Definition 4.10 ([BMS18, Definition 3.5]).

A commutative ring $R$ is perfectoid if there exists $\pi\in R$ such that
$p\in\pi^{p}R$, such that the ring $R$ is $(\pi)$-adically complete, such that
the $\mathbb{F}_{p}$-algebra $R/p$ is semiperfect, and such that the kernel of
$\theta:W(R^{\flat})\rightarrow R$ is a principal ideal.

######  Definition 4.11.

Let $R$ be a perfectoid ring. The special fiber, denoted by $\kappa$, is the
direct limit perfection of $R/p$, that is to say
$\kappa:=(R/p)_{\operatorname{perf}}=R/\sqrt{pR}$ since $R/p$ is semiperfect.

######  Notation 4.12.

Let $R$ be a perfectoid ring. We denote by $\xi$ a generator of Fontaine’s map
$\theta:W(R^{\flat})\rightarrow R$.

######  Proposition 4.13 ([BMS18, Lemma 3.13]).

Let $R$ be a perfectoid ring. Then the commutative diagram

$\begin{array}[]{ccc}W(R^{\flat})&\xrightarrow{\theta}&R\\\
{\mbox{\rotatebox[origin={c}]{-90.0}{$\longrightarrow$}}}&&{\mbox{\rotatebox[origin={c}]{-90.0}{$\longrightarrow$}}}\\\
W(\kappa)&\xrightarrow{\operatorname{mod}p}&\kappa\end{array}$

is a $\operatorname{Tor}$-independent pushout square.

######  Corollary 4.14.

Let $R$ be a perfectoid ring. For any generator $\xi\in\ker\theta$, there
exists an invertible element $u\in\operatorname{GL}_{1}(W(\kappa))$ such that
the image of $\xi\in W(R^{\flat})$ in $W(\kappa)$ is $pu$.

###### Proof.

By Proposition 4.13, the image of $u\in W(R^{\flat})$ in $W(\kappa)$ is a
generator of the ideal $pW(\kappa)$. Since $p\in W(\kappa)$ is not a zero
divisor, we deduce the result that we need. ∎

######  Proposition 4.15.

Let $R$ be a perfectoid ring. Then the kernel of the composition
$R^{\flat}\rightarrow R/p\rightarrow\kappa$ is $\sqrt{\xi R^{\flat}}$.

###### Proof.

The kernel of the composition $W(R^{\flat})\rightarrow R/p\rightarrow\kappa$
is $\sqrt{pW(R^{\flat})+\xi W(R^{\flat})}$ whose image under the canonical map
$W(R^{\flat})\rightarrow R^{\flat}$ is $\sqrt{\xi R^{\flat}}$. ∎

###  4.2. Universal properties of Fontaine’s map (and a spherical analogue)

The results of this subsection will not be used later. However, we find it
better to understand that Fontaine’s map $\theta:W(R^{\flat})\rightarrow R$
and its spherical analogue $W^{+}(R^{\flat})\rightarrow\tau_{\leq
0}(W^{+}(R^{\flat}))\simeq HW(R^{\flat})\xrightarrow{H\theta}HR$ satisfy a
universal property, which is related to the thickening defined in Definition
2.1. Roughly speaking, they are mixed characteristic “absolute” versions of
thickenings in Definition 2.1. The following proposition is essentially due to
Fontaine (see [Fon94], Theorem 1.2.1), rephrased in the modern language:

######  Proposition 4.16 ([HN19, Proposition 1.3.4]).

Let $R$ be a perfectoid ring. Then Fontaine’s map
$\theta:W(R^{\flat})\rightarrow R$ is initial among surjections
$\theta_{D}:D\rightarrow R$ of rings such that the ring $D$ is both
$p$-adically complete and $\ker\theta_{D}$-adically complete.

We will sketch the proof in [HN19] for the universal property, that is, assume
that the $p$-adic completeness and the $\xi$-adic completeness of
$W(R^{\flat})$ is already given, we show that Fontaine’s map
$\theta:W(R^{\flat})\rightarrow R$ is initial as claimed.

###### Proof.

Let $\theta_{D}:D\rightarrow R$ be a map of rings such that $D$ is both
$p$-adically complete and $\ker\theta_{D}$-adically complete. We need to show
that $\theta_{D}$ factors uniquely through $\theta:W(R^{\flat})\rightarrow R$.
In view of Example 2.5 and Definition 2.1, we have a bijection

$\operatorname{Hom}_{\operatorname{CAlg}_{\mathbb{Z}_{p}}^{\heartsuit}}(W(R^{\flat}),D)\xrightarrow{\cong}\operatorname{Hom}_{\operatorname{CAlg}_{\mathbb{F}_{p}}^{\heartsuit}}(R^{\flat},D/p)$

(here everything is discrete therefore classical, but in order to avoid
conflicts of notations with other parts of the article, we retain the
cumbersome notations $\operatorname{CAlg}_{\mathbb{Z}_{p}}^{\heartsuit}$ and
$\operatorname{CAlg}_{\mathbb{F}_{p}}^{\heartsuit}$) which is given as
follows: for any map $W(R^{\flat})\rightarrow D$ of discrete
$\mathbb{Z}_{p}$-algebras, we compose it with the canonical map $D\rightarrow
D/p$ to get the map $W(R^{\flat})\rightarrow D/p$, which factors uniquely
through $W(R^{\flat})\rightarrow W(R^{\flat})/p\cong R^{\flat}$ therefore
gives rise to a map $R^{\flat}\rightarrow D/p$. Note that
$\theta_{R}=\operatorname{id}_{R}:R\rightarrow R$ serves as a special choice
of $\theta_{D}$ since the perfectoid ring $R$ is $p$-adically complete by
Definition 4.10 and tautologically $\ker(\theta_{R})=(0)$-adically complete.
That is to say, we also have a bijection

$\operatorname{Hom}_{\operatorname{CAlg}_{\mathbb{Z}_{p}}^{\heartsuit}}(W(R^{\flat}),R)\xrightarrow{\cong}\operatorname{Hom}_{\operatorname{CAlg}_{\mathbb{F}_{p}}^{\heartsuit}}(R^{\flat},R/p)$

The map $\theta_{D}:D\rightarrow R$ gives rise to a commutative diagram

$\begin{array}[]{ccc}\operatorname{Hom}_{\operatorname{CAlg}_{\mathbb{Z}_{p}}^{\heartsuit}}(W(R^{\flat}),D)&\xrightarrow{\cong}&\operatorname{Hom}_{\operatorname{CAlg}_{\mathbb{F}_{p}}^{\heartsuit}}(R^{\flat},D/p)\\\
{\mbox{\rotatebox[origin={c}]{-90.0}{$\longrightarrow$}}}&&{\mbox{\rotatebox[origin={c}]{-90.0}{$\longrightarrow$}}}\\\
\operatorname{Hom}_{\operatorname{CAlg}_{\mathbb{Z}_{p}}^{\heartsuit}}(W(R^{\flat}),R)&\xrightarrow{\cong}&\operatorname{Hom}_{\operatorname{CAlg}_{\mathbb{F}_{p}}^{\heartsuit}}(R^{\flat},R/p)\end{array}$

So in order to show that the map $\theta_{D}:D\rightarrow R$ factors through
the canonical map $\theta$, or equivalently put, the preimage of the element
$\theta\in\operatorname{Hom}_{\operatorname{CAlg}_{\mathbb{Z}_{p}}^{\heartsuit}}(W(R^{\flat}),R)$
under the induced map
$\operatorname{Hom}_{\operatorname{CAlg}_{\mathbb{Z}_{p}}^{\heartsuit}}(W(R^{\flat}),D)\rightarrow\operatorname{Hom}_{\operatorname{CAlg}_{\mathbb{Z}_{p}}^{\heartsuit}}(W(R^{\flat}),R)$
is a singleton, it suffices to show that the preimage of the element
$(R^{\flat}\rightarrow
R/p)\in\operatorname{Hom}_{\operatorname{CAlg}_{\mathbb{F}_{p}}^{\heartsuit}}(R^{\flat},R/p)$
under the map
$\operatorname{Hom}_{\operatorname{CAlg}_{\mathbb{F}_{p}}^{\heartsuit}}(R^{\flat},D/p)\rightarrow\operatorname{Hom}_{\operatorname{CAlg}_{\mathbb{F}_{p}}^{\heartsuit}}(R^{\flat},R/p)$
is a singleton, or equivalently put, the canonical map
$\sigma:R^{\flat}\rightarrow R/p$ lifts uniquely through the map
$\sigma_{D}:D/p\rightarrow R/p$ induced by the map $\theta_{D}:D\rightarrow
R$. Note that the surjectivity of $\theta_{D}$ implies that of $\sigma_{D}$.
Since the ring $D$ is $(p,\ker\theta_{D})$-adically complete, the ring $D/p$
is $\ker\sigma_{D}$-adically complete.

We can conclude the existence and the uniqueness of lift of the map
$\sigma:R^{\flat}\rightarrow R/p$ along the surjection
$\sigma_{D}:D/p\rightarrow R/p$ simply by the fact that the
$\mathbb{F}_{p}$-algebra $R^{\flat}$ is perfect and thus the cotangent complex
$\mathbb{L}_{R^{\flat}/\mathbb{F}_{p}}$ is contractible, which implies the
existence and the uniqueness of such lift.

However, we prefer to give a direct argument: We set the
$\mathbb{F}_{p}$-algebra $A:=R^{\flat}$ to stress that we only depend on the
fact that $A$ is a perfect $\mathbb{F}_{p}$-algebra, but not on the properties
of the map $\sigma:A\rightarrow R/p$. Denote by $\varphi_{B}:B\rightarrow
B,x\mapsto x^{p}$ the Frobenius map on any $\mathbb{F}_{p}$-algebra $A$. Then
the Frobenius map $\varphi_{A}$ is an isomorphism by assumption.

For each $a\in A$, we choose a sequence
$(b_{n})_{n=0}^{\infty}\in(D/p)^{\mathbb{N}}$ such that for each
$n\in\mathbb{N}$, we have $\sigma_{D}(b_{n})=\sigma(\varphi_{A}^{-n}(a))$.

Note that the sequence $(\varphi_{D/p}^{n}(b_{n}))_{n=0}^{\infty}$ converges
$\ker\sigma_{D}$-adically:
$\sigma_{D}(b_{n}-b_{n+1}^{p})=\sigma_{D}(b_{n})-\sigma_{D}(b_{n+1})^{p}=\sigma(\varphi_{A}^{-n}(a))-\sigma(\varphi_{A}^{-(n+1)}(a))^{p}=\sigma(\varphi_{A}^{-n}(a))-\sigma(\varphi_{A}(\varphi_{A}^{-n}(a)))=0$
and therefore
$\varphi_{D/p}^{n}(b_{n})-\varphi_{D/p}^{n+1}(b_{n+1})=\varphi_{D/p}^{n}(b_{n}-b_{n+1}^{p})\in\varphi_{D/p}^{n}(\ker\sigma_{D})\subseteq(\ker\sigma_{D})^{p^{n}}$.
Let $b:=\lim_{n\rightarrow\infty}\varphi_{D/p}^{n}(b_{n})$.

We first note that $\sigma_{D}(b)=\sigma(a)$, since
$\sigma(\varphi_{D/p}^{n}(b_{n}))=\sigma(b_{n}^{p^{n}})=\sigma(b_{n})^{p^{n}}=\sigma(\varphi_{A}^{-n}(a))^{p^{n}}=\sigma(\varphi_{A}^{n}(\varphi_{A}^{-n}(a)))=\sigma(a)$
for all $n\in\mathbb{N}$.

Now the value $b\in D/p$ does not depend on the choice of $(b_{n})$, since for
any other choice $(c_{n})$, we have $c_{n}-b_{n}\in\ker\sigma_{D}$, thus
$\varphi_{D/p}^{n}(b_{n})-\varphi_{D/p}^{n}(c_{n})=\varphi_{D/p}^{n}(b_{n}-c_{n})\in\varphi_{D/p}^{n}(\ker\sigma_{D})\subseteq(\ker\sigma_{D})^{p^{n}}$
which implies that $\lim_{n\rightarrow\infty}\varphi_{D/p}^{n}(c_{n})=b$.

Combining the preceding discussions, we have shown that for each $a\in A$, we
can associate a $b\in D/p$ such that $\sigma_{D}(b)=\sigma(a)$. It is routine
to check that $a\mapsto b$ defines a map $A\rightarrow D/p$ of rings which
serves as a lift of $\sigma:A\rightarrow R/p$. Furthermore, the uniqueness
essentially follows from the above argument that the value $b\in D/p$ does not
depend on the choice of $(b_{n})$. ∎

######  Remark 4.17.

In fact, we can weaken our assumption on $D$ to be derived $p$-complete and
that the map $D\rightarrow R$ is Adams complete (due to [Car08] while the
terminology is coined in [Bha12]) by using some basic facts about Adams
complete surjective maps of animated rings.

Now we give a spherical version of Fontaine’s universal property:

######  Proposition 4.18.

Let $R$ be a perfectoid ring. We compose Fontaine’s map
$\theta:W(R^{\flat})\rightarrow R$ with the 0th Postnikov section
$W^{+}(R^{\flat})\rightarrow\tau_{\leq 0}(W^{+}(R^{\flat}))=HW(R^{\flat})$,
obtaining the map $\eta:W^{+}(R^{\flat})\rightarrow HR$. Then we have

  1. 1.

The $\mathbb{E}_{\infty}$-$\mathbb{S}_{p}^{\wedge}$-algebra $W^{+}(R^{\flat})$
of spherical Witt vectors is $(p,\ker\theta)$-complete. Furthermore, the
discrete ring $\pi_{0}(W^{+}(R^{\flat}))/p$ is
$\ker\theta/(p\pi_{0}(W^{+}(R^{\flat}))+\ker\theta)$-adically separated.

  2. 2.

The map $\eta:W^{+}(R^{\flat})\rightarrow HR$ is initial among all maps
$\eta_{D}:D\rightarrow HR$ surjective on $\pi_{0}$ where $D$ is an
$\mathbb{E}_{\infty}$-$\mathbb{S}_{p}^{\wedge}$-algebra such that $D$ is
$(p,\ker\eta_{D})$-complete and the discrete ring $\pi_{0}(D)/p$ is
$\ker\theta_{D}/(p\pi_{0}(D)+\ker\theta_{D})$-adically separated, where we
denote the map $\pi_{0}(\eta_{D}):\pi_{0}(D)\rightarrow R$ by $\theta_{D}$.

######  Remark 4.19.

In Proposition 4.18, the technical conditions imposed on the
$\mathbb{E}_{\infty}$-$\mathbb{S}_{p}^{\wedge}$-algebra $D$ are somewhat
complicated. However, we can restrict to the full subcategory of $\eta_{D}$
such that $\pi_{0}(D)$ is $(p,\ker\eta_{D})$-adically complete, where
$\eta:W^{+}(R^{\flat})\rightarrow HR$ lives (see the proof of Proposition
4.18) and hence $\eta$ is still an initial object in this full subcategory.

######  Remark 4.20.

Using Remark 4.17, we can drop the adic completeness of $\pi_{0}(D)/p$ in
Proposition 4.18.

Now we want to establish some computational results about homotopy groups of
the ring $W^{+}(k)$ of spherical Witt vectors of a perfect
$\mathbb{F}_{p}$-algebra $k$. First, we need the following proposition, which
follows from Serre’s computations of homotopy groups of spheres:

######  Proposition 4.21.

The sphere spectrum $\mathbb{S}$ is connective,
$\pi_{0}(\mathbb{S})=\mathbb{Z}$, and for all $n\in\mathbb{N}_{>0}$, the $n$th
(stable) homotopy group $\pi_{n}(\mathbb{S})$ is finite.

Thus for each $n\in\mathbb{N}$, the homotopy group $\pi_{n}(\mathbb{S})$ has
bounded $p$-torsion. Combined with Milnor sequence of homotopy groups, we have

######  Corollary 4.22.

The $p$-adic sphere spectrum $\mathbb{S}_{p}^{\wedge}$ is connective,
$\pi_{0}(\mathbb{S}_{p}^{\wedge})=\mathbb{Z}_{p}$ and for all
$n\in\mathbb{N}_{>0}$, the $n$th (stable) homotopy group
$\pi_{n}(\mathbb{S}_{p}^{\wedge})$ is a finite direct sum of finite abelian
groups of form $\mathbb{Z}/p^{r}\cong\mathbb{Z}_{p}/p^{r}$ for some positive
integer $r\in\mathbb{N}_{>0}$.

We need a result announced in [Lur18a, Example 5.2.7] the argument of which we
learn from Matthew Morrow:

######  Proposition 4.23.

Let $k$ be a perfect $\mathbb{F}_{p}$-algebra. Then the ring of spherical Witt
vectors $W^{+}(k)$ is a flat $\mathbb{S}_{p}^{\wedge}$-module.

###### Proof.

First, by Proposition 2.7, $\pi_{0}(W^{+}(k))\cong W(k)$ which is a torsion-
free $\mathbb{Z}_{p}$-module. Since $\mathbb{Z}_{p}$ is a valuation ring, we
deduce that $W(k)$ is a flat $\mathbb{Z}_{p}$-module (see [Sta21, Tag 0539]).
Now we consider the Postnikov tower $(\tau_{\geq
n}\mathbb{S}_{p}^{\wedge})_{n\in\mathbb{N}}$ of the $p$-adic sphere spectrum
$\mathbb{S}_{p}^{\wedge}$, which induces a tower $X_{n}:=(\tau_{\geq
n}\mathbb{S}_{p}^{\wedge})\otimes_{\mathbb{S}_{p}^{\wedge}}W^{+}(k)$. Note
that
$X_{n}/X_{n-1}\cong\Sigma^{n}H\pi_{n}(\mathbb{S}_{p}^{\wedge})\otimes_{\mathbb{S}_{p}^{\wedge}}W^{+}(k)$.
We have shown in Corollary 4.22 that $\pi_{n}(\mathbb{S}_{p}^{\wedge})$ is a
direct sum of finite abelian groups of form $\mathbb{Z}_{p}/p^{r}$, which
allows us to realize $H\pi_{n}(\mathbb{S}_{p}^{\wedge})$ as a direct sum of
spectra of form
$\operatorname{cofib}\left(H\mathbb{Z}_{p}\xrightarrow{p^{r}}H\mathbb{Z}_{p}\right)$.
Note that the smash product $-\otimes_{\mathbb{S}_{p}^{\wedge}}W^{+}(k)$
commutes with taking cofibers, we deduce that
$H\pi_{n}(\mathbb{S}_{p}^{\wedge})\otimes_{\mathbb{S}_{p}^{\wedge}}W^{+}(k)\cong
H\operatorname{Tor}_{0}^{\mathbb{Z}_{p}}(\pi_{n}(\mathbb{S}_{p}^{\wedge}),W(k))$.
Thus the tower $(X_{n})_{n\in\mathbb{N}}$ constitutes the Postnikov tower of
the spectrum $W^{+}(k)$, therefore
$\pi_{n}(W^{+}(k))\cong\operatorname{Tor}_{0}^{\pi_{0}(\mathbb{S}_{p}^{\wedge})}(\pi_{n}(\mathbb{S}_{p}^{\wedge}),W(k))$.
∎

######  Corollary 4.24.

Let $k$ be a perfect $\mathbb{F}_{p}$-algebra. Then the ring of spherical Witt
vectors $W^{+}(k)$ is connective, $\pi_{0}(W^{+}(k))=W(k)$, and for all
$n\in\mathbb{N}_{>0}$, the $n$th (stable) homotopy group $\pi_{n}(W^{+}(k))$
is a finite direct sum of $W(k)$-modules of form $W(k)/p^{r}$.

We are now ready to prove Proposition 4.18:

Proof of Proposition 4.18 We check two statements one by one:

  1. 1.

Proposition 4.16 tells us that the discrete ring
$\pi_{0}(W^{+}(R^{\flat}))\cong W(R^{\flat})$ is $(p,\ker\theta)$-adically
complete, therefore by Proposition A.29, it is $(p,\ker\theta)$-complete.
Furthermore, we deduce from $(p,\ker\theta)$-adic completeness that
$\pi_{0}(W^{+}(R^{\flat}))$ is
$\ker\theta/(p\pi_{0}(W^{+}(R^{\flat}))+\ker\theta)$-adically separated. In
view of Theorem A.25, it remains to show that for each $n\in\mathbb{N}_{>0}$,
the homotopy group $\pi_{n}(W^{+}(R^{\flat}))$ is (derived)
$(p,\ker\theta)$-complete as a discrete $W(R^{\flat})$-module. However, by
Corollary 4.24, we have realized $\pi_{n}(W^{+}(R^{\flat}))$ as a direct sum
of cofibers of $(p,\ker\theta)$-complete modules, therefore it is
$(p,\ker\theta)$-complete.

  2. 2.

This part is parallel to the proof of Proposition 4.16. We start with the
following commutative diagram induced by the map $\eta_{D}:D\rightarrow HR$:

$\begin{array}[]{ccc}\operatorname{Map}_{\operatorname{CAlg}_{\mathbb{S}_{p}^{\wedge}}}(W^{+}(R^{\flat}),D)&\xrightarrow{\simeq}&\operatorname{Hom}_{\operatorname{CAlg}_{\mathbb{F}_{p}}^{\heartsuit}}(R^{\flat},\pi_{0}(D)/p)\\\
{\mbox{\rotatebox[origin={c}]{-90.0}{$\longrightarrow$}}}&&{\mbox{\rotatebox[origin={c}]{-90.0}{$\longrightarrow$}}}\\\
\operatorname{Map}_{\operatorname{CAlg}_{\mathbb{S}_{p}^{\wedge}}}(W^{+}(R^{\flat}),HR)&\xrightarrow{\simeq}&\operatorname{Hom}_{\operatorname{CAlg}_{\mathbb{F}_{p}}^{\heartsuit}}(R^{\flat},R/p)\end{array}$

as in the proof of Proposition 4.16. It follows from Definition 2.1 and
Example 2.6 that the horizontal maps are homotopy equivalences, which implies
that the connected components of each space on the left are all contractible.
We pick the connected component of
$\operatorname{Map}_{\operatorname{CAlg}_{\mathbb{S}_{p}^{\wedge}}}(W^{+}(R^{\flat}),HR)$
corresponds to the map $\eta:W^{+}(R^{\flat})\rightarrow HR$. In order to show
that $\eta$ is an initial object, it suffices to show that there exists one
and only one connected component in
$\operatorname{Map}_{\operatorname{CAlg}_{\mathbb{S}_{p}^{\wedge}}}(W^{+}(R^{\flat}),D)$
which maps to the connect component corresponding to $\eta$. Note that the
image of $\eta$ in
$\operatorname{Hom}_{\operatorname{CAlg}_{\mathbb{F}_{p}}^{\heartsuit}}(R^{\flat},R/p)$
along the bottom horizontal map coincides with
$\sigma\in\operatorname{Hom}_{\operatorname{CAlg}_{\mathbb{F}_{p}}^{\heartsuit}}(R^{\flat},R/p)$
defined in the proof of Proposition 4.16. In view of the commutative diagram,
it remains to show that the preimage of
$\sigma\in\operatorname{Hom}_{\operatorname{CAlg}_{\mathbb{F}_{p}}^{\heartsuit}}(R^{\flat},R/p)$
in
$\operatorname{Hom}_{\operatorname{CAlg}_{\mathbb{F}_{p}}^{\heartsuit}}(R^{\flat},\pi_{0}(D)/p)$.
The rest of the proof is identical to that of Proposition 4.16.

$\Box$

##  5\. Proof of the main theorem

Fix a perfectoid ring $R$ and a generator $\xi$ of Fontaine’s map
$\theta:W(R^{\flat})\rightarrow R$, the goal of this section is to prove
Theorem 1.13. We first need a much weaker version which says that the 0th
homotopy group of the $\mathbb{E}_{2}$-Thom spectrum in question, as a ring,
is isomorphic to $R$:

######  Lemma 5.1.

The 0th homotopy group $\pi_{0}(Mf_{R,\xi})$ of the Thom spectrum associated
to $f_{R,\xi}$ is isomorphic to $R$ for any perfectoid ring $R$.

###### Proof.

We mimic a segment of the proof of Theorem A.1 in [KN]:

We note that $Mf_{R,\xi}$ is connective, so we have

$\pi_{0}(Mf_{R,\xi})\cong\pi_{0}(W^{+}(R^{\flat})_{h\Omega^{3}S^{3}})\cong\pi_{0}(W^{+}(R^{\flat}))_{\pi_{0}(\Omega^{3}S^{3})}$

where the $\pi_{0}(\Omega^{3}S^{3})\cong\mathbb{Z}$-action on
$\pi_{0}(W^{+}(R^{\flat}))\cong W(R^{\flat})$ is given by multiplication by
$1-\xi$, hence

$\pi_{0}(W^{+}(R^{\flat}))_{\pi_{0}(\Omega^{3}S^{3})}\cong
W(R^{\flat})/(1-(1-\xi))\cong R$

∎

In view of Lemma 5.1, in order to prove Theorem 1.13, it suffices to show that

######  Proposition 5.2.

The 0th Postnikov section $t_{R,\xi}:Mf_{R,\xi}\rightarrow\tau_{\leq
0}Mf_{R,\xi}\simeq HR$, being an $\mathbb{E}_{2}$-map a priori, is an
equivalence of spectra.

To begin with, we first note that the special case when $R$ is a perfect
$\mathbb{F}_{p}$-algebra is already covered by previous considerations:

######  Lemma 5.3.

The $t_{R,\xi}$ in question is an equivalence of spectra when $R$ is a perfect
$\mathbb{F}_{p}$-algebra.

###### Proof.

Theorem 3.1 tells us that there is an equivalence $Mf_{R,\xi}\rightarrow HR$.
The lemma follows from the fact that $HR$ lives in
$(\operatorname{Mod}_{W^{+}(R)})_{\leq 0}$ and that the 0th Postnikov section
is essentially unique. ∎

We first note that both $Mf_{R,\xi}$ and $HR$ admit canonical
$W^{+}(R^{\flat})$-module structures. Our strategy breaks up into several
steps:

  1. 1.

Prove some finiteness and completeness results of $Mf_{R,\xi}$ and $HR$ as
$W^{+}(R^{\flat})$-modules;

  2. 2.

Show that $t_{R,\xi}$ is an equivalence after the base change along
$W^{+}(R^{\flat})\rightarrow W^{+}(\kappa)$, and hence an equivalence after a
further base change along $W^{+}(\kappa)\rightarrow H\kappa$ to the special
fiber $H\kappa$;

  3. 3.

The composition $W^{+}(R^{\flat})\rightarrow W^{+}(\kappa)\rightarrow H\kappa$
coincides with the composition $W^{+}(R^{\flat})\rightarrow
HR^{\flat}\rightarrow H\kappa$, and a Nakayama-like argument shows that
$t_{R,\xi}$ is an equivalence after base change along
$W^{+}(R^{\flat})\rightarrow HR^{\flat}$;

  4. 4.

Deduce that $t_{R,\xi}$ is an equivalence by completeness.

To proceed, by Corollary 4.14, we also fix an invertible element
$u\in\operatorname{GL}_{1}(W(\kappa))$ associated to $\xi$ so that the image
of $\xi$ in $W(\kappa)$ is $pu$.

###  5.1. Finiteness and completeness of $Mf_{R,\xi}$ and $HR$ as
$W^{+}(R^{\flat})$-modules

######  Lemma 5.4.

$HW(k)$ is an almost perfect $W^{+}(k)$-module for any perfect
$\mathbb{F}_{p}$-algebra $k$.

###### Proof.

If $k=\mathbb{F}_{p}$, then
$W^{+}(\mathbb{F}_{p})\simeq\mathbb{S}_{p}^{\wedge}$ is a coherent ring as in
Definition A.10, and $HW(\mathbb{F}_{p})\simeq H\mathbb{Z}_{p}\simeq
H\pi_{0}(W^{+}(\mathbb{F}_{p}))$ is an almost perfect
$\mathbb{S}_{p}^{\wedge}$-module by Corollary A.12.

In general, by Proposition 2.7, we have $HW(k)\simeq
W^{+}(k)\otimes_{\mathbb{S}_{p}^{\wedge}}H\mathbb{Z}_{p}$, hence $HW(k)$ is
almost perfect by Proposition A.8. ∎

######  Corollary 5.5.

$HR$ is an almost perfect $W^{+}(R^{\flat})$-module.

###### Proof.

$HR$ is the cofiber of the multiplication map
$m_{\xi}:HW(R^{\flat})\rightarrow HW(R^{\flat})$ where the domain and the
codomain are almost perfect (Lemma 5.4), hence $HR$ is also almost perfect
(Proposition A.7). ∎

We need a nontrivial input from algebraic topology:

######  Proposition 5.6.

There exists a Kan complex $X_{\bullet}$ which is homotopy equivalent to the
double loop space $\Omega^{2}S^{3}$ of the 3-sphere such that $X_{n}$ is a
finite set for each $[n]\in\Delta^{\operatorname{op}}$.

###### Proof.

This is essentially due to [Wal65, Thm A and B] and Serre. We first note that,
the loop space $\Omega^{2}S^{3}$ is a loop space therefore simple [MP12, Cor
1.4.5]. Now we show that $\Omega^{2}S^{3}$ is of finite type, i.e. homotopy
equivalent to a CW-complex with finite skeleta. By [MP12, Thm 4.5.2], it
suffices to show that $H_{i}(\Omega^{2}S^{3};\mathbb{Z})$ are finitely
generated for all $i\in\mathbb{N}_{>0}$. The argument is standard (due to
Serre): we know that $H_{i}(S^{3};\mathbb{Z})$ are finitely generated for all
$i\in\mathbb{N}$. Applying [tD08, Thm 20.4.1] to the fiber sequence $\Omega
S^{3}\rightarrow\mathord{\ast}\rightarrow S^{3}$ in $\mathcal{S}$, we deduce
that $H_{i}(\Omega S^{3})$ are finitely generated for all $i\in\mathbb{N}$. We
apply again [tD08, Thm 20.4.1] to the fiber sequence
$\Omega^{2}S^{3}\rightarrow\mathord{\ast}\rightarrow\Omega S^{3}$, we deduce
that $H_{i}(\Omega^{2}S^{3})$ are finitely generated. Now the result follows
from the simplicial approximation theorem. ∎

######  Lemma 5.7.

$Mf_{R,\xi}$ is an almost perfect $W^{+}(R^{\flat})$-module.

###### Proof.

We first pick up a Kan complex $X_{\bullet}$ representing $\Omega^{2}S^{3}$
where each $X_{n}$ is a finite set as in Proposition 5.6. It follows from
Bousfield-Kan formula (see, for example, Corollary 12.3 in [Sha18]) that
$Mf_{R,\xi}$ could be written as the geometric realization of a simplicial
object $N_{\bullet}$ where each $N_{n}$ is a free $W^{+}(R^{\flat})$-module of
finite rank, hence almost perfect by Proposition A.7. ∎

######  Corollary 5.8.

$\operatorname{cofib}(t_{R,\xi})$ is an almost perfect
$W^{+}(R^{\flat})$-module.

###### Proof.

The subcategory of almost perfect modules are closed under taking cofibers and
base changes (Proposition A.7). The statement then follows from Corollary 5.5
and Lemma 5.7. ∎

######  Lemma 5.9.

The spectrum $HR$ is $p$-complete.

###### Proof.

By definition of perfectoid rings, $R$ is $p$-adically complete, therefore
$HR$ is $p$-complete by Proposition A.29. ∎

######  Lemma 5.10.

The spectrum $Mf_{R,\xi}$ is $p$-complete.

###### Proof.

We note that $W^{+}(R^{\flat})$ is $p$-complete by definition of spherical
Witt vectors, and $Mf_{R,\xi}$ is almost perfect, therefore $p$-complete by
Proposition A.27. ∎

######  Corollary 5.11.

The spectrum $\operatorname{cofib}(t_{R,\xi})$ is $p$-complete.

###### Proof.

It follows from Corollary 5.8 and Proposition A.27. ∎

###  5.2. $t_{R,\xi}$ is an equivalence after the base change along
$W^{+}(R^{\flat})\rightarrow W^{+}(\kappa)$

The proof is similar to that of Theorem 3.1, except that we need to be more
careful to identify the maps.

######  Lemma 5.12.

There is a canonical equivalence
$Mf_{\kappa,pu}\xrightarrow{\simeq}W^{+}(\kappa)\otimes_{W^{+}(R^{\flat})}Mf_{R,\xi}$
of $W^{+}(\kappa)$-modules.

###### Proof.

We first note that the image of the multiplication map
$m_{1-\xi}:W^{+}(R^{\flat})\rightarrow W^{+}(R^{\flat})$ under the base change
functor
$W^{+}(\kappa)\otimes_{W^{+}(R^{\flat})}-:\operatorname{Mod}_{W^{+}(R^{\flat})}\rightarrow\operatorname{Mod}_{W^{+}(\kappa)}$
is the multiplication map $m_{1-pu}:W^{+}(\kappa)\rightarrow W^{+}(\kappa)$.

Therefore $f_{\kappa,pu}$ coincides with the composition

$\Omega^{2}S^{3}\xrightarrow{f_{R,\xi}}\operatorname{BGL}_{1}(W^{+}(R^{\flat}))\xrightarrow{W^{+}(\kappa)\otimes_{W^{+}(R^{\flat})}-}\operatorname{BGL}_{1}(W^{+}(\kappa))$

Along with the fact that the functor
$W^{+}(\kappa)\otimes_{W^{+}(R^{\flat})}-:\operatorname{Mod}_{W^{+}(R^{\flat})}\rightarrow\operatorname{Mod}_{W^{+}(\kappa)}$
commutes with small colimits, or to be more precise, that the natural
transformation
$\operatorname{colim}_{i}(W^{+}(\kappa)\otimes_{W^{+}(R^{\flat})}M_{i})\rightarrow
W^{+}(\kappa)\otimes_{W^{+}(R^{\flat})}(\operatorname{colim}_{i}M_{i})$ is an
equivalence for any diagram $(M_{i})_{i}$ in
$\operatorname{Mod}_{W^{+}(R^{\flat})}$, we deduce that there is a canonical
equivalence
$Mf_{\kappa,pu}\xrightarrow{\simeq}W^{+}(\kappa)\otimes_{W^{+}(R^{\flat})}Mf_{R,\xi}$
as $W^{+}(\kappa)$-modules. ∎

######  Lemma 5.13.

Given a morphism of perfect $\mathbb{F}_{p}$-algebras $k\rightarrow K$, the
commutative diagram of $\mathbb{E}_{\infty}$-rings

$\begin{array}[]{ccc}W^{+}(k)&\longrightarrow&W^{+}(K)\\\
{\mbox{\rotatebox[origin={c}]{-90.0}{$\longrightarrow$}}}&&{\mbox{\rotatebox[origin={c}]{-90.0}{$\longrightarrow$}}}\\\
HW(k)&\longrightarrow&HW(K)\end{array}$

is a pushout square.

###### Proof.

Consider the commutative diagram of $\mathbb{E}_{\infty}$-rings

$\begin{array}[]{ccccc}\mathbb{S}_{p}^{\wedge}&\longrightarrow&W^{+}(k)&\longrightarrow&W^{+}(K)\\\
{\mbox{\rotatebox[origin={c}]{-90.0}{$\longrightarrow$}}}&&{\mbox{\rotatebox[origin={c}]{-90.0}{$\longrightarrow$}}}&&{\mbox{\rotatebox[origin={c}]{-90.0}{$\longrightarrow$}}}\\\
H\mathbb{Z}_{p}&\longrightarrow&HW(k)&\longrightarrow&HW(K)\end{array}$

By Proposition 2.7, we know that the left square and the outer square are
pushout squares, therefore so is the right square. ∎

######  Lemma 5.14.

There is a canonical equivalence
$W^{+}(\kappa)\otimes_{W^{+}(R^{\flat})}HR\rightarrow H\kappa$ of
$W^{+}(\kappa)$-modules.

###### Proof.

Combining two pushout squares in the category of $\mathbb{E}_{\infty}$-rings:

$\begin{array}[]{ccc}W^{+}(R^{\flat})&\longrightarrow&W^{+}(\kappa)\\\
{\mbox{\rotatebox[origin={c}]{-90.0}{$\longrightarrow$}}}&\sigma&{\mbox{\rotatebox[origin={c}]{-90.0}{$\longrightarrow$}}}\\\
HW(R^{\flat})&\longrightarrow&HW(\kappa)\\\
{\mbox{\rotatebox[origin={c}]{-90.0}{$\longrightarrow$}}}&\tau&{\mbox{\rotatebox[origin={c}]{-90.0}{$\longrightarrow$}}}\\\
HR&\longrightarrow&H\kappa\end{array}$

where $\sigma$ is a pushout square by Lemma 5.13 and $\tau$ is a pushout
square by Proposition 4.13. ∎

######  Lemma 5.15.

The map
$W^{+}(\kappa)\otimes_{W^{+}(R^{\flat})}t_{R,\xi}:W^{+}(\kappa)\otimes_{W^{+}(R^{\flat})}Mf_{R,\xi}\rightarrow
W^{+}(\kappa)\otimes_{W^{+}(R^{\flat})}HR$ is equivalent to
$t_{\kappa,pu}:Mf_{\kappa,pu}\rightarrow H\kappa$.

###### Proof.

In view of Lemma 5.12 and Lemma 5.14, we only need to show that
$t_{\kappa,pu}:Mf_{\kappa,pu}\rightarrow H\kappa$ coincides with the
composition of the equivalences $Mf_{\kappa,pu}\rightarrow
W^{+}(\kappa)\otimes_{W^{+}(R^{\flat})}Mf_{R,\xi}$,
$W^{+}(\kappa)\otimes_{W^{+}(R^{\flat})}t_{R,\xi}$ and
$W^{+}(\kappa)\otimes_{W^{+}(R^{\flat})}HR\rightarrow H\kappa$. In other
words, it suffices to show that the composition in question is the 0th
Postnikov section. We only need to check that the composition induces an
isomorphism on $\pi_{0}$ by basic properties of $t$-structures, since
$\tau_{\leq 0}Mf_{\kappa,pu}\simeq H\kappa$. It suffices to show that
$W^{+}(\kappa)\otimes_{W^{+}(R^{\flat})}t_{R,\xi}$ induces an isomorphism on
$\pi_{0}$, and this follows from the fact that all spectra in question are
connective and that $t_{R,\xi}$ induces an isomorphism on $\pi_{0}$ by Lemma
5.1. ∎

######  Corollary 5.16.

$H\kappa\otimes_{W^{+}(R^{\flat})}t_{R,\xi}$ is an equivalence of spectra.

###### Proof.

It follows from Lemma 5.15 and Lemma 5.3. ∎

###  5.3. $t_{R,\xi}$ is an equivalence after the base change along
$W^{+}(R^{\flat})\rightarrow HR^{\flat}$

######  Lemma 5.17.

Let $M$ be an $HR^{\flat}$-module which is bounded below and almost perfect.
If there exists an $r\in\mathbb{N}$ such that $\xi^{r}\pi_{n}(M)=0$ for all
$n\in\mathbb{Z}$, and $H\kappa\otimes_{HR^{\flat}}M\simeq 0$, then $M\simeq
0$.

###### Proof.

We show inductively on $n$ that $\pi_{n}M=0$.

  * $\bullet$

Since $M$ is bounded below, $\pi_{n}M=0$ for $n\ll 0$;

  * $\bullet$

Suppose that for $m<n$ we have $\pi_{m}M=0$. Then by unrolling Definition A.6,
$\pi_{n}M$ is a compact object in the category of discrete
$R^{\flat}$-modules, therefore is finitely presented and in particular
finitely generated. Now we have

$0=\pi_{n}(H\kappa\otimes_{HR^{\flat}}M)=\operatorname{Tor}_{0}^{R^{\flat}}(\kappa,\pi_{n}M).$

By Proposition 4.15 and that $\xi^{r}\pi_{n}(M)=0$, we have

$\operatorname{Tor}_{0}^{R^{\flat}}(\kappa,\pi_{n}M)=\operatorname{Tor}_{0}^{R^{\flat}/\xi^{r}}(\kappa,\pi_{n}M)$

We note that $\ker(R^{\flat}/\xi^{r}\rightarrow\kappa)$ lies in the (nil-
radical, therefore) Jacobson radical of $R^{\flat}/\xi^{r}$, thus
$\pi_{n}M=0$, by Nakayama’s lemma along with the fact that $\pi_{n}M$ is
finitely generated.

∎

######  Remark 5.18.

Matthew Morrow told us that in Lemma 5.17, the hypothesis
$\xi^{r}\pi_{n}(M)=0$ is redundant, since the kernel of the map
$R^{\flat}\rightarrow\kappa$ of $\mathbb{F}_{p}$-algebras lies in the radical
of the ideal $\xi R^{\flat}\subseteq\operatorname{Rad}(R^{\flat})$ where
$\operatorname{Rad}(R^{\flat})$ is the Jacobson radical of the
$\mathbb{F}_{p}$-algebra $R^{\flat}$ and the inclusion $\xi
R^{\flat}\subseteq\operatorname{Rad}(R^{\flat})$ is deduced from the $\xi
R^{\flat}$-adically completeness of the $\mathbb{F}_{p}$-algebra $R^{\flat}$.
Since the Jacobson radical is “radical”, the kernel of the map
$R^{\flat}\rightarrow\kappa$ also lies in the Jacobson radical
$\operatorname{Rad}(R^{\flat})$. We decide to preserve the original version to
reflect our real thoughts.

######  Corollary 5.19.

$HR^{\flat}\otimes_{W^{+}(R^{\flat})}t_{R,\xi}$ is an equivalence of spectra.

###### Proof.

Note that

$\pi_{0}(HR^{\flat}\otimes_{W^{+}(R^{\flat})}Mf_{R,\xi})=\operatorname{Tor}_{0}^{W(R^{\flat})}(R^{\flat},\pi_{0}(Mf_{R,\xi}))=R^{\flat}/\xi
R^{\flat}$

and

$\pi_{0}(HR^{\flat}\otimes_{W^{+}(R^{\flat})}HR)=\operatorname{Tor}_{0}^{W(R^{\flat})}(R^{\flat},R)=R^{\flat}/\xi
R^{\flat}$

and that $HR^{\flat}\otimes_{W^{+}(R^{\flat})}Mf_{R,\xi}$,
$HR^{\flat}\otimes_{W^{+}(R^{\flat})}HR$ are connective
$\mathbb{E}_{\infty}$-rings, we conclude that the homotopy groups of these
$\mathbb{E}_{\infty}$-rings are $\xi$-torsion groups, which implies that for
all $n\in\mathbb{Z}$,

$\xi^{2}\pi_{n}(\operatorname{cofib}(HR^{\flat}\otimes_{W^{+}(R^{\flat})}t_{R,\xi}))=0$

In addition, since the subcategory of almost perfect modules are closed under
base changes (Proposition A.8), we deduce from Corollary 5.8 that
$\operatorname{cofib}(HR^{\flat}\otimes_{W^{+}(R^{\flat})}t_{R,\xi})\simeq
HR^{\flat}\otimes_{W^{+}(R^{\flat})}\operatorname{cofib}(t_{R,\xi})$ is almost
perfect. On the other hand, being the cofiber of a map of connective spectra,
it is also connective. Then we invoke Lemma 5.17 along with Corollary 5.16 to
conclude that
$\operatorname{cofib}(HR^{\flat}\otimes_{W^{+}(R^{\flat})}t_{R,\xi})\simeq 0$.
∎

###  5.4. Conclude: $t_{R,\xi}$ is an equivalence

We are now at the final stage to conclude a proof of Proposition 5.2, and
consequently, Theorem 1.13.

Proof of Proposition 5.2 We recall that by Theorem 2.4 and Example 2.6, there
is a pushout square of $\mathbb{E}_{\infty}$-rings:

$\begin{array}[]{ccc}\mathbb{S}_{p}^{\wedge}&\longrightarrow&W^{+}(R^{\flat})\\\
{\mbox{\rotatebox[origin={c}]{-90.0}{$\longrightarrow$}}}&&{\mbox{\rotatebox[origin={c}]{-90.0}{$\longrightarrow$}}}\\\
H\mathbb{F}_{p}&\longrightarrow&HR^{\flat}\end{array}$

Therefore by Corollary 5.19 we have

$0\simeq\operatorname{cofib}(HR^{\flat}\otimes_{W^{+}(R^{\flat})}t_{R,\xi})\simeq
HR^{\flat}\otimes_{W^{+}(R^{\flat})}\operatorname{cofib}(t_{R,\xi})\simeq
H\mathbb{F}_{p}\otimes_{\mathbb{S}_{p}^{\wedge}}\operatorname{cofib}(t_{R,\xi})$

We then invoke Corollary A.33 with Corollary 5.11 to deduce that
$\operatorname{cofib}(t_{R,\xi})\simeq 0$. $\Box$

###  5.5. An intermezzo: Identifying
$\operatorname{THH}\left(-/W^{+}(k)\right)$ with
$\operatorname{THH}\left(-\right)$ after $p$-completion

In this subsection, we will show that Proposition 1.17 follows from
Proposition 1.16. It suffices to prove the following lemma:

######  Lemma 5.20.

Let $R$ be an $\mathbb{E}_{1}$-algebra over $W^{+}(k)$ where $k$ is a perfect
$\mathbb{F}_{p}$-algebra. Then the canonical map
$\operatorname{THH}(R)\rightarrow\operatorname{THH}(R/W^{+}(k))$ induced by
$\mathbb{S}\rightarrow\mathbb{S}_{p}^{\wedge}$ is an equivalence after
$p$-completion.

###### Proof.

Note that $\operatorname{THH}(R/W^{+}(k))\simeq
W^{+}(k)\otimes_{\operatorname{THH}(W^{+}(k))}\operatorname{THH}(R)$. We are
left to show that the canonical map $\operatorname{THH}(W^{+}(k))\rightarrow
W^{+}(k)$ is an equivalence after $p$-completion. In view of Corollary A.33,
we only need to check it after tensoring with $H\mathbb{F}_{p}$. We note that
the base changed map
$H\mathbb{F}_{p}\otimes\operatorname{THH}(W^{+}(k))\rightarrow
H\mathbb{F}_{p}\otimes W^{+}(k)$ fits into the commutative diagram

$\begin{array}[]{ccc}H\mathbb{F}_{p}\otimes\operatorname{THH}(W^{+}(k))&\longrightarrow&H\mathbb{F}_{p}\otimes
W^{+}(k)\\\
{\mbox{\rotatebox[origin={c}]{-90.0}{$\longrightarrow$}}}&\sigma&{\mbox{\rotatebox[origin={c}]{-90.0}{${=\\!\\!=}$}}}\\\
\operatorname{THH}(H\mathbb{F}_{p}\otimes
W^{+}(k)/H\mathbb{F}_{p})&\longrightarrow&H\mathbb{F}_{p}\otimes W^{+}(k)\\\
{\mbox{\rotatebox[origin={c}]{-90.0}{$\longrightarrow$}}}&\tau&{\mbox{\rotatebox[origin={c}]{-90.0}{$\longrightarrow$}}}\\\
\operatorname{THH}(Hk/H\mathbb{F}_{p})&\longrightarrow&Hk\end{array}$

where the commutativity of $\sigma$ follows from the functoriality of the base
change functor of $\operatorname{THH}$, and the commutative of $\tau$ follows
from the functoriality of the natural transformation
$\operatorname{THH}\left(-/H\mathbb{F}_{p}\right)\rightarrow(-)$. All vertical
maps are equivalences of spectra: the upper left map is the base change
equivalence, and the lower right map is the equivalence by Proposition 2.7,
and the lower left map is the image of this equivalence under the functor
$\operatorname{THH}\left(-/H\mathbb{F}_{p}\right)$ and hence also an
equivalence. The bottom horizontal map is an equivalence by the fact that $k$
is a perfect $\mathbb{F}_{p}$-algebra. ∎

##  6\. Analogues

It is worth to note that in Bhatt and Scholze’s recent work [BS19], they
introduced the concept of prisms $(A,I)$ which serves as a “non-perfect”
version of perfectoid rings. Especially, the category of perfect prisms
$(A,I)$ is equivalent to that of perfectoid rings $A/I$, and given a
perfectoid ring $R$, the corresponding perfect prism is given by
$(W(R^{\flat}),\ker\theta)$. It is interesting to know whether we can
generalize our description for general orientable prisms $(A,I)$, that is to
say,

######  Question 1.

Given an orientable prism $(A,I=(d))$ . When can we find an
$\mathbb{E}_{\infty}$-ring spectrum $A^{+}$ (which satisfies some hypotheses
related to $A$. A naive guess would be that $\pi_{0}(A^{+})=A$) and a map
$\Omega^{2}S^{3}\rightarrow\operatorname{BGL}_{1}(A^{+})$ to which the
associated $\mathbb{E}_{2}$-Thom spectrum (possibly after $p$-completion)
coincides with $A/I$.

We don’t know the answer in this generality. However, we will discuss another
special class of prism (related to Breuil-Kisin cohomology) for which an
analogue holds. This result is more-or-less known by experts. In fact, it is
essentially equivalent to Remark 3.4 in [KN19] of which no proof is presented.
In this section, we will first recall some basic facts about complete discrete
valuation rings, then we will indicate briefly how to adapt our proof above to
this special class.

###  6.1. Preparations

######  Definition 6.1 ([Ser79, Section I.1]).

A ring $A$ is called a discrete valuation ring, or a DVR, if it is a principal
ideal domain that has a unique non-zero prime ideal $\mathfrak{m}$. In this
case, since $A$ is local, we also denote the DVR $A$ by $(A,\mathfrak{m})$.
The field $A/\mathfrak{m}$ is called the residue field of $A$. A generator of
$\mathfrak{m}$, unique up to multiplication by an invertible element, is
called a uniformizer, usually denoted by $\varpi$.

######  Definition 6.2.

A DVR $(A,\mathfrak{m})$ is called of mixed characteristics $(0,p)$ if the
field of fraction $\operatorname{Frac}(A)$ of $A$ is of characteristics $0$
while the residue field $A/\mathfrak{m}$ is of characteristics $p$, which
implies that $0\neq p\in\mathfrak{m}$.

######  Definition 6.3 ([Ser79, Section I.1]).

Let $(A,\mathfrak{m})$ be a DVR. The valuation of an element $x\in A\setminus
0$ is defined to be the maximal integer $n\in\mathbb{N}$ such that
$x\in\mathfrak{m}^{n}$, which always exists, denoted by $v(x)\in\mathbb{N}$.

######  Definition 6.4 ([Ser79, Section II.5]).

Let $(A,\mathfrak{m})$ be a DVR of mixed characteristics $(0,p)$. Then the
integer $e=v(p)$ is called the absolute ramification index of $A$.

######  Definition 6.5 ([Ser79, Chapter II]).

A DVR $(A,\mathfrak{m})$ is called complete if it is complete with respect to
the $\mathfrak{m}$-adic topology, that is to say, the canonical map from $A$
to the limit of the tower

$\cdots\rightarrow A/\mathfrak{m}^{n}\rightarrow\cdots\rightarrow
A/\mathfrak{m}^{2}\rightarrow A/\mathfrak{m}$

is an isomorphism.

######  Proposition 6.6 ([Ser79, Section II.5, Theorem 4]).

Let $(A,\mathfrak{m})$ be a complete DVR of mixed characteristics $(0,p)$ with
residue field $k$ being perfect. Let $e$ be its absolute ramification index.
Let $\varpi\in\mathfrak{m}$ be a uniformizer. Then there exists an Eisenstein
$W(k)$-polynomial $E(u)\in W(k)[u]$ (that is, a $W(k)$-polynomial
$E(u)=u^{e}+\sum_{j=0}^{e-1}a_{j}u^{j}$ such that $p\mathrel{|}a_{j}$ for
$j=0,\ldots,e-1$ and $p^{2}\mathrel{\nmid}a_{0}$, where $W(k)$ is the ring of
Witt vectors as before) along with an isomorphism
$W(k)[u]/(E(u))\xrightarrow{\sim}A$ which maps $u$ to the uniformizer
$\varpi\in\mathfrak{m}$.

In the rest of this section, we will fix a complete DVR $(A,\mathfrak{m})$ of
mixed characteristics $(0,p)$ with residue field $k$ being perfect, absolute
ramification index $e$ and a uniformizer $\varpi\in\mathfrak{m}$. We also fix
a choice of an Eisenstein $W(k)$-polynomial $E(u)\in W(k)[u]$ as in
Proposition 6.6. We first note that

######  Proposition 6.7.

The element $1-E(u)\in W(k)[[u]]$ is invertible.

###### Proof.

Write $E(u)=u^{e}+\sum_{j=0}^{e-1}a_{j}u^{j}$ as in Proposition 6.6. Note that
$W(k)$ is $p$-adically complete, therefore $1-a_{0}$ is invertible in $W(k)$,
which implies that $1-E(u)\in W(k)[[u]]$ is invertible. ∎

Let $W^{+}(k)[u]$ be the “single variable polynomial $W^{+}(k)$-algebra”, that
is, the $\mathbb{E}_{\infty}$-$W^{+}(k)$-algebra
$W^{+}(k)\otimes_{\mathbb{S}}\mathbb{S}[\mathbb{N}]$. Since the space
$\mathbb{N}$ is endowed with discrete topology, we have

######  Proposition 6.8.

As a $W^{+}(k)$-module, $W^{+}(k)[u]$ is equivalent to the direct sum
$\bigoplus_{j=0}^{\infty}u^{j}W^{+}(k)$, a free $W^{+}(k)$-module. The graded
homotopy group $\pi_{\ast}(W^{+}(k)[u])$, as a (graded-commutative)
$\pi_{\ast}(W^{+}(k))$-algebra, is equivalent to $\pi_{\ast}(W^{+}(k))[u]$,
where $\deg u=0$.

Now let $W^{+}(k)[[u]]$ be the $(u)$-completion of the
$\mathbb{E}_{\infty}$-$W^{+}(k)$-algebra $W^{+}(k)[u]$. To study
$W^{+}(k)[[u]]$, we need some preparations.

######  Proposition 6.9.

Let $n\in\mathbb{N}$ be a natural number. Let
$m_{u^{n}}:W^{+}(k)[u]\rightarrow W^{+}(k)[u]$ be the multiplication map given
by $u^{n}\in\pi_{0}(W^{+}(k)[u])=W(k)[u]$. Then the $W^{+}(k)[u]$-module
$\operatorname{cofib}(m_{u^{n}})$ as a $W^{+}(k)$-module is a free
$W^{+}(k)$-module $\bigoplus_{j=0}^{n-1}u^{j}W^{+}(k)$ of rank $n$, which
admits an $\mathbb{E}_{\infty}$-$W^{+}(k)[u]$-algebra structure. In
particular, we have the cofiber sequence

$W^{+}(k)[u]\xrightarrow{m_{u}}W^{+}(k)[u]\rightarrow W^{+}(k)$

of $W^{+}(k)[u]$-modules.

###### Proof.

For any space $X\in\mathcal{S}$, we let $X_{+}\in\mathcal{S}_{\ast}$ denote
the pointed discrete space $\\{\ast\\}\cup X$. Especially,
$\mathbb{N}_{+}=\\{\ast\\}\cup\mathbb{N}$ and
$(\mathbb{N}_{<n})_{+}=\\{\ast\\}\cup\mathbb{N}_{<n}$. The addition map
$\mathbb{N}\rightarrow\mathbb{N},m\mapsto n+m$ induces a map of pointed spaces
$\alpha_{n}:\mathbb{N}_{+}\rightarrow\mathbb{N}_{+}$. Note that in the
$\infty$-category $\mathcal{S}$ of spaces, we have a pushout diagram

$\begin{array}[]{ccc}\mathbb{N}_{+}&\xrightarrow{\alpha_{n}}&\mathbb{N}_{+}\\\
{\mbox{\rotatebox[origin={c}]{-90.0}{$\longrightarrow$}}}&&{\mbox{\rotatebox[origin={c}]{-90.0}{$\longrightarrow$}}}\\\
\\{\ast\\}&\longrightarrow&(\mathbb{N}_{<n})_{+}\end{array}$

to which we apply the functor
$\Sigma^{\infty}:\mathcal{S}_{\ast}\rightarrow\operatorname{Sp}$, left adjoint
of the functor $\Omega_{\ast}^{\infty}:\operatorname{Sp}\rightarrow S_{\ast}$
therefore preserving colimits, we get a cofiber sequence
$\mathbb{S}[u]\xrightarrow{u^{n}}\mathbb{S}[u]\rightarrow\bigoplus_{j=0}^{n-1}u^{j}\mathbb{S}$.
A further base change to $W^{+}(k)$ gives rise to the result. In addition, the
multiplication structure could be seen from the fact that the addition map
$\mathbb{N}\rightarrow\mathbb{N},m\mapsto n+m$ in fact defines a monoidal
action. ∎

######  Corollary 6.10.

Let $n\in\mathbb{N}$ be a natural number. Let
$m_{u^{n}}:W^{+}(k)[u]\rightarrow W^{+}(k)[u]$ be the multiplication map. Then
homotopy groups $\pi_{\ast}(\operatorname{cofib}(m_{u^{n}}))$ of the cofiber
as $\pi_{\ast}(W^{+}(k))$ could be identified with
$\pi_{\ast}(W^{+}(k))[u]/(u^{n})$, and the long exact sequence of homotopy
groups associated to the cofiber sequence
$W^{+}(k)[u]\xrightarrow{m_{u^{n}}}W^{+}(k)[u]\rightarrow\operatorname{cofib}(m_{u^{n}})$
decomposes as short exact sequences, which assemble to a short exact sequence
of graded $\pi_{\ast}(W^{+}(k))[u]$-modules:

$0\rightarrow\pi_{\ast}(W^{+}(k))[u]\xrightarrow{u^{n}}\pi_{\ast}(W^{+}(k))[u]\rightarrow\pi_{\ast}(W^{+}(k))[u]/(u^{n})\rightarrow
0$

Furthermore, this sequence is functorial in $n\in(\mathbb{N},>)$.

######  Proposition 6.11.

The $\mathbb{E}_{\infty}$-$W^{+}(k)$-algebra $W^{+}(k)[[u]]$ is connective.
The zeroth homotopy group of $\pi_{0}(W^{+}(k)[[u]])$ is isomorphic to the
$(u)$-adic completion of the polynomial $W(k)$-algebra $W(k)[u]$, that is, the
formal power series $W(k)$-algebra $W(k)[[u]]$, as $W(k)$-algebras.

Our proof is incomplete: we only identify the $W(k)$-module structures on
homotopy groups. A formal identification of algebra structures would require
more rudiments about the symmetric monoidal structure on the completion
functor than we know.

###### Proof.

We reinterpret Proposition A.19 as follows: since the limit functor is exact,
it commutes with cofibers, therefore we can rewrite
$W^{+}(k)[[u]]=(W^{+}(k)[u])_{(u)}^{\wedge}$ as the limit of the tower

$\cdots\rightarrow\operatorname{cofib}\left(W^{+}(k)[u]\xrightarrow{u^{2}}W^{+}(k)[u]\right)\rightarrow\operatorname{cofib}\left(W^{+}(k)[u]\xrightarrow{u}W^{+}(k)[u]\right)$

After passage to homotopy groups, by Corollary 6.10, we get the tower of
graded $\pi_{\ast}(W^{+}(k))[u]$-modules

(1)
$\cdots\rightarrow\pi_{\ast}(W^{+}(k))[u]/(u^{n})\rightarrow\cdots\rightarrow\pi_{\ast}(W^{+}(k))[u]/(u^{2})\rightarrow\pi_{\ast}(W^{+}(k))[u]/(u)$

which is degree-wise a tower of surjective maps. It follows from Milnor’s
sequence that the graded $\pi_{\ast}(W^{+}(k))[u]$-module
${\pi_{\ast}}(W^{+}(k)[[u]])$ is isomorphic to the (ordinary) inverse limit of
the tower (1), that is, $\pi_{\ast}(W^{+}(k))[[u]]$. Take $\ast=0$, we get the
result. ∎

The following lemma serves as a key tool in our proof:

######  Lemma 6.12.

Let $M$ be a $W^{+}(k)[u]$-(or $W^{+}(k)[[u]]$-)module (spectrum). If the
spectrum $W^{+}(k)\otimes_{W^{+}(k)[u]}M$ (or
$W^{+}(k)\otimes_{W^{+}(k)[[u]]}M$ respectively) is contractible, then so is
the $(u)$-completion of the spectrum $M$. In particular, if furthermore
$W^{+}(k)[u]$-(or $W^{+}(k)[[u]]$-)module $M$ is assumed to be $(u)$-complete,
then the spectrum $M$ is contractible.

###### Proof.

We first assume that the spectrum $W^{+}(k)\otimes_{W^{+}(k)[u]}M$ is
contractible. In this case, we apply the exact functor
$-\otimes_{W^{+}(k)[u]}M$ to the cofiber sequence

(2) $W^{+}(k)[u]\xrightarrow{m_{u}}W^{+}(k)[u]\rightarrow W^{+}(k)$

indicated in Proposition 6.9 obtaining that the base-changed map
$M\xrightarrow{m_{u}\otimes_{W^{+}(k)[u]}M}M$ is an equivalence of spectra.
Note that this map is just the multiplication map, denoted by $m_{M,u}$. Now
we look at Proposition A.19: the $(u)$-completion of the $W^{+}(k)[u]$-module
$M$ is the cofiber of the canonical map $T(M)\rightarrow M$, where $T(M)$ is
the limit of the tower

$\cdots\xrightarrow{m_{M,u}}M\xrightarrow{m_{M,u}}M\xrightarrow{m_{M,u}}M$

Since all maps in the tower are equivalences of spectra, we deduce that the
canonical map $T(M)\rightarrow M$ is an equivalence of spectra, which implies
that the $(u)$-completion of the $W^{+}(k)[u]$-module $M$ is contractible. In
particular, the $W^{+}(k)[u]$-module $M$ is assumed to be $(u)$-complete,
therefore the spectrum $M$ is contractible.

If, on the other hand, $W^{+}(k)\otimes_{W^{+}(k)[[u]]}M$ is contractible,
then to adopt the proof above, it suffices to establish the cofiber sequence

(3) $W^{+}(k)[[u]]\xrightarrow{m_{u}}W^{+}(k)[[u]]\rightarrow W^{+}(k)$

We apply the $(u)$-complete functor to the cofiber sequence (2), and note that
the $W^{+}(k)[u]$-module $W^{+}(k)$ is $(u)$-nilpotent (in fact, multiplying
$u$ is the zero map on $W^{+}(k)$), therefore $W^{+}(k)$ is $(u)$-complete by
Corollary A.17, which leads to the cofiber sequence (3). The rest of the proof
is same as before. ∎

###  6.2. The Breuil-Kisin case

As before, we fix a complete DVR $(A,\mathfrak{m})$ of mixed characteristics
$(0,p)$ with residue field $k$ being perfect, absolute ramification index $e$,
a uniformizer $\varpi\in\mathfrak{m}$ and an Eisenstein $W(k)$-polynomial
$E(u)\in W(k)[u]$ which induces an isomorphism
$W(k)[u]/(E(u))\xrightarrow{\sim}A,u\mapsto\varpi$ as in Proposition 6.6. As
in Remark 1.9 and Remark 1.12, $1-E(u)\in
W(k)[[u]]=\pi_{1}(\operatorname{BGL}_{1}(W^{+}(k)[[u]]))$ gives rise to a map
$f_{E}:\Omega^{2}S^{3}\rightarrow\operatorname{BGL}_{1}(W^{+}(k)[[u]])$. The
proof of Lemma 5.1 results in the following analogue:

######  Lemma 6.13.

The zeroth homotopy group of the $\mathbb{E}_{2}$-Thom spectrum $Mf_{E}$
associated to the map $f_{E}$ is isomorphic to the $W(k)$-algebra
$W(k)[[u]]/(E(u))\cong W(k)[u]/(E(u))\cong A$.

The $W(k)[u]$-module structure on $A$ gives rise to a $W^{+}(k)[u]$-module
structure on $HA$. Since $A$ is $\mathfrak{m}=(\varpi)$-adically complete, the
$W(k)[u]$-module structure on $A$ also gives rise to a $W(k)[[u]]$-module
structure on $A$ and consequently a $W^{+}(k)[[u]]$-module structure on $HA$.
We readily [details needed] check that these structures are compatible, in the
sense that the $W^{+}(k)[u]$-module structure on $HA$ coincides with the image
of the $W^{+}(k)[[u]]$-module $HA$ under the forgetful functor
$\operatorname{Mod}_{W^{+}(k)[[u]]}\rightarrow\operatorname{Mod}_{W^{+}(k)[u]}$.
Matthew Morrow proposed the following analogue of the Hopkins-Mahowald
theorem:

######  Theorem 6.14.

The truncation map $t_{E}:Mf_{E}\rightarrow H\pi_{0}(Mf_{E})\cong HA$ of
$\mathbb{E}_{2}$-$W^{+}(k)[[u]]$-algebras is an equivalence of spectrum. Thus
the Eilenberg-Maclane spectrum $HA$ is the $\mathbb{E}_{2}$-Thom spectrum
$Mf_{E}$ associated to the map
$f_{E}:\Omega^{2}S^{3}\rightarrow\operatorname{BGL}_{1}(W^{+}(k)[[u]])$.

######  Corollary 6.15 (see [KN19, Remark 3.4]).

The $\mathbb{E}_{2}$-$HA$-algebra $HA\otimes_{W^{+}(k)[[u]]}HA$ is a free
$\mathbb{E}_{2}$-$HA$-algebra on a single generator in degree 1.

###### Proof.

The strategy is already covered in the proof of Lemma 3.2 and Lemma 5.12.
Since this pattern will appear again soon, we find it beneficial to present
again. Let’s recall that the $\mathbb{E}_{2}$-Thom spectrum $Mf_{E}$ is the
colimit of the composite functor

$\Omega^{2}S^{3}\xrightarrow{f_{E}}\operatorname{BGL}_{1}(W^{+}(k)[[u]])\rightarrow\operatorname{Mod}_{W^{+}(k)[[u]]}$

which by abuse of notation will be still denoted by $f_{E}$.

Since the base change functor
$HA\otimes_{W^{+}(k)[[u]]}-:\operatorname{Mod}_{W^{+}(k)[[u]]}\rightarrow\operatorname{Mod}_{HA}$
is a left adjoint, it commutes with colimits, we deduce that
$HA\otimes_{W^{+}(k)[[u]]}Mf_{E}\simeq M(f_{E}\otimes_{W^{+}(k)[[u]]}HA)$,
where $f_{E}\otimes_{W^{+}(k)[[u]]}HA$ is the map
$\Omega^{2}S^{3}\rightarrow\operatorname{BGL}_{1}(HA)$.

As in the proof of Lemma 3.2, we can identify map as follows: we pick the
image of $1-E(u)\in\operatorname{GL}_{1}(W(k)[[u]])$ under the map
$\operatorname{GL}_{1}(W(k)[[u]])\rightarrow\operatorname{GL}_{1}(A)$, that
is, the element
$1\in\operatorname{GL}_{1}(A)\cong\pi_{1}(\operatorname{BGL}_{1}(HA))$, which
gives rise to the constant map $S^{1}\rightarrow\operatorname{BGL}_{1}(HA)$
and consequently the constant map
$f_{A}:\Omega^{2}S^{3}\rightarrow\operatorname{BGL}_{1}(HA)$, as in Remark 1.9
and Remark 1.12.

In conclusion, the map
$f_{E}\otimes_{W^{+}(k)}HA:\Omega^{2}S^{3}\rightarrow\operatorname{BGL}_{1}(HA)$
coincides with the constant map $f_{A}$, and the $\mathbb{E}_{2}$-Thom
spectrum $Mf_{A}$ is thus the colimit of a constant map, which evaluates to
$HA\otimes\Omega^{2}S^{3}$, the free $\mathbb{E}_{2}$-$HA$-algebra on a single
generator in degree 1. ∎

Recall that $E(u)\in W(k)[u]$ is an Eisenstein $W(k)$-polynomial. Let $a_{0}$
denote the constant term of $E(u)$. By assumption, $p\mathrel{|}a_{0}$ but
$p^{2}\mathrel{\nmid}a_{0}$. Let $a_{0}=pb_{0}$ where $b_{0}\in W(k)$. Since
$p$ is not a zero-divisor in $W(k)$, we have $p\mathrel{\nmid}b_{0}$, which
implies that the image of $b_{0}$ in $W(k)/p\cong k$ is invertible since $k$
is a field. Now since $W(k)$ is $p$-adically complete, we have
$b_{0}\in\operatorname{GL}_{1}(W(k))$.

The strategy to prove Theorem 6.14 is similar to the approach to attack
Theorem 1.13. We first show that the base change of the truncation map $t_{E}$
along the map $W^{+}(k)[[u]]\rightarrow W^{+}(k)$ coincides with the
truncation map $t_{k,a_{0}}$, then it follows from Lemma 5.3 that the base
changed map $W^{+}(k)\otimes_{W^{+}(k)[[u]]}t_{E}\simeq t_{k,a_{0}}$ is an
equivalence of spectra, and by completeness, we deduce that the map $t_{E}$ is
also an equivalence of spectra by Lemma 6.12.

######  Lemma 6.16.

There is a canonical equivalence
$Mf_{k,a_{0}}\xrightarrow{\simeq}W^{+}(k)\otimes_{W^{+}(k)[[u]]}Mf_{E}$ of
$W^{+}(k)$-modules.

###### Proof.

We will duplicate the proof of Lemma 5.12. The image of the multiplication map
$m_{1-E(u)}:W^{+}(k)[[u]]\rightarrow W^{+}(k)[[u]]$ under the base change
functor
$W^{+}(k)\otimes_{W^{+}(k)[[u]]}-:\operatorname{Mod}_{W^{+}(k)[[u]]}\rightarrow\operatorname{Mod}_{W^{+}(k)}$
is the multiplication map $m_{1-a_{0}}:W^{+}(k)\rightarrow W^{+}(k)$. Note
also that the base change functor is symmetric monoidal. Now we conclude that
the map $f_{k,a_{0}}$ coincides with the composite map

$\Omega^{2}S^{3}\xrightarrow{f_{E}}\operatorname{BGL}_{1}(W^{+}(k)[[u]])\xrightarrow{W^{+}(k)\otimes_{W^{+}(k)[[u]]}-}\operatorname{BGL}_{1}(W^{+}(k))$

Thus by commuting the colimit and the base-change, we obtain

$\displaystyle Mf_{k,a_{0}}$ $\displaystyle=$
$\displaystyle\operatorname{colim}(W^{+}(k)\otimes_{W^{+}(k)[[u]]}f_{E})$
$\displaystyle\xrightarrow{\simeq}$ $\displaystyle
W^{+}(k)\otimes_{W^{+}(k)[[u]]}\operatorname{colim}f_{E}$ $\displaystyle=$
$\displaystyle W^{+}(k)\otimes_{W^{+}(k)[[u]]}Mf_{E}$

where by abuse of notation, the colimit of the maps $f_{k,a_{0}}$ (or $f_{E}$
respectively) are understood as the colimit of the maps $f_{k,a_{0}}$ (or
$f_{E}$ respectively) composed with the functor
$\operatorname{BGL}_{1}(W^{+}(k))\rightarrow\operatorname{Mod}_{W^{+}(k)}$ (or
$\operatorname{BGL}_{1}(W^{+}(k)[[u]])\rightarrow\operatorname{Mod}_{W^{+}(k)[[u]]}$
respectively) as in the definition of Thom spectra. ∎

######  Lemma 6.17.

There is a canonical equivalence
$W^{+}(k)\otimes_{W^{+}(k)[[u]]}HA\xrightarrow{\simeq}Hk$ of
$W^{+}(k)$-modules.

###### Proof.

As in the proof of Lemma 6.12, we identify $W^{+}(k)$ with the cofiber of the
multiplication map $m_{u}:W^{+}(k)[[u]]\rightarrow W^{+}(k)[[u]]$ which gives
us an equivalence

$W^{+}(k)\otimes_{W^{+}(k)[[u]]}HA\simeq\operatorname{cofib}\left(HA\xrightarrow{m_{HA,u}}HA\right)$

Now by the definition of the $W^{+}(k)[[u]]$-module structure on $HA$ and that
$u$ is not a zero-divisor in $A$, we have the equivalence
$\operatorname{cofib}\left(HA\xrightarrow{m_{HA,u}}HA\right)\simeq
H\left(\operatorname{coker}\left(A\xrightarrow{\varpi}A\right)\right)\simeq
Hk$. Thus we obtain an equivalence $W^{+}(k)\otimes_{W^{+}(k)[[u]]}HA\simeq
Hk$. We can readily check [details needed] that this equivalence could be
described as follows: consider the commutative diagram in the
$\infty$-category of $\mathbb{E}_{\infty}$-rings

$\begin{array}[]{ccc}W^{+}(k)[[u]]&\longrightarrow&W^{+}(k)\\\
{\mbox{\rotatebox[origin={c}]{-90.0}{$\longrightarrow$}}}&&{\mbox{\rotatebox[origin={c}]{-90.0}{$\longrightarrow$}}}\\\
HA&\longrightarrow&Hk\end{array}$

where the left vertical map is the composite map $W^{+}(k)[[u]]\rightarrow
H(\pi_{0}(W^{+}(k)[[u]]))\simeq H(W(k)[[u]])\xrightarrow{u\mapsto\varpi}HA$
(where the first map is the Postnikov section). The commutative diagram
induces a map $W^{+}(k)\otimes_{W^{+}(k)[[u]]}HA\rightarrow Hk$ (note that the
left hand side is a pushout of $\mathbb{E}_{\infty}$-rings), which coincides
with the equivalence obtained above. ∎

######  Lemma 6.18.

The equivalences in Lemma 6.16 and Lemma 6.17 assembles into a commutative
diagram:

$\begin{array}[]{ccc}Mf_{k,a_{0}}&\xrightarrow{t_{k,a_{0}}}&Hk\\\
{\mbox{\rotatebox[origin={c}]{-90.0}{$\longrightarrow$}}}\simeq&&{\mbox{\rotatebox[origin={c}]{90.0}{$\longrightarrow$}}}\simeq\\\
W^{+}(k)\otimes_{W^{+}(k)[[u]]}Mf_{E}&\longrightarrow&W^{+}(k)\otimes_{W^{+}(k)[[u]]}HA\end{array}$

where the top horizontal map is the 0th Postnikov section $t_{k,a_{0}}$
defined in Proposition 5.2 and the bottom horizontal map is the base-changed
0th Postnikov section $W^{+}(k)\otimes_{W^{+}(k)[[u]]}t_{E}$.

###### Proof.

As in the proof of Lemma 5.15, it suffices to show that the composite map on
the 0th homotopy group
$\pi_{0}(Mf_{k,a_{0}})\rightarrow\pi_{0}(W^{+}(k)\otimes_{W^{+}(k)[[u]]}Mf_{E})\rightarrow\pi_{0}(W^{+}(k)\otimes_{W^{+}(k)[[u]]}HA)\rightarrow\pi_{0}(Hk)\cong
k$ is an isomorphism, which follows from an explicit element chasing. ∎

Combined with Lemma 5.3, we obtain that

######  Corollary 6.19.

The base-changed map
$W^{+}(k)\otimes_{W^{+}(k)[[u]]}t_{E}:W^{+}(k)\otimes_{W^{+}(k)[[u]]}Mf_{E}\rightarrow
W^{+}(k)\otimes_{W^{+}(k)[[u]]}HA$ is an equivalence of $W^{+}(k)$-modules.

Apply Lemma 6.12 to the cofiber $\operatorname{cofib}(t_{E})$, we deduce that

######  Corollary 6.20.

The map $t_{E}:Mf_{E}\rightarrow HA$ is an equivalence of spectra after
$(u)$-completion.

As in Lemma 5.9, we deduce from Theorem A.25 that

######  Lemma 6.21.

The $W^{+}(k)[[u]]$-module $HA$ is $(u)$-complete.

Now, given the nontrivial topological input Proposition 5.6, as in Lemma 5.10
and Corollary 5.11, we deduce that

######  Lemma 6.22.

The $W^{+}(k)[[u]]$-module $Mf_{E}$ is $(u)$-complete.

######  Corollary 6.23.

The cofiber $\operatorname{cofib}(t_{E})$ is a $(u)$-complete
$W^{+}(k)[[u]]$-module, and thus the map $t_{E}$ is an equivalence of spectra
by Corollary 6.20.

This completes the proof of Theorem 6.14.

###  6.3. Complete regular local rings

Inspired by [KN19, Section 9], we will provide a Hopkins-Mahowald theorem for
complete regular local rings of mixed characteristic. We will show how to
modify our proof of Theorem 6.14 to deduce this. Note that this is also a
special case of Question 1, by [BS19, Remark 3.11].

We need some preparations in higher algebra:

Let $W^{+}(k)[u_{1},\ldots,u_{n}]$ be the “$n$-variate polynomial
$W^{+}(k)$-algebra”, that is, the $\mathbb{E}_{\infty}$-$W^{+}(k)$-algebra
$W^{+}(k)\otimes_{\mathbb{S}}\mathbb{S}[\mathbb{N}^{n}]$. Since the space
$\mathbb{N}^{n}$ is endowed with discrete topology, parallel to Proposition
6.8, we have

######  Proposition 6.24.

As a $W^{+}(k)$-module, $W^{+}(k)[u_{1},\ldots,u_{n}]$ is equivalent to the
direct sum $\bigoplus_{\alpha\in\mathbb{N}^{n}}u^{\alpha}W^{+}(k)$, a free
$W^{+}(k)$-module. The graded homotopy group
$\pi_{\ast}(W^{+}(k)[u_{1},\ldots,u_{n}])$, as a (graded-commutative)
$\pi_{\ast}(W^{+}(k))$-algebra, is equivalent to
$\pi_{\ast}(W^{+}(k))[u_{1},\ldots,u_{n}]$, where $\deg u_{1}=\cdots=\deg
u_{n}=0$.

Now let $W^{+}(k)[[u_{1},\ldots,u_{n}]]$ be the
$(u_{1},\ldots,u_{n})$-completion of the
$\mathbb{E}_{\infty}$-$W^{+}(k)$-algebra $W^{+}(k)[u_{1},\ldots,u_{n}]$. By
induction on $n\in\mathbb{N}_{>0}$ and argue as in Proposition 6.11 [details
needed], we obtain:

######  Proposition 6.25.

The $\mathbb{E}_{\infty}$-$W^{+}(k)$-algebra $W^{+}(k)[[u_{1},\ldots,u_{n}]]$
is connective. The zeroth homotopy group of
$\pi_{0}(W^{+}(k)[[u_{1},\ldots,u_{n}]])$ is isomorphic to the
$(u_{1},\ldots,u_{n})$-adic completion of the polynomial $W(k)$-algebra
$W(k)[u_{1},\ldots,u_{n}]$, that is, the formal power series $W(k)$-algebra
$W(k)[[u_{1},\ldots,u_{n}]]$, as $W(k)$-algebras.

Similarly, argue inductively on $n\in\mathbb{N}_{>0}$ as in Lemma 6.12, we
obtain:

######  Lemma 6.26.

Let $M$ be a $W^{+}(k)[u_{1},\ldots,u_{n}]$-(or
$W^{+}(k)[[u_{1},\ldots,u_{n}]]$-)module (spectrum). If the spectrum
$W^{+}(k)\otimes_{W^{+}(k)[u_{1},\ldots,u_{n}]}M$ (or
$W^{+}(k)\otimes_{W^{+}(k)[[u_{1},\ldots,u_{n}]]}M$ respectively) is
contractible, then so is the $(u_{1},\ldots,u_{n})$-completion of the spectrum
$M$. In particular, if furthermore $W^{+}(k)[u_{1},\ldots,u_{n}]$-(or
$W^{+}(k)[[u_{1},\ldots,u_{n}]]$-)module $M$ is assumed to be
$(u_{1},\ldots,u_{n})$-complete, then the spectrum $M$ is contractible.

We note that in these inductive arguments, we heavily depend on Proposition
A.23.

Now we are ready to formulate the Hopkins-Mahowald theorem for complete
regular local rings. We fix a positive integer $n\in\mathbb{N}_{>0}$, a
perfectoid ring $R$. As in Section 5, let $\theta:W(R^{\flat})\rightarrow R$
be Fontaine’s pro-infinitesimal thickening. Let $\phi\in
W(R^{\flat})[[u_{1},\ldots,u_{n}]]$ be formal power series such that
$\phi(0,\ldots,0)\in W(R^{\flat})$ is a generator of $\ker\theta$. We recall
that $\ker\theta$ is principal by definition. We note that the element
$1-\phi(u_{1},\ldots,u_{n})\in W(R^{\flat})[[u_{1},\ldots,u_{n}]]$ is
invertible, since $1-\phi(0,\ldots,0)\in W(R^{\flat})$ is invertible as the
ring $W(R^{\flat})$ is $\ker\theta$-adically complete. As in Remark 1.9 and
Remark 1.12, the element
$1-\phi(u_{1},\ldots,u_{n})\in\operatorname{GL}_{1}(W(R^{\flat})[[u_{1},\ldots,u_{n}]])$
gives rise to an $\mathbb{E}_{2}$-map
$f:\Omega^{2}S^{3}\rightarrow\operatorname{BGL}_{1}(W(R^{\flat})[[u_{1},\ldots,u_{n}]])$.
The proof of Lemma 5.1 results in the following analogue:

######  Lemma 6.27.

The zeroth homotopy group of the $\mathbb{E}_{2}$-Thom spectrum $Mf$
associated to the map $f$ is isomorphic to the $W(R^{\flat})$-algebra
$W(R^{\flat})[[u_{1},\ldots,u_{n}]]/(\phi(u_{1},\ldots,u_{n}))$.

We now phrase the following variant of the Hopkins-Mahowald theorem:

######  Theorem 6.28.

The truncation map $t:Mf\rightarrow H\pi_{0}(Mf)\cong
HW(R^{\flat})[[u_{1},\ldots,u_{n}]]/(\phi(u_{1},\ldots,u_{n}))$ of
$\mathbb{E}_{2}$-$W^{+}(R^{\flat})[[u_{1},\ldots,u_{n}]]$-algebras is an
equivalence of spectrum. Thus the Eilenberg-Maclane spectrum
$HW(R^{\flat})[[u_{1},\ldots,u_{n}]]/(\phi(u_{1},\ldots,u_{n}))$ is the
$\mathbb{E}_{2}$-Thom spectrum $Mf$ associated to the map
$f:\Omega^{2}S^{3}\rightarrow\operatorname{BGL}_{1}(W^{+}(R^{\flat})[[u_{1},\ldots,u_{n}]])$.

The proof is parallel to that of Theorem 6.14, which we will omit. Now let
$(A,\mathfrak{m})$ be a complete regular local ring with residue field
$k=A/\mathfrak{m}$ being perfect of characteristic $p$. We also assume that
$p\neq 0$ in $A$. Let $(a_{1},\ldots,a_{n})\subseteq\mathfrak{m}$ be a regular
sequence which generates the maximal ideal $\mathfrak{m}$. We need the
following lemma:

######  Lemma 6.29 ([KN19, Lemma 9.2]).

There exists a map $W(k)[[u_{1},\ldots,u_{n}]]\rightarrow A$ of rings given by
$u_{i}\mapsto a_{i}$ for $i=1,\ldots,n$, which is surjective with kernel being
principal, generated by a formal power series $\phi\in
W(k)[[u_{1},\ldots,u_{n}]]$ with $\phi(0,\ldots,0)=p$.

###### Proof.

First, the isomorphism $k\rightarrow A/\mathfrak{m}$ lifts to a map
$W(k)\rightarrow A$ since $A$ is $\mathfrak{m}$-adically complete, see Example
2.5 or [Ser79, Section II.5, Proposition 10]. The map
$W(k)[[u_{1},\ldots,u_{n}]]\rightarrow A$ is then well-defined since $A$ is
$\mathfrak{m}$-adic complete. Let $C,K$ be the cokernel and the kernel of the
map $W(k)[[u_{1},\ldots,u_{n}]]\rightarrow A$ of
$W(k)[[u_{1},\ldots,u_{n}]]$-modules. By right-exactness of classical tensor
products, we have

$\operatorname{Tor}_{0}^{W(k)[[u_{1},\ldots,u_{n}]]}(C,W(k))\cong\operatorname{coker}(W(k)\rightarrow
k)\cong 0$

Now by inspecting the exact sequence $0\rightarrow\mathfrak{m}\rightarrow
A\rightarrow k\rightarrow 0$, we deduce that $A$ is a finitely generated
$W(k)[[u_{1},\ldots,u_{n}]]$-module, therefore so is $C$. We deduce from
Nakayama’s lemma that $C\cong 0$, therefore the map
$W(k)[[u_{1},\ldots,u_{n}]]$ is surjective. Now we obtain a short exact
sequence of $W(k)[[u_{1},\ldots,u_{n}]]$-modules

$0\rightarrow K\rightarrow W(k)[[u_{1},\ldots,u_{n}]]\rightarrow A\rightarrow
0$

which gives rise to an exact sequence of $W(k)$-modules

$\operatorname{Tor}_{1}^{W(k)[[u_{1},\ldots,u_{n}]]}(A,W(k))\rightarrow\operatorname{Tor}_{0}^{W(k)[[u_{1},\ldots,u_{n}]]}(K,W(k))\rightarrow
W(k)\rightarrow k\rightarrow 0$

Since $(a_{1},\ldots,a_{n})$ is a regular sequence, it is also Koszul regular
[Sta21, Tag 062F], hence
$\operatorname{Tor}_{1}^{W(k)[[u_{1},\ldots,u_{n}]]}(A,W(k))\cong 0$. Thus

$\operatorname{Tor}_{0}^{W(k)[[u_{1},\ldots,u_{n}]]}(K,W(k))\cong\ker(W(k)\rightarrow
k)\cong pW(k)$

We pick a lift $\phi\in K$ of $p\in pW(k)$. By Nakayama’s lemma, the
$W(k)[[u_{1},\ldots,u_{n}]]$ module (and hence the ideal) $K$ is generated by
the element $\phi\in K$. Furthermore, by multiplying an invertible element in
$W(k)$, we can assume that the lift $\phi$ is so chosen that
$\phi(0,\ldots,0)=p$. ∎

######  Remark 6.30.

Our proof of Lemma 6.29 leads to a more general result: Let $A$ be a
commutative ring with an ideal $I\subseteq A$ generated by a (Koszul) regular
sequence $(a_{1},\ldots,a_{n})\subseteq I$. If $A$ is both $p$-adically
complete and $I$-adically complete, and $R:=A/I$ is a perfectoid ring, then by
Proposition 4.16, there exists a unique map $W(R^{\flat})\rightarrow A$ such
that the composite map $W(R^{\flat})\rightarrow A\rightarrow R$ coincides with
Fontaine’s map, which allows us to view $A$ as a $W(R^{\flat})$-algebra. Now
we consider the map $\varphi:W(R^{\flat})[[u_{1},\ldots,u_{n}]]\rightarrow A$
of $W(R^{\flat})$-algebras given by $u_{i}\mapsto a_{i}$ for $i=1,\ldots,n$.
Our proof of Lemma 6.29 implies that the map $\varphi$ is surjective with
kernel being principal, generated by a formal power series $\phi\in
W(R^{\flat})[[u_{1},\ldots,u_{n}]]$ such that $\phi(0,\ldots,0)$ generates the
kernel $\ker(\theta)$ of Fontaine’s map $\theta:W(R^{\flat})\rightarrow R$.

######  Corollary 6.31.

Let $\phi\in W(k)[[u_{1},\ldots,u_{n}]]$ be a power series as described in
Lemma 6.29. Let
$f:\Omega^{2}S^{3}\rightarrow\operatorname{BGL}_{1}(W^{+}(k)[[u_{1},\ldots,u_{n}]])$
be the map given by the element
$1-\phi(u_{1},\ldots,u_{n})\in\operatorname{GL}_{1}(W(k)[[u_{1},\ldots,u_{n}]])$.
Then the $\mathbb{E}_{2}$-Thom spectrum $Mf$ associated to the map $f$ is as
an $\mathbb{E}_{2}$-$W^{+}(k)[[u_{1},\ldots,u_{n}]]$-algebra equivalent to the
Eilenberg-Maclane spectrum $HA$ of the complete regular local ring $A$ (of
mixed characteristic).

###### Proof.

It follows from Theorem 6.28 by taking $R=k$ and Lemma 6.29. ∎

##  7\. Characterizing Thom spectra as quotients of free
$\mathbb{E}_{2}$-algebras

In this section, we will discuss an alternative characterization of Thom
spectra which we learn from [AB19]. This characterization will enable us to
peel off some redundant restraints in the definition of Thom spectra. We will
rephrase Question 1 more broadly, and give a toy example related to the
Breuil-Kisin case. We note that in fact, we have already used this
characterization in Lemma 3.3.

We first present a theorem which we learn from Antolín-Camarena and Barthel’s
paper:

######  Remark 7.1.

Let $R$ be an $\mathbb{E}_{\infty}$-ring. Let $R[\Omega^{2}S^{2}]$ be the free
$\mathbb{E}_{2}$-$R$-algebra on a single generator in degree 0. Then for all
$\mathbb{E}_{2}$-$R$-algebra $S$ and elements $x\in\pi_{0}(S)$, the universal
property of free $\mathbb{E}_{2}$-$R$-algebras gives rise to a map
$R[\Omega^{2}S^{2}]\rightarrow S$ which maps the generator (in fact, a
connected component) to $x$. We will call this map the evaluation map of
$R[\Omega^{2}S^{2}]$ at $x$.

######  Theorem 7.2 ([AB19, Theorem 4.10]).

Let $R$ be an $\mathbb{E}_{\infty}$-ring and
$\alpha\in\pi_{1}(\operatorname{BGL}_{1}(R))\cong\operatorname{GL}_{1}(\pi_{0}R)$.
Let $q:S^{1}\rightarrow\operatorname{BGL}_{1}(R)$ a loop representing
$\alpha\in\pi_{1}(\operatorname{BGL}_{1}(R))$. Let
$f:\Omega^{2}S^{3}\rightarrow\operatorname{BGL}_{1}(R)$ be the double loop map
associated to $q$ (see Remark 1.2). Then the $\mathbb{E}_{2}$-Thom spectrum
$Mf$ associated to the $\mathbb{E}_{2}$-map $f$ fits into a pushout diagram of
$\mathbb{E}_{2}$-$R$-algebras:

$\begin{array}[]{ccc}R[\Omega^{2}S^{2}]&\longrightarrow&R\\\
{\mbox{\rotatebox[origin={c}]{-90.0}{$\longrightarrow$}}}&&{\mbox{\rotatebox[origin={c}]{-90.0}{$\longrightarrow$}}}\\\
R&\longrightarrow&Mf\end{array}$

where $R[\Omega^{2}S^{2}]\cong R\otimes_{\mathbb{S}}\Sigma_{+}^{\infty}S^{2}$
is the free $\mathbb{E}_{2}$-$R$-algebra on a single generator in degree 0,
and two maps $R[\Omega^{2}S^{2}]\rightarrow R$ are evaluation maps of
$R[\Omega^{2}S^{3}]$ at $0\in\pi_{0}R$ and $1-\alpha\in\pi_{0}R$ respectively.

######  Remark 7.3.

Theorem 7.2 shows that the Thom spectrum description is equivalent to the
pushout-diagram description. However, we note that the pushout-diagram
description is more general in the sense that even if $\alpha\in\pi_{0}R$ is
not invertible, the pushout-diagram description is still valid while we can no
longer, at least superficially, give a Thom spectrum description. We find it
easier to write down proofs for Thom spectrum description so we adapted the
Thom spectrum description for perfectoid rings.

We can now rephrase Question 1 as follows:

######  Question 2.

Given an orientable prism $(A,I=(d))$ . When can we find an
$\mathbb{E}_{\infty}$-ring spectrum $A^{+}$ (which satisfies some hypotheses
related to $A$. A naive guess would be that $\pi_{0}(A^{+})=A$) so that the
Eilenberg-Maclane spectrum $H(A/I)$ as an $\mathbb{E}_{2}$-$A^{+}$-algebra
fits into a pushout diagram

$\begin{array}[]{ccc}A^{+}[\Omega^{2}S^{2}]&\longrightarrow&A^{+}\\\
{\mbox{\rotatebox[origin={c}]{-90.0}{$\longrightarrow$}}}&&{\mbox{\rotatebox[origin={c}]{-90.0}{$\longrightarrow$}}}\\\
A^{+}&\longrightarrow&H(A/I)\end{array}$

such that two maps $A^{+}[\Omega^{2}S^{2}]\rightarrow A^{+}$ are evaluation
maps of the free $\mathbb{E}_{2}$-$A^{+}$-algebra $A^{+}[\Omega^{2}S^{2}]$ at
$0\in\pi_{0}(A^{+})$ and $d\in\pi_{0}(A^{+})$ respectively.

######  Remark 7.4.

Theorem 7.2 shows that Theorem 1.13 answers this question affirmatively when
$(A,I)$ is a perfect prism $(W(R^{\flat}),\ker\theta)$, with
$A^{+}:=W^{+}(R^{\flat})$.

######  Remark 7.5.

Similarly, Theorem 6.14 answers this question affirmatively when $(A,I)$ is a
prism $(W(k)[[u]],(E(u)))$ associated to Breuil-Kisin cohomology where $k$ is
a perfect $\mathbb{F}_{p}$-algebra and $E(u)\in W(k)[u]$ is an Eisenstein
polynomial.

We now announce a toy example of a variant of Theorem 6.14. As there, we fix a
complete DVR $(A,\mathfrak{m})$ of mixed characteristics $(0,p)$ with residue
field $k$ being perfect, absolute ramification index $e$, a uniformizer
$\varpi\in\mathfrak{m}$ and an Eisenstein $W(k)$-polynomial $E(u)\in W(k)[u]$
which induces an isomorphism
$W(k)[u]/(E(u))\xrightarrow{\sim}A,u\mapsto\varpi$ as in Proposition 6.6.

######  Theorem 7.6.

The $(u)$-completion of the total cofiber of the commutative diagram of
$\mathbb{E}_{2}$-$W^{+}(k)[u]$-algebras

$\begin{array}[]{ccc}W^{+}(k)[u]\otimes_{\mathbb{S}}\mathbb{S}[\Omega^{2}S^{2}]&\longrightarrow&W^{+}(k)[u]\\\
{\mbox{\rotatebox[origin={c}]{-90.0}{$\longrightarrow$}}}&&{\mbox{\rotatebox[origin={c}]{-90.0}{$\longrightarrow$}}}\\\
W^{+}(k)[u]&\longrightarrow&HA\end{array}$

is contractible, where two maps
$W^{+}(k)[u]\otimes_{\mathbb{S}}\mathbb{S}[\Omega^{2}S^{3}]\rightarrow
W^{+}(k)[u]$ are given by evaluation maps at $0\in\pi_{0}(W^{+}(k)[u])$ and
$E(u)\in\pi_{0}(W^{+}(k)[u])$ respectively. Equivalently put, the commutative
diagram above induces an equivalence of $W^{+}(k)[u]$-modules from the
$\mathbb{E}_{2}$-pushout of the diagram $W^{+}(k)[u]\leftarrow
W^{+}(k)[u]\otimes_{\mathbb{S}}\mathbb{S}[\Omega^{2}S^{2}]\rightarrow
W^{+}(k)[u]$ to the Eilenberg-Maclane spectrum $HA$ after $(u)$-completion.

######  Corollary 7.7 ([KN19, Remark 3.4]).

The $\mathbb{E}_{2}$-$HA$-algebra $HA\otimes_{W^{+}(k)[u]}HA$ is the
$(p)$-completion of the free $\mathbb{E}_{2}$-$HA$-algebra on a single
generator in degree 1.

###### Proof.

Note that $E(u)$ vanishes after tensoring $HA$, and that $(u)$-completion
coincides with $(p)$-completion for $HA$ since $\varpi^{e}/p$ is an invertible
element, the result follows. ∎

We sketch a proof of Theorem 7.6, which is totally parallel to that of Theorem
6.14.

A sketch of a proof of Theorem 7.6 Let $X$ be the pushout of the diagram
$W^{+}(k)[u]\leftarrow
W^{+}(k)[u]\otimes_{\mathbb{S}}\mathbb{S}[\Omega^{2}S^{2}]\rightarrow
W^{+}(k)[u]$ in question. We first check that the induced map $X\rightarrow
HA$ is the 0th Postnikov section. Then we perform a base change
$W^{+}(k)\otimes_{W^{+}(k)[u]}-$. We show that after such a base change, the
induced map $X\rightarrow HA$ becomes an equivalence, given by Theorem 7.2 and
Lemma 5.3. We then conclude the result by Corollary 6.20. $\Box$

##  Appendix A Recollection of Higher Algebra

This appendix is devoted to a recollection of basic facts in Higher Algebra
needed in the main text. Our main reference is [Lur17], [Lur18b] and [Lur18a].

###  A.1. Finiteness properties of rings and modules

We will include some definitions and properties from [Lur17, Section 7.2.4].

######  Definition A.1 ([Lur17, Notation 7.1.1.10, Proposition 7.1.1.13]).

Given a connective $\mathbb{E}_{1}$-ring $R$, there is a canonical accessible
$t$-structure on $\operatorname{LMod}_{R}$ determined by subcategories
$(\operatorname{LMod}_{R})_{\geq 0}$ and $(\operatorname{LMod}_{R})_{\leq 0}$,
where $(\operatorname{LMod}_{R})_{\geq 0}$ is the full subcategory of
$\operatorname{LMod}_{R}$ spanned by those left $R$-modules $M$ for which
$\pi_{n}M\cong 0$ for $n<0$, and $(\operatorname{LMod}_{R})_{\leq 0}$ is the
full subcategory of $\operatorname{LMod}_{R}$ spanned by those left
$R$-modules $M$ for which $\pi_{n}M\cong 0$ for $n>0$.

######  Proposition A.2 ([Lur17, Proposition 7.1.1.13]).

Let $R$ be a connective $\mathbb{E}_{1}$-ring, then the subcategories
$(\operatorname{LMod}_{R})_{\geq 0},(\operatorname{LMod}_{R})_{\leq
0}\subseteq\operatorname{LMod}_{R}$ are stable under small products and small
filtered colimits.

######  Definition A.3 ([Lur17, Proposition 7.2.2.10]).

Let $M$ be a left module over an $\mathbb{E}_{1}$-ring $R$. We will say that
$M$ is flat if the following conditions are satisfied:

  1. (1)

The homotopy group $\pi_{0}M$ is flat as a left module over $\pi_{0}R$ in the
usual sense.

  2. (2)

For each $n\in\mathbb{Z}$, the natural map
$\operatorname{Tor}_{0}^{\pi_{0}R}(\pi_{n}R,\pi_{0}M)\rightarrow\pi_{n}M$ is
an isomorphism of abelian groups.

######  Definition A.4 ([Lur17, Definition 7.2.4.1]).

Let $R$ be an $\mathbb{E}_{1}$-ring. We let
$\operatorname{LMod}_{R}^{\operatorname{perf}}$ denote the smallest stable
subcategory of $\operatorname{LMod}_{R}$ which contains $R$ (regarded as a
left module over itself) and is closed under retracts. We will say that a left
$R$-module $M$ is perfect if it belongs to
$\operatorname{LMod}_{R}^{\operatorname{perf}}$.

######  Definition A.5 ([Lur17, Definition 7.2.4.8]).

Let $\mathcal{C}$ be a compactly generated $\infty$-category. We will say that
an object $C\in\mathcal{C}$ is almost compact if $\tau_{\leq n}C$ is a compact
object of $\tau_{\leq n}\mathcal{C}$ for all $n\geq 0$.

######  Definition A.6 ([Lur17, Definition 7.2.4.10]).

Let $R$ be a connective $\mathbb{E}_{1}$-ring. We will say that a left
$R$-module $M$ is almost perfect if there exists an integer $k$ such that
$M\in(\operatorname{LMod}_{R})_{\geq k}$ and is almost compact as an object of
$(\operatorname{LMod}_{R})_{\geq k}$. We let
$\operatorname{LMod}_{R}^{\operatorname{aperf}}$ denote the full subcategory
of $\operatorname{LMod}_{R}$ spanned by the almost perfect left $R$-modules.

######  Proposition A.7 ([Lur17, Proposition 7.2.4.11]).

Let $R$ be a connective $\mathbb{E}_{1}$-ring. Then:

  1. 1.

The full subcategory
$\operatorname{LMod}_{R}^{\operatorname{aperf}}\subseteq\operatorname{LMod}_{R}$
is closed under translation and finite colimits, and is therefore a stable
subcategory of $\operatorname{LMod}_{R}$;

  2. 2.

The full subcategory
$\operatorname{LMod}_{R}^{\operatorname{aperf}}\subseteq\operatorname{LMod}_{R}$
is closed under retracts;

  3. 3.

Every perfect left $R$-module is almost perfect;

  4. 4.

The full subcategory $(\operatorname{LMod}_{R}^{\operatorname{aperf}})_{\geq
0}\subseteq\operatorname{LMod}_{R}$ is closed under geometric realizations of
simplicial objects;

  5. 5.

Let $M$ be a left $R$-module which is connective and almost perfect. Then $M$
can be obtained as the geometric realization of a simplicial left $R$-module
$P_{\bullet}$ such that each $P_{n}$ is a free $R$-module of finite rank.

######  Proposition A.8.

Let $f:A\rightarrow A^{\prime}$ be a map of connective $\mathbb{E}_{1}$-rings.
Let $M$ be a connective left $A$-module and set
$M^{\prime}=A^{\prime}\otimes_{A}M$. If $M$ is an almost perfect left
$A$-module, then $M^{\prime}$ is an almost perfect left $A^{\prime}$-module.

###### Proof.

Since $M$ is connective and almost perfect, by Proposition A.7, there exists a
simplicial object $P_{\bullet}$ in $\operatorname{LMod}_{A}$ such that each
$P_{n}$ is a free $A$-module of finite rank and $M$ is equivalent to the
geometric realization of $P_{\bullet}$. Therefore $M^{\prime}$ is equivalent
to the geometric realization of $A^{\prime}\otimes_{A}P_{\bullet}$, by the
fact the tensor products commute with small colimits. On the other hand, each
$A^{\prime}\otimes_{A}P_{n}$ is a free $A^{\prime}$-module of finite rank,
hence perfect, thus almost perfect. Now $M^{\prime}$ is equivalent to the
geometric realization of almost perfect modules, therefore $M^{\prime}$ is
almost perfect by Proposition A.7. ∎

######  Definition A.9 ([Lur17, Definition 7.2.4.13]).

A discrete associative ring $R$ is left coherent if every finitely generated
left ideal of $R$ is finitely presented as a left $R$-module.

######  Definition A.10 ([Lur17, Definition 7.2.4.16]).

Let $R$ be an $\mathbb{E}_{1}$-ring. We will say that $R$ is left coherent if
the following conditions are satisfied:

  1. 1.

The $\mathbb{E}_{1}$-ring $R$ is connective;

  2. 2.

The discrete associative ring $\pi_{0}R$ is left coherent;

  3. 3.

For each $n\geq 0$, the homotopy group $\pi_{n}R$ is finitely presented as a
left module over $\pi_{0}R.$

######  Proposition A.11 ([Lur17, Proposition 7.2.4.17]).

Let $R$ be an $\mathbb{E}_{1}$-ring and $M$ a left $R$-module. Suppose that
$R$ is left coherent. Then $M$ is almost perfect if and only if the following
conditions are satisfied:

  1. i.

For $m\ll 0$, $\pi_{m}M=0$;

  2. ii.

For every integer $m$, $\pi_{m}M$ is finitely presented as a left
$\pi_{0}R$-module.

######  Corollary A.12.

Let $R$ be a left coherent $\mathbb{E}_{1}$-ring, then H $\pi_{0}(R)$ as a
left $R$-module is almost perfect.

###  A.2. Nilpotent, local and complete modules

We will include several definitions and propositions from [Lur18b], Chapter 7.

######  Definition A.13 ([Lur18b, Definition 7.1.1.1, Example 7.1.1.2]).

Let $R$ be a connective $\mathbb{E}_{\infty}$-ring and let $x\in\pi_{0}R$. An
$R$-module $M$ is $x$-nilpotent if the localization $M[1/x]$ vanishes.
Equivalently, $M$ is $x$-nilpotent if and only if the action of $x$ on
$\pi_{\ast}M$ is locally nilpotent, that is, if and only if for each
$y\in\pi_{j}M$, there exists an integer $n\gg 0$ such that $x^{n}y=0$ in
$\pi_{j}M$ for all $j\in\mathbb{Z}$.

######  Definition A.14 ([Lur18b, Definition 7.1.1.6]).

Let $R$ be a connective $\mathbb{E}_{\infty}$-ring and let
$I\subseteq\pi_{0}R$ be an ideal. We say that an $R$-module $M$ is
$I$-nilpotent if it is $x$-nilpotent for each $x\in I$.

######  Definition A.15 ([Lur18b, Definition 7.2.4.1]).

Let $R$ be a connective $\mathbb{E}_{\infty}$-ring and let
$I\subseteq\pi_{0}R$ be an ideal. We say that an $R$-module $M$ is $I$-local
if for every $I$-nilpotent $R$-module $N$, the mapping space
$\operatorname{Map}_{\operatorname{Mod}_{R}}(N,M)$ is contractible.

######  Definition A.16 ([Lur18b, Definition 7.3.1.1]).

Let $R$ be a connective $\mathbb{E}_{\infty}$-ring and let
$I\subseteq\pi_{0}R$ be an ideal. We will say that an $R$-module $M$ is
$I$-complete if for every $I$-local $R$-module $N$, the mapping space
$\operatorname{Map}_{\operatorname{Mod}_{R}}(N,M)$ is contractible.

######  Corollary A.17.

Let $R$ be a connective $\mathbb{E}_{\infty}$-ring and let
$I\subseteq\pi_{0}R$ be an ideal. If $M$ is an $I$-nilpotent $R$-module, then
it is also an $I$-complete $R$-module.

######  Proposition A.18 ([Lur18b, Proposition 7.3.1.4 and Notation 7.3.1.5]).

Let $R$ be a connective $\mathbb{E}_{\infty}$-ring and let
$I\subseteq\pi_{0}R$ be a finitely generated ideal. Then every left $R$-module
$M$ fits into an (essentially unique) fiber sequence $M^{\prime}\rightarrow
M\rightarrow M^{\prime\prime}$, where $M^{\prime}$ is $I$-local and
$M^{\prime\prime}$ is $I$-complete. Moreover, there is a functor, called the
$I$-completion functor,
$\operatorname{Mod}_{R}\rightarrow\operatorname{Mod}_{R}$, which maps $M$ to
$M^{\prime\prime}$. We denote by $M_{I}^{\wedge}$ the image of $M$ under the
$I$-completion functor.

We can compute the $I$-completion functor when $I$ is principal:

######  Proposition A.19 ([Lur18b, Proposition 7.3.2.1]).

Let $R$ be a connective $\mathbb{E}_{\infty}$-ring and let $x\in\pi_{0}R$ be
an element. For any $R$-module $M\in\operatorname{Mod}_{R}$, let $T(M)$ denote
the limit of the tower

$\cdots\xrightarrow{x}M\xrightarrow{x}M\xrightarrow{x}M\xrightarrow{x}M$

Then $T(M)$ is $(x)$-local and the $(x)$-completion of $M$ can be identified
with the cofiber of the canonical map $\theta:T(M)\rightarrow M$.

######  Corollary A.20 ([Lur18b, Corollary 7.3.2.2]).

Let $R$ be a connective $\mathbb{E}_{\infty}$-ring and let $x\in\pi_{0}R$ be
an element. The following conditions on an $R$-module
$M\in\operatorname{Mod}_{R}$ are equivalent:

  1. 1.

The module $M$ is $(x)$-complete.

  2. 2.

The limit of the tower

$\cdots\xrightarrow{x}M\xrightarrow{x}M\xrightarrow{x}M\xrightarrow{x}M$

vanishes.

######  Corollary A.21 ([Lur18b, Corollary 7.3.2.3]).

Let $R$ be a connective $\mathbb{E}_{\infty}$-ring, $I\subseteq\pi_{0}R$ an
ideal and $x\in\pi_{0}R$ an element. Then the $(x)$-completion functor
$\operatorname{Mod}_{R}\rightarrow\operatorname{Mod}_{R},M\mapsto
M_{(x)}^{\wedge}$ carries $I$-complete modules to $I$-complete modules.

######  Corollary A.22 ([Lur18b, Corollary 7.3.2.4]).

Let $R$ be a connective $\mathbb{E}_{\infty}$-ring, $x\in\pi_{0}R$ and let $M$
be an $R$-module.

  1. 1.

If the $R$-module $M$ is connective, then the $(x)$-completion
$M_{(x)}^{\wedge}$ is connective.

  2. 2.

If $M\in(\operatorname{Mod}_{R})_{\leq 0}$, then
$M_{(x)}^{\wedge}\in(\operatorname{Mod}_{R})_{\leq 1}$.

###### Proof.

Let $T(M)$ be the limit of the tower
$\left(\cdots\xrightarrow{x}M\xrightarrow{x}M\xrightarrow{x}M\right)$. Then by
Proposition A.19, we have the cofiber sequence $T(M)\rightarrow M\rightarrow
M_{(x)}^{\wedge}$ which gives rise to a long exact sequence

$\cdots\rightarrow\pi_{n}(T(M))\rightarrow\pi_{n}(M)\rightarrow\pi_{n}(M_{(x)}^{\wedge})\rightarrow\pi_{n-1}(T(M))\rightarrow\pi_{n-1}(M)\rightarrow\pi_{n-1}(M_{(x)}^{\wedge})\rightarrow\cdots$

Furthermore, let $T_{n}(M)_{\ast}$ be the tower

$\cdots\xrightarrow{x}\pi_{n}(M)\xrightarrow{x}\pi_{n}(M)\xrightarrow{x}\pi_{n}(M)$

Then there is a Milnor sequence

$0\rightarrow\lim^{1}T_{n+1}(M)_{\ast}\rightarrow\pi_{n}(T(M))\rightarrow\lim
T_{n}(M)_{\ast}\rightarrow 0$

Especially, if $M$ is assumed to be connective, then $T_{n}(M)_{\ast}$ is a
tower of $0$ for $n<0$, which implies that $\pi_{n-1}(T(M))$ vanishes when
$n<0$. We deduce from the long exact sequence that $\pi_{n}(M_{(x)}^{\wedge})$
vanishes when $n<0$. Similarly, if $M\in(\operatorname{Mod}_{R})_{\leq 0}$,
then $T_{n}(M)_{\ast}$ is a tower of $0$ for $n>0$, thus $\pi_{n}(T(M))$
vanishes when $n\geq 0$. We deduce from the long exact sequence that
$\pi_{n}(M_{(x)}^{\wedge})$ vanishes when $n>0$. ∎

######  Proposition A.23 ([Lur18b, Corollary 7.3.3.3]).

Let $R$ be a connective $\mathbb{E}_{\infty}$-ring and $I\subseteq\pi_{0}R$ be
a finitely generated ideal. Let $M$ be an $R$-module. Then the following
conditions on $M$ are equivalent:

  1. 1.

$M$ is $I$-complete;

  2. 2.

For each $x\in I$, $M$ is $(x)$-complete;

  3. 3.

There exists a set of generators $x_{1},\ldots,x_{n}$ for the ideal $I$ such
that $M$ is $(x_{i})$-complete for $i=1,\ldots,n$.

######  Remark A.24 ([Lur18b, Corollary 7.3.3.6]).

Let $\phi:R\rightarrow R^{\prime}$ be a morphism of connective
$\mathbb{E}_{\infty}$-rings, $I\subseteq\pi_{0}R$ a finitely generated ideal
and $I^{\prime}=\phi(I)\pi_{0}(R^{\prime})$ the ideal generated by the image
of $I$. Then

  1. 1.

An $R^{\prime}$-module $M$ is $I^{\prime}$-complete if and only if it is
$I$-complete as an $R$-module;

  2. 2.

For every $R^{\prime}$-module $M$, the canonical map $M\rightarrow
M_{I^{\prime}}^{\wedge}$ exhibits $M_{I^{\prime}}^{\wedge}$ as an
$I$-completion of $M$, regarded as a morphism of $R$-modules.

######  Theorem A.25 ([Lur18b, Theorem 7.3.4.1]).

Let $R$ be an $\mathbb{E}_{\infty}$-ring, let $I\subseteq\pi_{0}R$ be a
finitely generated ideal and let $M$ be an $R$-module. The following
conditions are equivalent:

  1. a)

The $R$-module $M$ is $I$-complete;

  2. b)

For every integer $k$, the homotopy group $\pi_{k}M$ satisfies the condition
that for each $x\in I$, we have
$\operatorname{Ext}_{A}^{0}(A[1/x],\pi_{k}M)=0=\operatorname{Ext}_{A}^{1}(A[1/x],\pi_{k}M)$
where $A=\pi_{0}R.$

######  Proposition A.26 ([Lur18b, Proposition 7.3.4.8]).

Let $R$ be a connective $\mathbb{E}_{\infty}$-ring, let $I\subseteq\pi_{0}R$
be a finitely generated ideal, and let $x\in\pi_{0}R$ be an element whose
image in $(\pi_{0}R)/I$ is invertible. If $M$ is an $I$-complete left
$R$-module, then multiplication by $x$ induces an equivalence from $M$ to
itself.

######  Proposition A.27 ([Lur18b, Proposition 7.3.5.7]).

Let $R$ be a connective $\mathbb{E}_{\infty}$-ring, let $I\subseteq\pi_{0}R$
be a finitely generated ideal, and let $M$ be an almost perfect $R$-module. If
$R$ is $I$-complete, then so is $M$.

######  Definition A.28 ([Lur18b, Section 7.3.6]).

Let $R$ be a discrete commutative ring and $M$ be a (discrete) $R$-module. Let
$I\subseteq R$ be a finitely generated ideal. The $I$-adic completion of $M$,
denoted by $\operatorname{Cpl}(M;I)$, is defined to be the limit of tower
$\lim M/I^{n}M$. There is a canonical map
$M\rightarrow\operatorname{Cpl}(M;I)$. An $R$-module $M$ is called
$I$-adically complete if the canonical map
$M\rightarrow\operatorname{Cpl}(M;I)$ is an isomorphism, and $M$ is called
$I$-adically separated if the canonical map
$M\rightarrow\operatorname{Cpl}(M;I)$ is injective.

######  Proposition A.29 ([Lur18b, Corollary 7.3.6.3]).

Let $R$ be a discrete commutative ring, let $I\subseteq R$ be a finitely
generated ideal, and let $M$ be a discrete $R$-module. The following
conditions are equivalent:

  1. a)

The module $M$ is $I$-adically complete;

  2. b)

The module $HM$ is $I$-complete and $M$ is $I$-adically separated.

######  Warning A.30.

By Proposition A.29, the concept of $I$-adic completeness does not coincide
with the concept of $I$-completeness for discrete modules over discrete
commutative rings. Rather, the former is stronger than the latter.

######  Definition A.31.

A spectrum $X$ is called $p$-complete if it is $(p)$-complete as an
$\mathbb{S}$-module. For any spectrum $X$, the $p$-completion of $X$, denoted
by $X_{p}^{\wedge}$, is the $(p)$-completion of $X$ as an $\mathbb{S}$-module.

######  Remark A.32.

When $M$ is an $R$-module for a connective $\mathbb{E}_{\infty}$-ring $R$,
$(p)$ is also an ideal of $\pi_{0}R$. In this case, it follows from Remark
A.24 that $M$ is $(p)$-complete as an $\mathbb{S}$-module if and only if it is
$(p)$-complete as an $R$-module, so there is completely no ambiguity to talk
about $p$-completeness. Similarly, Remark A.24 implies the $p$-completion of
an $R$-module $M$ is the underlying spectrum of the $(p)$-completion of $M$ as
an $R$-module.

######  Corollary A.33.

Let $X$ be a bounded below spectrum. If $X$ is $p$-complete and
$H\mathbb{F}_{p}\otimes X\simeq 0$, then $X\simeq 0$.

###### Proof.

We will show inductively on $n\in\mathbb{Z}$ that $\pi_{n}X=0$.

  1. 1.

Since $X$ is bounded below, $\pi_{n}X=0$ for $n\ll 0$.

  2. 2.

Suppose now that for every $m<n$, we have $\pi_{m}X=0$. We will show that
$\pi_{n}X=0$. In this case, we have $0=\pi_{n}(H\mathbb{F}_{p}\otimes
X)\cong\operatorname{Tor}_{0}^{\mathbb{Z}}(\mathbb{F}_{p},\pi_{n}X)$. Thus for
each $x\in\pi_{n}X$, there exists (by axiom of choice) a sequence
$(x_{j})_{j\in\mathbb{N}}\in(\pi_{n}X)^{\mathbb{N}}$ such that $x_{0}=x$ and
$x_{j}=px_{j+1}$ for all $j\in\mathbb{N}$, which gives rise to a map
$\varphi_{x}:\mathbb{Z}[1/p]\rightarrow\pi_{n}X$ of abelian groups given by
$\varphi(1/p^{j})=x_{j}$. Theorem A.25 tells us that $\varphi_{x}=0$, and
especially, $x=0$. In conclusion, we have proved that $x=0$ for each
$x\in\pi_{n}X$, thus $\pi_{n}X=0$.

∎

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