# A motivic analogue of
the $K(1)$-local sphere spectrum

William Balderrama  University of Virginia<EMAIL_ADDRESS>,  Kyle Ormsby
Reed College / University of Washington<EMAIL_ADDRESS>/<EMAIL_ADDRESS>and
J.D. Quigley  University of Oregon<EMAIL_ADDRESS>
###### Abstract.

We identify the motivic $KGL/2$-local sphere as the fiber of $\psi^{3}-1$ on
$(2,\eta)$-completed Hermitian $K$-theory, over any base scheme containing
$1/2$. This is a motivic analogue of the classical resolution of the
$K(1)$-local sphere, and extends to a description of the $KGL/2$-localization
of an arbitrary motivic spectrum. Our proof relies on a novel conservativity
argument that should be of broad utility in stable motivic homotopy theory.

###### Contents

  1. 1 Introduction
  2. 2 First reductions
  3. 3 Reduction to $S=\operatorname{Spec}(\mathbb{C})$
  4. 4 The case $S=\operatorname{Spec}(\mathbb{C})$

##  1\. Introduction

The motivic stable homotopy category $\operatorname{SH}(k)$ over a field $k$
was introduced by Morel and Voevodsky [MV99] in the 1990’s to apply powerful
tools from algebraic topology to problems in algebraic geometry. Its successes
include the resolution of the Milnor and Bloch–Kato conjectures [Voe03b,
Voe03a, OVV07, Voe11, HW19], new computations in algebraic and Hermitian
$K$-theory [HKO11, RØ16b], detailed analyses of algebraic cobordism [OOsr13],
contributions to the developing field of quadratic enumerative geometry [KW19,
Lev20], and computational input to the Asok–Fasel–Hopkins program on algebraic
vector bundles [AF14a, AF14b, AFH19].

There has been considerable interest in studying localizations of the motivic
stable homotopy category. Morel showed in [Mor06] that
$\operatorname{SH}(k)[\frac{1}{2}]$ splits as a product of its “positive” and
“negative” parts, and Cisinski–Déglise [CD19], Garkusha [Gar19], and
Déglise–Fasel–Khan–Jin [DFJK21] have analyzed the rationalization
$\operatorname{SH}(k)_{\mathbb{Q}}$. In particular, the positive part is
related to the theory of rational motives, while the negative part is related
to Witt theory and $\eta$-periodic phenomenon.

Here $\eta\in\pi_{1,1}(\mathbf{1}_{k})$ is represented by the canonical
$\mathbb{G}_{m}$-torsor $\mathbb{A}^{2}\smallsetminus 0\to\mathbb{P}^{1}$ and
may be viewed as a graded endomorphism of the unit object $\mathbf{1}_{k}$ of
$\operatorname{SH}(k)$. The coefficients of the $\mathbf{1}_{k}[\eta^{-1}]$
have been studied over various base fields ([AM17, CQ21, GI15, GI16, OR20,
Wil18]), and Bachmann–Hopkins [BH20, Thm. 1.1] recently showed that

$\mathbf{1}_{k}[\eta^{-1}]_{(2)}\simeq\operatorname{fib}\left(\varphi\colon
kw_{(2)}\to\Sigma^{4}kw_{(2)}\right).$

Here, $kw=KQ[\eta^{-1}]_{\geq 0}$ is the spectrum of connective Balmer–Witt
K-theory and $\varphi$ is a lift of $\psi^{3}-1$.

Chromatic localizations play a prominent role in the homotopy theory of
topological spectra. Readers unfamiliar with Morava $E$\- and $K$-theories and
their roles in stratifying the stable homotopy category via the moduli stack
of one-dimensional commutative formal group laws may consult [Rav92]. Many of
these constructions have been ported to the motivic context [Bor09, Hor18,
HO18, Joa20, Kra18], and their role in determining thick subcategories,
nilpotence, and periodicity in stable motivic homotopy theory remains a topic
of contemporary interest.

In this work, we study a motivic analogue of $K(1)$-localization at the prime
$2$. (The assumption $p=2$ is implicit henceforth.) In the topological
setting, $K(1)=KU/2$, and localization at $K(1)$ corresponds to the first
monochromatic layer. Famously, there is a short “resolution” of the
$K(1)$-local sphere in terms of real $K$-theory $KO$ (see [Ada74], [Bou79,
Theorem 4.3], [Rav84, Lemma 8.7], and [HMS94, Lemma 2.3]). Define the spectrum
$J$ to be the fiber of $\psi^{3}-1$ acting on $2$-complete real $K$-theory

$J\longrightarrow KO^{\wedge}_{2}\xrightarrow{\psi^{3}-1}KO^{\wedge}_{2};$

then $L_{K(1)}X=(X\otimes J)^{\wedge}_{2}$ for any spectrum $X$. (Here
$\psi^{3}$ is the third Adams operation on $KO^{\wedge}_{2}$; it may be
replaced with any $\psi^{g}$ for $g$ a topological generator of
$\mathbb{Z}_{2}^{\times}/\\{\pm 1\\}$.) This result provides tremendous
computational control over $K(1)$-localizations. Indeed, the induced long
exact sequence demonstrates that the groups $\pi_{*}L_{K(1)}X$ only depend on
the $KO^{\wedge}_{2}$-homology of $X$, the action of $\psi^{3}$ thereon, and
potential additive extensions.

We have used the traditional name $J$ for $L_{K(1)}S$ because of the tight
relationship between this spectrum and classes in the image of the
$J$-homomorphism. First constructed by Mahowald, the _connective image-of- $J$
spectrum_ $j$ is the fiber of a lift of $\psi^{3}-1$ to a map
$ko\to\Sigma^{4}ksp$. (Here $ksp$ is connective quaternionic $K$-theory, and
$\Sigma^{4}ksp\simeq bspin$ is the $3$-connected cover of $KO$.) The spectrum
$j$ admits a unit $\mathbf{1}=S^{0}\to j$ that detects the image of the
classical $J$-homorphism $\pi_{*+1}KO\to\pi_{*}S^{0}$ and the simple
$2$-torsion Adams classes in degrees $8k+1$ and $8k+2$ [LM87]. It is also the
case that the natural map $j\to J$ induces an isomorphism on $\pi_{n}$ for
$n\geq 2$.

Our motivic analogue of $J$ follows a similar pattern but replaces $KO$ with
the Hermitian $K$-theory $\mathbb{P}^{1}$-spectrum $KQ$ representing the
$K$-theory of symmetric bilinear forms. To be more precise, fix a base scheme
$S$ containing $1/2$. The $S$-motivic spectra $KGL$ and $KQ$ of algebraic and
Hermitian $K$-theory are motivic refinements of the classical spectra $KU$ and
$KO$ of complex and real $K$-theory. (For convenience, we use the definitions
of [BH20, §3.2.5] for $KGL$ and $KQ$ over $S$, and note that they represent
algebraic and Hermitian $K$-theory, respectively, as long as $S$ is Noetherian
and regular.) We define the motivic spectrum $JQ$ (the _quadratic
$J$-spectrum_) to be the fiber of $\psi^{3}-1$ acting on
$KQ^{\wedge}_{(2,\eta)}$ so that there is a fiber sequence

$JQ\longrightarrow
KQ_{(2,\eta)}^{\wedge}\xrightarrow{\psi^{3}-1}KQ_{(2,\eta)}^{\wedge}.$

######  Theorem A.

Let $X$ be a motivic spectrum over a base scheme $S$ containing $1/2$. Then
there is a fiber sequence

$L_{KGL/2}X\to(X\otimes
KQ)_{(2,\eta)}^{\wedge}\xrightarrow{\psi^{3}-1}(X\otimes
KQ)_{(2,\eta)}^{\wedge},$

or equivalently,

$L_{KGL/2}X\simeq(X\otimes JQ)^{\wedge}_{(2,\eta)}.$

######  Remark 1.1.

Belmont–Isaksen–Kong [BIK22] and Kong and the third author [KQ22] studied a
“connective” analogue of $JQ$ over a variety of base fields $k$, there denoted

$L=\operatorname{fib}\left(\psi^{3}-1\colon kq_{(2,\eta)}^{\wedge}\rightarrow
kq_{(2,\eta)}^{\wedge}\right).$

Here, $kq=\tilde{f}_{0}KQ$ is the very effective cover of Hermitian K-theory
[ARØ20]. The very effective cover functor is not triangulated, so the
computations in [BIK22, KQ22] do not immediately yield the homotopy groups of
$\tilde{f}_{0}L_{KGL/2}\mathbf{1}_{k}$; the precise relationship between these
groups will be described in a forthcoming revision of [KQ22].

######  Remark 1.2.

The $C_{2}$-equivariant Betti realization functor
$\operatorname{Be}\colon\operatorname{SH}(\mathbb{R})\rightarrow\operatorname{Sp}^{C_{2}}$
sends $KGL$ and $KQ$ to the $C_{2}$-equivariant spectra $K\mathbb{R}$ and
$KO_{C_{2}}$ respectively. Thus the $\mathbb{R}$-motivic $KGL/2$-local sphere
may be viewed as a lift of the $C_{2}$-equivariant $K\mathbb{R}/2$-local
sphere
$L_{K\mathbb{R}/2}S_{C_{2}}\simeq\operatorname{fib}\left(\psi^{3}-1\colon(KO_{C_{2}})_{2}^{\wedge}\rightarrow(KO_{C_{2}})_{2}^{\wedge}\right)$
analyzed by the first author in [Bal21].

After fixing some notation in Subsection 1.1, we outline the proof of A in
Subsection 1.2 below.

###  1.1. Conventions

We maintain the following conventions throughout the paper.

  1. (1)

We work over a base scheme $S$ containing $1/2$.

  2. (2)

We write $\operatorname{SH}(S)$ for the category of $S$-motivic spectra.

  3. (3)

We write $\mathbf{1}_{S}\in\operatorname{SH}(S)$ for the $S$-motivic sphere
spectrum.

  4. (4)

We abbreviate
$\operatorname{SH}(\operatorname{Spec}(k))=\operatorname{SH}(k)$,
$\mathbf{1}_{k}=\mathbf{1}_{\operatorname{Spec}(k)}$, and so forth.

  5. (5)

We write $KGL$ and $KQ$ for the $S$-motivic algebraic and Hermitian $K$-theory
spectra respectively (see [BH20, §3.2], where $KQ$ is denoted $KO$).

  6. (6)

For a field $k$ containing $1/2$, we write $k^{M}_{n}(k)=K^{M}_{n}(k)/2$ for
the $n$th mod $2$ Milnor $K$-groups of $k$.

  7. (7)

We abbreviate $\rho:=[-1]\in\pi_{-1,-1}\mathbf{1}_{S}$ for the class
represented by the unstable map
$\operatorname{Spec}(S)_{+}\to\mathbb{A}^{1}\smallsetminus 0$ induced by the
map $S[x]\to S$ evaluating at $-1$.

  8. (8)

We abbreviate $JQ=\operatorname{fib}\left(\psi^{3}-1\colon
KQ_{(2,\eta)}^{\wedge}\to KQ_{(2,\eta)}^{\wedge}\right)$, so that in general
$(X\otimes
JQ)_{(2,\eta)}^{\wedge}=\operatorname{fib}\left(\psi^{3}-1\colon(X\otimes
KQ)_{(2,\eta)}^{\wedge}\to(X\otimes KQ)_{(2,\eta)}^{\wedge}\right)$.

  9. (9)

We write $\operatorname{map}(X,Y)$ for the spectrum of maps between objects
$X$ and $Y$.

###  1.2. Outline of proof

We now outline the proof of A. Using the above notation, the theorem asserts
that if $X$ is an $S$-motivic spectrum then the natural map
$X\rightarrow(X\otimes JQ)^{\wedge}_{(2,\eta)}$ realizes $(X\otimes
JQ)^{\wedge}_{(2,\eta)}$ as the $KGL/2$-localization of $X$. By definition, to
prove this we must verify that $(X\otimes JQ)_{(2,\eta)}^{\wedge}$ is
$KGL/2$-local, and that $X\rightarrow(X\otimes JQ)_{(2,\eta)}^{\wedge}$ is a
$KGL/2$-equivalence.

The fact that $(X\otimes JQ)_{(2,\eta)}^{\wedge}$ is $KGL/2$-local follows
from a formal argument using the fact that $KGL\simeq KQ/\eta$. The same line
of reasoning reduces checking that $X\rightarrow(X\otimes
JQ)_{(2,\eta)}^{\wedge}$ is a $KGL/2$-equivalence for any $X$ to just the case
$X=\mathbf{1}_{S}$. We give these simple reductions in Section 2. The bulk of
our work is then to verify that the map $\mathbf{1}_{S}\rightarrow JQ$ is a
$KGL/2$-equivalence. The proof of this splits naturally into two parts: the
reduction to $S=\operatorname{Spec}(\mathbb{C})$, carried out in Section 3,
and the case $S=\operatorname{Spec}(\mathbb{C})$, carried out in Section 4.

These proceed as follows. First, the assertion that $\mathbf{1}_{S}\rightarrow
JQ$ is a $KGL/2$-equivalence is stable under base change, and this allows us
to reduce to considering just
$S=\operatorname{Spec}(\mathbb{Z}[\tfrac{1}{2}])$. To then reduce to
$S=\operatorname{Spec}(\mathbb{C})$, we prove the following general
conservativity result: base change
$\operatorname{SH}(\mathbb{Z}[\tfrac{1}{2}])\rightarrow\operatorname{SH}(\mathbb{C})$
is conservative when restricted to the full subcategory of $2$-torsion
cellular motivic spectra with vanishing real Betti realization and convergent
effective slice tower (Theorem 3.1). Once we have reduced to
$S=\operatorname{Spec}(\mathbb{C})$, we make use of the close relation between
$\mathbb{C}$-motivic homotopy theory and the classical Adams–Novikov spectral
sequence to further reduce to the classical description of the non-motivic
$K(1)$-local sphere.

######  Remark 1.3.

Pelaez and Weibel show in [PW14] that the $KGL$-cooperation algebra
$KGL_{\star}KGL$ admits a decomposition mimicking the classical
$KU$-cooperation algebra $KU_{\ast}KU$. We do not use this, but do use the
simpler fact that $KGL\otimes KGL$ is a free $KGL$-module. It could be
interesting to pursue an alternate approach which uses their theorem to give a
more computational proof of A. The authors hope that the reduction-
to-$\mathbb{C}$ method pioneered here will prove useful in other applications
where explicit computations may be inaccessible over general base schemes but
feasible over $\mathbb{C}$.

###  1.3. Acknowledgements

The authors thank Jeremiah Heller for explaining the extension of A from
fields to Dedekind domains, and thank Tom Bachmann for pointing out that
cellularity is not necessary. They thank the anonymous referees for multiple
helpful suggestions. The first author was supported by NSF RTG grant
DMS-1839968; the second author was partially supported by NSF grant
DMS-2204365; the third author was supported by NSF grants DMS-2039316 and
DMS-2314082.

##  2\. First reductions

By definition, to show that $L_{KGL/2}X\simeq(X\otimes
JQ)_{(2,\eta)}^{\wedge}$, we must show that

  1. (1)

$(X\otimes JQ)_{(2,\eta)}^{\wedge}$ is $KGL/2$-local, in the sense that if $C$
is $KGL/2$-acyclic then $\operatorname{map}(C,(X\otimes
JQ)_{(2,\eta)}^{\wedge})=0$;

  2. (2)

$X\to(X\otimes JQ)_{(2,\eta)}^{\wedge}$ is a $KGL/2$-equivalence.

In this section, we prove (1), and show that (2) holds for all $X$ as soon as
it holds for $X=\mathbf{1}_{S}$. We will make frequent use of the following
fact, due to Röndigs and Østvær.

######  Lemma 2.1 ([RØ16a, Theorem 3.4]).

There is an equivalence $KGL\simeq KQ/\eta$. ∎

######  Proposition 2.2.

Let $X$ be an $S$-motivic spectrum. Then $(X\otimes JQ)_{(2,\eta)}^{\wedge}$
is $KGL/2$-local.

###### Proof.

As the subcategory of $KGL/2$-local spectra is closed under fibers, it
suffices to show that $(X\otimes KQ)_{(2,\eta)}^{\wedge}$ is $KGL/2$-local.
Note that $KQ$ is an $\mathbb{E}_{\infty}$ ring spectrum [BH20, Section 3.2].
So to simplify the notation, let us show more generally that if $M$ is any
$(2,\eta)$-complete $KQ$-module, then the underlying motivic spectrum of $M$
is $KGL/2$-local.

By Lemma 2.1, we may identify $KGL/2\simeq KQ/(2,\eta)$. It follows that if
$C$ is $KGL/2$-acyclic, then $(KQ\otimes C)_{(2,\eta)}^{\wedge}\simeq 0$. From
here we compute

$\operatorname{map}(C,M)\simeq\operatorname{Mod}_{KQ}(KQ\otimes
C,M)\simeq\operatorname{Mod}_{KQ}((KQ\otimes C)_{(2,\eta)}^{\wedge},M)\simeq
0,$

so that $M$ is $KGL/2$-local as claimed. ∎

######  Proposition 2.3.

Suppose that $\mathbf{1}_{S}\to JQ$ is a $KGL/2$-equivalence. Then
$X\to(X\otimes JQ)_{(2,\eta)}^{\wedge}$ is a $KGL/2$-equivalence for any
$S$-motivic spectrum $X$.

###### Proof.

As $KGL/2\simeq KQ/(2,\eta)$, we have

$(X\otimes KQ)_{(2,\eta)}^{\wedge}\otimes KGL/2\simeq X\otimes KQ\otimes
KGL/2$

for any $X$. Thus $X\to(X\otimes JQ)_{(2,\eta)}^{\wedge}$ is a
$KGL/2$-equivalence if and only if the natural map

$X\otimes KGL/2\rightarrow X\otimes JQ\otimes KGL/2$

is an equivalence. This map is obtained from the case $X=\mathbf{1}_{S}$ by
smashing with $X$, so if it is an equivalence for $X=\mathbf{1}_{S}$ then it
is an equivalence for any $X$. ∎

##  3\. Reduction to $S=\operatorname{Spec}(\mathbb{C})$

It remains to show that $\mathbf{1}_{S}\to JQ$ is a $KGL/2$-equivalence. In
this section, we show how one may reduce this statement over an arbitrary base
scheme $S$ containing $1/2$ to just the case
$S=\operatorname{Spec}(\mathbb{C})$, which will then be carried out in Section
4.

###  3.1. A conservativity result

The heart of our argument is the following.

######  Theorem 3.1.

Base change
$\operatorname{SH}(\mathbb{Z}[\tfrac{1}{2}])\rightarrow\operatorname{SH}(\mathbb{C})$
is conservative when restricted to the full subcategory of $2$-torsion
cellular motivic spectra with vanishing real Betti realization and convergent
effective slice tower.

Here, we refer to the effective slice tower in the sense of Voevodsky [Voe98,
Section 2]. The rest of this subsection is dedicated to the proof of Theorem
3.1. In Subsection 3.2, we verify that this implies the desired reduction. The
starting point for the proof is the following conservativity result of
Bachmann and Hoyois.

######  Lemma 3.2.

For $X\in\operatorname{SH}(\mathbb{Z}[\tfrac{1}{2}])$ to vanish, it suffices
that the base change of $X$ to $\operatorname{Spec}(k)$ vanishes for $k$ a
prime field other than $\mathbb{F}_{2}$.

###### Proof.

This is a special case of [BH21, Proposition B.3]. ∎

This would reduce us to considering fields, only if $k$ has positive
characteristic then there is no obvious comparison between
$\operatorname{SH}(k)$ and $\operatorname{SH}(\mathbb{C})$. To handle this we
make use of the following Lefschetz principle.

######  Lemma 3.3.

Let $K$ be an algebraically closed field containing $1/2$. Then there is a
zigzag of base change functors between $\operatorname{SH}(K)$ and
$\operatorname{SH}(\mathbb{C})$ which is an equivalence on full subcategories
of $2$-torsion cellular obects.

###### Proof.

This follows from [BCQ21, Proposition 5.2.1] and its proof upon restricting to
$2$-torsion objects. ∎

Now fix a prime field $k$ other than $\mathbb{F}_{2}$, and fix an algebraic
closure $p\colon\operatorname{Spec}(K)\rightarrow\operatorname{Spec}(k)$. The
core of our argument is to show that
$p^{\ast}\colon\operatorname{SH}(k)\rightarrow\operatorname{SH}(K)$ is
conservative when restricted to the full subcategory of $2$-torsion cellular
motivic spectra with convergent slice tower, and with vanishing real Betti
realization when $k=\mathbb{Q}$. As $p^{\ast}$ is exact, it is equivalent to
show that if $X\in\operatorname{SH}(k)$ is such an object and $p^{\ast}X=0$,
then $X=0$. First let us reformulate the condition on the Betti realization of
$X$ when $k=\mathbb{Q}$ to one which is uniform in $k$. Recall that
$\rho=[-1]\in\pi_{-1,-1}\mathbf{1}_{k}$ (see Subsection 1.1), and write
$C(\rho)$ for its cofiber.

######  Lemma 3.4.

Let $X\in\operatorname{SH}(k)$, and when $k=\mathbb{Q}$ suppose that the real
Betti realization of $X$ vanishes. If $C(\rho)\otimes X=0$ then $X=0$.

###### Proof.

We claim that $X$ is $\rho$-torsion, that is that $X[\rho^{-1}]=0$. The claim
follows as smashing with $C(\rho)$ is conservative on $\rho$-torsion objects,
as can be seen using the cofiber sequence
$\mathbf{1}_{S}\rightarrow\mathbf{1}_{S}[\rho^{-1}]\rightarrow\mathbf{1}_{S}/(\rho^{\infty})$,
where
$\mathbf{1}_{S}/(\rho^{\infty})=\operatorname{colim}_{n}\Sigma^{n,n}C(\rho^{n})$
is built out of copies of $C(\rho)$.

First consider $k=\mathbb{F}_{p}$ ($p\neq 2$). Here $\rho$ is already
nilpotent in $\pi_{\ast,\ast}\mathbf{1}_{S}$ [Mil70, Example 1.5], and so
$X[\rho^{-1}]\simeq 0$ for any $X$.

Next consider $k=\mathbb{Q}$. As $\mathbb{Q}$ has a unique ordering, work of
Bachmann [Bac18, Theorem 35, Proposition 36] shows that
$\operatorname{SH}(\mathbb{Q})[\rho^{-1}]\simeq\operatorname{SH}$ via real
Betti realization. Thus the condition that the real Betti realization of $X$
vanishes exactly says that $X[\rho^{-1}]=0$. ∎

We now turn to considering slices. For a motivic spectrum $X$, write $f_{q}X$
for the $q$-effective cover of $X$, and write $s_{q}X=f_{q}X/f_{q+1}X$ for the
$q$th slice of $X$.

######  Lemma 3.5.

Let $k$ be any field containing $1/2$. Then the assignment $X\mapsto
s_{\ast}(X/2)$ lifts to an exact functor
$\operatorname{SH}(k)\rightarrow\smash{\operatorname{Mod}_{H\mathbb{F}_{2}^{k}}}$.
Moreover, if $X\in\operatorname{SH}(k)$ is cellular then $s_{\ast}(X/2)$ is
cellular as an $H\mathbb{F}_{2}^{k}$-module.

###### Proof.

Exactness follows from the definitions, see [Voe98, Theorem 2.2]. The
assignment $X\mapsto s_{\ast}(X)$ defines a lax monoidal functor from
$\operatorname{SH}(k)$ to $\mathbb{Z}$-graded objects in
$\operatorname{SH}(k)$, see [GRSØ12]. In particular, $s_{\ast}(X)$ is a module
over $s_{0}(\mathbf{1}_{k})$. Theorems of Voevodsky [Voe04, Thm. 6.6] and
Levine [Lev08, Theorem 10.5.1] identify $s_{0}(\mathbf{1}_{k})\simeq
H\mathbb{Z}^{k}$, and so exactness of $s_{\ast}$ implies that

$s_{\ast}(X/2)\simeq s_{\ast}(X)/2\simeq
H\mathbb{F}_{2}^{k}\otimes_{H\mathbb{Z}^{k}}s_{\ast}(X)$

is naturally an $H\mathbb{F}_{2}^{k}$-module as claimed. Finally, to verify
that if $X$ is cellular then $s_{\ast}(X/2)$ is cellular as an
$H\mathbb{F}_{2}^{k}$-module, it suffices to verify that
$s_{\ast}(\mathbf{1}_{k})_{(2)}$ is cellular as an
$H\mathbb{Z}_{(2)}^{k}$-module. As $2$ is invertible in $k$, this follows from
Röndigs–Spitzweck–Østvær’s computation of the slices
$s_{\ast}(\mathbf{1}_{k})_{(2)}$ in [RSØ19, Theorem 2.12], applying their
theorem in the case $S=\operatorname{Spec}(k)$ and
$\Lambda=A=\mathbb{Z}_{(2)}$. ∎

Our last main ingredient in the proof of Theorem 3.1 is to show that
$\operatorname{Mod}_{H\mathbb{F}_{2}^{k}}\rightarrow\operatorname{Mod}_{H\mathbb{F}_{2}^{K}}$
is conservative upon restriction to cellular $\rho$-torsion objects. The proof
of this is a variant of [Mat18, Example 3.5]. Write $\operatorname{Cell}$ for
the cellularization functor, see [BS20, Section 4]. Note that
$\operatorname{Cell}$ does not affect homotopy groups, and set

$E=\operatorname{Cell}(\operatorname{Spec}(K)_{+}\otimes
H\mathbb{F}_{2}^{k})\in\operatorname{Mod}_{H\mathbb{F}_{2}^{k}}.$

Recall that $\pi_{\ast,\ast}H\mathbb{F}_{2}^{k}=k_{\ast}^{M}(k)[\tau]$, where
$\tau\in\pi_{0,-1}H\mathbb{F}_{2}^{k}$ and $x\in k_{n}^{M}(k)$ lives in
$\pi_{-n,-n}H\mathbb{F}_{2}^{k}$, and likewise
$\pi_{\ast,\ast}E=\mathbb{F}_{2}[\tau]$. Say that an
$H\mathbb{F}_{2}^{k}$-module $M$ is $\tau$-free if $\pi_{\ast,\ast}M$ is free
as an $\mathbb{F}_{2}[\tau]$-module.

######  Lemma 3.6.

Let $M$ be a $\tau$-free $H\mathbb{F}_{2}^{k}$-module. Then there exists a
$\tau$-free $H\mathbb{F}_{2}^{k}$-module $M_{\geq n}$ receiving a map
$M\rightarrow M_{\geq n}$ with the following properties:

  1. (1)

$\pi_{i,\ast}M_{\geq n}=0$ for $i<n$;

  2. (2)

$\pi_{i,\ast}M\rightarrow\pi_{i,\ast}M_{\geq n}$ is an isomorphism for $i\geq
n$.

###### Proof.

The proof of [DGI06, Proposition 3.3] applies. For $m\in\mathbb{Z}$, choose a
basis $\\{x_{i}:i\in I_{m}\\}$ for $\pi_{m,\ast}M$ as an
$\mathbb{F}_{2}[\tau]$-module, and define

$V_{m}M=\bigoplus_{x\in I_{m}}\Sigma^{|x_{i}|}H\mathbb{F}_{2}^{k}.$

Then there is a map $V_{m}M\rightarrow M$ inducing an isomorphism on
$\pi_{m,\ast}$. Let

$C(M)=\operatorname{cof}\left(\bigoplus_{m<n}V_{m}(M)\rightarrow M\right).$

Coconnectivity of $H\mathbb{F}_{2}^{k}$ ensures that $M\rightarrow C(M)$
induces an isomorphism on $\pi_{i,\ast}$ for $i\geq n$ and is zero on
$\pi_{i,\ast}$ for $i<n$. In particular, $C(M)$ is $\tau$-free: if $i\geq n$
then $\pi_{i,\ast}C(M)\cong\pi_{i,\ast}M$, so this follows from
$\tau$-freeness of $M$, and if $i<n$ then
$\pi_{i,\ast}C(M)\subset\pi_{i-1,\ast}\bigoplus_{m<n}V_{m}(M)$, so this
follows from $\tau$-freeness of each $V_{m}(M)$. Thus iterating this
construction and setting $M_{\geq n}=\operatorname{colim}_{k}C^{k}(M)$ does
the job. ∎

######  Lemma 3.7.

Let $M$ be a $\tau$-free cellular $H\mathbb{F}_{2}^{k}$-module. Fix
$n\in\mathbb{Z}$ and suppose $\pi_{i,\ast}M=0$ for $i\neq n$. Then $M$ is
equivalent to a sum of copies of $\Sigma^{n,\ast}E$.

###### Proof.

Without loss of generality we may suppose $n=0$. First we claim that if
$x\in\pi_{0,w}M$, then there is a map $g\colon\Sigma^{0,w}E\rightarrow M$
satisfying $g(1)=x$. Without loss of generality we may suppose $w=0$. Note
that $\pi_{\ast,\ast}E\cong\mathbb{F}_{2}[\tau]$, and consider the universal
coefficient spectral sequence

$E_{2}^{s,f,w}=\operatorname{Ext}_{\pi_{\ast,\ast}H\mathbb{F}_{2}^{k}}^{f}(\mathbb{F}_{2}[\tau],\pi_{\ast-s,\ast-w}M)\Rightarrow[E,\Sigma^{s+f,w}M],\quad
d_{r}\colon E_{r}^{s,f,w}\rightarrow E_{r}^{s+1,f+r,w}.$

The proposed map $g$ defines a class in $E_{2}^{0,0,0}$. The only way this
could fail to survive the spectral sequence is if there is some nontrivial
differential $d_{r}(g)\neq 0$. This lives in a subquotient of
$\operatorname{Ext}^{r}_{\pi_{\ast,\ast}H\mathbb{F}_{2}^{k}}(\mathbb{F}_{2}[\tau],\pi_{\ast-1,\ast}M)$,
which vanishes because $M$ is concentrated in nonnegative stems and
$\pi_{\ast,\ast}H\mathbb{F}_{2}^{k}$ and $\mathbb{F}_{2}[\tau]$ are
concentrated in nonpositive stems. Thus $g$ survives to a map
$g\colon\Sigma^{0,w}E\rightarrow M$ satisfying $g(1)=x$ as claimed.

Now choose a basis $\\{x_{i}:i\in I\\}$ for $\pi_{0,\ast}M$ as an
$\mathbb{F}_{2}[\tau]$-module. Then the above argument provides a map

$(x_{i})_{i\in I}\colon\bigoplus_{i\in I}E\rightarrow M$

inducing an isomorphism in $\pi_{\ast,\ast}$, which is then an equivalence as
$E$ and $M$ are cellular. ∎

The above lemmas hold in general over any base field $k$ containing $1/2$.
However, the following relies on $k$ being a prime field other than
$\mathbb{F}_{2}$ in order to ensure that $\pi_{\ast,\ast}C(\rho)\otimes
H\mathbb{F}_{2}^{k}$ is concentrated in finitely many degrees.

######  Proposition 3.8.

Base change
$p^{\ast}\colon\operatorname{Mod}_{H\mathbb{F}_{2}^{k}}\rightarrow\operatorname{Mod}_{H\mathbb{F}_{2}^{K}}$
is conservative when restricted to the full subcategory of cellular
$\rho$-torsion $H\mathbb{F}_{2}^{k}$-modules.

###### Proof.

Write $p\colon\operatorname{Spec}(K)\rightarrow\operatorname{Spec}(k)$. By the
definition of $E$, the composite

${\operatorname{Mod}_{H\mathbb{F}_{2}^{k}}^{\text{cell}}}$${\operatorname{Mod}_{H\mathbb{F}_{2}^{k}}}$${\operatorname{Mod}_{H\mathbb{F}_{2}^{K}}}$${\operatorname{Mod}_{H\mathbb{F}_{2}^{k}}}$${\operatorname{Mod}_{H\mathbb{F}_{2}^{k}}^{\text{cell}}}$$\scriptstyle{\subset}$$\scriptstyle{p^{\ast}}$$\scriptstyle{p_{\ast}}$$\scriptstyle{\operatorname{Cell}}$

sends $\Sigma^{i,j}H\mathbb{F}_{2}^{k}$ to $\Sigma^{i,j}E$. As
$\operatorname{Mod}_{H\mathbb{F}_{2}^{k}}^{\text{cell}}$ is generated under
colimits by the modules $\Sigma^{i,j}H\mathbb{F}_{2}^{k}$, and each functor in
this composition preserves colimits, this identifies the composite as smashing
with $E$. It therefore suffices to show that smashing with $E$ is conservative
on the full subcategory of $\rho$-torsion $H\mathbb{F}_{2}^{k}$-modules.

Examination of the long exact sequence shows that $C(\rho)$ is $\tau$-free for
any field $k$. When $k$ is a prime field other than $\mathbb{F}_{2}$, the
kernel and cokernel of $\rho$ acting on $k_{\ast}^{M}(k)$ is concentrated in
degrees $0\leq\ast\leq 2$ (convenient references are [IØ20, Section 2.1] and
[OØ13, Section 5]), and further examination of the long exact sequence then
shows that $\pi_{i,\ast}C(\rho)\neq 0$ only for $-3\leq i\leq 0$. Combining
Lemma 3.6 and Lemma 3.7, we find that $C(\rho)$ admits a finite filtration
with associated graded equivalent to a direct sum of copies of
$\Sigma^{\ast,\ast}E$. In particular $C(\rho)$ lies in the thick
$\otimes$-ideal of $\operatorname{Mod}_{H\mathbb{F}_{2}^{k}}$ generated by
$E$.

Smashing with $C(\rho)$ is conservative on the full subcategory of
$\rho$-torsion $H\mathbb{F}_{2}^{k}$-modules. As $C(\rho)$ lies in the thick
$\otimes$-ideal generated by $E$, it follows that smashing with $E$ is
conservative on the full subcategory of $\rho$-torsion
$H\mathbb{F}_{2}^{k}$-modules, proving the proposition. ∎

We can now give the following.

###### Proof of Theorem 3.1.

Combining Lemma 3.2, Lemma 3.3, and Lemma 3.4, we reduce to verifying the
following assertion:

> Let $k$ be a prime field other than $\mathbb{F}_{2}$ and
> $p\colon\operatorname{Spec}(K)\rightarrow\operatorname{Spec}(k)$ be an
> algebraic closure. Let $X\in\operatorname{SH}(k)$ be a cellular motivic
> spectrum with convergent slice tower, and suppose $p^{\ast}(C(\rho)\otimes
> X/2)=0$. Then $C(\rho)\otimes X/2=0$.

Indeed, as $X$ is slice complete, so is $C(\rho)\otimes X/2$. It therefore
suffices to verify that $s_{\ast}(C(\rho)\otimes X/2)=0$. By exactness
$s_{\ast}(C(\rho)\otimes X/2)\simeq C(\rho)\otimes s_{\ast}(X/2)$, and so by
Lemma 3.5 and Proposition 3.8 it suffices to verify that
$p^{\ast}(s_{\ast}(C(\rho)\otimes X/2))=0$. As slices of cellular spectra are
preserved by base change [RSØ19, Corollary 2.17], we have
$p^{\ast}(s_{\ast}(C(\rho)\otimes X/2))\simeq s_{\ast}(p^{\ast}(C(\rho)\otimes
X/2))$, and this vanishes by assumption. ∎

###  3.2. Proof of the reduction to $S=\operatorname{Spec}(\mathbb{C})$

We now show that Theorem 3.1 allows us to reduce from verifying that
$\mathbf{1}_{S}\rightarrow JQ$ is a $KGL/2$-equivalence to just the case
$S=\operatorname{Spec}(\mathbb{C})$.

######  Lemma 3.9.

The following $S$-motivic spectra are cellular and compatible with base change
between schemes containing $1/2$:

$KQ,\qquad KGL,\qquad KGL/2,\qquad KGL/2\otimes JQ.$

###### Proof.

Recall that we have taken $KQ$ and $KGL$ to be defined as in [BH20, §3.2].
They are cellular and compatible with base change as discussed there. As
$KGL/2\simeq KQ/(2,\eta)$, we may identify

$KGL/2\otimes JQ\simeq KGL/2\otimes\operatorname{fib}\left(\psi^{3}-1\colon
KQ[\tfrac{1}{3}]\rightarrow KQ[\tfrac{1}{3}]\right).$

This is then cellular, and is compatible with base change between schemes
containing $1/2$ by [BH20, Theorem 3.1(1)]. ∎

######  Lemma 3.10.

The motivic spectra $KGL/2$ and $KGL/2\otimes JQ$ are slice complete.

###### Proof.

As the functors $f_{q}$ are exact, slice complete motivic spectra form a thick
subcategory of $\operatorname{SH}(S)$. As $KGL/2\simeq KQ/(2,\eta)$, there is
an equivalence $KGL/2\otimes(KQ)_{(2,\eta)}^{\wedge}\simeq KGL/2\otimes KQ$;
moreover, $\eta$ acts trivially on $KGL\otimes KQ$, and thus $KGL\otimes KQ$
is a retract of $(KGL\otimes KQ)/\eta\simeq KGL\otimes KGL$. Thus we reduce to
verifying that $KGL$ and $KGL\otimes KGL$ are slice complete. Following the
proof of [ARØ20, Lemma 2.6], $f_{q}KGL$ is $q$-connected in the sense of
[RSØ19, Definition 3.16]. As $\infty$-connected objects are contractible, it
follows that $KGL$ is slice complete. By [PW14, Corollary 7.5], $KGL\otimes
KGL$ is a free $KGL$-module. As $f_{q}$ commutes with direct sums [PW14,
Proposition 6.1], it follows that $f_{q}(KGL\otimes KGL)$ is also
$q$-connected, and thus $KGL\otimes KGL$ is slice complete. ∎

######  Lemma 3.11.

The real Betti realizations of $KGL/2$ and $KGL/2\otimes JQ$ vanish.

###### Proof.

As real Betti realization is exact and monoidal, it suffices to verify that
the real Betti realization of $KGL$ vanishes. This is proved by Bachmann and
Hopkins in [BH20, Lemma 3.9]. ∎

As far as the proof of A is concerned, the following is the main result of
this section.

######  Proposition 3.12.

To show that $\mathbf{1}_{S}\rightarrow JQ$ is a $KGL/2$-equivalence over any
scheme $S$ containing $1/2$, it suffices to consider just
$S=\operatorname{Spec}(\mathbb{C})$.

###### Proof.

Lemma 3.9 reduces the case of an arbitrary scheme $S$ containing $1/2$ to just
$S=\operatorname{Spec}(\mathbb{Z}[\tfrac{1}{2}])$. By Lemma 3.10 and Lemma
3.11, the hypotheses of Theorem 3.1 apply to then reduce to
$S=\operatorname{Spec}(\mathbb{C})$. ∎

##  4\. The case $S=\operatorname{Spec}(\mathbb{C})$

It remains to show that $\mathbf{1}_{\mathbb{C}}\rightarrow JQ$ is a
$KGL/2$-equivalence. To do this, we will make use of the close relationship
between $\mathbb{C}$-motivic homotopy theory and the classical Adams–Novikov
spectral sequence. There are (at least) two closely related approaches to
formalizing this relation: the synthetic spectra of Pstragowski [Pst23], and
the $\Gamma_{\star}$-construction of Gheorghe–Isaksen–Krause–Ricka [GIKR22].
For our argument it is convenient to use the latter.

In [GIKR22], properties of the classical and motivic Adams–Novikov spectral
sequences are used to construct:

  1. (1)

A lax symmetric monoidal functor
$\Gamma_{\star}\colon\operatorname{Sp}\rightarrow\operatorname{Fun}((\mathbb{Z},<)^{\text{op}},\operatorname{Sp})$,
and

  2. (2)

A lax symmetric monoidal functor
$\Omega^{0,\star}\colon\operatorname{SH}(\mathbb{C})\rightarrow\operatorname{Mod}_{\Gamma_{\star}(S)}$,

and to prove that $\Omega^{0,\star}$ induces an equivalence between the
category of $2$-complete cellular $\mathbb{C}$-motivic spectra and the
category of $2$-complete $\Gamma_{\star}(S)$-modules. Moreover, under this
correspondence one has

(4.1) $KGL\simeq\Gamma_{\star}(KU),\qquad KQ\simeq\Gamma_{\star}(KO),$

up to $2$-completion. We are now ready to prove the following.

######  Proposition 4.1.

The map $\mathbf{1}_{\mathbb{C}}\rightarrow JQ$ is a $KGL/2$-equivalence.

###### Proof.

It is equivalent to show that

(4.2) ${KGL/2}$${KGL/2\otimes KQ}$${KGL/2\otimes
KQ}$$\scriptstyle{\psi^{3}-1}$

is a fiber sequence of $\mathbb{C}$-motivic spectra. Consider the classical
fiber sequence

${KU/2}$${KU/2\otimes KO}$${KU/2\otimes KO}$$\scriptstyle{\psi^{3}-1}$

.

As these spectra have $MU$-homology concentrated in even degrees, [GIKR22,
Proposition 3.17] implies that $\Gamma_{\star}$ sends this to a fiber sequence

(4.3) ${\Gamma_{\star}(KU/2)}$${\Gamma_{\star}(KU/2\otimes
KO)}$${\Gamma_{\star}(KU/2\otimes KO)}$$\scriptstyle{\psi^{3}-1}$

of $\Gamma_{\star}(S)$-modules. We claim that
$\Gamma_{\star}(KU/2)\simeq\Gamma_{\star}(KU)/2$ and
$\Gamma_{\star}(KU/2\otimes
KO)\simeq\Gamma_{\star}(KU/2)\otimes_{\Gamma_{\star}(S)}\Gamma_{\star}(KO)$.
In light of the equivalence between $2$-complete cellular motivic spectra and
$2$-complete $\Gamma_{\star}(S)$-modules, this would give an equivalence
between the two sequences Eq. 4.2 and Eq. 4.3, and thus the former would be a
fiber sequence as claimed.

That $\Gamma_{\star}(KU/2)\simeq\Gamma_{\star}(KU)/2$ follows by again
applying [GIKR22, Proposition 3.17], this time to the cofiber sequence
$KU\rightarrow KU\rightarrow KU/2$; likewise one has
$\Gamma_{\star}(KU/2\otimes KO)\simeq\Gamma_{\star}(KU\otimes KO)/2$. Recall
that $KU$ is a filtered colimit of finite complexes with only even cells
[HS99, Proposition 2.12]. As $KU\simeq KO/(\eta)$ and $\eta$ acts trivially on
$MU$, it follows that $KO$ has $MU$-homology concentrated in even degrees.
Thus [GIKR22, Proposition 3.25] implies that $\Gamma_{\star}(KU\otimes
KO)\simeq\Gamma_{\star}(KU)\otimes_{\Gamma_{\star}(S)}\Gamma_{\star}(KO)$.
Combining these observations provides the requires equivalences. ∎

Combining Proposition 2.2, Proposition 2.3, Proposition 3.12, and Proposition
4.1 now proves A.

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