# On the infinite loop spaces of algebraic cobordism
and the motivic sphere

Tom Bachmann  Mathematisches Institut, LMU München, Theresienstr. 39, 80333
München, Germany<EMAIL_ADDRESS>,  Elden Elmanto  Department of
Mathematics, Harvard University, 1 Oxford St., Cambridge, MA 02138, USA
<EMAIL_ADDRESS>,  Marc Hoyois  Fakultät für Mathematik, Universität
Regensburg, Universitätsstr. 31, 93040 Regensburg, Germany<EMAIL_ADDRESS>,  Adeel A. Khan  IHES, 35 route de Chartres, 91440 Bures-sur-Yvette, France;
Institute of Mathematics, Academia Sinica, Taipei 10617, Taiwan<EMAIL_ADDRESS>,  Vladimir Sosnilo  Laboratory “Modern Algebra and Applications”, St.
Petersburg State University, 14th line, 29B, 199178 Saint Petersburg, Russia
<EMAIL_ADDRESS>and  Maria Yakerson  Institute for Mathematical Research
(FIM), ETH Zürich, Rämistr. 101, 8092 Zürich, Switzerland
<EMAIL_ADDRESS>
  *     * scAbstract. scWe obtain geometric models for the infinite loop spaces of the motivic spectra $\mathrm{MGL}$, $\mathrm{MSL}$, and $\mathbf{1}$ over a field. They are motivically equivalent to $\mathbf{Z}\times\mathrm{Hilb}_{\infty}^{\mathrm{lci}}(\mathbf{A}^{\infty})^{+}$, $\mathbf{Z}\times\mathrm{Hilb}_{\infty}^{\mathrm{or}}(\mathbf{A}^{\infty})^{+}$, and $\mathbf{Z}\times\mathrm{Hilb}_{\infty}^{s}c\operatorname{fr}$(A^∞)^+$,respectively,where$Hilb_d^lci(A^n)$(resp.\ $Hilb_d^$\mathrm{or}$(A^n),Hilb_d^scfr(A^n)$)istheHilbertschemeoflcipoints(resp.\ orientedpoints,framedpoints)ofdegree$d$in$A^n$,and$+$isQuillen^{s}c\prime$s plus construction. Moreover, we show that the plus construction is redundant in positive characteristic.

scKeywords. Motivic homotopy theory; algebraic cobordism

sc2020 Mathematics Subject Classification. 14F42 (primary); 19D06, 55P47
(secondary)

sc[Français]

scSur les espaces de lacets infinis du cobordisme algébrique et de la sphère
motivique

scRésumé. scNous obtenons des modèles géométriques des espaces de lacets
infinis des spectres motiviques $\mathrm{MGL}$, $\mathrm{MSL}$ et $\mathbf{1}$
sur un corps. Ils sont motiviquement équivalents à
$\mathbf{Z}\times\mathrm{Hilb}_{\infty}^{\mathrm{lci}}(\mathbf{A}^{\infty})^{+}$,
$\mathbf{Z}\times\mathrm{Hilb}_{\infty}^{\mathrm{or}}(\mathbf{A}^{\infty})^{+}$
et
$\mathbf{Z}\times\mathrm{Hilb}_{\infty}^{s}c\operatorname{fr}$(A^∞)^+$,respectivement,o\\`{u}$Hilb_d^lci(A^n)$(resp.\
$Hilb_d^$\mathrm{or}$(A^n),Hilb_d^scfr(A^n)$)estlesch\'{e}madeHilbertdespointslocalementintersectioncompl\\`{e}te(resp.\
pointsorient\'{e}s,pointscadr\'{e}s)dedegr\'{e}$d$dans$A^n$,et$+$estlaconstructionplusdeQuillen.Enoutre,nousmontronsquelaconstructionplusestredondanteencaract\'{e}ristiquepositive.$

  *
cJanuary 25, 2021Received by the Editors on June 19, 2020\.
Accepted on February 26, 2021.

Mathematisches Institut, LMU München, Theresienstr. 39, 80333 München, Germany

sce-mail<EMAIL_ADDRESS>
Department of Mathematics, Harvard University, 1 Oxford St., Cambridge, MA
02138, USA

sce-mail<EMAIL_ADDRESS>
Fakultät für Mathematik, Universität Regensburg, Universitätsstr. 31, 93040
Regensburg, Germany

sce-mail<EMAIL_ADDRESS>
IHES, 35 route de Chartres, 91440 Bures-sur-Yvette, France; Institute of
Mathematics, Academia Sinica, Taipei 10617, Taiwan

sce-mail<EMAIL_ADDRESS>
Laboratory “Modern Algebra and Applications”, St. Petersburg State University,
14th line, 29B, 199178 Saint Petersburg, Russia

sce-mail<EMAIL_ADDRESS>
Institute for Mathematical Research (FIM), ETH Zürich, Rämistr. 101, 8092
Zürich, Switzerland

sce-mail<EMAIL_ADDRESS>

© by the author(s) This work is licensed under
http://creativecommons.org/licenses/by-sa/4.0/

###### Contents

  1. 1 Introduction
  2. 2 A moving lemma
  3. 3 Quillen’s plus construction and group completion
  4. 4 Proofs of the main results
  5. 5 Removing the plus construction in positive characteristic

## 1\. Introduction

Let $k$ be a field, let $\mathrm{MGL}$ be Voevodsky’s algebraic cobordism
spectrum over $k$, and let $\mathcal{FS}\mathrm{yn}$ be the moduli stack of
finite syntomic $k$-schemes. In [EHK+20b, Corollary 3.4.2], we showed that
there is an equivalence

${}^{\infty}_{\mathbf{T}}\mathrm{MGL}\simeq
L_{\mathrm{mot}}(\mathcal{FS}\mathrm{yn}^{\mathrm{gp}}),$

where $\mathcal{FS}\mathrm{yn}^{\mathrm{gp}}$ is the group completion of
$\mathcal{FS}\mathrm{yn}$ under disjoint union and $L_{\mathrm{mot}}$ is the
motivic localization functor. This can be regarded as an algebro-geometric
analogue of the identification of ${}^{\infty}\mathrm{MO}$ with the cobordism
space of $0$-dimensional compact smooth manifolds. However, unlike in
topology, the group completion is necessary in the above equivalence. The goal
of this paper is to obtain a group-completion-free description of
${}^{\infty}_{\mathbf{T}}\mathrm{MGL}$, which is more accessible for some
purposes. Let us write $\mathcal{FS}\mathrm{yn}_{\infty}$ for the colimit of
the sequence

$\dotsb\to\mathcal{FS}\mathrm{yn}_{d}\xrightarrow{+1}\mathcal{FS}\mathrm{yn}_{d+1}\to\dotsb,$

where $\mathcal{FS}\mathrm{yn}_{d}\subset\mathcal{FS}\mathrm{yn}$ is the
moduli stack of finite syntomic $k$-schemes of degree $d$. Our main result is
the following:

######  Theorem 1.1.

Let $k$ be a field. Then there is an equivalence

${}^{\infty}_{\mathbf{T}}\mathrm{MGL}\simeq
L_{\mathrm{mot}}(\mathbf{Z}\times\mathcal{FS}\mathrm{yn}_{\infty})^{+}$

in $\mathbf{H}(k)$, where $+$ denotes Quillen’s plus construction (in the
$\infty$-topos of Nisnevich sheaves). Moreover, there is an equivalence

$({}^{\infty}_{\mathbf{T}}\mathrm{MGL})(k)\simeq\mathbf{Z}\times(L_{\mathbf{A}^{1}}\mathcal{FS}\mathrm{yn}_{\infty})(k)^{+}.$

If $k$ has positive characteristic, the same equivalences hold without the
plus construction.

Here, $L_{\mathbf{A}^{1}}$ is the “naive” $\mathbf{A}^{1}$-localization
functor given by the formula

$(L_{\mathbf{A}^{1}}\mathcal{F})(X)=\operatorname*{colim}_{n\in{}^{\mathrm{op}}}\mathcal{F}(X\times\mathbf{A}^{n}).$

We recall that $\mathcal{FS}\mathrm{yn}_{d}$ is motivically equivalent to the
Hilbert scheme

$\mathrm{Hilb}_{d}^{\mathrm{lci}}(\mathbf{A}^{\infty})=\operatorname*{colim}_{n\to\infty}\mathrm{Hilb}_{d}^{\mathrm{lci}}(\mathbf{A}^{n})$

of finite local complete intersections of degree $d$ in $\mathbf{A}^{\infty}$,
which is a smooth ind-scheme [EHK+20b, Lemma 3.5.1]. Taking the colimit along
the obvious closed immersions

$\mathrm{Hilb}_{d}^{\mathrm{lci}}(\mathbf{A}^{\infty})\to\mathrm{Hilb}_{d+1}^{\mathrm{lci}}(\mathbf{A}^{1}\times\mathbf{A}^{\infty})\simeq\mathrm{Hilb}_{d+1}^{\mathrm{lci}}(\mathbf{A}^{\infty}),$

we obtain a smooth ind-scheme
$\mathrm{Hilb}_{\infty}^{\mathrm{lci}}(\mathbf{A}^{\infty})$ such that the
canonical map (forgetting the embedding into $\mathbf{A}^{\infty}$)

$\mathrm{Hilb}_{\infty}^{\mathrm{lci}}(\mathbf{A}^{\infty})\to\mathcal{FS}\mathrm{yn}_{\infty}$

is a motivic equivalence, and in fact an $\mathbf{A}^{1}$-equivalence on
affine schemes. Thus, we can rewrite Theorem 1.1 as follows:

######  Theorem 1.2.

Let $k$ be a field. Then there is an equivalence

${}^{\infty}_{\mathbf{T}}\mathrm{MGL}\simeq
L_{\mathrm{mot}}(\mathbf{Z}\times\mathrm{Hilb}_{\infty}^{\mathrm{lci}}(\mathbf{A}^{\infty}))^{+}$

in $\mathbf{H}(k)$. Moreover, there is an equivalence

$({}^{\infty}_{\mathbf{T}}\mathrm{MGL})(k)\simeq\mathbf{Z}\times(L_{\mathbf{A}^{1}}\mathrm{Hilb}_{\infty}^{\mathrm{lci}}(\mathbf{A}^{\infty}))(k)^{+}.$

If $k$ has positive characteristic, the same equivalences hold without the
plus construction.

######  Remark 1.3.

Unfortunately, we do not know if the plus construction can be removed in
characteristic zero. As we will see, this is the case if and only if the
cyclic permutation of three points in $\mathcal{FS}\mathrm{yn}_{3}(k)$ becomes
homotopic to the identity in
$(L_{\mathbf{A}^{1}}\mathcal{FS}\mathrm{yn}_{\infty})(k)$.

We have similar results for the motivic sphere spectrum $\mathbf{1}$ and for
the special linear algebraic cobordism spectrum $\mathrm{MSL}$. Recall that a
_framing_ of a finite syntomic morphism $f\colon Y\to X$ is a trivialization
of the cotangent complex $\mathcal{L}_{f}$ in $K(Y)$, and an _orientation_ of
$f$ is a trivialization of the dualizing sheaf
$\omega_{f}=\det(\mathcal{L}_{f})$ in $\mathrm{Pic}(Y)$. We denote by
$\mathcal{FS}\mathrm{yn}^{\operatorname{fr}}$ and
$\mathcal{FS}\mathrm{yn}^{\mathrm{or}}$ the moduli stacks of framed and
oriented finite syntomic $k$-schemes, respectively. There are forgetful
morphisms

$\mathcal{FS}\mathrm{yn}^{\operatorname{fr}}\to\mathcal{FS}\mathrm{yn}^{\mathrm{or}}\to\mathcal{FS}\mathrm{yn}.$

By [EHK+19, Theorem 3.5.18] and [EHK+20b, Corollary 3.4.4], there are
equivalences

${}^{\infty}_{\mathbf{T}}\mathbf{1}\simeq
L_{\mathrm{mot}}(\mathcal{FS}\mathrm{yn}^{{\operatorname{fr}},\mathrm{gp}})\quad\text{and}\quad{}^{\infty}_{\mathbf{T}}\mathrm{MSL}\simeq
L_{\mathrm{mot}}(\mathcal{FS}\mathrm{yn}^{\mathrm{or},\mathrm{gp}}).$

Let $h\in\mathcal{FS}\mathrm{yn}^{\operatorname{fr}}(k)$ be the framed finite
syntomic $k$-scheme consisting of two $k$-points, one with trivial framing and
the other with the opposite framing. We define

$\displaystyle\mathcal{FS}\mathrm{yn}^{\operatorname{fr}}_{\infty}=\operatorname*{colim}(\dotsb\to\mathcal{FS}\mathrm{yn}_{2d}^{\operatorname{fr}}\xrightarrow{+h}\mathcal{FS}\mathrm{yn}_{2d+2}^{\operatorname{fr}}\to\dotsb),$
$\displaystyle\mathcal{FS}\mathrm{yn}^{\mathrm{or}}_{\infty}=\operatorname*{colim}(\dotsb\to\mathcal{FS}\mathrm{yn}_{2d}^{\mathrm{or}}\xrightarrow{+h}\mathcal{FS}\mathrm{yn}_{2d+2}^{\mathrm{or}}\to\dotsb).$

######  Theorem 1.4.

Let $k$ be a field. Then there are equivalences

$\displaystyle{}^{\infty}_{\mathbf{T}}\mathbf{1}\simeq
L_{\mathrm{mot}}(\mathbf{Z}\times\mathcal{FS}\mathrm{yn}_{\infty}^{\operatorname{fr}})^{+},$
$\displaystyle{}^{\infty}_{\mathbf{T}}\mathrm{MSL}\simeq
L_{\mathrm{mot}}(\mathbf{Z}\times\mathcal{FS}\mathrm{yn}_{\infty}^{\mathrm{or}})^{+}$

in $\mathbf{H}(k)$. Moreover, there are equivalences

$\displaystyle({}^{\infty}_{\mathbf{T}}\mathbf{1})(k)\simeq\mathbf{Z}\times(L_{\mathbf{A}^{1}}\mathcal{FS}\mathrm{yn}_{\infty}^{\operatorname{fr}})(k)^{+},$
$\displaystyle({}^{\infty}_{\mathbf{T}}\mathrm{MSL})(k)\simeq\mathbf{Z}\times(L_{\mathbf{A}^{1}}\mathcal{FS}\mathrm{yn}_{\infty}^{\mathrm{or}})(k)^{+}.$

If $k$ has positive characteristic, the same equivalences hold without the
plus construction.

Finally, we also have a Hilbert scheme version of Theorem 1.4: the moduli
stacks $\mathcal{FS}\mathrm{yn}^{\operatorname{fr}}_{d}$ and
$\mathcal{FS}\mathrm{yn}^{\mathrm{or}}_{d}$ are motivically equivalent to the
smooth ind-schemes
$\mathrm{Hilb}_{d}^{\operatorname{fr}}(\mathbf{A}^{\infty})$ and
$\mathrm{Hilb}_{d}^{\mathrm{or}}(\mathbf{A}^{\infty})$ (see [EHK+19, §5.1.6]
and [EHK+20b, §3.5]). We define
$\mathrm{Hilb}_{\infty}^{\operatorname{fr}}(\mathbf{A}^{\infty})$ and
$\mathrm{Hilb}_{\infty}^{\mathrm{or}}(\mathbf{A}^{\infty})$ in an obvious way
so as to match the definitions of
$\mathcal{FS}\mathrm{yn}^{\operatorname{fr}}_{\infty}$ and
$\mathcal{FS}\mathrm{yn}^{\mathrm{or}}_{\infty}$.

######  Theorem 1.5.

Let $k$ be a field. Then there are equivalences

$\displaystyle{}^{\infty}_{\mathbf{T}}\mathbf{1}\simeq
L_{\mathrm{mot}}(\mathbf{Z}\times\mathrm{Hilb}_{\infty}^{\operatorname{fr}}(\mathbf{A}^{\infty}))^{+},$
$\displaystyle{}^{\infty}_{\mathbf{T}}\mathrm{MSL}\simeq
L_{\mathrm{mot}}(\mathbf{Z}\times\mathrm{Hilb}_{\infty}^{\mathrm{or}}(\mathbf{A}^{\infty}))^{+}$

in $\mathbf{H}(k)$. Moreover, there are equivalences

$\displaystyle({}^{\infty}_{\mathbf{T}}\mathbf{1})(k)\simeq\mathbf{Z}\times(L_{\mathbf{A}^{1}}\mathrm{Hilb}_{\infty}^{\operatorname{fr}}(\mathbf{A}^{\infty}))(k)^{+},$
$\displaystyle({}^{\infty}_{\mathbf{T}}\mathrm{MSL})(k)\simeq\mathbf{Z}\times(L_{\mathbf{A}^{1}}\mathrm{Hilb}_{\infty}^{\mathrm{or}}(\mathbf{A}^{\infty}))(k)^{+}.$

If $k$ has positive characteristic, the same equivalences hold without the
plus construction.

Our interest in these group-completion-free models is motivated in part by the
following conjecture of Mike Hopkins, called the “Wilson space hypothesis”:

######  Conjecture 1.6 (M. Hopkins).

For every $n\in\mathbf{Z}$, the motive
$M({}^{\infty-n}_{\mathbf{T}}\mathrm{MGL})\in\mathbf{DM}(k)$ is pure Tate.

The analog of this conjecture for the complex cobordism spectrum $\mathrm{MU}$
is a theorem of Wilson [Wil73], and the analog in $C_{2}$-equivariant homotopy
theory was recently proved by Hill and Hopkins [HH18]; these results are
subsumed by Conjecture 1.6 for $k=\mathbf{C}$ and $k=\mathbf{R}$,
respectively. Since the plus construction is invisible to motives, Theorem 1.1
implies that

$M({}^{\infty}_{\mathbf{T}}\mathrm{MGL})\simeq\bigoplusop\displaylimits_{d\in\mathbf{Z}}M(\mathrm{Hilb}^{\mathrm{lci}}_{\infty}(\mathbf{A}^{\infty})).$

We thus obtain the following geometric reformulation of the case $n=0$ of the
above conjecture:

######  Corollary 1.7.

Conjecture 1.6 holds for $n=0$ if and only if
$M(\mathrm{Hilb}^{\mathrm{lci}}_{\infty}(\mathbf{A}^{\infty}))$ is a pure Tate
motive.

######  Remark 1.8.

The motive of $\mathrm{Hilb}^{\mathrm{lci}}(\mathbf{A}^{n})$ is _not_ pure
Tate for $2\leqslant n<\infty$. Indeed, one can show that
$M(\mathrm{Hilb}_{3}^{\mathrm{lci}}(\mathbf{A}^{n}))$ cannot be pure Tate by
analyzing the restriction to the lci locus of the Białynicki-Birula
stratification of the Hilbert scheme $\mathrm{Hilb}_{3}(\mathbf{A}^{n})$,
which is smooth. For example,

$M(\mathrm{Hilb}_{3}^{\mathrm{lci}}(\mathbf{A}^{2}))\simeq\mathbf{Z}\oplus\mathbf{Z}(1)[2]\oplus\mathbf{Z}(2)[4]\oplus\mathbf{Z}(4)[7].$

Nevertheless, the motive of
$\mathrm{Hilb}_{3}^{\mathrm{lci}}(\mathbf{A}^{\infty})$ turns out to be pure
Tate, as shown in [HJN+20, Theorem 7.1].

######  Acknowledgments.

We would like to thank Marc Levine for explaining to us the principle of
moving via an étale correspondence; this was the main inspiration of
Proposition 2.3.

The last five authors would like to acknowledge the hospitality of the
Institute for Advanced Study in Princeton, where part of this work was done in
July 2019.

## 2\. A moving lemma

######  Lemma 2.1.

Let $k$ be a field, $X$ a smooth geometrically connected quasi-projective
$k$-scheme of dimension $\geqslant 2$, and $A\subset X$ a subscheme étale over
$k$. Then there exists a smooth geometrically connected closed subscheme
$H\subset X$ of codimension $1$ containing $A$.

###### Proof.

If $k$ is infinite, a generic hypersurface section of $X$ of large enough
degree containing $A$ is geometrically irreducible and smooth of codimension
$1$, by [AK79, Theorems 1 and 7].

If $k$ is finite, choose an embedding $X\subset\mathbf{P}^{n}_{k}$ and a
first-order thickening $\tilde{A}$ of $A$ in $\mathbf{P}^{n}_{k}$ that is
transverse to $X$. Let $P_{d}=\Gamma(\mathbf{P}^{n}_{k},\mathcal{O}(d))$ and
let

$\displaystyle Q_{d}$ $\displaystyle=\\{f\in P_{d}\;|\;\text{$Z(f)\cap(X-A)$
is smooth of codimension $1$ in $X$ and $f|\tilde{A}=0$}\\},$ $\displaystyle
R_{d}$ $\displaystyle=\\{f\in P_{d}\;|\;\text{$Z(f)\cap X$ is geometrically
irreducible}\\}.$

Note that $Z(f)\cap X$ is smooth for any $f\in Q_{d}$, so it suffices to show
that $Q_{d}\cap R_{d}$ is nonempty for some $d$. We have

$\operatorname*{lim}_{d\to\infty}\frac{\\#Q_{d}}{\\#P_{d}}>0\quad\text{and}\quad\operatorname*{lim}_{d\to\infty}\frac{\\#R_{d}}{\\#P_{d}}=1$

by [Poo04, Theorem 1.2] and [CP16, Theorem 1.1], respectively. Hence,
$Q_{d}\cap R_{d}$ is nonempty for all large enough $d$. ∎

######  Lemma 2.2.

Let $k$ be a field, $X\subset\mathbf{P}^{n}_{k}$ a quasi-projective
$k$-scheme, and $A,B\subset X$ disjoint finite subschemes. Suppose $A$ étale
over $k$ and $X-B$ smooth of dimension $m\geqslant 1$ over $k$. Then, for
every large enough $d$, there exist global sections
$f,g\in\Gamma(\mathbf{P}^{n}_{k},\mathcal{O}(d))$ such that:

  1. (1)

$f|B=g|B$;

  2. (2)

$f|A=0$;

  3. (3)

$g$ does not vanish on $A\cup B$;

  4. (4)

$Z(f)\cap X$ and $Z(g)\cap X$ are smooth over $k$ of dimension $m-1$.

###### Proof.

Suppose first that $k$ is finite. Let $h\in\Gamma(A\cup B,\mathcal{O}(1))$ be
a nonvanishing section. Applying [Poo04, Theorem 1.2] with an auxiliary
thickening of $A$ as in the proof of Lemma 2.1, we deduce that for every large
enough $d$ there exists a global section $f$ of $\mathcal{O}(d)$ satisfying
conditions (2) and (4) such that $f|B=h^{\otimes d}|B$. By the same theorem,
there also exists a global section $g$ of $\mathcal{O}(d)$ satisfying
condition (4) such that $g|(A\cup B)=h^{\otimes d}$. Then $f$ and $g$ satisfy
all conditions.

If $k$ is infinite, take $d$ such that $\mathcal{I}_{A}(d-1)$ is generated by
its global sections and such that the vanishing of
$H^{1}(\mathbf{P}^{n}_{k},\mathcal{I}_{A\cup B}(d))$ holds, where
$\mathcal{I}_{Z}\subset\mathcal{O}_{\mathbf{P}^{n}_{k}}$ denotes the ideal
defining a closed subscheme $Z$. Let $f$ and $h$ be generic sections of
$\mathcal{O}(d)$ with $f|A=0$ and $h|B=0$, and let $g=f+h$. By choice of $d$,
every global section of $\mathcal{O}(d)$ is the sum of one vanishing on $A$
and one vanishing on $B$. Hence, $g$ is a generic section of $\mathcal{O}(d)$,
and in particular does not vanish on $A\cup B$. Moreover, $Z(f)\cap(X-B)$ and
$Z(g)\cap(X-B)$ are smooth of dimension $m-1$ by [AK79, Theorem 7]. The
sections $f$ and $g$ thus satisfy conditions (1)–(4). ∎

######  Proposition 2.3.

Let $k$ be a field, $X$ a smooth $k$-scheme, and $U\subset X$ a dense open
subscheme. For any $k$-point $x\colon\operatorname{Spec}k\to X$, there exist
framed correspondences
$\alpha,\beta\in\mathrm{Corr}^{\operatorname{fr}}_{k}(\operatorname{Spec}k,U)$
with finite étale support and an $\mathbf{A}^{1}$-homotopy $x+\alpha\sim\beta$
in $\mathrm{Corr}^{\operatorname{fr}}_{k}(\mathbf{A}^{1}_{k},X)$.

###### Proof.

Since the question is local around $x$, we can assume $X$ quasi-projective and
geometrically connected [Gro65, corollaire 4.5.14]. By Lemma 2.1, we can find
a smooth connected subscheme $C\subset X$ of dimension $1$ such that $x\in C$
and such that $C\cap U$ is nonempty (hence cofinite in $C$). Let
$\overline{C}$ be a projective closure of $C$ and $\mathcal{L}$ an ample
invertible sheaf on $\overline{C}$. Shrinking $C$ if necessary, we may assume
that $C$ is affine and that C/k and $\mathcal{L}|C$ are trivial. Let
$A=\\{x\\}$, and let $B$ be the complement of $C\cap U$ in $\overline{C}$ with
$x$ removed. Then $A$ and $B$ are disjoint finite subschemes of $\overline{C}$
such that $A$ is étale and $\overline{C}-B$ is smooth. By Lemma 2.2, we can
find an integer $d$ and global sections $f$ and $g$ of $\mathcal{L}^{\otimes
d}$ with étale vanishing loci, such that $f(x)=0$, $f=g$ on $B$, and
$Z(f)-\\{x\\},Z(g)\subset C\cap U$. Let $h$ be the section $(1-t)f+tg$ over
$\mathbf{A}^{1}\times\overline{C}$ and let $H=Z(h)$. Then $H$ is a proper
local complete intersection of relative virtual dimension $0$ over
$\mathbf{A}^{1}_{k}$. Moreover, since $f$ and $g$ agree and do not vanish on
$B$, $h$ does not vanish on $\mathbf{A}^{1}\times B$, so
$H\subset\mathbf{A}^{1}\times C$ and $H$ is finite over $\mathbf{A}^{1}_{k}$
(hence syntomic by [EHK+19, Proposition 2.1.16]). Thus, $H$ defines a finite
syntomic correspondence from $\mathbf{A}^{1}_{k}$ to $X$. Since C/k and
$\mathcal{L}^{\otimes d}|C$ are trivial, there exists a framing
$\tau\colon\mathcal{L}_{H/\mathbf{A}^{1}_{k}}\simeq 0$ in $K(H)$. Multiplying
$\tau$ by a unit of $k$, we can assume that $\tau$ restricts to the trivial
loop in $K(\\{x\\})$. Then $(H,\tau)$ is a framed correspondence from
$\mathbf{A}^{1}_{k}$ to $X$ with the desired properties. ∎

## 3\. Quillen’s plus construction and group completion

We shall say that a morphism $f\colon X\to Y$ in an $\infty$-topos $T$ is
_acyclic_ if it is an epimorphism in the categorical sense, i.e., if the
square

${X}$${Y}$${Y}$${Y}$$\scriptstyle{f}$$\scriptstyle{f}$$\scriptstyle{\mathrm{id}}$$\scriptstyle{\mathrm{id}}$

is cocartesian. Recall that a _modality_ on $T$ is a factorization system on
$T$ (in the sense of Joyal and as explained in [Lur17b, §5.2.8]) whose left
class of morphisms is stable under base change [ABFJ20, §2].

######  Lemma 3.1.

In any $\infty$-topos $T$, acyclic morphisms form the left class of a
modality. In particular, acyclic morphisms are stable under composition,
retracts, colimits, cobase change, base change, and finite products.

###### Proof.

Acyclic morphisms are stable under base change since cocartesian squares are
(by universality of colimits). By [Lur17b, Proposition 5.5.5.7], it remains to
show that the class of acyclic maps is of small generation as a saturated
class. For any $X\in T$, the full subcategory of $T_{/X}$ spanned by the
acyclic morphisms is accessible by [Lur17b, Proposition 5.4.6.6], being the
fiber of the suspension functor. It is thus generated under filtered colimits
by a small subcategory. Let $C\subset\mathrm{Fun}({}^{1},T)$ be the union of
these small subcategories of $T_{/X}$ as $X$ ranges over a small set of
generators of $T$. Using that acyclic maps are stable under base change, we
immediately deduce that $C$ generates the class of acyclic maps under
colimits. ∎

For $X\in T$, we denote by $X\to X^{+}$ the final object in the
$\infty$-category of acyclic maps out of $X$, which exists by Lemma 3.1. The
functor $X\mapsto X^{+}$ is called the _plus construction_. Note that acyclic
morphisms are connected and become equivalences after a single suspension; it
follows that the canonical map $X\to X^{+}$ induces an isomorphism on
$\pi_{0}$, and that it is an equivalence whenever $X$ admits a structure of
$\mathcal{E}_{1}$-group (i.e., when $X$ is a loop space [Lur17a, Theorem
5.2.6.15]).

In the $\infty$-topos $\mathrm{Spc}$ of spaces, the above construction
coincides with Quillen’s plus construction which was used to define algebraic
$K$-theory [Qui72, §12]. More precisely, $X\to X^{+}$ is the initial map that
kills the maximal perfect subgroups of the fundamental groups of $X$, and
hence it is an equivalence if and only if the fundamental groups of $X$ are
hypoabelian (i.e., have no nontrivial perfect subgroups). We refer to [Rap19]
for a discussion of acyclic morphisms in $\mathrm{Spc}$ and for a proof of
this fact.

Let $M$ be a commutative monoid in $T$ and let $m\colon*\to M$ be a global
section. We denote by $\mathbf{Mod}_{M}(T)$ the $\infty$-category of
$M$-modules, i.e., objects of $T$ with an action of $M$ (which is again an
$\infty$-topos). The full subcategory of $M$-modules on which $m$ acts
invertibly is reflective, and we denote by $E\mapsto E[m^{-1}]$ the associated
localization functor. It is easy to check that this functor preserves finite
products. In particular, $M[m^{-1}]$ is again a commutative monoid and $M\to
M[m^{-1}]$ is the initial morphism of commutative monoids sending $m$ to an
invertible global section. For $E\in\mathbf{Mod}_{M}(T)$, we define the
telescope $\mathrm{tel}{}_{m}(E)\in\mathbf{Mod}_{M}(T)$ as the colimit of the
sequence

$\dotsb\to E\xrightarrow{\cdot m}E\to\dotsb.$

There is then a canonical map of $M$-modules $\mathrm{tel}{}_{m}(E)\to
E[m^{-1}]$, but unlike in the case of $1$-topoi, it is not invertible in
general. For example, if $T=\mathrm{Spc}$ and
$F=\coprodop\displaylimits_{n\geqslant 0}B{}_{n}$ is the free commutative
monoid on a single element $a$, then $F[a^{-1}]$ is the group completion of
$F$, but $\mathrm{tel}{}_{a}(F)\simeq\mathbf{Z}\times B{}_{\infty}$ does not
admit a monoid structure since its fundamental groups are not abelian.

######  Proposition 3.2.

Let $T$ be an $\infty$-topos, let $M$ be a commutative monoid in $T$, and let
$m\colon*\to M$ be a global section. For any $M$-module $E$, the canonical map
$\mathrm{tel}{}_{m}(E)\to E[m^{-1}]$ is acyclic.

###### Proof.

Since acyclic maps are closed under colimits and finite products, it suffices
to prove the result for $E=M$. The classifying $\infty$-topos for pointed
commutative monoids is a presheaf $\infty$-topos (namely, presheaves on the
pushout of
$(\mathrm{Spc}_{*}^{\mathrm{fin}})^{\mathrm{op}}\leftarrow(\mathrm{Spc}^{\mathrm{fin}})^{\mathrm{op}}\to(\mathrm{CMon}{}(\mathrm{Spc})^{\mathrm{fin}})^{\mathrm{op}}$
in the $\infty$-category of $\infty$-categories with finite limits), so we are
immediately reduced to the case $T=\mathrm{Spc}$. Let $F$ be the free
commutative monoid on an element $a$. The element $m$ induces a morphism of
commutative monoids $F\to M$ sending $a$ to $m$, and we have
$\mathrm{tel}{}_{m}(M)=M\otimes_{F}\mathrm{tel}{}_{a}(F)$ and $M[m^{-1}]\simeq
M\otimes_{F}F^{\mathrm{gp}}$, where $\otimes_{F}$ is the tensor product of
spaces with $F$-action. Since acyclic maps are closed under colimits and
finite products, $M\otimes_{F}(-)$ preserves acyclic maps. We can therefore
replace $(M,m)$ by $(F,a)$, and in particular we can assume that
$\pi_{0}(M)[m^{-1}]$ is a group. In this case we must show that the canonical
map $\mathrm{tel}{}_{m}(M)^{+}\to M^{\mathrm{gp}}$ is an equivalence, which is
the classical McDuff–Segal group completion theorem [MS76]. We recall a proof
due to Nikolaus [Nik17]. Note that the plus construction preserves finite
products and hence commutative monoids. Consider the commutative square

${\mathrm{tel}{}_{m}(M)^{+}}$${M^{\mathrm{gp}}}$${\mathrm{tel}{}_{m}(M^{+})^{+}}$${(M^{+})^{\mathrm{gp}}\hbox
to0.0pt{.\hss}}$

The left vertical is an equivalence since the plus construction is a left
localization of $\mathrm{Spc}$. The lower horizontal map is an equivalence by
Proposition 5.1, since the cyclic permutation of order $5$ becomes trivial in
the hypoabelianization of 5. The top horizontal map is a stable equivalence by
the localization theory of ${\mathcal{E}_{\infty}}$-ring spectra. Hence the
right vertical map is a stable equivalence. Since
${\mathcal{E}_{\infty}}$-groups are simple, the right vertical map is in fact
an equivalence, so the top horizontal map is an equivalence as well. ∎

######  Remark 3.3.

In the proof of Proposition 3.2, the reduction to the McDuff–Segal theorem
only uses that $M$ is an $\mathcal{E}_{2}$-monoid, and the given proof of that
theorem only uses that $M$ is $\mathcal{E}_{3}$. In fact, the McDuff–Segal
theorem holds more generally for homotopy commutative
$\mathcal{E}_{1}$-monoids [RW13], so Proposition 3.2 holds for $M$ an
$\mathcal{E}_{2}$-monoid.

If $S$ is an arbitrary set of global sections of $M$, we can define more
generally the telescope functor
$\mathrm{tel}{}_{S}\colon\mathbf{Mod}_{M}(T)\to\mathbf{Mod}_{M}(T)$ by

$\mathrm{tel}{}_{S}(E)=\operatorname*{colim}_{\begin{subarray}{c}F\subset S\\\
F\text{ finite}\end{subarray}}\mathrm{tel}{}_{\prodop\displaylimits_{m\in
F}m}(E)$

(see [Hoy17, §6.1] for a precise construction). Proposition 3.2 immediately
implies that the canonical map $\mathrm{tel}{}_{S}(E)\to E[S^{-1}]$ is
acyclic. In particular:

######  Corollary 3.4.

Let $T$ be an $\infty$-topos, $M$ a commutative monoid in $T$, and $S$ a set
of global sections of $M$ such that $\pi_{0}(M)[S^{-1}]$ is a group. Then
there is an equivalence $\mathrm{tel}{}_{S}(M)^{+}\simeq M^{\mathrm{gp}}$.

In what follows we will use the plus construction in $\mathrm{Spc}$ and in the
$\infty$-topos of Nisnevich sheaves on $\mathrm{Sm}_{k}$.

## 4\. Proofs of the main results

Recall that
$\mathcal{FS}\mathrm{yn}_{\infty}=\operatorname*{colim}_{d\to\infty}\mathcal{FS}\mathrm{yn}_{d}$.
In the notation of Section 3, there is a canonical map

$\mathbf{Z}\times\mathcal{FS}\mathrm{yn}_{\infty}\to\mathrm{tel}{}_{1}(\mathcal{FS}\mathrm{yn}),$

which is an equivalence on connected schemes (in particular, it is a Zariski-
local equivalence).

######  Proposition 4.1.

Let $k$ be a field. Then
$L_{\mathrm{mot}}\mathcal{FS}\mathrm{yn}_{\infty}\in\mathbf{H}(k)$ is
connected in the Nisnevich topology and
$(L_{\mathbf{A}^{1}}\mathcal{FS}\mathrm{yn}_{\infty})(k)$ is connected.

###### Proof.

For the first statement, by [Mor05, Lemma 6.1.3], it suffices to show that the
stalks of $L_{\mathrm{mot}}\mathcal{FS}\mathrm{yn}_{\infty}$ at separable
finitely generated field extensions of $k$ are connected. Since there is a
surjection

$\pi_{0}(L_{\mathbf{A}^{1}}\mathcal{FS}\mathrm{yn}_{\infty})(k)\to\pi_{0}(L_{\mathrm{mot}}\mathcal{FS}\mathrm{yn}_{\infty})(k)$

by [MV99, §2, Corollary 3.22], we are reduced to proving the second statement.
We will show more precisely that for any $T\in\mathcal{FS}\mathrm{yn}_{d}(k)$
there exists $n\geqslant 0$ and a cobordism $T+n\sim d+n$ in
$\mathcal{FS}\mathrm{yn}_{d+n}$. Choosing an embedding of $T$ in
$\mathbf{A}^{m}$ defines a $k$-point of the smooth $k$-scheme
$\mathrm{Hilb}_{d}^{\mathrm{lci}}(\mathbf{A}^{m})$. We now apply Proposition
2.3 with $X=\mathrm{Hilb}_{d}^{\mathrm{lci}}(\mathbf{A}^{m})$ and $U$ the open
subscheme classifying finite étale subschemes of $\mathbf{A}^{m}$ of degree
$d$. We obtain in particular a finite syntomic correspondence
$\mathbf{A}^{1}\leftarrow
H\to\mathrm{Hilb}_{d}^{\mathrm{lci}}(\mathbf{A}^{m})$ such that the fibers
$H_{0}$ and $H_{1}$ are étale over $k$, $H_{0}=\\{x\\}\sqcup H_{0}^{\prime}$,
and $H_{0}^{\prime}$ and $H_{1}$ map to $U$. The map
$H\to\mathrm{Hilb}_{d}^{\mathrm{lci}}(\mathbf{A}^{m})$ classifies a finite
syntomic morphism $W\to H$, and the composite $W\to H\to\mathbf{A}^{1}_{k}$ is
a cobordism between $W_{0}\simeq T\sqcup W_{0}^{\prime}$ and $W_{1}$ such that
$W_{0}^{\prime}$ and $W_{1}$ are finite étale over $k$. Using [EHK+19,
Proposition B.1.4], both $W_{0}^{\prime}$ and $W_{1}$ are further cobordant to
their degree, which concludes the proof. ∎

######  Corollary 4.2.

Let $k$ be a field. Then the canonical maps

$\displaystyle
L_{\mathrm{mot}}(\mathbf{Z}\times\mathcal{FS}\mathrm{yn}_{\infty})^{+}\to
L_{\mathrm{mot}}(\mathcal{FS}\mathrm{yn}^{\mathrm{gp}})$
$\displaystyle\mathbf{Z}\times(L_{\mathbf{A}^{1}}\mathcal{FS}\mathrm{yn}_{\infty})(k)^{+}\to(L_{\mathbf{A}^{1}}\mathcal{FS}\mathrm{yn})(k)^{\mathrm{gp}}$

are equivalences.

###### Proof.

We first observe that
$L_{\mathrm{mot}}(\mathcal{FS}\mathrm{yn}^{\mathrm{gp}})$ is the group
completion of $L_{\mathrm{mot}}(\mathcal{FS}\mathrm{yn})$ in Nisnevich
sheaves. To prove this we may replace $k$ by a perfect subfield, and the claim
follows from [EHK+19, Theorem 3.4.11] since the objectwise group completion of
$L_{\mathrm{mot}}(\mathcal{FS}\mathrm{yn})$ is an $\mathbf{A}^{1}$-invariant
presheaf with framed transfers. By Corollary 3.4, it remains to show that the
monoids
$\pi_{0}^{\mathrm{nis}}L_{\mathrm{mot}}(\mathcal{FS}\mathrm{yn}_{\infty})$ and
$\pi_{0}(L_{\mathbf{A}^{1}}\mathcal{FS}\mathrm{yn}_{\infty})(k)$ are groups,
which follows from Proposition 4.1. ∎

Except for the statement about positive characteristic, which we shall prove
in Section 5, Theorem 1.1 follows from Corollary 4.2 and [EHK+20b, Corollary
3.4.2(i)].

If $S$ is a scheme and $a\in\mathcal{O}(S)^{\times}$, we denote by $\langle
a\rangle\in\mathcal{FS}\mathrm{yn}^{\operatorname{fr}}(S)$ the finite syntomic
$S$-scheme $S$ framed by the image of $a$ under the canonical map
$\mathcal{O}(S)^{\times}\to\Omega K(S)$. We write $n_{\epsilon}$ for the
alternating sum

$\langle 1\rangle+\langle-1\rangle+\langle 1\rangle+\dotsb$

with $n$ terms, and we write $h$ for $2_{\epsilon}=\langle
1\rangle+\langle-1\rangle$. Recall that we have
$\mathcal{FS}\mathrm{yn}_{\infty}^{\operatorname{fr}}=\operatorname*{colim}_{d\to\infty}\mathcal{FS}\mathrm{yn}_{2d}^{\operatorname{fr}}$
and
$\mathcal{FS}\mathrm{yn}_{\infty}^{\mathrm{or}}=\operatorname*{colim}_{d\to\infty}\mathcal{FS}\mathrm{yn}_{2d}^{\mathrm{or}}$,
where the transition maps are given by adding $h$. In the notation of Section
3, we have equivalences

$\mathbf{Z}\times\mathcal{FS}\mathrm{yn}_{\infty}^{\operatorname{fr}}\simeq\mathrm{tel}{}_{h}(\mathcal{FS}\mathrm{yn}^{\operatorname{fr}})\quad\text{and}\quad\mathbf{Z}\times\mathcal{FS}\mathrm{yn}_{\infty}^{\mathrm{or}}\simeq\mathrm{tel}{}_{h}(\mathcal{FS}\mathrm{yn}^{\mathrm{or}})$

on connected schemes.

We recall a basic construction from [EHK+19, §B.1.1]. Let $S$ be a scheme and
$f(x)\in\mathcal{O}(S)[x]$ a polynomial with invertible leading coefficient.
Then the zero locus $Z(f)\subset\mathbf{A}^{1}_{S}$ is a finite syntomic
$S$-scheme. Moreover, $f$ generates the conormal sheaf
$\mathcal{N}_{Z(f)/\mathbf{A}^{1}_{S}}=(f)/(f^{2})$, hence defines a framing
of $Z(f)$ over $S$. We denote this framed finite syntomic $S$-scheme by
$\varphi(f)$. By [EHK+19, Proposition B.1.4], $\varphi(f)$ is framed cobordant
to $\langle a\rangle d_{\epsilon}$, where $a$ is the leading coefficient of
$f(x)$ and $d$ is its degree.

######  Lemma 4.3.

Let $S$ be a scheme and $a\in\mathcal{O}(S)^{\times}$. Then there exists an
$\mathbf{A}^{1}$-homotopy in $\mathcal{FS}\mathrm{yn}_{S}^{\operatorname{fr}}$
between $\langle a\rangle+\langle-a\rangle$ and $h=\langle
1\rangle+\langle-1\rangle$.

###### Proof.

We have $\langle a\rangle+\langle-a\rangle\simeq\varphi(x(x-a))$, because
$x(x-a)$ is congruent to $a(x-a)$ modulo ${(x-a)^{2}}$ and to $-ax$ modulo
${x^{2}}$. Thus, if $h(t)=1+t(a-1)$, then $\varphi(x(x-h(t)))$ is the desired
$\mathbf{A}^{1}$-homotopy. ∎

######  Remark 4.4.

The main point of Lemma 4.3 is that the relation $\langle
a\rangle+\langle-a\rangle=\langle 1\rangle+\langle-1\rangle$ in the
Grothendieck–Witt group holds in
$L_{\mathbf{A}^{1}}\mathcal{FS}\mathrm{yn}_{S}^{\operatorname{fr}}$ prior to
group completion. This is not the case of some other relations, such as
$\langle ab^{2}\rangle=\langle a\rangle$ (indeed,
$\mathcal{FS}\mathrm{yn}^{\operatorname{fr}}_{1}\simeq\Omega K$ is already
$\mathbf{A}^{1}$-invariant on regular schemes).

We denote by $\mathrm{GW}$ the presheaf of Grothendieck–Witt rings [Kne77],
and by $\underline{\mathrm{GW}}=\underline{K}^{MW}_{0}$ the presheaf of
unramified Grothendieck–Witt rings [Mor12, §3.2]. Note that if $K$ is a field
then $\underline{GW}(K)=\mathrm{GW}(K)$ [Mor12, Lemma 3.10], whereas in
general the relationship between $\underline{GW}(R)$ and $\mathrm{GW}(R)$ is
more subtle.

######  Lemma 4.5.

Let $R$ be a regular local ring over a field. Then the group
$\underline{\mathrm{GW}}(R)$ is generated by $\langle a\rangle$ for $a\in
R^{\times}$.

###### Proof.

If $R$ has characteristic $\neq 2$, we know that
$\underline{\mathrm{GW}}(R)=\mathrm{GW}(R)$ [BH21, Theorem 10.12], and
$\mathrm{GW}(R)$ has the claimed property by [HM73, Corollary I.3.4]. Suppose
therefore that $R$ has characteristic $2$. By Popescu’s theorem [Stacks, Tag
07GC], we can assume $R$ essentially smooth over a perfect field. The result
now follows from Lemma 4.6(iii) below. ∎

######  Lemma 4.6.

Let $k$ be a perfect field of characteristic $2$. Let
$I\subset\underline{\mathrm{GW}}$ denote the kernel of the rank map
$\underline{\mathrm{GW}}\to\mathbf{Z}$, and write
$I^{n}\subset\underline{\mathrm{GW}}$ for the $n$th power Zariski subsheaf of
ideals.

  * (i)

Each sheaf $I^{n}$ is strictly $\mathbf{A}^{1}$-invariant $($in particular a
Nisnevich sheaf $)$.

  * (ii)

We have $L_{\mathrm{zar}}(I^{n}/I^{n+1})\simeq\nu^{n}$, where $\nu^{n}$ is the
sheaf of logarithmic differential $n$-forms [GL00, §2].

  * (iii)

For $n\geqslant 1$, the sheaf $I^{n}$ is Zariski locally generated by $n$-fold
Pfister forms $\langle\\!\langle a_{1},\dots,a_{n}\rangle\\!\rangle$ with
$a_{i}\in\mathcal{O}^{\times}$. The sheaf $\underline{\mathrm{GW}}$ is Zariski
locally generated by $\langle a\rangle$ for $a\in\mathcal{O}^{\times}$.

###### Proof.

Denote by $\underline{I}^{n}\subset\underline{\mathrm{GW}}$ the sheaf defined
in [Mor12, Example 3.34]. Thus each $\underline{I}^{n}$ is a strictly
$\mathbf{A}^{1}$-invariant subsheaf of ideals,
$\underline{I}^{n}\underline{I}^{m}\subset\underline{I}^{n+m}$, and
$\underline{I}^{n}(K)=I^{n}(K)$ for finitely generated field extensions $K/k$.
Moreover $\underline{I}^{1}$ is the kernel of the rank map, so that
$\underline{I}^{1}=I$. For $X\in\mathrm{Sm}_{k}$, write
$N^{n}(X)\subset\nu^{n}(X)$ for the subgroup generated by global logarithmic
differentials, i.e., expressions of the form
$[a_{1},\dotsc,a_{n}]=da_{1}/a_{1}\wedge\dotsb\wedge da_{n}/a_{n}$ with
$a_{i}\in\mathcal{O}(X)^{\times}$. It is clear that $N^{n}$ is a subpresheaf
of $\nu^{n}$. Let $X\in\mathrm{Sm}_{k}$ be connected with generic point
$\eta$, and consider the following diagram

${N^{n}(X)}$${(\underline{I}^{n}/\underline{I}^{n+1})(X)}$${\nu^{n}(\eta)}$${(\underline{I}^{n}/\underline{I}^{n+1})(\eta)\hbox
to0.0pt{,\hss}}$$\scriptstyle{\tilde{f}}$$\scriptstyle{f}$$\scriptstyle{\simeq}$

where $\underline{I}^{n}/\underline{I}^{n+1}$ is the quotient in the Nisnevich
topology. It follows from [GL00, Theorem 8.3] that $\nu^{n}$ is strictly
$\mathbf{A}^{1}$-invariant, and hence the left-hand vertical map is injective.
Since strictly $\mathbf{A}^{1}$-invariant sheaves form an abelian subcategory
of Nisnevich sheaves, $\underline{I}^{n}/\underline{I}^{n+1}$ is strictly
$\mathbf{A}^{1}$-invariant and the right-hand vertical map is an injection. By
[Kat82], there is an isomorphism $f$ as displayed, which is determined by
$f([a_{1},\dotsc,a_{n}])=\langle\\!\langle
a_{1},\dotsc,a_{n}\rangle\\!\rangle$. This formula implies that $f$ restricts
to a morphism $\tilde{f}$, and that the maps $\tilde{f}$ (for various $X$)
assemble into a morphism of presheaves. By [Mor15, Theorem 1.2],
$L_{\mathrm{zar}}N^{n}=\nu^{n}$ and hence we have constructed a map
$L_{\mathrm{zar}}\tilde{f}\colon\nu^{n}\to\underline{I}^{n}/\underline{I}^{n+1}$.
Since $L_{\mathrm{zar}}\tilde{f}$ induces an isomorphism on fields, it is an
isomorphism.

Let $R$ be an essentially smooth local $k$-algebra. For $n\geqslant 1$, let
$J^{n}\subset I^{n}(R)$ denote the subgroup generated by $n$-fold Pfister
forms. Consider the exact sequence

$0\to\underline{I}^{n+1}(R)\to\underline{I}^{n}(R)\to(\underline{I}^{n}/\underline{I}^{n+1})(R)\simeq\nu^{n}(R).$

The map $J^{n}\subset\underline{I}^{n}(R)\to\nu^{n}(R)$ is surjective, since
it sends $\langle\\!\langle a_{1},\dotsc,a_{n}\rangle\\!\rangle$ to
$[a_{1},\dotsc,a_{n}]$. It follows that
$\nu^{n}(R)\simeq\underline{I}^{n}(R)/\underline{I}^{n+1}(R)$. It also follows
that $\underline{I}^{n}(R)=J^{n}+\underline{I}^{n+1}(R)$ and hence by
induction that $\underline{I}^{n}(R)=J^{n}+\underline{I}^{m}(R)$ for any
$m>n$. By [HM73, Theorem III.5.10 and preceeding paragraph] and the
unramifiedness of $\underline{I}^{m}$, we have $\underline{I}^{m}(R)=0$ for
$m>\dim R$. Thus $J^{n}=I^{n}(R)=\underline{I}^{n}(R)$, and so
$I^{n}=\underline{I}^{n}$. If $G\subset\underline{GW}(R)$ denotes the subgroup
generated by $\langle a\rangle$ for $a\in R^{\times}$, then $G\to\mathbf{Z}$
is surjective and we similarly get $\underline{GW}(R)=G+I(R)=G$. All claims
follow. ∎

######  Remark 4.7.

One may prove that for any local ring $R$, $\mathrm{GW}(R)$ is generated by
elements of the form $\langle a\rangle$ for $a\in R^{\times}$. It follows from
Lemma 4.6(iii) that for any regular local $k$-algebra $R$ with fraction field
$K$, the ring $\underline{GW}(R)\subset\underline{GW}(K)=\mathrm{GW}(K)$
coincides with the image of $\mathrm{GW}(R)\to\mathrm{GW}(K)$.

######  Proposition 4.8.

Let $k$ be a field. Then the canonical maps

$\displaystyle
L_{\mathrm{mot}}(\mathbf{Z}\times\mathcal{FS}\mathrm{yn}_{\infty}^{\operatorname{fr}})^{+}\to
L_{\mathrm{mot}}(\mathcal{FS}\mathrm{yn}^{{\operatorname{fr}},\mathrm{gp}})$
$\displaystyle\mathbf{Z}\times(L_{\mathbf{A}^{1}}\mathcal{FS}\mathrm{yn}^{\operatorname{fr}}_{\infty})(k)^{+}\to(L_{\mathbf{A}^{1}}\mathcal{FS}\mathrm{yn}^{\operatorname{fr}})(k)^{\mathrm{gp}}$

are equivalences. The same holds for $\mathcal{FS}\mathrm{yn}^{\mathrm{or}}$.

###### Proof.

Let $f\colon
L_{\mathrm{mot}}(\mathbf{Z}\times\mathcal{FS}\mathrm{yn}_{\infty}^{\operatorname{fr}})\to
L_{\mathrm{mot}}(\mathcal{FS}\mathrm{yn}^{{\operatorname{fr}},\mathrm{gp}})$
be the canonical map. As in the proof of Corollary 4.2,
$L_{\mathrm{mot}}(\mathcal{FS}\mathrm{yn}^{{\operatorname{fr}},\mathrm{gp}})$
is the group completion of
$L_{\mathrm{mot}}(\mathcal{FS}\mathrm{yn}^{{\operatorname{fr}}})$ in Nisnevich
sheaves. For the first equivalence, it suffices by Corollary 3.4 to show that
$f$ induces an isomorphism on $\pi_{0}^{\mathrm{nis}}$. The
$\pi_{0}^{\mathrm{nis}}$ of the right-hand side is isomorphic to the
unramified Grothendieck–Witt sheaf $\underline{\mathrm{GW}}$, and this
isomorphism sends $\langle a\rangle$ to $\langle a\rangle$ [EHK+20a, Corollary
3.3.11]. If $X$ is an essentially smooth henselian local scheme, then
$\underline{\mathrm{GW}}(X)$ is generated by $\langle a\rangle$ for
$a\in\mathcal{O}(X)^{\times}$ by Lemma 4.5, and since $\langle a\rangle$ is
invertible in the left-hand side by Lemma 4.3, we deduce that $f$ is an
effective epimorphism. Note that a surjective map of discrete monoids whose
codomain is a group is injective if and only if its kernel is trivial. It
therefore remains to show that the fiber of $f$ over $0$ is connected. By
[Mor05, Lemma 6.1.3], it suffices to check this on finitely generated
separable field extensions of $k$. Thus, it suffices to show that
$\pi_{0}L_{\mathrm{mot}}(\mathbf{Z}\times\mathcal{FS}\mathrm{yn}_{\infty}^{\operatorname{fr}})(k)$
is a group for any field $k$; indeed, by Corollary 3.4 this implies that $f$
is acyclic on $k$-points, and in particular connected. Since we have a
surjection

$\pi_{0}L_{\mathbf{A}^{1}}(\mathbf{Z}\times\mathcal{FS}\mathrm{yn}_{\infty}^{\operatorname{fr}})(k)\to\pi_{0}L_{\mathrm{mot}}(\mathbf{Z}\times\mathcal{FS}\mathrm{yn}_{\infty}^{\operatorname{fr}})(k),$

it suffices to show that the left-hand side is a group, which also implies the
second equivalence (by Corollary 3.4 again).

Let $T\in\mathcal{FS}\mathrm{yn}^{\operatorname{fr}}_{d}(k)$ be a framed
finite syntomic $k$-scheme of degree $d$. Choosing an embedding of $T$ in
$\mathbf{A}^{m}$ and lifting the framing of $T$ to a trivialization of its
conormal sheaf, we obtain a $k$-point of the smooth $k$-scheme
$\mathrm{Hilb}^{\operatorname{fr}}_{d}(\mathbf{A}^{m})$. We now apply
Proposition 2.3 with $X=\mathrm{Hilb}^{\operatorname{fr}}_{d}(\mathbf{A}^{m})$
and $U\subset X$ the finite étale locus: we obtain framed finite étale
$k$-schemes $A$ and $B$ with a framed cobordism $T\sqcup A\sim B$. Now $B$ is
a sum of framed finite étale schemes of the form
$(\operatorname{Spec}L,\langle a\rangle)$ with $L/k$ a finite separable field
extension and $a\in L^{\times}$. We are therefore reduced to proving that
$(\operatorname{Spec}L,\langle a\rangle)$ is invertible in
$\pi_{0}L_{\mathbf{A}^{1}}(\mathbf{Z}\times\mathcal{FS}\mathrm{yn}_{\infty}^{\operatorname{fr}})(k)$.
By Lemma 4.3, it is enough to prove that, for _some_ $a\in L^{\times}$, both
$(\operatorname{Spec}L,\langle a\rangle)$ and
$(\operatorname{Spec}L,\langle-a\rangle)$ are invertible. If $f(x)$ is a monic
polynomial such that $L\simeq k[x]/(f(x))$, then
$(\operatorname{Spec}L,\langle\pm f^{\prime}(x)\rangle)\simeq\varphi(\pm f)$.
By [EHK+19, Proposition B.1.4], $\varphi(\pm f)$ is framed cobordant to
$\langle\pm 1\rangle[L:k]_{\epsilon}$, which is clearly invertible in
$\pi_{0}(\mathbf{Z}\times\mathcal{FS}\mathrm{yn}_{\infty}^{\operatorname{fr}})(k)$.
This completes the proof for $\mathcal{FS}\mathrm{yn}^{\operatorname{fr}}$.
The proof for $\mathcal{FS}\mathrm{yn}^{\mathrm{or}}$ is exactly the same,
using that the unit map
${}^{\infty}_{\mathbf{T}}\mathbf{1}\to{}^{\infty}_{\mathbf{T}}\mathrm{MSL}$ is
a $\pi_{0}^{\mathrm{nis}}$-isomorphism (see [BH21, Example 16.35] and [Yak19,
Corollary 3]). ∎

Except for the statement about positive characteristic, Theorem 1.4 follows
from Proposition 4.8, [EHK+19, Theorem 3.5.18], and [EHK+20b, Corollary
3.4.4(i)].

## 5\. Removing the plus construction in positive characteristic

The following proposition is an elaboration of [Rob15, Proposition 2.19].

######  Proposition 5.1.

Let $C$ be an $\mathcal{E}_{k}$-monoidal $\infty$-category with $2\leqslant
k\leqslant\infty$ and let $x\in C$. Let $C[x^{-1}]$ be the
$\mathcal{E}_{k}$-monoidal $\infty$-category obtained from $C$ by inverting
$x$, and let $\mathrm{tel}{}_{x}(C)$ be the $C$-module colimit of the sequence

$\dotsb\to C\xrightarrow{\otimes x}C\to\dotsb.$

Consider the following assertions:

  1. (1)

The cyclic permutation of $x^{\otimes 3}$ becomes homotopic to the identity in
$\mathrm{tel}{}_{x}(C)$.

  2. (2)

For some $n\geqslant 2$, the cyclic permutation of $x^{\otimes n}$ becomes
homotopic to the identity in $\mathrm{tel}{}_{x}(C)$.

  3. (3)

The canonical map $\mathrm{tel}{}_{x}(C)\to C[x^{-1}]$ is an equivalence.

Then $(1)\Rightarrow(2)\Rightarrow(3)$. If $k\geqslant 3$, then
$(3)\Rightarrow(1)$.

###### Proof.

The implication $(1)\Rightarrow(2)$ is trivial. Note that assertion $(3)$ is
equivalent to the assertion that $x$ acts invertibly on the telescope
$\mathrm{tel}{}_{x}(C)$. Consider the commutative diagram

${C}$${C}$${\dotsb}$${C}$${C}$${\dotsb}$$\scriptstyle{x^{\otimes(n-1)}}$$\scriptstyle{x}$$\scriptstyle{x^{\otimes(n-1)}}$$\scriptstyle{x}$$\scriptstyle{x^{\otimes(n-1)}}$$\scriptstyle{x^{\otimes(n-1)}}$

where each square commutes via the cyclic permutation of $x^{\otimes n}$. Let
$\mathrm{Seq}$ be the $1$-skeleton of the nerve of the poset $\mathbf{N}$ and
let
$\sigma\colon\mathrm{Seq}{}^{\triangleright}\to\infty\textnormal{-}\mathrm{Cat}$
be the cone given by the first row mapping to the colimit of the second row.
Then the action of $x$ on $\mathrm{tel}{}_{x}(C)$ is the induced map
$\operatorname*{colim}(\sigma|\mathrm{Seq})\to\sigma(\infty)$. Under
assumption (2), we obtain an equivalent cone if we replace each cyclic
permutation in the above diagram by the identity, which is trivially a
colimiting cone. This proves $(2)\Rightarrow(3)$.

If $C$ is $\mathcal{E}_{3}$-monoidal and $x\in C$ is invertible, then the
cyclic permutation of $x^{\otimes 3}$ is homotopic to the identity. Indeed,
$\pi_{0}\operatorname{\mathrm{Aut}}(x^{\otimes 3})$ is an abelian group and
the cyclic permutation of order $3$ becomes trivial in the abelianization of
3. This proves $(3)\Rightarrow(1)$. ∎

######  Example 5.2.

Let $R$ be a derived commutative ring. The cyclic permutation of $R^{3}$ is
induced by a matrix in $SL_{3}(\mathbf{Z})$ and hence is
$\mathbf{A}^{1}$-homotopic to the identity. Applying Proposition 5.1 to the
${\mathcal{E}_{\infty}}$-space $(L_{\mathbf{A}^{1}}\mathrm{Vect})(R)$, we
deduce the well-known fact that the canonical map

$\underline{\mathbf{Z}}\times\mathrm{Vect}_{\infty}\to K$

is an $\mathbf{A}^{1}$-equivalence on derived commutative rings, where
$\underline{\mathbf{Z}}$ is the constant sheaf with value $\mathbf{Z}$ and
$\mathrm{Vect}_{\infty}=\operatorname*{colim}_{n}\mathrm{Vect}_{n}$. This
explains why the plus construction is not needed in the equivalences (over a
regular base)

${}^{\infty}_{\mathbf{T}}\mathrm{KGL}\simeq
L_{\mathrm{mot}}(\mathbf{Z}\times\mathrm{Vect}_{\infty})\simeq
L_{\mathrm{mot}}(\mathbf{Z}\times\mathrm{Gr}_{\infty}(\mathbf{A}^{\infty})).$

######  Lemma 5.3.

Let $p$ be a prime number. Over ${\mathbf{F}}_{p}$, there exists a sequence of
$C_{p}$-equivariant framed cobordisms between $p$ with nontrivial
$C_{p}$-action and $\langle-1\rangle p_{\epsilon}$ with trivial
$C_{p}$-action. In particular, for any ${\mathbf{F}}_{p}$-scheme $S$ and any
$\alpha\in\mathcal{FS}\mathrm{yn}^{\operatorname{fr}}(S)$, the action of
$C_{p}$ on $p\alpha$ in
$(L_{\mathbf{A}^{1}}\mathcal{FS}\mathrm{yn}^{\operatorname{fr}})(S)$ is
trivial.

###### Proof.

Let $f(t,x)=x(x-t)(x-2t)\dotsb(x-(p-1)t)$. Consider the finite syntomic
${\mathbf{F}}_{p}[t]$-scheme

$X=\operatorname{Spec}{\mathbf{F}}_{p}[t,x]/(f(t,x)),$

with the framing $\tau$ induced by $-f(t,x)$. The group $C_{p}$ acts on $X$
over ${\mathbf{F}}_{p}[t]$ by $x\mapsto x+t$. As this action fixes $-f(t,x)$,
we obtain an action of $C_{p}$ on $(X,\tau)$. The fiber $X_{1}$ consists of
$p$ points cyclically permuted and trivially framed: indeed, the framing on
the point $\operatorname{Spec}{\mathbf{F}}_{p}[x]/(x-i)$ is induced by
$(-1)^{p}(p-1)!(x-i)=x-i$. On the other hand, the fiber $X_{0}$ is
$\operatorname{Spec}{\mathbf{F}}_{p}[x]/(x^{p})$ with trivial action and
framing induced by $-x^{p}$. By [EHK+19, Proposition B.1.4], $X_{0}$ is framed
cobordant to $\langle-1\rangle p_{\epsilon}$. ∎

######  Remark 5.4.

Let $S$ be an affine ${\mathbf{F}}_{p}$-scheme, $\mathcal{M}$ an étale sheaf
of ${\mathcal{E}_{\infty}}$-spaces on $\mathrm{Sm}_{S}$, and
$m\in\mathcal{M}(S)$. Then the action of $C_{p}$ on $m^{p}$ in
$(L_{\mathbf{A}^{1}}\mathcal{M})(S)$ is trivial, because
$B_{\mathrm{\acute{e}t}}C_{p}$ is $\mathbf{A}^{1}$-contractible on affine
${\mathbf{F}}_{p}$-schemes by Lemma 5.5 below. This does not apply directly to
$\mathcal{FS}\mathrm{yn}^{\operatorname{fr}}$, which is not an étale sheaf.
However, by [EHK+19, §4.2.36], there is an ${\mathcal{E}_{\infty}}$-map
$\mathcal{FE}\mathrm{t}\to\mathcal{FS}\mathrm{yn}^{\operatorname{fr}}$ where
$\mathcal{FE}\mathrm{t}$ is the moduli stack of finite étale schemes, which is
an étale sheaf. This gives an alternative proof of the last statement of Lemma
5.3.

######  Lemma 5.5.

The presheaf $B_{\mathrm{\acute{e}t}}C_{p}$ is $\mathbf{A}^{1}$-contractible
on affine ${\mathbf{F}}_{p}$-schemes.

###### Proof.

By the Artin–Schreier sequence [Stacks, Tag 0A3J], we have a fiber sequence of
presheaves

$B_{\mathrm{\acute{e}t}}C_{p}\rightarrow
B_{\mathrm{\acute{e}t}}\mathbf{G}_{a}\rightarrow
B_{\mathrm{\acute{e}t}}\mathbf{G}_{a},$

where $\mathbf{G}_{a}$ is the additive group scheme. Since
$L_{\mathbf{A}^{1}}B_{\mathrm{\acute{e}t}}\mathbf{G}_{a}\simeq\ast$, it
suffices to prove that this sequence remains a fiber sequence after applying
$L_{\mathbf{A}^{1}}$. This follows from a general criterion for the geometric
realization of a cartesian square to remain cartesian [Lur17a, Lemma
5.5.6.17], which applies because
$\pi_{0}(B_{\mathrm{\acute{e}t}}\mathbf{G}_{a}(X))=H^{1}_{\mathrm{\acute{e}t}}(X,\mathcal{O})=*$
for $X$ an affine scheme [Stacks, Tags 03P2 and 01XB]. ∎

The following proposition completes the proofs of all the theorems in the
introduction.

######  Proposition 5.6.

Let $k$ be a field of positive characteristic. Then the canonical maps

$\displaystyle L_{\mathrm{mot}}(\mathcal{FS}\mathrm{yn}_{\infty})\to
L_{\mathrm{mot}}(\mathcal{FS}\mathrm{yn}_{\infty})^{+}$
$\displaystyle(L_{\mathbf{A}^{1}}\mathcal{FS}\mathrm{yn}_{\infty})(k)\to(L_{\mathbf{A}^{1}}\mathcal{FS}\mathrm{yn}_{\infty})(k)^{+}$

are equivalences. The same holds for
$\mathcal{FS}\mathrm{yn}^{\operatorname{fr}}_{\infty}$ and
$\mathcal{FS}\mathrm{yn}^{\mathrm{or}}_{\infty}$.

###### Proof.

By Lemma 5.3 and Proposition 5.1, the left-hand sides are commutative monoids,
and they are grouplike by Corollary 4.2 and Proposition 4.8. Hence they
coincide with their plus constructions. ∎

######  Remark 5.7.

Let $C$ be a smooth curve over a field $k$ of characteristic zero. If $X$ is a
finite flat $C$-scheme with an action of the cyclic group $C_{n}$, then the
locus of points $c\in C$ such that $C_{n}$ acts trivially on the fiber $X_{c}$
is clopen. Indeed, that locus is the equalizer of a pair of sections of a
finite flat $C$-subgroup scheme of $\operatorname{\mathrm{Aut}}(X/C)$, which
is necessarily étale. In particular, unlike in positive characteristic, there
cannot exist a $C_{n}$-equivariant cobordism between a finite syntomic
$k$-scheme with nontrivial $C_{n}$-action and one with trivial $C_{n}$-action.
The group homomorphism
$C_{n}\to\pi_{1}(L_{\mathbf{A}^{1}}\mathcal{FS}\mathrm{yn}_{\infty})(k)$ could
nevertheless be trivial for other reasons.

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