Viktor Balch Barth William Hornslien Gereon Quick Glen Matthew Wilson

# Making the motivic group structure on the endomorphisms of the projective
line explicit

Viktor Balch Barth<EMAIL_ADDRESS>Department of Mathematics, UiO,
Norway.  William Hornslien<EMAIL_ADDRESS>Department of
Mathematics, NTNU, Norway.  Gereon Quick<EMAIL_ADDRESS>Department of
Mathematics, NTNU, Norway.  Glen Matthew Wilson<EMAIL_ADDRESS>Department
of Mathematics and Statistics, UiT, Norway.

###### Abstract

We construct a group structure on the set of pointed naive homotopy classes of
scheme morphisms from the Jouanolou device to the projective line. The group
operation is defined via matrix multiplication on generating sections of line
bundles and only requires basic algebraic geometry. In particular, it is
completely independent of the construction of the motivic homotopy category.
We show that a particular scheme morphism, which exhibits the Jouanolou device
as an affine torsor bundle over the projective line, induces a monoid morphism
from Cazanave’s monoid to this group. Moreover, we show that this monoid
morphism is a group completion to a subgroup of the group of scheme morphisms
from the Jouanolou device to the projective line. This subgroup is generated
by a set of morphisms that are simple to describe.

###### Contents

  1. 1 Introduction
  2. 2 The Jouanolou device and morphisms to $\mathbb{P}^{1}$
    1. 2.1 Convenient coordinates for $\mathcal{J}$
    2. 2.2 Invertible sheaves on $\mathcal{J}$
    3. 2.3 Pointed morphisms $\mathbb{P}^{1}\to\mathbb{P}^{1}$ and $\mathcal{J}\to\mathbb{P}^{1}$
    4. 2.4 The pointed naive homotopy relation
    5. 2.5 Morphisms $\mathcal{J}\to\mathbb{A}^{2}\setminus\\{0\\}$
  3. 3 Operations on naive homotopy classes of morphisms
    1. 3.1 Group structure on maps of degree $0$
    2. 3.2 Action of degree $0$ maps on degree $n$ maps
  4. 4 The group structure on $[\mathcal{J},\mathbb{P}^{1}]^{\mathrm{N}}$
  5. 5 Compatibility with Cazanave’s monoid structure
    1. 5.1 Compatibility with certain degree $0$ maps
    2. 5.2 The map $\pi_{\mathrm{N}}^{*}$ is a monoid morphism
  6. 6 Group completion
    1. 6.1 Motivic Brouwer degree
    2. 6.2 Group completion of naive homotopy classes
    3. 6.3 Milnor–Witt K-theory and morphisms in degree 0
  7. A Affine representability for pointed spaces and homotopies
  8. B Facts about the resultant
  9. C Testing compatibility via real realization and signatures

##  1 Introduction

The $\mathbb{A}^{1}$-homotopy category over a field $k$ introduced by Morel
and Voevodsky [23] makes it possible to define a group structure on the
$\mathbb{A}^{1}$-homotopy classes of pointed maps
$\mathbb{P}^{1}\to\mathbb{P}^{1}$. This construction plays the role of the
fundamental group of the circle in motivic homotopy theory. Although the group
operation mimics the usual construction in algebraic topology, the set of
$\mathbb{A}^{1}$-homotopy classes of maps $\mathbb{P}^{1}\to\mathbb{P}^{1}$ is
not simply the set of morphisms $\mathbb{P}^{1}\to\mathbb{P}^{1}$ modulo an
equivalence relation. It is unsettling that such an important group does not
arise easily in some elementary way as a set of morphisms up to a homotopy
relation with some geometrically defined group operation. Thankfully, the work
of Asok, Hoyois, and Wendt in [6], along with Cazanave’s investigation in [13]
lays a foundation to build off of.

In [13] Cazanave defines an operation $\oplus^{\mathrm{N}}$ which turns the
set $[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathrm{N}}$ of pointed naive homotopy
classes into a monoid and shows that the canonical map
$\nu_{\mathbb{P}^{1}}:[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathrm{N}}\to[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathbb{A}^{1}}$
is a group completion. However, this approach cannot yield candidates for
scheme morphisms which represent inverses of $\mathbb{A}^{1}$-homotopy
classes.

In [6] Asok, Hoyois, and Wendt show that the set
$[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathbb{A}^{1}}$ of motivic homotopy classes
is in bijection to an explicit set of maps modulo the naive homotopy relation
by using the larger set of maps $\mathrm{Sm}_{k}(\mathcal{J},\mathbb{P}^{1})$
where $\mathcal{J}$ denotes the Jouanolou device of $\mathbb{P}^{1}$, which we
consider equipped with a morphism $\pi:\mathcal{J}\to\mathbb{P}^{1}$ that
exhibits $\mathcal{J}$ as an affine torsor bundle. This resolves the problem
of a lack of candidates of morphisms which may represent inverses in
$[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathbb{A}^{1}}$, but it is not clear at all
how the group operation on $[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathbb{A}^{1}}$
or the operation of [13] may be lifted.

In the present paper we define an explicit group structure on the set
$[\mathcal{J},\mathbb{P}^{1}]^{\mathrm{N}}$ of pointed naive homotopy classes.
The construction of this group operation is independent of the general
machinery of motivic homotopy theory and only uses basic algebraic geometry.
We then show that the induced map
$\pi_{\mathrm{N}}^{*}:[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathrm{N}}\to[\mathcal{J},\mathbb{P}^{1}]^{\mathrm{N}}$
is a morphism of monoids where $[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathrm{N}}$
has the monoid structure of [13, §3]. Moreover, we show that
$\pi_{\mathrm{N}}^{*}$ has image in a concrete subgroup $\mathbf{G}$ and that
the map
$\pi_{\mathrm{N}}^{*}:[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathrm{N}}\to\mathbf{G}$
is a group completion. Hence there are canonical isomorphisms between
$\mathbf{G}$ and $[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathbb{A}^{1}}$ which are
compatible with $\pi_{\mathrm{N}}^{*}$ and $\nu_{\mathbb{P}^{1}}$. A key
feature of the group $\mathbf{G}$ is that is defined by explicit generating
scheme morphisms $\mathcal{J}\to\mathbb{P}^{1}$ that are defined in terms of
very simple $(2\times 2)$-matrices.

We will now describe our results in more detail. First we recall the
conventional group operation $\oplus^{\mathbb{A}^{1}}$ on
$[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathbb{A}^{1}}$, the set of maps in the
pointed $\mathbb{A}^{1}$-homotopy category over a field $k$. It is a simple
exercise to produce an $\mathbb{A}^{1}$-weak equivalence $\mathbb{P}^{1}\simeq
S^{1}\wedge\mathbb{G}_{m}$ using the standard covering of $\mathbb{P}^{1}$ by
two affine lines with intersection $\mathbb{G}_{m}$. The simplicial circle
$S^{1}$ (or some suitable homotopy equivalent model of it, like
$\partial\Delta^{2}$) admits the structure of an h-cogroup, or just a cogroup
in the homotopy category. Explicitly, the pointed simplicial set
$\partial\Delta^{2}\simeq S^{1}$ admits two maps: a pinch map $\mu:S^{1}\to
S^{1}\vee S^{1}$ and an inverse map $S^{1}\to S^{1}$. These operations fit
into homotopy commutative diagrams that give the expected algebraic
properties, like associativity and the definition of the inverse [24, Chapter
2]. These two observations together allow us to define a group operation on
$[S^{1}\wedge\mathbb{G}_{m},\mathbb{P}^{1}]^{\mathbb{A}^{1}}$ as follows.
Given two maps $f,g:S^{1}\wedge\mathbb{G}_{m}\to\mathbb{P}^{1}$ in the
$\mathbb{A}^{1}$-homotopy category, the composition below represents the sum
$f\oplus^{\mathbb{A}^{1}}g$ of the maps $f$ and $g$.

(1)
$\textstyle{S^{1}\wedge\mathbb{G}_{m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu\wedge
1}$$\textstyle{(S^{1}\vee
S^{1})\wedge\mathbb{G}_{m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{(S^{1}\wedge\mathbb{G}_{m})\vee(S^{1}\wedge\mathbb{G}_{m})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f\vee
g}$$\textstyle{\mathbb{P}^{1}}$

Note that the morphism $f\vee g$ exists by the universal property of wedge
sums. One must take the time to verify that the operation defined above does
indeed make $[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathbb{A}^{1}}$ into a group,
but the pleasant properties of the $\mathbb{A}^{1}$-homotopy category make
this doable. This operation seems quite explicit, however, it is not evident
how to describe maps $S^{1}\wedge\mathbb{G}_{m}\to\mathbb{P}^{1}$ as the
domain is only defined as a simplicial presheaf on $\mathrm{Sm}/k$. This is
where the work of Asok, Hoyois, and Wendt [5, 6, 7] can be used to get a
geometric description of the set of maps $\mathbb{P}^{1}\to\mathbb{P}^{1}$.

The Jouanolou device of $\mathbb{P}^{1}$ over a field $k$ is the smooth affine
scheme $\mathcal{J}=\operatorname{Spec}(R)$ where

$R=\frac{k[x,y,z,w]}{\left(x+w-1,xw-yz\right)}.$

The ring $R$ is used to represent $(2\times 2)$-matrices with trace $1$ and
determinant $0$. Namely, a ring homomorphism $R\to S$ is equivalent to a
$(2\times 2)$-matrix over $S$ with trace $1$ and determinant $0$. The
Jouanolou device of $\mathbb{P}^{1}$ is $\mathbb{A}^{1}$-homotopy equivalent
to $\mathbb{P}^{1}$ and also an affine scheme. These good properties are
exploited by Asok, Hoyois, and Wendt [6]. Briefly, their work can be used to
show that there is a bijection
$\xi\colon[\mathcal{J},\mathbb{P}^{1}]^{\mathrm{N}}\xrightarrow{\cong}[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathbb{A}^{1}}$,
where $[\mathcal{J},\mathbb{P}^{1}]^{\mathrm{N}}$ is the set of naive homotopy
classes of pointed maps $\mathcal{J}\to\mathbb{P}^{1}$. The map $\xi$ is the
composite of the canonical map
$\nu\colon[\mathcal{J},\mathbb{P}^{1}]^{\mathrm{N}}\to[\mathcal{J},\mathbb{P}^{1}]^{\mathbb{A}^{1}}$
and the inverse of the map
$\pi_{\mathbb{A}^{1}}^{*}\colon[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathbb{A}^{1}}\to[\mathcal{J},\mathbb{P}^{1}]^{\mathbb{A}^{1}}$
induced by a scheme morphism $\pi\colon\mathcal{J}\to\mathbb{P}^{1}$. The set
$[\mathcal{J},\mathbb{P}^{1}]^{\mathrm{N}}$ is concrete in the following
sense: it is the set of pointed scheme morphisms
$\mathcal{J}\to\mathbb{P}^{1}$ modulo an equivalence relation generated by
naive homotopies. A naive homotopy between two pointed maps
$f,g:\mathcal{J}\to\mathbb{P}^{1}$ is given by a map
$H:\mathcal{J}\times\mathbb{A}^{1}\to\mathbb{P}^{1}$ satisfying the evident
restrictions $H_{0}=f$ and $H_{1}=g$. We note that $H$ must be pointed in the
sense that $*\times\mathbb{A}^{1}$ maps to the basepoint of $\mathbb{P}^{1}$.
We explain in appendix A how the unpointed results of [5] and [6] imply that
the canonical map
$[\mathcal{J},\mathbb{P}^{1}]^{\mathrm{N}}\rightarrow[\mathcal{J},\mathbb{P}^{1}]^{\mathbb{A}^{1}}$
is a bijection.

The bijection
$\xi:[\mathcal{J},\mathbb{P}^{1}]^{\mathrm{N}}\xrightarrow{\cong}[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathbb{A}^{1}}$
gets us started on our quest for an elementary description of the group
structure. There are at least two possible approaches: We first describe the
one we did not take in this paper. One can hope to construct a cogroup
structure on $\mathcal{J}$. This is not so easy. However, Asok and Fasel have
done much of the work to make this possible. In [3], Asok and Fasel give an
explicit construction of a smooth scheme
$\widetilde{\mathcal{J}\vee\mathcal{J}}$ that is $\mathbb{A}^{1}$-weak
equivalent to the wedge sum $\mathcal{J}\vee\mathcal{J}$. We have constructed
an explicit map $\mu:\mathcal{J}\to\widetilde{\mathcal{J}\vee\mathcal{J}}$
that conjecturally represents the pinch map
$\mathbb{P}^{1}\to\mathbb{P}^{1}\vee\mathbb{P}^{1}$. We also have a candidate
for a map representing the inverse map $\mathcal{J}\to\mathbb{P}^{1}$, but
unfortunately both of these claims have proven too difficult to verify. We
therefore decided not to include this construction in this paper.

The approach we take is more explicit. Recall that a morphism
$f\colon\mathcal{J}\to\mathbb{P}^{1}$ is determined by an invertible sheaf
$\mathcal{L}$ over $\mathcal{J}$ and a choice of two generating sections
$s_{0},s_{1}\in\Gamma(\mathcal{L},\mathcal{J})$. The invertible sheaf
$\mathcal{L}$ is the pullback $f^{*}\mathcal{O}(1)$. We say that a morphism
$f\colon\mathcal{J}\to\mathbb{P}^{1}$ has degree 0 if $f^{*}\mathcal{O}(1)$ is
the structure sheaf on $\mathcal{J}$. As we will show in section 2.5, the maps
$\mathcal{J}\to\mathbb{P}^{1}$ of degree 0 are exactly the maps which factor
through the Hopf map $\eta:\mathbb{A}^{2}\setminus\\{0\\}\to\mathbb{P}^{1}$. A
little work revealed that the set of maps
$[\mathcal{J},\mathbb{A}^{2}\setminus\\{0\\}]^{\mathrm{N}}$ has an apparent
group structure. Seen one way, a map
$\mathcal{J}\to\mathbb{A}^{2}\setminus\\{0\\}$ is given by a unimodular row
$(A,B)$ in $R^{2}$, i.e., there exist $U,V\in R$ for which $AU+BV=1$. Any such
unimodular row can be completed to a $(2\times 2)$-matrix over $R$, and the
product of these matrices defines a group operation on
$[\mathcal{J},\mathbb{A}^{2}\setminus\\{0\\}]^{\mathrm{N}}$. Seen another way,
the punctured plane $\mathbb{A}^{2}\setminus\\{0\\}$ is $\mathbb{A}^{1}$-weak
equivalent to $\mathrm{SL}_{2}$, the scheme representing $(2\times
2)$-matrices with determinant $1$. The scheme $\mathrm{SL}_{2}$ is a group
scheme, and so has the structure of a group object in the
$\mathbb{A}^{1}$-homotopy category. It is standard algebraic topology then to
use the structure of $\mathrm{SL}_{2}$ to make the set of maps
$[\mathcal{J},\mathrm{SL}_{2}]^{\mathrm{N}}$ into a group [24]. Passing to the
$\mathbb{A}^{1}$-homotopy classes of maps, it follows that this is the correct
group structure, because
$[\mathcal{J},\mathrm{SL}_{2}]^{\mathrm{N}}\cong[S^{1}\wedge\mathbb{G}_{m},\mathrm{SL}_{2}]^{\mathbb{A}^{1}}$
has the domain space a cogroup object and the codomain space a group object,
and the Eckmann–Hilton argument implies that these two structures define
isomorphic group operations on
$[S^{1}\wedge\mathbb{G}_{m},\mathrm{SL}_{2}]^{\mathbb{A}^{1}}$.

What remains then? The subgroup of degree 0 maps
$\mathcal{J}\to\mathbb{P}^{1}$ is quite large. Once we have verified that
$[\mathcal{J},\mathbb{P}^{1}]^{\mathrm{N}}$ is a group, we will show that the
quotient group
$[\mathcal{J},\mathbb{P}^{1}]^{\mathrm{N}}/[\mathcal{J},\mathbb{A}^{2}\setminus\\{0\\}]^{\mathrm{N}}\cong\mathbb{Z}$
is the group of integers. This isomorphism serves as our guiding principle to
determine the group operation on $[\mathcal{J},\mathbb{P}^{1}]^{\mathrm{N}}$.
More precisely, the bijection $\xi$ fits into the commutative diagram below in
which the bottom row is an exact sequence of groups by the work of Morel [22].

(6)

Since we have an explicit group structure on
$[\mathcal{J},\mathbb{A}^{2}\setminus\\{0\\}]^{\mathrm{N}}$, it then suffices
to pick representative lifts for each integer
$n\in\mathbb{Z}\cong\mathrm{Pic}(\mathcal{J})$ and to understand how the group
$[\mathcal{J},\mathbb{A}^{2}\setminus\\{0\\}]^{\mathrm{N}}$ acts on them. More
concretely, for any integer $n$, let $n\pi$ denote a morphism
$n\pi\colon\mathcal{J}\to\mathbb{P}^{1}$ which represents the
$\mathbb{A}^{1}$-homotopy class
$n[\operatorname{id}:\mathbb{P}^{1}\to\mathbb{P}^{1}]$ under the bijection
$\xi\colon[\mathcal{J},\mathbb{P}^{1}]^{\mathrm{N}}\to[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathbb{A}^{1}}$.
Let $f\colon\mathcal{J}\to\mathbb{P}^{1}$ and
$g\colon\mathcal{J}\to\mathbb{P}^{1}$ be morphisms of degrees $n$ and $m$
respectively. We will show that there are degree $0$ maps
$f_{0}:\mathcal{J}\to\mathbb{A}^{2}\setminus\\{0\\}\to\mathbb{P}^{1}$ and
$g_{0}:\mathcal{J}\to\mathbb{A}^{2}\setminus\\{0\\}\to\mathbb{P}^{1}$ for
which $f\simeq f_{0}\oplus n\pi$ and $g\simeq g_{0}\oplus m\pi$. We then
define the sum of $[f]$ and $[g]$ to be

$\displaystyle[f]\oplus[g]$
$\displaystyle:=([f_{0}]\oplus[n\pi])\oplus([g_{0}]\oplus[m\pi])$
$\displaystyle=([f_{0}]\oplus[g_{0}])\oplus[(n+m)\pi].$

The term $[f_{0}]\oplus[g_{0}]$ is calculated by matrix multiplication in the
group $[\mathcal{J},\mathbb{A}^{2}\setminus\\{0\\}]^{\mathrm{N}}$. The key
idea for how $[\mathcal{J},\mathbb{A}^{2}\setminus\\{0\\}]^{\mathrm{N}}$ acts
on the set $\\{[n\pi]:n\in\Z\\}$ is that any map $n\pi$ is given by the choice
of a line bundle together with two generating sections. We can then let a
morphism $\mathcal{J}\to\mathbb{A}^{2}\setminus\\{0\\}$ given by a $(2\times
2)$-matrix act on the sections via matrix multiplication. We explain the
details of this operation in section 3.2.

In section 4 we prove the following result, see theorem 84:

######  Theorem 7.

The operation $\oplus$ of definition 77, which is described above, makes
$\left([\mathcal{J},\mathbb{P}^{1}]^{\mathrm{N}},\oplus\right)$ an abelian
group. There is a group isomorphism

$\phi\colon\left([\mathcal{J},\mathbb{P}^{1}]^{\mathrm{N}},\oplus\right)\xrightarrow{\cong}\left([\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathbb{A}^{1}},\oplus^{\mathbb{A}^{1}}\right)$

which restricts to $\xi_{0}$ on the subgroup
$[\mathcal{J},\mathbb{A}^{2}\setminus\\{0\\}]^{\mathrm{N}}$.

The step we are not able to prove yet is that the bijection
$\xi\colon[\mathcal{J},\mathbb{P}^{1}]^{\mathrm{N}}\to[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathbb{A}^{1}}$
is compatible with $\oplus$ on $[\mathcal{J},\mathbb{P}^{1}]^{\mathrm{N}}$ and
the conventional group structure $\oplus^{\mathbb{A}^{1}}$ on
$[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathbb{A}^{1}}$. Nevertheless, we hope to
ultimately prove the following conjecture for which we report on further
evidence in appendix C.

######  Conjecture 8.

The bijection
$\xi\colon[\mathcal{J},\mathbb{P}^{1}]^{\mathrm{N}}\to[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathbb{A}^{1}}$
is a group isomorphism and equals $\phi$.

The isomorphism $\phi$ is produced by rather formal arguments and does not yet
provide a concrete description of the group
$[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathbb{A}^{1}}$. We will therefore now
explain how we do obtain a very explicit expression of the latter group in
terms of concrete scheme morphisms $\mathcal{J}\to\mathbb{P}^{1}$.

Recall that Cazanave shows in [13, Theorem 3.22] that there is an operation
$\oplus^{\mathrm{N}}$ which provides
$[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathrm{N}}$ with the structure of a
commutative monoid and that the canonical map
$[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathrm{N}}\to[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathbb{A}^{1}}$
is a group completion. We show that the morphism
$\pi\colon\mathcal{J}\to\mathbb{P}^{1}$ also induces a group completion in the
following way. First we prove in section 5 that the operation $\oplus$ is
compatible with $\oplus^{\mathrm{N}}$, see theorem 103:

######  Theorem 9.

The morphism $\pi\colon\mathcal{J}\to\mathbb{P}^{1}$ induces a morphism of
commutative monoids

$\pi_{\mathrm{N}}^{*}\colon\left([\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathrm{N}},\oplus^{\mathrm{N}}\right)\to\left([\mathcal{J},\mathbb{P}^{1}]^{\mathrm{N}},\oplus\right)$

where the left-hand side denotes the monoid of [13, §3].

The proof of theorem 9 is based on the following observation. Let $u\in
k^{\times}$. As in [13] we identify a rational function $X/u$ in the
indeterminate $X$ with the morphism $\mathbb{P}^{1}\to\mathbb{P}^{1}$ defined
by $[x_{0}:x_{1}]\mapsto[x_{0}:ux_{1}]$. For $u,v\in k^{\times}$, we let
$g_{u,v}\colon\mathcal{J}\to\mathbb{A}^{2}\setminus\\{0\\}$ denote the
morphism given by the unimodular row $\left(x+\frac{v}{u}w,(u-v)y\right)$ in
$R^{2}$. For the rational functions $X/u$ and $X/v$ we then have the identity

(10)
$g_{u,v}\oplus\pi_{\mathrm{N}}^{*}\left(X/v\right)=\pi_{\mathrm{N}}^{*}\left(X/u\right)$

which we emphasize is an identity of morphisms not just homotopy classes. In
particular, for $v=1$ and $\pi_{\mathrm{N}}^{*}\left(X/1\right)=\pi$, formula
(10) reads $g_{u,1}\oplus\pi=\pi_{\mathrm{N}}^{*}\left(X/u\right)$ and reduces
computations for $\pi_{\mathrm{N}}^{*}\left(X/u\right)$ to computations for
$g_{u,1}$ and $\pi$. The key technical result needed to prove theorem 9 is
that, for every pointed morphism $f\colon\mathbb{P}^{1}\to\mathbb{P}^{1}$, we
have an explicit naive homotopy

(11)
$\displaystyle\pi_{\mathrm{N}}^{*}\left(X/u\oplus^{\mathrm{N}}f\right)\simeq
g_{u,1}\oplus\left(\pi_{\mathrm{N}}^{*}\left(X/1\oplus^{\mathrm{N}}f\right)\right).$

The construction of the concrete homotopy in formula (11) is based on
computations of the resultants of certain morphisms which we provide in
section 5.1 and appendix B. Theorem 9 then follows from the fact that the set
of homotopy classes $[X/u]$ for all $u\in k^{\times}$ generates
$[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathrm{N}}$ and a successive application of
formula (11).

Identity (10) also implies that the image of $\pi_{\mathrm{N}}^{*}$ is
contained in the subgroup
$\mathbf{G}\subseteq[\mathcal{J},\mathbb{P}^{1}]^{\mathrm{N}}$ generated by
the homotopy classes $[g_{u,v}]$ for all $u,v\in k^{\times}$ and $[\pi]$.
Theorem 9 and the work of Cazanave [13, Theorem 3.22] then imply that there is
a unique group homomorphism
$\psi\colon[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathbb{A}^{1}}\to\mathbf{G}$ such
that $\psi\circ\nu_{\mathbb{P}^{1}}=\pi_{\mathrm{N}}^{*}$. In section 6 we
show the following key result, see theorem 129:

######  Theorem 12.

The monoid morphism
$\pi_{\mathrm{N}}^{*}\colon[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathrm{N}}\to\mathbf{G}$
is a group completion. There is a unique isomorphism
$\chi\colon\mathbf{G}\to[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathbb{A}^{1}}$ such
that the diagram below commutes. Moreover, $\chi$ and $\psi$ are mutual
inverses to each other.

Theorem 12 gives a very concrete description of all pointed endomorphisms of
$\mathbb{P}^{1}$ in the unstable $\mathbb{A}^{1}$-homotopy category in the
following sense: the group $\mathbf{G}$ is given by a simple set of generating
morphisms, and the group operation $\oplus$ in $\mathbf{G}$ inherited from
$[\mathcal{J},\mathbb{P}^{1}]^{\mathrm{N}}$ is defined in basic algebro-
geometric terms. The only thing we are missing is a concrete morphism
$\mathcal{J}\to\mathbb{P}^{1}$ that is sent to the homotopy class
$-[\operatorname{id}\colon\mathbb{P}^{1}\to\mathbb{P}^{1}]$. We speculate that
the map $\tilde{\pi}$ defined in the beginning of section 2 corresponds with
$-[\operatorname{id}]$. In theorem 121 we prove that the image of the naive
homotopy class of $\tilde{\pi}$ under the motivic Brouwer degree is the class
$-\langle 1\rangle$ in $\mathrm{GW}(k)$. This shows that the image of the
class of $\tilde{\pi}$ represents $-[\operatorname{id}]$ in the stable motivic
homotopy category. However, this is not sufficient to determine the class of
$\tilde{\pi}$ unstably. We elaborate on this claim in section 4 and report on
our evidence in section 6.1 and appendix C.

Finally, we note that the isomorphisms
$\mathbf{G}\xrightarrow{\chi}[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathbb{A}^{1}}\xleftarrow{\phi}[\mathcal{J},\mathbb{P}^{1}]^{\mathrm{N}}$
do not imply that $\mathbf{G}$ equals
$[\mathcal{J},\mathbb{P}^{1}]^{\mathrm{N}}$. However, we conjecture that the
inclusion $\mathbf{G}\subseteq[\mathcal{J},\mathbb{P}^{1}]^{\mathrm{N}}$ is an
equality and we show in section 6.3 that this is true for all finite fields by
computing the first Milnor–Witt K-theory $K_{1}^{MW}(\mathbb{F}_{q})$, which
is isomorphic to
$[\mathbb{P}^{1},\mathbb{A}^{2}\setminus\\{0\\}]^{\mathbb{A}^{1}}$ and to the
subgroup generated by all classes $[g_{u,v}]$ in
$[\mathcal{J},\mathbb{A}^{2}\setminus\\{0\\}]^{\mathrm{N}}$.

To conclude this discussion we remark that Cazanave speculates in his thesis
[12, page 31] whether $[\mathcal{J},\mathcal{J}]^{\mathrm{N}}$ may be used to
give a concrete model to study the motivic homotopy group
$[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathbb{A}^{1}}$. Our cursory attempts at
equipping this set with a group operation did not succeed. The obvious choices
for an operation on matrices do not yield the conventional group structure,
and we were unable to find a suitable candidate for an operation. We did not
include our findings in this paper. We refer to remark 92 for some additional
comments.

We thank Aravind Asok, Christophe Cazanave, Marc Hoyois, Marc Levine, Kirsten
Wickelgren, Ben Williams, and Paul Arne Østvær for helpful comments,
suggestions and clarifications. The first-named author is supported by the RCN
grant no. 300814 Young Research Talents of Trung Tuyen Truong. The second-
named author has received support from the project _Pure Mathematics in
Norway_ funded by the Trond Mohn Foundation. The third- and fourth-named
author gratefully acknowledge the partial support by the RCN Project no.
313472 Equations in Motivic Homotopy. The authors would like to thank the
Centre for Advanced Study in Oslo for its hospitality where parts of the work
on the paper were carried out.

##  2 The Jouanolou device and morphisms to $\mathbb{P}^{1}$

In this section we work out the details needed about the Jouanolou device
$\mathcal{J}$, morphisms $\mathcal{J}\to\mathbb{P}^{1}$, and the pointed naive
homotopy relation. We keep the terminology as elementary as possible and hope
that the details provided help making our approach accessible.

Throughout this paper $k$ will always denote a field. All schemes are schemes
over $\operatorname{Spec}{k}$. The letter $R$ will always denote the following
ring.

######  Definition 13.

We let $R$ denote the ring

$R=\frac{k[x,y,z,w]}{\left(x+w-1,xw-yz\right)}.$

We will frequently identify $R$ with the ring $k[x,y,z]/(x(1-x)-yz)$ where it
is convenient. The Jouanolou device of $\mathbb{P}^{1}$ is the smooth affine
$k$-scheme $\mathcal{J}=\operatorname{Spec}{R}$. We consider $\mathcal{J}$ to
be pointed at $\mathbf{j}=(x-1,y,z,w)$.

The Jouanolou device may also be considered as the ring representing $(2\times
2)$-matrices with trace $1$ and determinant $0$.

While we will discuss morphisms $\mathcal{J}\to\mathbb{P}^{1}$ in more detail
later, we point out that there are two evident morphisms that exhibit
$\mathcal{J}$ as an affine torsor bundle over $\mathbb{P}^{1}$. The matrices
over $R$

$p_{1}=\begin{pmatrix}x&y\\\ z&w\end{pmatrix}\,\text{ and
}\,q_{1}=\begin{pmatrix}x&z\\\ y&w\end{pmatrix}$

are idempotent matrices which thus define projective modules
$\mathcal{P}_{1}=\operatorname{Im}(p_{1})$ and
$\mathcal{Q}_{1}=\operatorname{Im}(q_{1})$ as the image of the associated map
of $R$-modules. Both $\mathcal{P}_{1}$ and $\mathcal{Q}_{1}$ have rank 1, and
so they yield invertible sheaves on $\mathcal{J}$. We obtain a map
$\pi\colon\mathcal{J}\to\mathbb{P}^{1}$ by selecting the invertible sheaf
associated to $\mathcal{P}_{1}$ and the generating sections

$s_{0}=\begin{pmatrix}x\\\ z\end{pmatrix}\,\text{ and
}\,s_{1}=\begin{pmatrix}y\\\ w\end{pmatrix}.$

We intuitively understand this map as sending a point in $\mathcal{J}$
corresponding to a matrix $\begin{pmatrix}x&y\\\ z&w\end{pmatrix}$ to either
the point with homogeneous coordinates $[x:y]$ or $[z:w]$, depending on which
is defined. When both points make sense in $\mathbb{P}^{1}$, they agree, so
the map is well-defined. Similarly, we obtain a map
$\tilde{\pi}\colon\mathcal{J}\to\mathbb{P}^{1}$ by using $\mathcal{Q}_{1}$ and
the choice of generating sections

$s_{0}=\begin{pmatrix}x\\\ y\end{pmatrix}\text{ and
}s_{1}=-\begin{pmatrix}z\\\ w\end{pmatrix}.$

Both $\pi$ and $\tilde{\pi}$ exhibit $\mathcal{J}$ as an affine torsor bundle
over $\mathbb{P}^{1}$, hence they are $\mathbb{A}^{1}$-homotopy equivalences.
It follows that

(14)
$\pi_{\mathbb{A}^{1}}^{*}\colon\left[\mathbb{P}^{1},\mathbb{P}^{1}\right]^{\mathbb{A}^{1}}\to\left[\mathcal{J},\mathbb{P}^{1}\right]^{\mathbb{A}^{1}}$

is a bijection. We show in proposition 146 in appendix A that the canonical
map
$\nu\colon[\mathcal{J},\mathbb{P}^{1}]^{\mathrm{N}}\to[\mathcal{J},\mathbb{P}^{1}]^{\mathbb{A}^{1}}$
is a bijection because $\mathcal{J}$ is affine and $\mathbb{P}^{1}$ is
$\mathbb{A}^{1}$-naive. Thus, the composition of the bijection $\nu$ and the
inverse of $\pi^{*}$ is a bijection

(15)
$\xi\colon\left[\mathcal{J},\mathbb{P}^{1}\right]^{\mathrm{N}}\rightarrow\left[\mathbb{P}^{1},\mathbb{P}^{1}\right]^{\mathbb{A}^{1}}.$

This bijection may be described as follows. A naive pointed homotopy class of
maps $[f]$ represented by the pointed scheme morphism
$f\colon\mathcal{J}\to\mathbb{P}^{1}$ is sent to
$\xi([f])=[f\circ\pi^{-1}]^{\mathbb{A}^{1}}$, the pointed
$\mathbb{A}^{1}$-homotopy class of the zig-zag
$\mathbb{P}^{1}\xleftarrow{\pi}\mathcal{J}\xrightarrow{f}\mathbb{P}^{1}$.

In the following sections we will investigate the domain of $\xi$, i.e., the
set $[\mathcal{J},\mathbb{P}^{1}]^{\mathrm{N}}$ of pointed naive homotopy
classes of pointed morphisms $\mathcal{J}\to\mathbb{P}^{1}$.

###  2.1 Convenient coordinates for $\mathcal{J}$

The map $\pi\colon\mathcal{J}\to\mathbb{P}^{1}$ encourages the choice of a
convenient set of coordinate charts for $\mathcal{J}$. For $\mathbb{P}^{1}$,
we use the standard notation $U_{0}=\mathbb{P}^{1}\setminus\\{[0:1]\\}$ and
$U_{1}=\mathbb{P}^{1}\setminus\\{[1:0]\\}$. It is straightforward to verify
that the preimages under $\pi$ of $U_{0}$ and $U_{1}$ are
$\pi^{-1}(U_{0})=D(x)\cup D(z)$ and $\pi^{-1}(U_{1})=D(y)\cup D(w)$. Both of
these open sets are isomorphic to $\mathbb{A}^{2}$ under the following maps.

######  Lemma 16.

The open set $D(x)\cup D(z)\subseteq\mathcal{J}$ is isomorphic to
$\operatorname{Spec}(k[a,b])$ under the map
$\phi_{0}\colon\mathbb{A}^{2}\to\mathcal{J}$ given by $x\mapsto 1-ab$,
$y\mapsto a(1-ab)$, $z\mapsto b$, and $w\mapsto ab$.

Similarly, the open set $D(y)\cup D(w)\subseteq\mathcal{J}$ is isomorphic to
$\operatorname{Spec}(k[s,t])$ under the map
$\phi_{1}\colon\mathbb{A}^{2}\to\mathcal{J}$ given by $x\mapsto st$, $y\mapsto
t$, $z\mapsto s(1-st)$, and $w\mapsto 1-st$.

###### Proof.

The proof proceeds by studying the map locally. For instance, $\phi_{0}$
induces an isomorphism of rings $k[a,b][(1-ab)^{-1}]\to R[x^{-1}]$ and also of
$k[a,b][b^{-1}]\to R[z^{-1}]$. The open sets $D(1-ab)$ and $D(b)$ cover
$\mathbb{A}^{2}$, so it follows that $\phi_{0}$ maps surjectively onto
$D(x)\cup D(z)$. The inverse map is obtained by gluing the maps that are
defined on $D(x)$ and $D(z)$, giving the result. A similar argument works for
$\phi_{1}$. ∎

######  Remark 17.

The open affine subschemes $D(x)\cup D(z)$ and $D(y)\cup D(w)$ of
$\operatorname{Spec}(R)$ have the odd property that their ring of global
sections is not a localization of $R$.

###  2.2 Invertible sheaves on $\mathcal{J}$

By [17, Theorem II.7.1], a morphism $\mathcal{J}\to\mathbb{P}^{1}$ is
determined by an invertible sheaf $\mathcal{L}$ on $\mathcal{J}$ and two
generating global sections of $\mathcal{L}$. We now take the time to study the
invertible sheaves on $\mathcal{J}$ to enable our study of the morphisms
$\mathcal{J}\to\mathbb{P}^{1}$. We will assume familiarity with the basic
terminology presented in for example [25, Chapter 1] and [8].

Since $\mathcal{J}=\operatorname{Spec}(R)$ is an irreducible affine scheme,
the invertible sheaves on $\mathcal{J}$ correspond to projective $R$-modules
of rank 1. We have already seen the projective modules $\mathcal{P}_{1}$ and
$\mathcal{Q}_{1}$ used to define $\pi$ and $\tilde{\pi}$ above. Since the map
$\pi\colon\mathcal{J}\to\mathbb{P}^{1}$ is an $\mathbb{A}^{1}$-weak
equivalence and the Picard group functor is homotopy invariant, the induced
map on Picard groups is an isomorphism
$\pi^{*}\colon\mathrm{Pic}(\mathbb{P}^{1})\to\mathrm{Pic}(\mathcal{J})$. Since
$\pi^{*}(\mathcal{O}(1))=\mathcal{P}_{1}$ and
$\mathrm{Pic}(\mathcal{J})\cong\mathbb{Z}$, it follows that $\mathcal{P}_{1}$
generates the Picard group of $\mathcal{J}$. For future reference, we state
this as a lemma.

######  Lemma 18.

The Picard group of $\mathcal{J}$ is isomorphic to $\mathbb{Z}$ and
$\mathcal{P}_{1}$ is a generator.

Furthermore, $\mathcal{Q}_{1}=-\mathcal{P}_{1}$ in $\mathrm{Pic}(\mathcal{J})$
as the following proposition shows.

######  Proposition 19.

There is an isomorphism $\mathcal{P}_{1}\otimes\mathcal{Q}_{1}\cong R$.

###### Proof.

The $R$-module $\mathcal{P}_{1}\otimes\mathcal{Q}_{1}$ is generated by

$\left\\{\begin{bmatrix}x\\\ z\end{bmatrix}\otimes\begin{bmatrix}x\\\
y\end{bmatrix},\begin{bmatrix}x\\\ z\end{bmatrix}\otimes\begin{bmatrix}z\\\
w\end{bmatrix},\begin{bmatrix}y\\\ w\end{bmatrix}\otimes\begin{bmatrix}x\\\
y\end{bmatrix},\begin{bmatrix}y\\\ w\end{bmatrix}\otimes\begin{bmatrix}z\\\
w\end{bmatrix}\right\\}.$

Consider the module homomorphism $\mu\colon R^{2}\otimes R^{2}\longrightarrow
R^{2}$ induced by component-wise multiplication $\mu\left(\begin{bmatrix}a\\\
b\end{bmatrix}\otimes\begin{bmatrix}c\\\
d\end{bmatrix}\right)=\begin{bmatrix}ac\\\ bd\end{bmatrix}$. We restrict $\mu$
to $\mathcal{P}_{1}\otimes\mathcal{Q}_{1}$ and observe that the image of
$\mathcal{P}_{1}\otimes\mathcal{Q}_{1}$ under $\mu$ is the submodule
$\left\langle\begin{bmatrix}x\\\ w\end{bmatrix}\right\rangle\subseteq R^{2}$
(use $x+w=1$). This is a free $R$-module of rank 1. As
$\mu\colon\mathcal{P}_{1}\otimes\mathcal{Q}_{1}\to\left\langle\begin{bmatrix}x\\\
w\end{bmatrix}\right\rangle$ is surjective, it follows that it is locally an
isomorphism at all maximal ideals $\mathfrak{m}\subseteq R$. Hence the map
$\mu$ itself restricted to $\mathcal{P}_{1}\otimes\mathcal{Q}_{1}$ is an
isomorphism onto its image. ∎

We would like to understand the tensor powers of $\mathcal{P}_{1}$ and
$\mathcal{Q}_{1}$. It turns out they can be described by the following
$R$-modules that are constructed by open patching data. We will investigate
these modules before returning to the claim that they are indeed tensor powers
of $\mathcal{P}_{1}$ and $\mathcal{Q}_{1}$.

######  Definition 20.

As in [25, 2.5 on page 14], we define projective $R$-modules of rank $1$ by
open patching data for the open sets $D(x)$, $D(w)$ as $x+w=1$. For $n\geq 0$,
we define the $R$-modules

$\displaystyle\mathcal{OP}_{n}$
$\displaystyle=\left\\{\left(f_{x},f_{w}\right)\in R[x^{-1}]\times
R[w^{-1}]\,|\,f_{w}=\left(\frac{z}{x}\right)^{n}f_{x}\right\\},$
$\displaystyle\mathcal{OQ}_{n}$
$\displaystyle=\left\\{\left(f_{x},f_{w}\right)\in R[x^{-1}]\times
R[w^{-1}]\,|\,f_{w}=\left(\frac{y}{x}\right)^{n}f_{x}\right\\}.$

######  Definition 21.

We denote by $\mathcal{P}_{n}$ and $\mathcal{Q}_{n}$ the submodules of $R^{2}$
generated as follows

$\displaystyle\mathcal{P}_{n}$
$\displaystyle=\left\langle\begin{bmatrix}x^{n}\\\
z^{n}\end{bmatrix},\begin{bmatrix}x^{n-1}y\\\
z^{n-1}w\end{bmatrix},\ldots,\begin{bmatrix}y^{n}\\\
w^{n}\end{bmatrix}\right\rangle,~{}\text{and}~{}\mathcal{Q}_{n}=\left\langle\begin{bmatrix}x^{n}\\\
y^{n}\end{bmatrix},\begin{bmatrix}x^{n-1}z\\\
y^{n-1}w\end{bmatrix},\ldots,\begin{bmatrix}z^{n}\\\
w^{n}\end{bmatrix}\right\rangle.$

The following lemma is useful for simplifying proofs. It shows that what we
prove about $\mathcal{P}_{n}$ by symmetry holds for $\mathcal{Q}_{n}$.

######  Lemma 22.

We define the involutive automorphism $\tau\colon R\rightarrow R$ by

$\displaystyle\tau:x$ $\displaystyle\mapsto x$ $\displaystyle y$
$\displaystyle\mapsto z$ $\displaystyle z$ $\displaystyle\mapsto y$
$\displaystyle w$ $\displaystyle\mapsto w.$

Pulling back along $\tau$ gives $R$-module isomorphisms
$\tau^{*}\mathcal{P}_{n}\cong\mathcal{Q}_{n}$ and
$\tau^{*}\mathcal{Q}_{n}\cong\mathcal{P}_{n}$.

###### Proof.

To more easily distinguish between them, we give the domain and codomain of
$\tau$ different names and write $\tau\colon R\to R^{\prime}$. Pulling back
the $R^{\prime}$-module $\mathcal{P}_{n}$, we get the $R$-module
$\tau^{*}\mathcal{P}_{n}$, where the multiplication is defined by
$r\cdot_{R}p=\tau(r)\cdot_{R^{\prime}}p$. The map
$\tau^{*}\mathcal{P}_{n}\rightarrow\mathcal{Q}_{n}$ is defined on basis
elements by

$\begin{bmatrix}x^{n-i}y^{i}\\\
z^{n-i}w^{i}\end{bmatrix}\mapsto\begin{bmatrix}x^{n-i}z^{i}\\\
y^{n-i}w^{i}\end{bmatrix}.$

It is easily checked that $f$ is bijective and $R$-linear and hence an
$R$-module isomorphism.

To see that $\tau^{*}\mathcal{Q}_{n}\cong\mathcal{P}_{n}$, we pull back the
isomorphism $\mathcal{Q}_{n}\cong\tau^{*}\mathcal{P}_{n}$, which we just
proved, along $\tau$ on both sides. Since $\tau\circ\tau=\operatorname{id}$,
this simplifies to
$\tau^{*}\mathcal{Q}_{n}\cong\tau^{*}\tau^{*}\mathcal{P}_{n}=\mathcal{P}_{n}$.
∎

######  Proposition 23.

The $R$-modules $\mathcal{P}_{n}$ and $\mathcal{Q}_{n}$ are also generated in
the following way

$\displaystyle\mathcal{P}_{n}=\left\langle\begin{bmatrix}x^{n}\\\
z^{n}\end{bmatrix},\begin{bmatrix}y^{n}\\\
w^{n}\end{bmatrix}\right\rangle~{}~{}\text{and}~{}~{}\mathcal{Q}_{n}=\left\langle\begin{bmatrix}x^{n}\\\
y^{n}\end{bmatrix},\begin{bmatrix}z^{n}\\\ w^{n}\end{bmatrix}\right\rangle.$

###### Proof.

We only prove the claim for $\mathcal{P}_{n}$, as the proof for
$\mathcal{Q}_{n}$ is analogous by lemma 22. Containment in one direction is
clear by definition of $\mathcal{P}_{n}$. Now fix $n$ and pick a number $0\leq
i\leq n$. We then have

$\begin{bmatrix}x^{n-i}y^{i}\\\
z^{n-i}w^{i}\end{bmatrix}=\left(x+w\right)^{n}\begin{bmatrix}x^{n-i}y^{i}\\\
z^{n-i}w^{i}\end{bmatrix}=\sum_{d=0}^{n}\binom{n}{d}x^{n-d}w^{d}\begin{bmatrix}x^{n-i}y^{i}\\\
z^{n-i}w^{i}\end{bmatrix}.$

For each $d$, one of the following hold, which completes the proof

$\begin{array}[]{ll}x^{n-d}w^{d}\begin{bmatrix}x^{n-i}y^{i}\\\
z^{n-i}w^{i}\end{bmatrix}=x^{n-i-d}y^{i}w^{d}\begin{bmatrix}x^{n}\\\
z^{n}\end{bmatrix}&\mbox{if }i+d\leq n,\\\ &\\\
x^{n-d}w^{d}\begin{bmatrix}x^{n-i}y^{i}\\\
z^{n-i}w^{i}\end{bmatrix}=x^{n-d}z^{n-i}w^{d+i-n}\begin{bmatrix}y^{n}\\\
w^{n}\end{bmatrix}&\mbox{if }i+d>n.\end{array}$

∎

######  Proposition 24.

There are isomorphisms $\mathcal{P}_{n}\cong\mathcal{OP}_{n}$ and
$\mathcal{Q}_{n}\cong\mathcal{OQ}_{n}$ for all natural numbers $n$. Hence
$\mathcal{P}_{n}$ and $\mathcal{Q}_{n}$ are algebraic line bundles over $R$
(that is, finitely generated $R$-modules of constant rank 1) and determine
invertible sheaves on $\mathcal{J}$.

###### Proof.

The canonical projections $p_{1}\colon\mathcal{P}_{n}[x^{-1}]\to R[x^{-1}]$
and $p_{2}\colon\mathcal{P}_{n}[w^{-1}]\to R[w^{-1}]$ are isomorphisms. For
$\begin{bmatrix}f\\\ g\end{bmatrix}\in\mathcal{P}_{n}[x^{-1},w^{-1}]$, one
checks that $(z/x)^{n}f=g$, which is the same open patching data for
$\mathcal{OP}_{n}$. The evident map $\mathcal{P}_{n}\to\mathcal{OP}_{n}$ is
thus an isomorphism. In the same way, the evident map
$\mathcal{Q}_{n}\to\mathcal{OQ}_{n}$ is an isomorphism. ∎

######  Remark 25.

Proposition 24 shows us how to interpret an element

$s_{0}=a_{0}\begin{bmatrix}x^{n}\\\
z^{n}\end{bmatrix}+a_{1}\begin{bmatrix}y^{n}\\\
w^{n}\end{bmatrix},~{}\text{with}~{}a_{0},a_{1}\in R,$

which is a global section of the invertible sheaf associated to
$\mathcal{P}_{n}$. Namely, the global section $s_{0}$ restricted to $D(x)$ is
described by $a_{0}x^{n}+a_{1}y^{n}$, while on $D(w)$ the section is
$a_{0}z^{n}+a_{1}w^{n}$. On the overlap, the two sections agree when compared
using the appropriate transition functions.

Note that proposition 24 also gives a canonical form for the elements
$(f_{x},f_{w})\in\mathcal{OP}_{n}$. Any such element is described as

$\begin{bmatrix}f_{x}\\\ f_{w}\end{bmatrix}=a_{0}\begin{bmatrix}x^{n}\\\
z^{n}\end{bmatrix}+a_{1}\begin{bmatrix}y^{n}\\\
w^{n}\end{bmatrix},~{}\text{for some}~{}a_{0},a_{1}\in R.$

The algebraic line bundles $\mathcal{P}_{n}$ and $\mathcal{Q}_{n}$ may also be
described as the image of an idempotent $(2\times 2)$-matrix of rank 1. For
$n\geq 1$, let $A=\sum_{k=0}^{n-1}\binom{2n-1}{k}x^{n-1-k}w^{k}$ and
$B=\sum_{k=n}^{2n-1}\binom{2n-1}{k}x^{2n-1-k}w^{k-n}$. Then
$x^{n}A+w^{n}B=(x+w)^{2n-1}=1$. Define

(26) $M_{n}=\begin{pmatrix}x^{n}A&y^{n}B\\\ z^{n}A&w^{n}B\end{pmatrix}\text{
and }M_{n}^{\prime}=\begin{pmatrix}x^{n}A&z^{n}B\\\
y^{n}A&w^{n}B\end{pmatrix}.$

######  Proposition 27.

For every $n\geq 1$, the matrices $M_{n}$ and $M_{n}^{\prime}$ are idempotent
of rank 1 and have image $\mathcal{P}_{n}$ and $\mathcal{Q}_{n}$ respectively.

###### Proof.

It is straightforward to verify that $M_{n}$ is idempotent using the relation
$1=x^{n}A+w^{n}B$ and that $\operatorname{Im}(M_{n})\subset\mathcal{P}_{n}$.
Note that

$x^{n}\begin{bmatrix}x^{n}A\\\
z^{n}A\end{bmatrix}+z^{n}\begin{bmatrix}y^{n}B\\\
w^{n}B\end{bmatrix}=(x^{n}A+w^{n}B)\begin{bmatrix}x^{n}\\\
z^{n}\end{bmatrix}=\begin{bmatrix}x^{n}\\\ z^{n}\end{bmatrix}$

and similarly,

$y^{n}\begin{bmatrix}x^{n}A\\\
z^{n}A\end{bmatrix}+w^{n}\begin{bmatrix}y^{n}B\\\
w^{n}B\end{bmatrix}=(x^{n}A+w^{n}B)\begin{bmatrix}y^{n}\\\
w^{n}\end{bmatrix}=\begin{bmatrix}y^{n}\\\ w^{n}\end{bmatrix}.$

So $\mathcal{P}_{n}\subset\operatorname{Im}(M_{n})$, and the image is equal to
$\mathcal{P}_{n}$. The argument for $M_{n}^{\prime}$ and $\mathcal{Q}_{n}$ is
similar. ∎

######  Proposition 28.

The morphisms $\mu\colon\mathcal{P}_{1}^{\otimes n}\to\mathcal{P}_{n}$ and
$\mu^{\prime}\colon\mathcal{Q}_{1}^{\otimes n}\to\mathcal{Q}_{n}$ obtained
from component-wise multiplication are isomorphisms.

###### Proof.

Consider the $R$-module map $\mu\colon R^{2}\otimes R^{2}\to R^{2}$ induced by
component-wise multiplication. By the description of the generators of the
modules $\mathcal{P}_{n}$, it is clear that $\mu$ restricts to a map
$\mu\colon\mathcal{P}_{n}\otimes\mathcal{P}_{1}\to\mathcal{P}_{n+1}$ and this
map is surjective. As both $\mathcal{P}_{n}\otimes\mathcal{P}_{1}$ and
$\mathcal{P}_{n+1}$ are algebraic line bundles, the map $\mu$ is surjective
locally at every maximal ideal $\mathfrak{m}\subseteq R$ and hence an
isomorphism. Thus
$\mu\colon\mathcal{P}_{n}\otimes\mathcal{P}_{1}\to\mathcal{P}_{n+1}$ is itself
an isomorphism by [8, Proposition 3.9]. The claim now follows by induction. ∎

We now have a complete description of the isomorphism
$\mathbb{Z}\cong\mathrm{Pic}(\mathcal{J})$.

######  Proposition 29.

Under the isomorphism
$\mathrm{Pic}(\mathcal{J})\cong\mathrm{Pic}(\mathbb{P}^{1})\cong\mathbb{Z}$
arising from the $\mathbb{A}^{1}$-homotopy equivalence
$\pi:\mathcal{J}\to\mathbb{P}^{1}$, the modules $\mathcal{P}_{n}$ and
$\mathcal{Q}_{n}$ correspond to $n$ and $-n$, respectively, while the trivial
invertible sheaf $\mathcal{O}$ corresponds to $0$.

###### Proof.

By lemma 18, $\mathrm{Pic}(\mathcal{J})=\mathbb{Z}$, and $\mathcal{P}_{1}$
generates the Picard group. By proposition 19, the inverse of
$\mathcal{P}_{1}$ is$\mathcal{Q}_{1}$. By proposition 28, the modules
$\mathcal{P}_{n}$ and $\mathcal{Q}_{n}$ correspond to $n$ and $-n$ in the
Picard group. ∎

###  2.3 Pointed morphisms $\mathbb{P}^{1}\to\mathbb{P}^{1}$ and
$\mathcal{J}\to\mathbb{P}^{1}$

We will now study morphisms to $\mathbb{P}^{1}$ in more detail. By [17,
Theorem II.7.1], for a smooth $k$-scheme $X$, the data needed to give a
morphism $f:X\to\mathbb{P}^{1}$ are an invertible sheaf $\mathcal{L}$ over $X$
and the choice of two global sections $s_{0},s_{1}\in\Gamma(X,\mathcal{L})$
that generate the invertible sheaf $\mathcal{L}$. That is, at every point
$p\in X$, the stalks of the sections $(s_{0})_{p}$ and $(s_{1})_{p}$ generate
the local ring $\mathcal{L}_{p}$. We then write $[s_{0},s_{1}]$ for the map
$X\to\mathbb{P}^{1}$ given by the data above, where we usually omit the
invertible sheaf $\mathcal{L}$ from the notation.

The scheme $\mathbb{P}^{1}$ is pointed at $\infty=[1:0]$. A pointed map
$f:\mathbb{P}^{1}\to\mathbb{P}^{1}$ by definition is a map satisfying
$f(\infty)=\infty$. A pointed morphism
$f\colon\mathbb{P}^{1}\to\mathbb{P}^{1}$ given by the invertible sheaf
$\mathcal{O}(n)$ on $\mathbb{P}^{1}$ with two generating sections
$\sigma_{0},\sigma_{1}\in k[x_{0},x_{1}]_{n}$ has the following special form
by work of Cazanave [13].

######  Proposition 30.

[13, Proposition 2.3] A pointed $k$-scheme morphism
$f\colon\mathbb{P}^{1}\to\mathbb{P}^{1}$ corresponds uniquely to the data of a
natural number $n$ and a choice of two polynomials,
$A=\sum_{i=0}^{n}a_{i}X^{i}$ and $B=\sum_{i=0}^{n-1}b_{i}X^{i}$ in $k[X]$ for
which $a_{n}=1$ and the resultant $\mathrm{res}_{n,n}(A,B)$ is non-zero. The
integer $n$ is called the degree of $f$ and is denoted $\deg(f)$; the scalar
$\mathrm{res}(f)=\mathrm{res}_{n,n}(A,B)\in k^{\times}$ is called the
resultant of $f$.

One easily translates from the morphism given by $n$, $A$, and $B$ in
proposition 30 to the morphism given by the invertible sheaf $\mathcal{O}(n)$
and the choice of global sections
$\sigma_{0}=\sum_{i=0}^{n}a_{i}x_{0}^{i}x_{1}^{n-i}$ and
$\sigma_{1}=\sum_{i=0}^{n}b_{i}x_{0}^{i}x_{1}^{n-i}$ where we understand
$b_{n}=0$. The resultant condition guarantees that these global sections
generate $\mathcal{O}(n)$. The condition $a_{n}=1$ is a normalizing condition
to give a bijective correspondence between morphisms and the data $n$, $A$,
and $B$. We will find it more convenient to use the data
$[\sigma_{0},\sigma_{1}]\colon\mathbb{P}^{1}\to\mathbb{P}^{1}$ and
$\mathcal{O}(n)$ to describe a pointed map in what follows.

######  Proposition 31.

Consider a pointed map
$[\sigma_{0},\sigma_{1}]\colon\mathbb{P}^{1}\to\mathbb{P}^{1}$ with invertible
sheaf $\mathcal{O}(n)$, $\sigma_{0}=\sum_{i=0}^{n}a_{i}x_{0}^{i}x_{1}^{n-i}$
and $\sigma_{1}=\sum_{i=0}^{n}b_{i}x_{0}^{i}x_{1}^{n-i}$. The composition
$[\sigma_{0},\sigma_{1}]\circ\pi$ is the map
$[s_{0},s_{1}]\colon\mathcal{J}\to\mathbb{P}^{1}$ with invertible sheaf
$\mathcal{P}_{n}$ and global sections

(32) $\displaystyle s_{0}=\sum_{i=0}^{n}a_{i}\begin{bmatrix}x^{i}y^{n-i}\\\
z^{i}w^{n-i}\end{bmatrix}~{}\text{and}~{}s_{1}$
$\displaystyle=\sum_{i=0}^{n}b_{i}\begin{bmatrix}x^{n}y^{n-i}\\\
z^{i}w^{n-i}\end{bmatrix}.$

###### Proof.

This is a straightforward calculation. The resultant condition ensures that
the sections $s_{0}$ and $s_{1}$ generate $\mathcal{P}_{n}$. ∎

######  Remark 33.

We note that the difference between a general map
$[s_{0},s_{1}]\colon\mathcal{J}\to\mathbb{P}^{1}$ with invertible sheaf
$\mathcal{P}_{n}$ and a map $\mathcal{J}\to\mathbb{P}^{1}$ which factors as
$f\circ\pi$ with $f\colon\mathbb{P}^{1}\to\mathbb{P}^{1}$ is that the
coefficients $a_{i}$ and $b_{i}$ in the expressions of the sections in
equation (32) are in the field $k$ when the map factors, but in general the
coefficients are in $R$.

For later purposes, we extend the definition of the resultant to homogeneous
polynomials in two variables. We collect some further facts about resultants
in appendix B.

######  Definition 34.

Let $R[\alpha,\beta]$ be a polynomial ring over $R$ in two variables and let
$R[\alpha,\beta]_{n}$ denote the subgroup of homogeneous polynomials of degree
$n$. For every $n\geq 1$, the map $\sigma\colon
R[\alpha,\beta]_{n}\to\mathcal{P}_{n}$, defined by
$\sigma(\alpha^{i}\beta^{n-i})=\begin{bmatrix}x^{i}y^{n-i}\\\
z^{i}w^{n-i}\end{bmatrix}$ for all $0\leq i\leq n$ is a surjective morphism of
$R$-modules. For a pair of homogeneous polynomials $(S_{0},S_{1})$, we write
$\sigma((S_{0},S_{1}))$ for the pair $(\sigma(S_{0}),\sigma(S_{1}))$ in
$(\mathcal{P}_{n})^{2}$ by slight abuse of notation. For a pair of homogeneous
polynomials
$(S_{0},S_{1})=(a_{n}\alpha^{n}+\ldots+a_{0}\beta^{n},b_{n}\alpha^{n}+\ldots+b_{0}\beta^{n})\in(R[\alpha,\beta]_{n})^{2}$,
we divide by $\beta^{n}$ to form univariate polynomials
$(\widehat{S}_{0},\widehat{S}_{1})=(\sum_{i=0}^{n}a_{i}\mathscr{X}^{i},\sum_{i=0}^{n}b_{i}\mathscr{X}^{i})$
in $\mathscr{X}:=\alpha/\beta$. We define the resultant
$\mathrm{res}(S_{0},S_{1})$ of the pair $(S_{0},S_{1})$ to be

$\mathrm{res}(S_{0},S_{1}):=\mathrm{res}\left(\sum_{i=0}^{n}a_{i}\mathscr{X}^{i},\sum_{i=0}^{n}b_{i}\mathscr{X}^{i}\right)=\det\left(\mathrm{Syl}(\widehat{S}_{0},\widehat{S}_{1})\right),$

where $\mathrm{Syl}(\widehat{S}_{0},\widehat{S}_{1})$ is the Sylvester matrix
of the pair of polynomials $(\widehat{S}_{0},\widehat{S}_{1})$ in
$R[\mathscr{X}]$.

######  Lemma 35.

Consider a pair of homogeneous polynomials of degree $n\geq 1$ in
$R[\alpha,\beta]_{n}$, denoted by
$S_{0}=\sum_{i=0}^{n}a_{i}\alpha^{n-i}\beta^{i}$ and
$S_{1}=\sum_{i=0}^{n}b_{i}\alpha^{n-i}\beta^{i}$. If
$\mathrm{res}(S_{0},S_{1})$ is a unit, then the pair of sections
$(\sigma(S_{0}),\sigma(S_{1}))$ generates $\mathcal{P}_{n}$ and defines a
morphism $[\sigma(S_{0}),\sigma(S_{1})]:\mathcal{J}\to\mathbb{P}^{1}$.

###### Proof.

It suffices to show that $(s_{0},s_{1})$ generate $\mathcal{P}_{n}$ on the
open patches $D(x)$ and $D(w)$. On $D(x)$, this requires showing that the
ideal $(a_{n}x^{n}+\ldots a_{0}y^{n},b_{n}x^{n}+\ldots+b_{0}y^{n})$ is the
unit ideal in $R[x^{-1}]$. The ideal is the same as the ideal $(a_{n}+\ldots
a_{0}\frac{y^{n}}{x^{n}},b_{n}+\ldots+b_{0}\frac{y^{n}}{x^{n}})$ which
corresponds to a pair of polynomials of degree $n$ in the variable
$\frac{y}{x}$. By lemma 148, this pair of polynomials has unit resultant.
Since the resultant is a unit, there exists $U_{x},V_{x}\in R[x^{-1}]$ by
lemma 147 giving a Bézout relation $U_{x}(a_{n}+\ldots
a_{0}\frac{y^{n}}{x^{n}})+V_{x}(b_{n}+\ldots+b_{0}\frac{y^{n}}{x^{n}})=1$. On
$D(w)$ we need to prove that the ideal $(a_{n}z^{n}+\ldots
a_{0}w^{n},b_{n}z^{n}+\ldots+b_{0}w^{n})$ is the unit ideal in $R[w^{-1}]$.
The ideal is equal to the ideal $(a_{n}\frac{z^{n}}{w^{n}}+\ldots
a_{0},b_{n}\frac{z^{n}}{w^{n}}+\ldots+b_{0})$. This pair of polynomials has
unit resultant by assumption. By lemma 147, unit resultant implies existence
of a Bézout relation $U_{w}(a_{n}\frac{z^{n}}{w^{n}}+\ldots
a_{0})+V_{x}(b_{n}\frac{z^{n}}{w^{n}}+\ldots+b_{0})=1$ in $R[w^{-1}]$. This
proves that $[s_{0},s_{1}]$ defines a morphism $\mathcal{J}\to\mathbb{P}^{1}$.
∎

######  Remark 36.

Let $[\sigma_{0},\sigma_{1}]\colon\mathbb{P}^{1}\to\mathbb{P}^{1}$ be a
pointed map given by invertible sheaf $\mathcal{O}(n)$, and sections
$\sigma_{0}=\sum_{i=0}^{n}a_{i}x_{0}^{i}x_{1}^{n-i}$ and
$\sigma_{1}=\sum_{i=0}^{n}b_{i}x_{0}^{i}x_{1}^{n-i}$. Then the pair of
homogeneous polynomials $S_{0}=\sum_{i=0}^{n}a_{i}\alpha^{n-i}\beta^{i}$ and
$S_{1}=\sum_{i=0}^{n}b_{i}\alpha^{n-i}\beta^{i}$ in $R[\alpha,\beta]_{n}$ has
unit resultant. By lemma 35, the pair $(\sigma(S_{0}),\sigma(S_{1}))$ defines
a morphism $[\sigma(S_{0}),\sigma(S_{1})]:\mathcal{J}\to\mathbb{P}^{1}$. This
morphism is equal to the morphism $[s_{0},s_{1}]$, constructed from
$[\sigma_{0},\sigma_{1}]$ in proposition 31.

######  Remark 37.

We note that there exist pairs of polynomials
$(S_{0},S_{1}),(S^{\prime}_{0},S^{\prime}_{1})$ such that
$\sigma((S_{0},S_{1}))=\sigma((S^{\prime}_{0},S^{\prime}_{1}))$, while
$\mathrm{res}(S_{0},S_{1})\neq\mathrm{res}(S^{\prime}_{0},S^{\prime}_{1})$. An
example is given by $(x\alpha+z\beta,\beta)$ and $(\alpha,\beta)$. We
calculate

$\sigma((x\alpha+z\beta,\beta))=\left(x\begin{bmatrix}x\\\
z\end{bmatrix}+z\begin{bmatrix}y\\\ w\end{bmatrix},\begin{bmatrix}y\\\
w\end{bmatrix}\right)=\left(\begin{bmatrix}x\\\
z\end{bmatrix},\begin{bmatrix}y\\\
w\end{bmatrix}\right)=\sigma((\alpha,\beta)).$

Their resultants are

$\mathrm{res}(x\alpha+z\beta,\beta)=x\neq 1=\mathrm{res}(\alpha,\beta).$

We now look at the data needed to describe a general morphism
$\mathcal{J}\to\mathbb{P}^{1}$ and also see what condition pointedness
imposes.

######  Construction 38.

A morphism $f\colon\mathcal{J}\to\mathbb{P}^{1}$ is determined by the
following data: an invertible sheaf $\mathcal{L}$ on $\mathcal{J}$ and the
choice of two global sections $s_{0},s_{1}\in\Gamma(\mathcal{J},\mathcal{L})$
that generate $\mathcal{L}$ [17, Theorem II.7.1]. Since
$\mathrm{Pic}(\mathcal{J})\cong\mathbb{Z}$, the invertible sheaf $\mathcal{L}$
may be chosen to be either $\mathcal{P}_{n}$, $\mathcal{Q}_{n}$, or
$\mathcal{O}$. We call the integer corresponding to the class of $\mathcal{L}$
in $\mathrm{Pic}(\mathcal{J})\cong\mathbb{Z}$ the degree of $f$.

We will now make the assignment $(\mathcal{L},s_{0},s_{1})\mapsto f$ explicit.
We will study only the case of $\mathcal{P}_{n}$, as $\mathcal{Q}_{n}$ is
handled in the same way by proposition 23, and the case of $\mathcal{O}$ is
discussed later in section 2.5.

In the case of $\mathcal{P}_{n}$, two generating sections
$s_{0},s_{1}\in\Gamma(\mathcal{J},\mathcal{P}_{n})$ may be chosen to be of the
form

$s_{0}=a_{0}\begin{bmatrix}x^{n}\\\
z^{n}\end{bmatrix}+a_{1}\begin{bmatrix}y^{n}\\\ w^{n}\end{bmatrix}\qquad
s_{1}=b_{0}\begin{bmatrix}x^{n}\\\
z^{n}\end{bmatrix}+b_{1}\begin{bmatrix}y^{n}\\\ w^{n}\end{bmatrix}.$

Define
$D(s_{i})=\\{p\in\mathcal{J}\,|\,(s_{i})_{p}\not\in\mathfrak{m}_{p}(\mathcal{P}_{n})_{p}\\}$.
The map $[s_{0},s_{1}]$ is defined on the open set $D(s_{i})$ to map into
$U_{i}=\left\\{[x_{0},x_{1}]\,|\,x_{i}\neq 0\right\\}$. Here
$U_{0}\cong\operatorname{Spec}(k[y_{1}])$ and
$U_{1}\cong\operatorname{Spec}(k[y_{0}])$, where $y_{0}=x_{0}/x_{1}$ and
$y_{1}=x_{1}/x_{0}$. The map $D(s_{i})\to U_{i}$ is given by the corresponding
map of rings $k[y_{j}]\to\mathcal{P}_{n}[s_{i}^{-1}]$ determined by
$y_{j}\mapsto s_{j}/s_{i}$. This requires some explanation due to the
description of the sheaf $\mathcal{P}_{n}$. Proposition 24 shows that the
components of each section $s_{i}$ describe the section on the open sets
$D(x)$ and $D(w)$. Hence there are four cases to consider to get a description
of the map in concrete terms of affine open sets.

  1. 1.

$D(x)\cap D(s_{0})$: Here $s_{0}$ is described by $a_{0}x^{n}+a_{1}y^{n}$ in
the ring $R[x^{-1}]$ and $s_{1}$ is given by $b_{0}x^{n}+b_{1}y^{n}$ in the
ring $R[x^{-1}]$. Hence on $D(s_{0})$ the corresponding ring map $k[y_{1}]\to
R[x^{-1},(a_{0}x^{n}+a_{1}y^{n})^{-1}]$ is given by
$y_{1}\mapsto\frac{b_{0}x^{n}+b_{1}y^{n}}{a_{0}x^{n}+a_{1}y^{n}}$.

  2. 2.

$D(x)\cap D(s_{1})$: Here $s_{0}$ is described by $a_{0}x^{n}+a_{1}y^{n}$ in
the ring $R[x^{-1}]$ and $s_{1}$ is given by $b_{0}x^{n}+b_{1}y^{n}$ in the
ring $R[x^{-1}]$. Hence on $D(s_{1})$ the corresponding ring map $k[y_{0}]\to
R[x^{-1},(b_{0}x^{n}+b_{1}y^{n})^{-1}]$ is given by
$y_{0}\mapsto\frac{a_{0}x^{n}+a_{1}y^{n}}{b_{0}x^{n}+b_{1}y^{n}}$.

  3. 3.

$D(w)\cap D(s_{0})$: Here $s_{0}$ is described by $a_{0}z^{n}+a_{1}w^{n}$ in
the ring $R[w^{-1}]$ and $s_{1}$ is given by $b_{0}z^{n}+b_{1}w^{n}$ in the
ring $R[w^{-1}]$. Hence on $D(s_{0})$ the corresponding ring map $k[y_{1}]\to
R[w^{-1},(a_{0}z^{n}+a_{1}w^{n})^{-1}]$ is given by
$y_{1}\mapsto\frac{b_{0}z^{n}+b_{1}w^{n}}{a_{0}z^{n}+a_{1}w^{n}}$.

  4. 4.

$D(w)\cap D(s_{1})$: Here $s_{0}$ is described by $a_{0}z^{n}+a_{1}w^{n}$ in
the ring $R[w^{-1}]$ and $s_{1}$ is given by $b_{0}z^{n}+b_{1}w^{n}$ in the
ring $R[w^{-1}]$. Hence on $D(s_{1})$ the corresponding ring map $k[y_{0}]\to
R[w^{-1},(b_{0}z^{n}+b_{1}w^{n})^{-1}]$ is given by
$y_{0}\mapsto\frac{a_{0}z^{n}+a_{1}w^{n}}{b_{0}z^{n}+b_{1}w^{n}}$.

This information can be consolidated into the two maps $D(x)\to\mathbb{P}^{1}$
and $D(w)\to\mathbb{P}^{1}$ given in terms of the pair of sections
$[a_{0}x^{n}+a_{1}y^{n},b_{0}x^{n}+b_{1}y^{n}]$ and
$[a_{0}z^{n}+a_{1}w^{n},b_{0}z^{n}+b_{1}w^{n}]$ respectively. Written in this
form, we see that a map $\mathcal{J}\to\mathbb{P}^{1}$ given by the invertible
sheaf $\mathcal{P}_{n}$ with two generating sections $s_{0},s_{1}$ should be
interpreted as giving a map to $\mathbb{P}^{1}$ on the open sets $D(x)$ and
$D(w)$ according to the first component of the sections $s_{0},s_{1}$ on
$D(x)$ and according to the second component of the sections $s_{0},s_{1}$ on
$D(w)$.

######  Remark 39.

Recall that $\mathcal{J}$ is pointed at $\mathbf{j}=(x-1,y,z,w)$ and
$\mathbb{P}^{1}$ is pointed at $\infty=[1:0]$. A map
$f:\mathcal{J}\to\mathbb{P}^{1}$ is pointed if $f(\mathbf{j})=\infty$. If
$f=[s_{0},s_{1}]$ with line bundle $\mathcal{L}$ and generating sections
$s_{0}$, $s_{1}$, pointedness can be verified by checking that the stalk
$s_{1}(\mathbf{j})$ satisfies $s_{1}(\mathbf{j})=0$ in the local ring
$\mathcal{L}_{\mathbf{j}}$. For us, it suffices to work on $D(x)$ where our
line bundles are trivial, and verify that modulo $\mathbf{j}$ the section
$s_{1}$ vanishes.

We give a concrete criterion for checking pointedness of a map
$f:\mathcal{J}\to\mathbb{P}^{1}$ with line bundle $\mathcal{P}_{n}$. The case
of $\mathcal{Q}_{n}$ is similar.

######  Proposition 40.

A map $[s_{0},s_{1}]:\mathcal{J}\to\mathbb{P}^{1}$ with invertible sheaf
$\mathcal{P}_{n}$ and generating sections

$s_{0}=a_{0}\begin{bmatrix}x^{n}\\\
z^{n}\end{bmatrix}+a_{1}\begin{bmatrix}y^{n}\\\ w^{n}\end{bmatrix},\qquad
s_{1}=b_{0}\begin{bmatrix}x^{n}\\\
z^{n}\end{bmatrix}+b_{1}\begin{bmatrix}y^{n}\\\ w^{n}\end{bmatrix}$

is pointed if and only if $b_{0}\in\mathbf{j}$, i.e., $b_{0}(\mathbf{j})=0$.

###### Proof.

First, assume the map $[s_{0},s_{1}]$ is pointed. Construction 38 gives a
description of the map in local coordinates. Note that for $\mathbf{j}$ to map
to $\infty\in U_{0}$, it is necessary that $\mathbf{j}\in D(s_{0})$. Since
$\mathbf{j}\in D(x)\cap D(s_{0})$, the map in local coordinates is obtained by
taking $\operatorname{Spec}$ of the ring map $\sigma\colon k[y_{1}]\to
R[x^{-1},(a_{0}x^{n}+a_{1}y^{n})^{-1}]$ which is given by
$\sigma(y_{1})=\frac{b_{0}x^{n}+b_{1}y^{n}}{a_{0}x^{n}+a_{1}y^{n}}$. The
condition for pointedness is then that the preimage of $\mathbf{j}$ under
$\sigma$ is the maximal ideal $(y_{1})$. This is equivalent to the condition
that $y_{1}$ maps into the ideal $(x-1,y,z,w)\subseteq
R[x^{-1},(a_{0}x^{n}+a_{1}y^{n})^{-1}]$. By the definition of $\sigma$, the
requirement is that
$\frac{b_{0}x^{n}+b_{1}y^{n}}{a_{0}x^{n}+a_{1}y^{n}}\in(x-1,y,z,w)$, which is
equivalent to $b_{0}x^{n}+b_{1}y^{n}\in(x-1,y,z,w)$. Since $y\in(x-1,y,z,w)$
and $x$ is invertible, this condition is met when $b_{0}\in\mathbf{j}$. Thus
when $[s_{0},s_{1}]$ is pointed, $\mathbf{j}\in D(s_{0})$ and
$b_{0}\in\mathbf{j}$.

Now assume that $b_{0}\in\mathbf{j}$. This implies $\mathbf{j}\in D(s_{0})$,
since the sections $s_{0},s_{1}$ generate $(\mathcal{P}_{n})_{\mathbf{j}}$,
and $b_{0}\in\mathbf{j}$ implies $s_{1}(\mathbf{j})=0$. Here we can use the
same construction above, since $\mathbf{j}\in D(x)\cap D(s_{0})$. The algebra
above shows that when $b_{0}\in\mathbf{j}$ the preimage of $\mathbf{j}$ under
$\sigma$ is $(y_{1})$, i.e., the map $[s_{0},s_{1}]$ is pointed. ∎

######  Proposition 41.

Let $f=[s_{0},s_{1}]\colon\mathcal{J}\to\mathbb{P}^{1}$ be a pointed map with
invertible sheaf $\mathcal{P}_{n}$. If $\alpha=s_{0}(\mathbf{j})$, then
$\left[\frac{s_{0}}{\alpha},\frac{s_{1}}{\alpha}\right]\colon\mathcal{J}\to\mathbb{P}^{1}$
is a pointed map that is equal to $f$. Thus any pointed map with line bundle
$\mathcal{P}_{n}$ may be represented by a pair of generating global sections
$[s_{0},s_{1}]$ where $s_{0}(\mathbf{j})=1$ and $s_{1}(\mathbf{j})=0$.

###### Proof.

Proposition 40 has established that $s_{1}(\mathbf{j})=0$ and
$s_{0}(\mathbf{j})=\alpha$ is a unit. We verify that the maps $[s_{0},s_{1}]$
and $\left[\frac{s_{0}}{\alpha},\frac{s_{1}}{\alpha}\right]$ are equal in
local coordinates by construction 38, where the constants $\frac{1}{\alpha}$
cancel out in every local coordinate chart. ∎

######  Proposition 42.

Let $s_{0}$ and $s_{1}$ be the following sections in $\mathcal{P}_{n}$

$s_{0}=a_{0}\begin{bmatrix}x^{n}\\\
z^{n}\end{bmatrix}+a_{1}\begin{bmatrix}y^{n}\\\ w^{n}\end{bmatrix},\quad
s_{1}=b_{0}\begin{bmatrix}x^{n}\\\
z^{n}\end{bmatrix}+b_{1}\begin{bmatrix}y^{n}\\\ w^{n}\end{bmatrix}.$

The sections $s_{0},s_{1}$ generate $\mathcal{P}_{n}$ if and only if there
exist $U_{x},V_{x},U_{w},V_{w}\in R$ such that

$U_{x}(x^{n}a_{0}+y^{n}a_{1})+V_{x}(x^{n}b_{0}+y^{n}b_{1})+U_{w}(z^{n}a_{0}+w^{n}a_{1})+V_{w}(z^{n}b_{0}+w^{n}b_{1})=1.$

Employing similar notation, sections $s_{0},s_{1}$ generate $\mathcal{Q}_{n}$
if and only if there exist $U_{x},V_{x},U_{w},V_{w}\in R$ such that

$U_{x}(x^{n}a_{0}+z^{n}a_{1})+V_{x}(x^{n}b_{0}+z^{n}b_{1})+U_{w}(y^{n}a_{0}+w^{n}a_{1})+V_{w}(y^{n}b_{0}+w^{n}b_{1})=1.$

###### Proof.

By lemma 22, it suffices to prove this for $\mathcal{P}_{n}$. Assume
$s_{0},s_{1}$ generate $\mathcal{P}_{n}$. Then there exist $U,V$ such that
$Us_{0}+Vs_{1}=\begin{bmatrix}x^{n}\\\ z^{n}\end{bmatrix}$. The first
component of this identity gives

$\left(U(x^{n}a_{0}+y^{n}a_{1})+V(x^{n}b_{0}+y^{n}b_{1})\right)=x^{n}.$

Similarly, there exist $U^{\prime},V^{\prime}$ such that
$U^{\prime}s_{0}+V^{\prime}s_{1}=\begin{bmatrix}y^{n}\\\ w^{n}\end{bmatrix}$.
This gives the relation

$\left(U^{\prime}(z^{n}a_{0}+w^{n}a_{1})+V^{\prime}(z^{n}b_{0}+w^{n}b_{1})\right)=w^{n}.$

Since the equation $1=(x+w)^{2n}$ demonstrates that $1$ can be expressed as a
linear combination of $x^{n}$ and $w^{n}$, we are done. Now we assume that
there exist elements $U_{x},V_{x},U_{w},V_{w}\in R$ such that

$U_{x}(x^{n}a_{0}+y^{n}a_{1})+V_{x}(x^{n}b_{0}+y^{n}b_{1})+U_{w}(z^{n}a_{0}+w^{n}a_{1})+V_{w}(z^{n}b_{0}+w^{n}b_{1})=1.$

A straight forward computation yields
$(U_{x}x^{n}+U_{w}z^{n})s_{0}+(V_{x}x^{n}+V_{w}z^{n})s_{1}=\begin{bmatrix}x^{n}\\\
z^{n}\end{bmatrix}$, and
$(U_{x}y^{n}+U_{w}w^{n})s_{0}+(V_{x}y^{n}+V_{w}w^{n})s_{1}=\begin{bmatrix}y^{n}\\\
w^{n}\end{bmatrix}.$ These two elements generate $\mathcal{P}_{n}$, thus
$[s_{0},s_{1}]$ do as well. ∎

For brevity, we write maps $\mathcal{J}\to\mathbb{P}^{1}$ of nonzero degree
using the following notation.

######  Definition 43.

Let $n$ be a positive integer. We write $(a_{0},a_{1};b_{0},b_{1})_{n}$,
respectively $(a_{0},a_{1};b_{0},b_{1})_{-n}$, for the map
$\mathcal{J}\to\mathbb{P}^{1}$ with invertible sheaf $\mathcal{P}_{n}$,
respectively $\mathcal{Q}_{n}$, and generating sections

$s_{0}=a_{0}\begin{bmatrix}x^{n}\\\
z^{n}\end{bmatrix}+a_{1}\begin{bmatrix}y^{n}\\\ w^{n}\end{bmatrix},\quad
s_{1}=b_{0}\begin{bmatrix}x^{n}\\\
z^{n}\end{bmatrix}+b_{1}\begin{bmatrix}y^{n}\\\
w^{n}\end{bmatrix}~{}\text{with}~{}a_{0},a_{1},b_{0},b_{1}\in R,$

respectively with invertible sheaf $\mathcal{Q}_{n}$ and generating sections

$s_{0}=a_{0}\begin{bmatrix}x^{n}\\\
y^{n}\end{bmatrix}+a_{1}\begin{bmatrix}z^{n}\\\ w^{n}\end{bmatrix},\quad
s_{1}=b_{0}\begin{bmatrix}x^{n}\\\
y^{n}\end{bmatrix}+b_{1}\begin{bmatrix}z^{n}\\\
w^{n}\end{bmatrix}~{}\text{with}~{}a_{0},a_{1},b_{0},b_{1}\in R.$

###  2.4 The pointed naive homotopy relation

Naive homotopy theory for schemes is a generalization of the homotopy theory
of rings in classical algebra, see [16] for a definition. Naive homotopy
classes of maps between schemes do not generally have the good properties one
expects from a homotopy theory, but in our case, thanks to work of Asok,
Hoyois, and Wendt, it is sufficiently good [6].

######  Definition 44.

Let $X$ and $Y$ be smooth schemes over the field $k$. For $a\in k$, let
$i_{a}=\operatorname{id}_{X}\times a$ be the map obtained by taking the
Cartesian product of $\operatorname{id}_{X}$ and the inclusion map
$a\colon\operatorname{Spec}k\to\mathbb{A}^{1}$ given by the ring map $k[t]\to
k$ sending $t$ to $a$. An elementary homotopy between two morphisms $f\colon
X\to Y$ and $g\colon X\to Y$ is given by a morphism $H(T)\colon
X\times\mathbb{A}^{1}\to Y$ satisfying $H(0)=f$ and $H(1)=g$, i.e.,
$H(0)=H(T)\circ i_{0}$ and $H(1)=H(T)\circ i_{1}$. We say that $f$ and $g$ are
elementarily homotopic and write $f\sim g$.

The relation of morphisms being elementarily homotopic is symmetric and
reflexive, but not transitive. To obtain an equivalence relation on the set of
morphisms $\mathrm{Sm}_{k}(X,Y)$, we take the transitive closure of $\sim$.
That is, we define two morphisms $f,g\in\mathrm{Sm}_{k}(X,Y)$ to be naively
homotopic if there is a finite sequence of elementary homotopies
$H_{i}(T)\colon X\times\mathbb{A}^{1}\to Y$, for $0\leq i\leq n$ with
$H_{0}(0)=f$, $H_{n}(1)=g$, and for all $0\leq i<n$ $H_{i}(1)=H_{i-1}(0)$. We
write $f\simeq g$ in this case. The relation $\simeq$ is now an equivalence
relation on $\mathrm{Sm}_{k}(X,Y)$, so we can study the set of naive homotopy
classes of morphisms from $X$ to $Y$.

For our constructions, we will work with pointed maps and pointed naive
homotopies.

######  Definition 45.

If $X$ and $Y$ are smooth $k$-schemes, pointed at $k$-points $x$ and $y$
respectively, we say that an elementary homotopy $H(T)\colon
X\times\mathbb{A}^{1}\to Y$ is pointed if the generic point of
$\\{x\\}\times\mathbb{A}^{1}$ maps to $y$. Said another way, the points $x$
and $y$ correspond to morphisms $x\colon\operatorname{Spec}(k)\to X$ and
$y\colon\operatorname{Spec}(k)\to Y$, and we require that
$H(T)\circ(x\times\operatorname{id}_{\mathbb{A}^{1}})=y\circ p_{1}$ where
$p_{1}\colon\operatorname{Spec}(k)\times\mathbb{A}^{1}\to\operatorname{Spec}(k)$
is the projection onto the first factor.

As in the unpointed case, the relation on the set of pointed morphisms
$\mathrm{Sm}_{k}(X,Y)_{*}$ given by pointed elementary homotopies is not an
equivalence relation. We say that pointed morphisms
$f,g\in\mathrm{Sm}_{k}(X,Y)_{*}$ are naively homotopic, and write $f\simeq g$,
if there is a chain of pointed elementary homotopies from $f$ to $g$. The
naive homotopy relation is an equivalence relation on pointed morphisms. We
write $[X,Y]^{\mathrm{N}}=\mathrm{Sm}_{k}(X,Y)_{*}/\simeq$ for the set of
equivalence classes.

For us, the most important case is when $X=\mathcal{J}=\operatorname{Spec}(R)$
with basepoint $\mathbf{j}=(x-1,y,z,w)$. This ideal extends to
$\mathbf{j}^{\prime}=(x-1,y,z,w)\subseteq R[T]$. The condition that a homotopy
$H(T)\colon\mathcal{J}\times\mathbb{A}^{1}\to Y$ be pointed is simply that
$H(T)(\mathbf{j}^{\prime})=y$, where $y$ is the basepoint of $Y$.

###  2.5 Morphisms $\mathcal{J}\to\mathbb{A}^{2}\setminus\\{0\\}$

We write
$\deg\colon[\mathcal{J},\mathbb{P}^{1}]^{\mathrm{N}}\to\mathrm{Pic}(\mathcal{J})\cong\mathbb{Z}$
for the map that sends a map $f$ to $f^{*}\mathcal{O}(1)$. Our choices thus
far set $\deg(\pi)=1$ and $\deg(\tilde{\pi})=-1$. Write
$[\mathcal{J},\mathbb{P}^{1}]^{\mathrm{N}}_{n}$ for the set of naive homotopy
classes of maps $\mathcal{J}\to\mathbb{P}^{1}$ with degree $n$. Our goal for
this section is to describe the maps $\mathcal{J}\to\mathbb{P}^{1}$ of degree
$0$. We consider the scheme
$\mathbb{A}^{2}\setminus\\{0\\}=\operatorname{Spec}\left(k[s,t]\right)\setminus\\{(s,t)\\}$
to be pointed at $(s-1,t)$ and write
$\eta\colon\mathbb{A}^{2}\setminus\\{0\\}\to\mathbb{P}^{1}$ for the Hopf map
given by the trivial algebraic line bundle
$\mathcal{O}_{\mathbb{A}^{2}\setminus\\{0\\}}$ with the choice of sections
$\eta_{0}=s$, $\eta_{1}=t$. Let $\mathrm{SL}_{2}$ denote the affine scheme
$\operatorname{Spec}k[a,b,c,d]/(ad-bc-1)$ pointed at the ideal
$(a-1,b,c,d-1)$. Intuitively, this is the scheme of $(2\times 2)$-matrices
with determinant 1, pointed at the identity matrix.

######  Proposition 46.

Consider a map $f\colon\mathcal{J}\to\mathbb{P}^{1}$. Then we have $\deg(f)=0$
if and only if $f$ factors through the Hopf map
$\eta\colon\mathbb{A}^{2}\setminus\\{0\\}\to\mathbb{P}^{1}$.

###### Proof.

If $f^{*}\mathcal{O}(1)=\mathcal{O}_{R}$, the map is given by sections
$s_{0},s_{1}\in R$ that generate $\mathcal{O}_{R}$, i.e., $(s_{0},s_{1})=R$.
Hence there are $U,V\in R$ for which $s_{0}U+s_{1}V=1$. This is exactly the
data needed to construct a scheme morphism
$\mathcal{J}\to\mathbb{A}^{2}\setminus\\{0\\}$, as such a morphism is given by
a morphism $\mathcal{J}\to\mathbb{A}^{2}$ that does not have $\\{0\\}$ in the
image. Since $\mathrm{Pic}(\mathbb{A}^{2}\setminus\\{0\\})=0$, it follows that
any map that factors as
$\mathcal{J}\to\mathbb{A}^{2}\setminus\\{0\\}\xrightarrow{\eta}\mathbb{P}^{1}$
has degree 0. ∎

######  Corollary 47.

Let $f\colon\mathcal{J}\to\mathbb{P}^{1}$ be a pointed map of degree 0. Then
there exists a unique pointed map
$f^{\prime}\colon\mathcal{J}\to\mathbb{A}^{2}\setminus\\{0\\}$ such that
$f=f^{\prime}\circ\eta$.

###### Proof.

Let $(s_{0},s_{1})\colon\mathcal{J}\to\mathbb{A}^{2}\setminus\\{0\\}$ be a
factorization of $f$ through the Hopf map. Note that
$\alpha=s_{0}(\mathbf{j})$ need not be $1$, although $\alpha$ is a unit of
$k$. Instead, the map
$f^{\prime}=\left(\frac{1}{\alpha}s_{0},\frac{1}{\alpha}s_{1}\right)$ is
pointed and satisfies $f=f^{\prime}\circ\eta$.

To show uniqueness, let
$(s_{0}^{\prime},s_{1}^{\prime})\colon\mathcal{J}\to\mathbb{A}^{2}\setminus\\{0\\}$
be another pointed map that factors $f$ through $\eta$. That is, we assume
$[s_{0},s_{1}]=[s_{0}^{\prime},s_{1}^{\prime}]$. Note that in this case,
$D(s_{0})=D(s_{0}^{\prime})=D(f^{*}x_{0})$ and
$D(s_{1})=D(s_{1}^{\prime})=D(f^{*}x_{1})$. Working locally in
$D(s_{1})=D(s_{1}^{\prime})$, we have
$s_{0}/s_{1}=s_{0}^{\prime}/s_{1}^{\prime}$ in $R[s_{1}^{-1}]$ by
construction. We may write $s_{0}^{\prime}=c_{0}s_{0}$ for
$c_{0}=s_{1}^{\prime}/s_{1}\in R[s_{1}^{-1}]$. Similarly, in
$D(s_{0})=D(s_{0}^{\prime})$, we obtain $s_{1}^{\prime}=c_{1}s_{1}$ for
$c_{1}=s_{0}^{\prime}/s_{0}$. In the intersection $D(s_{0})\cap D(s_{1})$ we
have $s_{0}^{\prime}/s_{1}^{\prime}=s_{0}/s_{1}$, which implies
$s_{0}^{\prime}/s_{0}=s_{1}^{\prime}/s_{1}$. This is exactly the equation
$c_{0}=c_{1}$. The elements $c_{1}\in R[s_{0}^{-1}]$ and $c_{0}\in
R[s_{1}^{-1}]$ therefore glue together to an element $c\in R$. Hence $c$
satisfies $s_{0}^{\prime}=cs_{0}$ and $s_{1}^{\prime}=cs_{1}$. Observe that
$c(s_{0}u^{\prime}+s_{1}v^{\prime})=1$, that is, $c\in R^{\times}=k^{\times}$.
The pointedness assumption forces $c(\mathbf{j})=1$, hence, $c=1$ with which
we conclude $(s_{0},s_{1})=(s_{0}^{\prime},s_{1}^{\prime})$. ∎

######  Remark 48.

The previous proposition says, in other words, that a map
$\mathcal{J}\to\mathbb{A}^{2}\setminus\\{0\\}$ is equivalent to a unimodular
row $(A,B)$ of length two in $R$. Furthermore, a pointed map
$\mathcal{J}\to\mathbb{A}^{2}\setminus\\{0\\}$ is equivalent to a unimodular
row $(A,B)$ of length two in $R$ that also satisfies $A(\mathbf{j})=1$ and
$B(\mathbf{j})=0$.

Pointed elementary homotopies between maps of degree $0$ can also be lifted to
a pointed elementary homotopy of maps
$\mathcal{J}\to\mathbb{A}^{2}\setminus\\{0\\}$.

######  Proposition 49.

Let
$H(T)=[s_{0}(T),s_{1}(T)]\colon\mathcal{J}\times\mathbb{A}^{1}\to\mathbb{P}^{1}$
be a pointed elementary homotopy between maps $H(0)$ and $H(1)$ which have
degree 0. There is a pointed elementary homotopy
$H^{\prime}(T)\colon\mathcal{J}\times\mathbb{A}^{1}\to\mathbb{A}^{2}\setminus\\{0\\}$
between the lifts $H^{\prime}(0)$ and $H^{\prime}(1)$.

###### Proof.

Since $H(0)$ and $H(1)$ have degree 0, the homotopy $H(T)$ is degree 0 too,
that is, the line bundle it determines is the trivial one
$\mathcal{O}_{\mathcal{J}\times\mathbb{A}^{1}}$. We can use the two generating
global sections $s_{0}(T),s_{1}(T)\in R[T]$ to build a map
$(s_{0}(T),s_{1}(T))\colon\mathcal{J}\times\mathbb{A}^{1}\to\mathbb{A}^{2}\setminus\\{0\\}$.
Note that since $s_{0}(T)$ and $s_{1}(T)$ generate $R[T]$, there are
$u(T),v(T)\in R[T]$ for which $s_{0}(T)u(T)+s_{1}(T)v(T)=1$. Since $H(T)$ is
pointed, $s_{1}(T)(\mathbf{j}^{\prime})=0$ in $R[T]/\mathbf{j}^{\prime}$. This
implies that $s_{0}(T)(\mathbf{j}^{\prime})u(T)(\mathbf{j}^{\prime})=1$ in
$R[T]/\mathbf{j}^{\prime}$. The ring $R[T]/\mathbf{j}^{\prime}$ is easily seen
to be isomorphic to $k[T]$. Hence $\alpha=s_{0}(T)(\mathbf{j}^{\prime})$ is a
unit of $k[T]$, and the units of $k[T]$ are exactly the units of $k$. With
this, the map
$\left(\frac{1}{\alpha}s_{0}(T),\frac{1}{\alpha}s_{1}(T)\right)\colon\mathcal{J}\times\mathbb{A}^{1}\to\mathbb{A}^{2}\setminus\\{0\\}$
is a pointed homotopy between $H^{\prime}(0)$ and $H^{\prime}(1)$. ∎

Let $(A,B)$ be a unimodular row in $R$. That is, there exist $U,V\in R$ for
which $AU+BV=1$. Thus the data of a map
$\mathcal{J}\to\mathbb{A}^{2}\setminus\\{0\\}$ can be used to produce a matrix
$\begin{pmatrix}A&-V\\\ B&U\end{pmatrix}\in\mathrm{SL}_{2}(R)$, in other
words, a map $\mathcal{J}\to\mathrm{SL}_{2}$.

######  Lemma 50.

A pointed map $(A,B)\colon\mathcal{J}\to\mathbb{A}^{2}\setminus\\{0\\}$ can be
lifted to a pointed map $\begin{pmatrix}A&-V\\\
B&U\end{pmatrix}\colon\mathcal{J}\to\mathrm{SL}_{2}$.

###### Proof.

Let $\begin{pmatrix}A&-V_{1}\\\ B&U_{1}\end{pmatrix}$ be an arbitrary lift of
$(A,B)$. Note that $A(\mathbf{j})=1$, $B(\mathbf{j})=0$, and
$U_{1}(\mathbf{j})=1$, but $V_{1}(\mathbf{j})=v$ for some $v\in k$. For any
$d\in R$, we can construct a different lift by setting $U_{2}=U_{1}+Bd$ and
$V_{2}=V_{1}-Ad$. Set $d=v$. Then $U_{2}(\mathbf{j})=1$, and
$V_{2}(\mathbf{j})=0$, so $\begin{pmatrix}A&-V_{2}\\\ B&U_{2}\end{pmatrix}$ is
pointed. ∎

######  Construction 51.

The pointed lift is not unique in general. For example, the unimodular row
$(1,0)$ lifts to the pointed maps

$\begin{pmatrix}1&0\\\
0&1\end{pmatrix}\quad\text{or}\quad\begin{pmatrix}1&y\\\ 0&1\end{pmatrix}.$

Generally, any two pointed lifts of a pointed unimodular row $(A,B)$ are
naively homotopic. Let

$\tilde{f}_{i}=\begin{pmatrix}A&-V_{i}\\\ B&U_{i}\end{pmatrix}$

for $i\in\\{1,2\\}$ be two pointed lifts of $(A,B)$. A pointed elementary
homotopy between $\tilde{f}_{1}$ and $\tilde{f}_{2}$ is given by

(52) $\tilde{f}_{t}=\begin{pmatrix}A&-(TV_{1}+(1-T)V_{2})\\\
B&TU_{1}+(1-T)U_{2}\end{pmatrix}.$

######  Proposition 53.

Every pointed elementary homotopy

$H(T)=(s_{0}(T),s_{1}(T))\colon\mathcal{J}\times\mathbb{A}^{1}\to\mathbb{A}^{2}\setminus\\{0\\}$

can be lifted to a pointed elementary homotopy

$\begin{pmatrix}s_{0}(T)&-V(T)\\\
s_{1}(T)&U(T)\end{pmatrix}\colon\mathcal{J}\times\mathbb{A}^{1}\to\mathrm{SL}_{2}.$

###### Proof.

Recall $\mathbf{j}^{\prime}=(x-1,y,z,w)$ must map to the basepoint for the
homotopy $H(T)$ to be pointed. The sections $s_{0}(T)$ and $s_{1}(T)$ generate
the unit ideal, hence there exist $u(T)$ and $v(T)$ in $R[T]$ for which
$s_{0}(T)u(T)+s_{1}(T)v(T)=1$. The pointedness assumption gives the relation
among ideals $(s_{0}(T)-1,s_{1}(T))\subseteq\mathbf{j}^{\prime}\subseteq
R[T]$.

With these data, we construct the matrix

$\begin{pmatrix}s_{0}(T)&-v(T)\\\
s_{1}(T)&u(T)\end{pmatrix}\in\mathrm{SL}_{2}(R[T]).$

This matrix determines a map
$\mathcal{J}\times\mathbb{A}^{1}\to\mathrm{SL}_{2}$ that lifts the unimodular
row $(s_{0}(T),s_{1}(T))$ we started with. This homotopy need not be a pointed
homotopy. However, for any choice of $d(T)\in R[T]$, the matrix

$\begin{pmatrix}s_{0}(T)&-v(T)+s_{0}(T)d(T)\\\
s_{1}(T)&u(T)+s_{1}(T)d(T)\end{pmatrix}$

is also a lift of $(s_{0}(T),s_{1}(T))$. We will now show that, for
$d(T)=v(T)$, the map this matrix determines is a pointed homotopy. Write
$u_{2}(T)=u(T)+s_{1}(T)v(T)$ and $v_{2}(T)=v(T)-s_{0}(T)v(T)$. Our assumption
that $(s_{0}(T),s_{1}(T))$ is pointed gives us
$(s_{0}(T)-1,s_{1}(T))\subseteq\mathbf{j}^{\prime}\subseteq R[T]$. We must
show that $(v_{2}(T),u_{2}(T)-1)\subseteq\mathbf{j}^{\prime}$ too. Since
$(s_{0}(T)-1)\in\mathbf{j}^{\prime}$, we have
$v_{2}(T)=-v(T)(s_{0}(T)-1)\in\mathbf{j}^{\prime}$. Observe that
$u_{2}(T)-1\in\mathbf{j}^{\prime}$ if $u(T)-1\in\mathbf{j}^{\prime}$ since
$s_{1}(T)\in\mathbf{j}^{\prime}$. Since $s_{0}(T)u(T)+s_{1}(T)v(T)=1$, it
follows that $s_{0}(T)u(T)-1\in\mathbf{j}^{\prime}$. This can be rewritten as
$s_{0}(T)u(T)-1=(s_{0}(T)-1)u(T)+u(T)-1$. Since
$s_{0}(T)-1\in\mathbf{j}^{\prime}$ it follows that
$u(T)-1\in\mathbf{j}^{\prime}$ too. ∎

######  Definition 54.

Let $\phi\colon\mathrm{SL}_{2}\to\mathbb{A}^{2}\setminus\\{0\\}$ be the
morphism determined by the ring map $f\colon k[s,t]\to k[a,b,c,d]/(ad-bc-1)$
given by $f(s)=a$ and $f(t)=c$. Intuitively, this is the morphism that
extracts the first column from a matrix in $\mathrm{SL}_{2}$. As given, this
map has codomain $\mathbb{A}^{2}$, but it is clear from the relation $ad-bc=1$
that $\phi$ maps into $\mathbb{A}^{2}\setminus\\{0\\}$.

######  Proposition 55.

The maps $\phi\colon\mathrm{SL}_{2}\to\mathbb{A}^{2}\setminus\\{0\\}$ and
$\eta\colon\mathbb{A}^{2}\setminus\\{0\\}\to\mathbb{P}^{1}$ induce bijections
of naive homotopy classes of pointed maps

$[\mathcal{J},\mathrm{SL}_{2}]^{\mathrm{N}}\xrightarrow{\phi_{*}}[\mathcal{J},\mathbb{A}^{2}\setminus\\{0\\}]^{\mathrm{N}}\xrightarrow{\eta_{*}}[\mathcal{J},\mathbb{P}^{1}]^{\mathrm{N}}_{0}.$

###### Proof.

The map $\phi_{*}$ is surjective by lemma 50. Construction 51 shows that
$\phi_{*}$ is injective. Corollary 47 shows that $\eta_{*}$ is bijective on
the level of pointed morphisms. This shows that $\eta_{*}$ is surjective. To
show that $\eta_{*}$ is injective, it suffices to show that a pointed
elementary homotopy
$H(T)\colon\mathcal{J}\times\mathbb{A}^{1}\to\mathbb{P}^{1}$ between degree 0
maps lifts to a pointed elementary homotopy
$H^{\prime}(T)\colon\mathcal{J}\times\mathbb{A}^{1}\to\mathbb{A}^{2}\setminus\\{0\\}$,
which is done in proposition 49. ∎

##  3 Operations on naive homotopy classes of morphisms

###  3.1 Group structure on maps of degree $0$

We may now define a binary operation on naive homotopy classes of morphisms
$\mathcal{J}\rightarrow\mathbb{A}^{2}\setminus\\{0\\}$. This is analogous to
Cazanave’s naive sum of pointed rational functions. The group structure is
obtained by lifting maps
$f,g\colon\mathcal{J}\to\mathbb{A}^{2}\setminus\\{0\\}$ to
$\tilde{f},\tilde{g}\colon\mathcal{J}\to\mathrm{SL}_{2}$, multiplying the two
resulting maps using the group structure on $\mathrm{SL}_{2}$, then mapping
back down to $\mathbb{A}^{2}\setminus\\{0\\}$ via the map
$\phi\colon\mathrm{SL}_{2}\to\mathbb{A}^{2}\setminus\\{0\\}$.

######  Definition 56.

A morphism $f\colon\mathcal{J}\to\mathrm{SL}_{2}$ is equivalent to the data of
a matrix $M\in\mathrm{SL}_{2}(R)$. A matrix $M\in\mathrm{SL}_{2}(R)$
corresponds to a pointed morphism if upon evaluation at $\mathbf{j}$, the
resulting matrix is the identity matrix. The set of pointed maps corresponds
to a subgroup of $\mathrm{SL}_{2}(R)$. The operation of matrix multiplication
respects the naive homotopy relation for pointed maps and therefore defines a
group operation on $[\mathcal{J},\mathrm{SL}_{2}]^{\mathrm{N}}$, the set of
pointed naive homotopy classes of morphisms. It suffices to prove the
following proposition, given that the naive homotopy relation is the
transitive closure of pointed elementary homotopies.

######  Proposition 57.

Let $M(T)\in\mathrm{SL}_{2}(R[T])$ be a pointed elementary homotopy between
the matrices $M_{0}=M(0)\in\mathrm{SL}_{2}(R)$ and
$M_{1}=M(1)\in\mathrm{SL}_{2}(R)$ corresponding to pointed morphisms. Let
$N(T)\in\mathrm{SL}_{2}(R[T])$ be another elementary homotopy where similar
notation is employed. The pointed morphisms corresponding to $M_{0}\cdot
N_{0}$ and $M_{1}\cdot N_{1}$ are elementarily homotopic.

###### Proof.

All that needs to be verified is that the morphism corresponding to the matrix
product $M(T)\cdot N(T)$ is pointed. Since both $M(T)$ and $N(T)$ are pointed,
evaluation at $\mathbf{j}^{\prime}$ gives the identity matrix in
$\mathrm{SL}_{2}(k[T])$. It’s clear then that the product $M(T)\cdot N(T)$
will evaluate to the identity matrix at $\mathbf{j}^{\prime}$ too. ∎

######  Definition 58.

Consider two pointed naive homotopy classes
$[(A_{i},B_{i})\colon\mathcal{J}\to\mathbb{A}^{2}\setminus\\{0\\}]$ for
$i=1,2$ represented by the unimodular rows $(A_{i},B_{i})\in R^{2}$. Pick
completions of the unimodular rows to matrices corresponding to pointed maps,
as guaranteed by lemma 50:

$\begin{pmatrix}A_{1}&-V_{1}\\\
B_{1}&U_{1}\end{pmatrix},\begin{pmatrix}A_{2}&-V_{2}\\\
B_{2}&U_{2}\end{pmatrix}\in\mathrm{SL}_{2}(R).$

We define $[(A_{1},B_{1})]\oplus[(A_{2},B_{2})]$ to be the naive homotopy
class $[(A_{3},B_{3})]$ where $(A_{3},B_{3})$ is the unimodular row obtained
from the matrix product

$\begin{pmatrix}A_{3}&-V_{3}\\\
B_{3}&U_{3}\end{pmatrix}=\begin{pmatrix}A_{1}&-V_{1}\\\
B_{1}&U_{1}\end{pmatrix}\cdot\begin{pmatrix}A_{2}&-V_{2}\\\
B_{2}&U_{2}\end{pmatrix}.$

######  Proposition 59.

The operation $\oplus$ of definition 58 is well-defined and gives the set
$[\mathcal{J},\mathbb{A}^{2}\setminus\\{0\\}]^{\mathrm{N}}$ the structure of a
group.

###### Proof.

We first show that the operation does not depend on the particular completion
to a matrix in $\mathrm{SL}_{2}(R)$. Let $M_{1}$ and $M_{1}^{\prime}$ be two
pointed completions of $(A_{1},B_{1})$, and similarly let $M_{2}$ and
$M_{2}^{\prime}$ be two pointed completions of $(A_{2},B_{2})$. There are two
representatives for the product $[(A_{1},B_{1})]\oplus[(A_{2},B_{2})]$ from
these choices. They are $(A_{3},B_{3})$, taken from the first column of
$M_{1}\cdot M_{2}$ and $(A_{3}^{\prime},B_{3}^{\prime})$, the first column of
$M_{1}^{\prime}\cdot M_{2}^{\prime}$. Any two completions of a unimodular row
to a matrix in $\mathrm{SL}_{2}(R)$ are homotopic by construction 51, hence
$[M_{1}]=[M_{1}^{\prime}]$ and $[M_{2}]=[M_{2}^{\prime}]$. By the proof of
proposition 53 there is an elementary pointed naive homotopy between
$M_{1}\cdot M_{2}$ and $M_{1}^{\prime}\cdot M_{2}^{\prime}$. Extracting the
first column of this homotopy gives a homotopy between the resulting
unimodular rows defining the resulting map.

We now show that the operation does not depend on the representative of the
naive homotopy class chosen. Let $(A_{1}(T),B_{1}(T))$ and
$(A_{2}(T),B_{2}(T))$ be pointed elementary homotopies. These can be completed
to matrices $M_{1}(T)\in\mathrm{SL}_{2}(R[T])$ and
$M_{2}(T)\in\mathrm{SL}_{2}(R[T])$ by proposition 53. The first column of the
product $M_{1}(T)\cdot M_{2}(T)$ provides the homotopy between the two
possible representations of the product. We conclude that the operation is
well-defined on the set
$[\mathcal{J},\mathbb{A}^{2}\setminus\\{0\\}]^{\mathrm{N}}$ of pointed naive
homotopy classes.

The identity for the operation is given by the unimodular row
$(1,0)\colon\mathcal{J}\to\mathbb{A}^{2}\setminus\\{0\\}$. Associativity of
$\oplus$ follows from the associativity of matrix multiplication. Finally, let
$(A,B)\colon\mathcal{J}\to\mathbb{A}^{2}\setminus\\{0\\}$ be given by the
unimodular row $(A,B)\in R^{2}$ and complete it to a matrix
$\begin{pmatrix}A&-V\\\ B&U\end{pmatrix}\in\mathrm{SL}_{2}(R)$ giving a
pointed map. The inverse of this matrix in $\mathrm{SL}_{2}(R)$ is the matrix
$\begin{pmatrix}U&V\\\ -B&A\end{pmatrix}$, and the first column of this matrix
represents the inverse of $(A,B)$ in
$[\mathcal{J},\mathbb{A}^{2}\setminus\\{0\\}]^{\mathrm{N}}$ for $\oplus$. That
is, $-[(A,B)]=[(U,-B)]$. ∎

######  Lemma 60.

The map $\phi\colon\mathrm{SL}_{2}\to\mathbb{A}^{2}\setminus\\{0\\}$ induces
an isomorphism of groups

$\phi_{*}\colon[\mathcal{J},\mathrm{SL}_{2}]^{\mathrm{N}}\xrightarrow{\cong}[\mathcal{J},\mathbb{A}^{2}\setminus\\{0\\}]^{\mathrm{N}}.$

###### Proof.

The map $\phi_{*}$ is a group homomorphism by our definition of $\oplus$ on
$[\mathcal{J},\mathbb{A}^{2}\setminus\\{0\\}]^{\mathrm{N}}$ in terms of matrix
multiplication. We have shown in proposition 55 that $\phi_{*}$ is bijective,
hence the result. ∎

For the next result, we recall that, as described before equation (1), the
cogroup structure of $\mathbb{P}^{1}\cong S^{1}\wedge\mathbb{G}_{m}$ in the
pointed $\mathbb{A}^{1}$-homotopy category endows
$[\mathbb{P}^{1},X]^{\mathbb{A}^{1}}$ with a group operation for any motivic
space $X$. We refer to this structure as the _conventional group structure_.

######  Definition 61.

We let
$\xi_{0}\colon[\mathcal{J},\mathbb{A}^{2}\setminus\\{0\\}]^{\mathrm{N}}\to[\mathbb{P}^{1},\mathbb{A}^{2}\setminus\\{0\\}]^{\mathbb{A}^{1}}$
denote the composition of the natural map
$\nu\colon[\mathcal{J},\mathbb{A}^{2}\setminus\\{0\\}]^{\mathrm{N}}\to[\mathcal{J},\mathbb{A}^{2}\setminus\\{0\\}]^{\mathbb{A}^{1}}$
and the bijection
$(\pi_{\mathbb{A}^{1}}^{*})^{-1}\colon[\mathcal{J},\mathbb{A}^{2}\setminus\\{0\\}]^{\mathbb{A}^{1}}\to[\mathbb{P}^{1},\mathbb{A}^{2}\setminus\\{0\\}]^{\mathbb{A}^{1}}$
that is given by the inverse of the bijection $\pi_{\mathbb{A}^{1}}^{*}$. We
note that $\nu$ is a bijection by proposition 143 since
$\mathbb{A}^{2}\setminus\\{0\\}$ is $\mathbb{A}^{1}$-naive.

######  Theorem 62.

The map $\xi_{0}$ is an isomorphism of groups between the group
$[\mathcal{J},\mathbb{A}^{2}\setminus\\{0\\}]^{\mathrm{N}}$ with operation
$\oplus$ and the group
$[\mathbb{P}^{1},\mathbb{A}^{2}\setminus\\{0\\}]^{\mathbb{A}^{1}}$ with the
conventional group operation.

###### Proof.

Let
$\xi^{\prime}_{0}\colon[\mathcal{J},\mathrm{SL}_{2}]^{\mathrm{N}}\to[\mathbb{P}^{1},\mathrm{SL}_{2}]^{\mathbb{A}^{1}}$
denote the composition of the canonical map
$[\mathcal{J},\mathrm{SL}_{2}]^{\mathrm{N}}\to[\mathcal{J},\mathrm{SL}_{2}]^{\mathbb{A}^{1}}$
and the bijection
$[\mathcal{J},\mathrm{SL}_{2}]^{\mathbb{A}^{1}}\to[\mathbb{P}^{1},\mathrm{SL}_{2}]^{\mathbb{A}^{1}}$
which is given by the inverse of the bijection $\pi_{\mathbb{A}^{1}}^{*}$.
Since both groups $[\mathbb{P}^{1},\mathrm{SL}_{2}]^{\mathbb{A}^{1}}$ and
$[\mathbb{P}^{1},\mathbb{A}^{2}\setminus\\{0\\}]^{\mathbb{A}^{1}}$ inherit
their operation from the cogroup structure of $\mathbb{P}^{1}$, the
$\mathbb{A}^{1}$-weak equivalence
$\mathrm{SL}_{2}\xrightarrow{\phi}\mathbb{A}^{2}\setminus\\{0\\}$ induces an
isomorphism of groups
$\phi_{*}\colon[\mathbb{P}^{1},\mathrm{SL}_{2}]^{\mathbb{A}^{1}}\to[\mathbb{P}^{1},\mathbb{A}^{2}\setminus\\{0\\}]^{\mathbb{A}^{1}}$.
By lemma 60 the map
$\phi_{*}\colon[\mathcal{J},\mathrm{SL}_{2}]^{\mathrm{N}}\to[\mathcal{J},\mathbb{A}^{2}\setminus\\{0\\}]^{\mathrm{N}}$
is a group isomorphism. We then have the following commutative diagram.

Hence, in order to establish that $\xi_{0}$ is a group isomorphism, it
suffices to show that $\xi^{\prime}_{0}$ is a group isomorphism. Since
$\mathcal{J}$ is affine and $\mathrm{SL}_{2}$ is $\mathbb{A}^{1}$-naive by [6,
Theorem 4.2.1], we know that $\xi^{\prime}_{0}$ is a bijection by proposition
143. Hence it suffices to show that $\xi^{\prime}_{0}$ is a group
homomorphism.

Again, because $\mathrm{SL}_{2}$ is $\mathbb{A}^{1}$-naive, the canonical map
$[\mathcal{J},\mathrm{SL}_{2}]^{\mathrm{N}}\to[\mathcal{J},\mathrm{SL}_{2}]^{\mathbb{A}^{1}}$
is a bijection by proposition 143. This bijection is a group isomorphism
because the operation on both sets is defined using the same construction,
that is, the sum of two maps is given by

$\mathcal{J}\xrightarrow{\Delta}\mathcal{J}\times\mathcal{J}\xrightarrow{f\times
g}\mathrm{SL}_{2}\times\mathrm{SL}_{2}\xrightarrow{m}\mathrm{SL}_{2},$

where $m\colon\mathrm{SL}_{2}\times\mathrm{SL}_{2}\to\mathrm{SL}_{2}$ is the
multiplication on $\mathrm{SL}_{2}$. In other words, the group structure is
induced by the group object structure on $\mathrm{SL}_{2}$.

Similarly, the set $[\mathbb{P}^{1},\mathrm{SL}_{2}]^{\mathbb{A}^{1}}$ also
obtains the structure of a group using that $\mathrm{SL}_{2}$ is a group
object in the pointed $\mathbb{A}^{1}$-homotopy category. The Eckmann–Hilton
argument given in [24, Proposition 2.25] can be applied in this scenario to
show that this group structure coincides with the conventional group
structure, see also [2, Proposition 2.2.12]. Hence we may assume that the
group operation on $[\mathbb{P}^{1},\mathrm{SL}_{2}]^{\mathbb{A}^{1}}$ is
induced by the group object structure on $\mathrm{SL}_{2}$. Combining these
observations shows that the composition

$[\mathcal{J},\mathrm{SL}_{2}]^{\mathrm{N}}\to[\mathcal{J},\mathrm{SL}_{2}]^{\mathbb{A}^{1}}\to[\mathbb{P}^{1},\mathrm{SL}_{2}]^{\mathbb{A}^{1}}$

is a group homomorphism. This is the map $\xi^{\prime}_{0}$ which proves the
assertion. ∎

######  Corollary 63.

The group $[\mathcal{J},\mathbb{A}^{2}\setminus\\{0\\}]^{\mathrm{N}}$ is
abelian.

###### Proof.

Since $[\mathcal{J},\mathbb{A}^{2}\setminus\\{0\\}]^{\mathrm{N}}$ is
isomorphic to $[\mathbb{P}^{1},\mathrm{SL}_{2}]^{\mathbb{A}^{1}}$, the
Eckmann–Hilton argument shows that this group is abelian. ∎

######  Remark 64.

Morel shows in [22, §7.3] that the group
$[\mathbb{P}^{1},\mathbb{A}^{2}\setminus\\{0\\}]^{\mathbb{A}^{1}}$ is
isomorphic to $K_{1}^{MW}(k)$, the first Milnor–Witt $K$-theory group of the
field $k$. In short, the computation [22, Theorem 7.13] and the
$\mathbb{A}^{1}$-weak equivalence between $\mathrm{SL}_{2}$ and
$\mathbb{A}^{2}\setminus\\{0\\}$ gives
$\pi_{1}^{\mathbb{A}^{1}}(\mathbb{A}^{2}\setminus\\{0\\})\cong\underline{\mathbf{K}}^{MW}_{2}$.
The contraction of this sheaf evaluated at $\operatorname{Spec}(k)$ then
computes $[\mathbb{P}^{1},\mathbb{A}^{2}\setminus\\{0\\}]^{\mathbb{A}^{1}}$.

$[\mathbb{P}^{1},\mathbb{A}^{2}\setminus\\{0\\}]^{\mathbb{A}^{1}}\cong\pi_{1}^{\mathbb{A}^{1}}(\mathbb{A}^{2}\setminus\\{0\\})_{-1}(\operatorname{Spec}(k))\cong(\underline{\mathbf{K}}^{MW}_{2})_{-1}(\operatorname{Spec}(k))\cong
K^{MW}_{1}(k)$

######  Remark 65.

For any two pointed matrices $M,N\in\mathrm{SL}_{2}(R)$, which represent
pointed morphisms $\mathcal{J}\to\mathrm{SL}_{2}$, there is a chain of
elementary homotopies connecting $M\cdot N$ and $N\cdot M$. We do not know of
a general algorithm to construct this chain of homotopies explicitly.

The following explicit naive homotopies will be of use in the later sections.

######  Lemma 66.

Consider a matrix $M=\begin{pmatrix}A&-V\\\
B&U\end{pmatrix}\in\mathrm{SL}_{2}(R)$. Then $M$ and $(M^{-1})^{T}$ are
naively homotopic. Thus, the unimodular rows $(A,B)$ and $(U,V)$ are naively
homotopic.

###### Proof.

Consider the matrix $H=\begin{pmatrix}1-T^{2}&-T\\\
T(2-T^{2})&1-T^{2}\end{pmatrix}\in\mathrm{SL}_{2}(R[T])$. The matrix $H$
defines an unpointed homotopy from the identity matrix to
$\begin{pmatrix}0&-1\\\ 1&0\end{pmatrix}$. It is straightforward to verify
that the product $HMH^{-1}$ is a pointed homotopy between $M$ and
$(M^{-1})^{T}$ as claimed. ∎

######  Lemma 67.

Consider a matrix $\begin{pmatrix}A&-V\\\ B&U\end{pmatrix}\in SL_{2}(R)$ and
let $u\in k^{\times}$. Then there is an elementary homotopy

(68) $\begin{pmatrix}A&-V\\\
B&U\end{pmatrix}\simeq\begin{pmatrix}A&-\frac{1}{u^{2}}V\\\
u^{2}B&U\end{pmatrix}.$

Thus, the unimodular row $(A,B)$ is naively homotopic to the unimodular row
$(A,u^{2}B)$.

###### Proof.

The matrix on the right-hand side of (68) can be written as the following
product

$\begin{pmatrix}A&-\frac{1}{u^{2}}V\\\
u^{2}B&U\end{pmatrix}=\begin{pmatrix}\frac{1}{u}&0\\\
0&u\end{pmatrix}\begin{pmatrix}A&-V\\\ B&U\end{pmatrix}\begin{pmatrix}u&0\\\
0&\frac{1}{u}\end{pmatrix}.$

The diagonal matrices can be decomposed to a product of elementary matrices,
which are all homotopic to the identity. ∎

###  3.2 Action of degree $0$ maps on degree $n$ maps

Recall that we write $[\mathcal{J},\mathbb{P}^{1}]^{\mathrm{N}}_{n}$ for the
set of naive homotopy classes of maps $\mathcal{J}\to\mathbb{P}^{1}$ with
degree $n$. We define a group action of
$[\mathcal{J},\mathbb{A}^{2}\setminus\\{0\\}]^{\mathrm{N}}\cong[\mathcal{J},\mathrm{SL}_{2}]^{\mathrm{N}}\cong[\mathcal{J},\mathbb{P}^{1}]^{\mathrm{N}}_{0}$
on $[\mathcal{J},\mathbb{P}^{1}]^{\mathrm{N}}_{n}$ for all $n\neq 0$. We start
by first defining an operation on actual morphisms, and then show that the
operation respects the naive homotopy equivalence relation.

######  Definition 69.

Let $M\colon\mathcal{J}\to\mathrm{SL}_{2}$ be a morphism with corresponding
matrix

$\begin{pmatrix}A&-V\\\ B&U\end{pmatrix}$

and consider a map $[s_{0},s_{1}]\colon\mathcal{J}\to\mathbb{P}^{1}$
determined by $n\in\mathbb{N}$, the algebraic line bundle $\mathcal{P}_{n}$ or
$\mathcal{Q}_{n}$, and generating global sections $s_{0},s_{1}$.

We define $M\oplus[s_{0},s_{1}]\colon\mathcal{J}\to\mathbb{P}^{1}$ to be the
morphism determined by the same algebraic line bundle with the generating
global sections $M\oplus[s_{0},s_{1}]=[As_{0}-Vs_{1},Bs_{0}+Us_{1}]$ which are
obtained from the following matrix multiplication

$\begin{pmatrix}A&-V\\\ B&U\end{pmatrix}\begin{pmatrix}s_{0}\\\
s_{1}\end{pmatrix}=\begin{pmatrix}As_{0}-Vs_{1}\\\
Bs_{0}+Us_{1}\end{pmatrix}.$

######  Proposition 70.

Given a map $[s_{0},s_{1}]\colon\mathcal{J}\to\mathbb{P}^{1}$ with algebraic
line bundle $\mathcal{L}$ (either $\mathcal{P}_{n}$ or $\mathcal{Q}_{n}$) and
a map $M\colon\mathcal{J}\to\mathrm{SL}_{2}$, the construction
$M\oplus[s_{0},s_{1}]$ is a morphism from $\mathcal{J}$ to $\mathbb{P}^{1}$.
If both maps are pointed, the result is also pointed. Furthermore, the
operation is a left group action.

###### Proof.

The morphism $M\colon\mathcal{J}\to\mathrm{SL}_{2}$ is described by a matrix

$\begin{pmatrix}A&-V\\\ B&U\end{pmatrix}\in\mathrm{SL}_{2}(R).$

We observe that $U(As_{0}-Vs_{1})+V(Bs_{0}+Us_{1})=s_{0}$, and
$-B(As_{0}-Vs_{1})+A(Bs_{0}+Us_{1})=s_{1}$. By assumption, the sections
$s_{0}$, $s_{1}$ generate the algebraic line bundle $\mathcal{L}$. Hence the
pair of sections $As_{0}-Vs_{1}$, $Bs_{0}+Us_{1}$ generate $\mathcal{L}$ as
well. This proves the first assertion.

That the map $[s_{0},s_{1}]$ is pointed means that
$s_{1}\in\mathbf{j}\subseteq R$, or equivalently, $s_{1}(\mathbf{j})=0$ in
$R/\mathbf{j}$. That $M$ is pointed means $M(\mathbf{j})$ is the identity
matrix. To verify $M\oplus[s_{0},s_{1}]$ is pointed, we must check that
$B(\mathbf{j})s_{0}(\mathbf{j})+U(\mathbf{j})s_{1}(\mathbf{j})=0$, but this is
clear as $B(\mathbf{j})=0$ and $s_{1}(\mathbf{j})=0$ from our assumptions.

The fact that the operation is a left group action follows from the
associativity of matrix multiplication and the definition of the group
structure on maps $\mathcal{J}\to\mathrm{SL}_{2}$. ∎

The next theorem employs the notation of definition 43 for morphisms
$\mathcal{J}\to\mathbb{P}^{1}$.

######  Theorem 71.

Let $f\colon\mathcal{J}\to\mathbb{P}^{1}$ be a map of degree $n$. Then there
exists a matrix $M\in\mathrm{SL}_{2}(R)$ such that $f=M\oplus(1,0:0,1)_{n}$.

###### Proof.

Let $f=(a_{0},a_{1}:b_{0},b_{1})_{n}$. For $c,c^{\prime},d,d^{\prime}\in R$,
we consider the matrix

(72)
$M=\begin{pmatrix}a_{0}+y^{n}c+w^{n}c^{\prime}&a_{1}-x^{n}c-z^{n}c^{\prime}\\\
b_{0}-y^{n}d-w^{n}d^{\prime}&b_{1}+x^{n}d+z^{n}d^{\prime}\end{pmatrix}.$

Note that for any choice of $c,c^{\prime},d,d^{\prime}\in R$ we have

$M\oplus\left[\begin{bmatrix}x^{n}\\\
z^{n}\end{bmatrix},\begin{bmatrix}y^{n}\\\
w^{n}\end{bmatrix}\right]=(a_{0},a_{1}:b_{0},b_{1})_{n}.$

We now show that there always exist $c,c^{\prime},d,d^{\prime}$ such that
$M\in SL_{2}(R)$. The determinant of $M$ is given by the formula

$\det(M)=a_{0}b_{1}-a_{1}b_{0}+c(x^{n}b_{0}+y^{n}b_{1})+c^{\prime}(z^{n}b_{0}+w^{n}b_{1})+d(x^{n}a_{0}+y^{n}a_{1})+d^{\prime}(z^{n}a_{0}+w^{n}a_{1}).$

Since $(a_{0},a_{1}:b_{0},b_{1})_{n}$ determines a morphism of schemes, it
follows from proposition 42 that the ideal
$I:=(x^{n}a_{0}+y^{n}a_{1},z^{n}a_{0}+w^{n}a_{1},x^{n}b_{0}+y^{n}b_{1},z^{n}b_{0}+w^{n}b_{1})$
is the unit ideal. Thus $1-a_{0}b_{1}-a_{1}b_{0}$ is in $I$, and there exist
elements $c,c^{\prime},d,d^{\prime}$ such that $\det(M)=1$. This shows $M\in
SL_{2}(R)$ and proves the assertion by definition of the operation $\oplus$. ∎

######  Corollary 73.

Let $f,g\colon\mathcal{J}\to\mathbb{P}^{1}$ be two morphisms of degree $n$.
Then there exists a matrix $M\in\mathrm{SL}_{2}(R)$ such that $M\oplus f=g$.

###### Proof.

By theorem 71, there exist $M^{\prime}$ and $M^{\prime\prime}$ such that
$M^{\prime}\oplus(1,0:0,1)_{n}=f$ and $M^{\prime\prime}\oplus(1,0:0,1)_{n}=g$.
The desired matrix is now given by $M=M^{\prime\prime}\cdot(M^{\prime})^{-1}$.
∎

######  Remark 74.

The $\mathrm{SL}_{2}(R)$-matrix $M$ constructed in the proof of theorem 71 is
not always pointed, even if the map $f$ we started with is pointed. For
example, following the construction for the map $f=(1,1:0,1)_{1}$ yields the
matrix $M=\begin{pmatrix}1&1\\\ 0&1\end{pmatrix}$ which is not pointed, since
$M(\mathbf{j})$ is not the identity matrix.

Remark 74 shows that we have to improve our argument in order to get an action
on pointed homotopy classes. We will now prove the necessary adjustments.

######  Proposition 75.

Let $f\colon\mathcal{J}\to\mathbb{P}^{1}$ be a pointed map of degree $n\neq
0$. Then there is a pointed naive homotopy between $f$ and a map of the form
$M_{f}\oplus(1,0:0,1)_{n}$ for some pointed matrix
$M_{f}\in\mathrm{SL}_{2}(R)$.

###### Proof.

Let $f=(a_{0},a_{1}:b_{0},b_{1})_{n}$, where we may assume
$a_{0}(\mathbf{j})=1$ by proposition 41. By theorem 71 we can find a matrix
$M^{\prime}\in\mathrm{SL}_{2}(R)$ such that $M^{\prime}\oplus(1,0:0,1)_{n}=f$.
However, $M^{\prime}$ may not be pointed. We can replace $M^{\prime}$ with a
pointed map $M_{f}$ as follows. Assuming $M^{\prime}$ is of the form (72) we
get $b_{1}(\mathbf{j})+d(\mathbf{j})=1$. Moreover, this implies that there is
an element $e\in k$ such that $M^{\prime}(\mathbf{j})=\begin{pmatrix}1&e\\\
0&1\end{pmatrix}$ and $e=a_{1}(\mathbf{j})-c(\mathbf{j})$. Define $M_{f}$ to
be $M_{f}=\begin{pmatrix}1&-e\\\ 0&1\end{pmatrix}M^{\prime}$. We compute
$M_{f}\oplus(1,0:0,1)_{n}=(a_{0}-eb_{0},a_{1}-eb_{1}:b_{0},b_{1})_{n}$. The
assertion now follows from the fact that the morphism
$(a_{0}-Teb_{0},a_{1}-Teb_{1}:b_{0},b_{1})_{n}$ is a pointed homotopy between
$M_{f}\oplus(1,0:0,1)_{n}$ and $f$. ∎

######  Corollary 76.

Let $f,g\colon\mathcal{J}\to\mathbb{P}^{1}$ be two pointed morphisms of degree
$n$. There exists a pointed map $M\colon\mathcal{J}\to\mathrm{SL}_{2}$ such
that $M\oplus f$ is pointed naively homotopic to $g$.

######  Definition 77.

Let
$[(A,B)]\in[\mathcal{J},\mathbb{A}^{2}\setminus\\{0\\}]^{\mathrm{N}}\cong[\mathcal{J},\mathbb{P}^{1}]^{\mathrm{N}}_{0}$
be a pointed naive homotopy class represented by the map with unimodular row
$(A,B)$ in $R$. Let $[f]\in[\mathcal{J},\mathbb{P}^{1}]^{\mathrm{N}}_{n}$ be a
pointed naive homotopy class of degree $n$ with $n\neq 0$ represented by a
pointed morphism $f\colon\mathcal{J}\to\mathbb{P}^{1}$. We define
$[(A,B)]\oplus[f]:=[M\oplus f]$ where $M$ is a completion of $(A,B)$ to a
matrix in $\mathrm{SL}_{2}(R)$ corresponding to a pointed map.

######  Theorem 78.

The operation of definition 77 is well-defined and for each $n\in\Z$ provides
the set $[\mathcal{J},\mathbb{P}^{1}]^{\mathrm{N}}_{n}$ with a left-action by
the group $[\mathcal{J},\mathbb{A}^{2}\setminus\\{0\\}]^{\mathrm{N}}$.

###### Proof.

First, consider a pointed map $f=[s_{0},s_{1}]$. We show that
$[(A,B)]\oplus[f]$ is independent of the choice of completion of $(A,B)$ to a
matrix in $\mathrm{SL}_{2}(R)$. So let

$\displaystyle M=\begin{pmatrix}A&-V\\\
B&U\end{pmatrix},~{}M^{\prime}=\begin{pmatrix}A&-V^{\prime}\\\
B&U^{\prime}\end{pmatrix}$

be two completions to matrices in $\mathrm{SL}_{2}(R)$ which correspond to
pointed maps. Then the naive homotopy
$H(T)=\begin{pmatrix}A&-(TV+(1-T)V^{\prime}\\\
B&TU+(1-T)U^{\prime}\end{pmatrix}$ is pointed independently of $T$, and
$H(T)\oplus f$ is a pointed homotopy between $M\oplus f$ and $M^{\prime}\oplus
f$.

Now we show that $[(A,B)]\oplus[f]$ is independent of the choice of the
representing unimodular row $(A,B)$. Suppose we have a pointed elementary
homotopy between two unimodular rows, $(A(T),B(T))$. Proposition 53 shows that
we can lift it to a pointed elementary homotopy
$M(T)=\begin{pmatrix}A(T)&-V(T)\\\
B(T)&U(T)\end{pmatrix}\in\mathrm{SL}_{2}(R[T])$. Then $M(T)\oplus f$ is a
pointed homotopy between $(A,B)\oplus f$ and $(A^{\prime},B^{\prime})\oplus
f$. Now we consider a unimodular row $(A,B)$ and let $M=\begin{pmatrix}A&-V\\\
B&U\end{pmatrix}$ be a lift to a matrix in $\mathrm{SL}_{2}(R)$. Let
$f_{0},f_{1}\colon\mathcal{J}\to\mathbb{P}^{1}$ be two pointed morphisms which
are homotopic via a pointed naive homotopy. Let
$f(T)\colon\mathcal{J}\times\mathbb{A}^{1}\to\mathbb{P}^{1}$ be a pointed
naive homotopy. We define the map $H(T):=M\oplus
f(T)\colon\mathcal{J}\times\mathbb{A}^{1}\to\mathbb{P}^{1}$ with the same
algebraic line bundle $\mathcal{L}^{\prime}$ on
$\mathcal{J}\times\mathbb{A}^{1}$ and global sections

$\begin{pmatrix}A&-V\\\ B&U\end{pmatrix}\cdot\begin{pmatrix}s_{0}(T)\\\
s_{1}(T)\end{pmatrix}=\begin{pmatrix}As_{0}(T)-Vs_{1}(T)\\\
Bs_{0}(T)+Us_{1}(T)\end{pmatrix}.$

We note that $H(T)$ thus defined is in fact a morphism
$\mathcal{J}\times\mathbb{A}^{1}\to\mathbb{P}^{1}$, since we have
$U(As_{0}(T)-Vs_{1}(T))+V(Bs_{0}(T)+Us_{1}(T))=s_{0}(T)$, and
$-B(As_{0}(T)-Vs_{1}(T))+A(Bs_{0}(T)+Us_{1}(T))=s_{1}(T)$. By assumption, the
sections $s_{0}(T)$, $s_{1}(T)$ generate the line bundle
$\mathcal{L}^{\prime}$. Hence $(As_{0}(T)-Vs_{1}(T),Bs_{0}(T)+Us_{1}(T))$
generate $\mathcal{L}^{\prime}$ as well. This shows that $H(T)$ defines a
morphism. We now verify that $H(T)$ is pointed by showing
$Bs_{0}(T)+Us_{1}(T)\in\mathbf{j}^{\prime}$ . Pointedness of
$[s_{0}(T),s_{1}(T)]$ means that $s_{1}(T)(\mathbf{j}^{\prime})=0$ in
$R[T]/\mathbf{j}^{\prime}$. Pointedness of $(A,B)$ means $M(\mathbf{j})$ is
the identity matrix. We calculate

$B(\mathbf{j}^{\prime})s_{0}(T)(\mathbf{j}^{\prime})+U(\mathbf{j}^{\prime})s_{1}(T)(\mathbf{j}^{\prime})=0\cdot
s_{0}(T)(\mathbf{j}^{\prime})+1\cdot 0=0$

which completes the verification. This shows that $\oplus$ is independent of
the choice of representatives in both naive homotopy classes and completes the
proof of the first assertion. The second assertion then follows from
proposition 70. ∎

######  Remark 79.

There are several variations to the operation given in definition 77 that
produce valid group actions. For $M\in\mathrm{SL}_{2}(R)$, the operation in
definition 69 is given by the matrix multiplication
$M\cdot(s_{0}\;s_{1})^{T}$. We could have taken equally well either
$M^{T}\cdot(s_{0}\;s_{1})^{T}$ or $M^{-1}\cdot(s_{0}\;s_{1})^{T}$, although
this would give a right-action rather than a left-action on maps. Up to
homotopy, the latter two choices in fact agree, since $M^{T}$ is homotopic to
$M^{-1}$ by lemma 66. Thus there are two natural choices for this action, one
of which applies the inverse operation to the morphism in $\mathrm{SL}_{2}(R)$
before acting. In appendix C we will use real realization to check which of
these operations can represent the group operation on
$[\mathcal{J},\mathbb{P}^{1}]^{\mathbb{A}^{1}}$ induced from Morel’s group
structure on $[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathbb{A}^{1}}$ via
$\pi_{\mathbb{A}^{1}}^{*}$. In fact, in example 169 we show that only the
choice of definitions 69 and 77 can be compatible.

##  4 The group structure on $[\mathcal{J},\mathbb{P}^{1}]^{\mathrm{N}}$

We now state an explicit group structure on
$[\mathcal{J},\mathbb{P}^{1}]^{\mathrm{N}}$.

######  Definition 80.

Let $-[\operatorname{id}]$ denote the additive inverse of
$[\operatorname{id}:\mathbb{P}^{1}\to\mathbb{P}^{1}]$ under the conventional
group structure on $[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathbb{A}^{1}}$. Define
$-\pi:\mathcal{J}\to\mathbb{P}^{1}$ to be a morphism which represents the
$\mathbb{A}^{1}$-homotopy class
$-[\operatorname{id}:\mathbb{P}^{1}\to\mathbb{P}^{1}]\in[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathbb{A}^{1}}$
under the bijection
$\xi:[\mathcal{J},\mathbb{P}^{1}]^{\mathrm{N}}\to[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathbb{A}^{1}}$
of equation (15). More generally, for any integer $n$, let $n\pi$ denote a
morphism $n\pi:\mathcal{J}\to\mathbb{P}^{1}$ which represents the
$\mathbb{A}^{1}$-homotopy class
$n[\operatorname{id}:\mathbb{P}^{1}\to\mathbb{P}^{1}]$ under the bijection
$\xi:[\mathcal{J},\mathbb{P}^{1}]^{\mathrm{N}}\to[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathbb{A}^{1}}$.

We are now ready to define a group operation on
$[\mathcal{J},\mathbb{P}^{1}]^{\mathrm{N}}$.

######  Definition 81.

Let $f\colon\mathcal{J}\to\mathbb{P}^{1}$ and
$g\colon\mathcal{J}\to\mathbb{P}^{1}$ be morphisms of degrees $n$ and $m$
respectively. By corollary 76 there are degree $0$ maps
$f_{0}\colon\mathcal{J}\to\mathbb{P}^{1}$ and
$g_{0}\colon\mathcal{J}\to\mathbb{P}^{1}$ for which $f\simeq f_{0}\oplus n\pi$
and $g\simeq g_{0}\oplus m\pi$. We define the sum of $[f]$ and $[g]$ to be

$\displaystyle[f]\oplus[g]$
$\displaystyle=([f_{0}]\oplus[n\pi])\oplus([g_{0}]\oplus[m\pi])$
$\displaystyle=([f_{0}]\oplus[g_{0}])\oplus[(n+m)\pi].$

The term $[f_{0}]\oplus[g_{0}]$ is calculated by matrix multiplication via
definition 58. The group action of definition 77 is used to compute
$(f_{0}\oplus g_{0})\oplus(n+m)\pi$.

######  Remark 82.

It follows from theorem 78 that the operation $\oplus$ of definition 81 is
well-defined. We also note that, for $n>0$, the proofs of theorem 71 and
proposition 75 may be used to write down a concrete algorithm to find a map
$f_{0}$ such that $f\simeq f_{0}\oplus n\pi$ for any degree $n$ map $f$.

######  Remark 83.

For $n>0$, we may construct morphisms $n\pi$ by using Cazanave’s group
operation on morphisms $[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathrm{N}}$ and lift
it to an element in $[\mathcal{J},\mathbb{P}^{1}]^{\mathrm{N}}$. A recursive
description of the maps $n\pi$ for $n>0$ can be given as follows: We set
$F_{0}=1$ and $F_{1}=\begin{bmatrix}x\\\ z\end{bmatrix}$. For $n>0$, we define
$F_{n+1}$ recursively by setting

$\displaystyle F_{n+1}=\begin{bmatrix}x\\\ z\end{bmatrix}\cdot
F_{n}-\begin{bmatrix}y^{2}\\\ w^{2}\end{bmatrix}\cdot F_{n-1}.$

For $n>0$, the morphism $n\pi$ is given by sections

$\left[F_{n},\begin{bmatrix}y\\\ w\end{bmatrix}\cdot F_{n-1}\right].$

Note that $[(1,0:0,1)_{n}]$ is in general not equal to $[n\pi]$ for $n>1$.

We are now ready to prove one of our main results.

######  Theorem 84.

The operation $\oplus$ turns the set
$[\mathcal{J},\mathbb{P}^{1}]^{\mathrm{N}}$ into an abelian group. Moreover,
there is an isomorphism of groups
$\phi\colon\left([\mathcal{J},\mathbb{P}^{1}]^{\mathrm{N}},\oplus\right)\xrightarrow{\cong}\left([\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathbb{A}^{1}},\oplus^{\mathbb{A}^{1}}\right)$.

###### Proof.

We observe that the set $\\{[n\pi]:n\in\Z\\}$ inherits the structure of an
abelian group from $\Z$. By definition,
$\left([\mathcal{J},\mathbb{P}^{1}]^{\mathrm{N}},\oplus\right)$ is isomorphic
to the direct product of the two groups $\\{[n\pi]:n\in\Z\\}$ and
$[\mathcal{J},\mathbb{A}^{2}\setminus\\{0\\}]^{\mathrm{N}}$. Both are abelian
by corollary 63. This implies the first assertion.

By definition of the operation $\oplus$, the group
$\left([\mathcal{J},\mathbb{P}^{1}]^{\mathrm{N}},\oplus\right)$ fits into the
short exact sequence displayed in the top row of diagram (89) below. By the
work of Morel in [22, §7.3], the group
$\left([\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathbb{A}^{1}},\oplus^{\mathbb{A}^{1}}\right)$
fits into the short exact sequence displayed in the bottom row.

(89)

By theorem 62, the vertical map $\xi_{0}$ on the left-hand side is an
isomorphism. The vertical map $q$ on the right-hand side is an isomorphism as
well. We define $\phi$ to be the unique group homomorphism satisfying
$\phi([\pi])=[\operatorname{id}]$ and $\phi([f_{0}])=\xi_{0}([f_{0}])$ for all
$[f_{0}]\in[\mathcal{J},\mathbb{A}^{2}\setminus\\{0\\}]^{\mathrm{N}}$. The
diagram commutes by our definition of $\phi$. Since $\xi_{0}$ and $q$ are
isomorphisms, $\phi$ is an isomorphism of groups as well by the five-lemma. ∎

Unfortunately, theorem 84 does not give a satisfactory way to describe an
$\mathbb{A}^{1}$-homotopy class
$\gamma\in[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathbb{A}^{1}}$ as a concrete
homotopy class of a scheme morphism $g\colon\mathcal{J}\to\mathbb{P}^{1}$.
However, since $\xi$ restricts to an isomorphism on the subgroups
$[\mathcal{J},\mathbb{A}^{2}\setminus\\{0\\}]^{\mathrm{N}}$ and
$\\{[n\pi]\,|\,n\in\mathbb{Z}\\}$, we do believe that the bijection $\xi$ is a
group isomorphism, which we state as a conjecture below.

######  Conjecture 90.

The bijection
$\xi\colon[\mathcal{J},\mathbb{P}^{1}]^{\mathrm{N}}\to[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathbb{A}^{1}}$
is a group isomorphism and equals $\phi$.

One obstacle to prove conjecture 90 is that, for $n<0$, we do not know which
morphism $\mathcal{J}\to\mathbb{P}^{1}$ is sent to $n[\operatorname{id}]$
under $\xi$. In particular, we do not know which morphism
$\mathcal{J}\to\mathbb{P}^{1}$ is mapped to the motivic homotopy class
$-[\operatorname{id}:\mathbb{P}^{1}\to\mathbb{P}^{1}]$. A potential candidate
for $-\pi$ may be the map $\tilde{\pi}=(1,0:0,-1)_{-1}$ determined by the line
bundle $\mathcal{Q}_{1}$ and generating sections

$s_{0}=\begin{pmatrix}x\\\ y\end{pmatrix}\text{ and
}~{}s_{1}=-\begin{pmatrix}z\\\ w\end{pmatrix}.$

######  Question 91.

Is $\tilde{\pi}$ the inverse of $\pi$ for $\oplus$, i.e., is $\tilde{\pi}$
naively homotopic to $-\pi$?

In appendix C we present further evidence for conjecture 90. We use the real
realization functor for fields $k\subset\R$ and Morel’s theorem which states
that the signature of the motivic Brouwer degree equals the topological
Brouwer degree under real realization. This provides a potential obstruction
to the compatibility of $\oplus^{\mathbb{A}^{1}}$ and the action of
$[\mathcal{J},\mathbb{A}^{2}\setminus\\{0\\}]^{\mathrm{N}}$ on
$[\mathcal{J},\mathbb{P}^{1}]^{\mathrm{N}}$. We then compute concrete examples
and show that other choices for the action of
$[\mathcal{J},\mathbb{A}^{2}\setminus\\{0\\}]^{\mathrm{N}}$ on
$[\mathcal{J},\mathbb{P}^{1}]^{\mathrm{N}}$ are not compatible with
$\oplus^{\mathbb{A}^{1}}$, while our choice of operation in definition 77 is
compatible with $\oplus^{\mathbb{A}^{1}}$ after real realization in the chosen
examples.

In proposition 120 we show that the naive homotopy class of $\pi$ is mapped to
the class $(\langle 1\rangle,1)$ in
$\mathrm{GW}(k)\times_{k^{\times}/k^{\times 2}}k^{\times}$ as expected if
$\xi$ is a group homomorphism. Based on the computations in appendix C we
prove in theorem 121 that the image of $[\tilde{\pi}]$ under the motivic
Brouwer degree is the class $-\langle 1\rangle$ in $\mathrm{GW}(k)$. This
brings us very close to a positive answer to question 91. We are, however, not
able to compute the image of $\tilde{\pi}$ in $k^{\times}$.

We end this section with a comment on a question by Cazanave in [12]:

######  Remark 92.

Since $\pi$ is an $\mathbb{A}^{1}$-weak equivalence, it induces a bijection
$\pi_{*}\colon[\mathcal{J},\mathcal{J}]^{\mathrm{N}}\to[\mathcal{J},\mathbb{P}^{1}]^{\mathrm{N}}$.
Hence there is a bijection between
$[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathbb{A}^{1}}$ and the set of pointed
naive homotopy classes $[\mathcal{J},\mathcal{J}]^{\mathrm{N}}$. In [12, page
31] Cazanave speculates whether $[\mathcal{J},\mathcal{J}]^{\mathrm{N}}$ can
be used to study the group structure on
$[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathbb{A}^{1}}$. A morphism
$\mathcal{J}\to\mathcal{J}$ corresponds to a ring homomorphism $R\to R$, or
equivalently, the data of a $(2\times 2)$-matrix with entries in $R$ and with
trace $1$ and determinant $0$. For every map
$f\colon\mathcal{J}\to\mathbb{P}^{1}$ we can find a map
$F\colon\mathcal{J}\to\mathcal{J}$ such that $f=\pi\circ F$. We will refer to
such a map $F$ as a lift of $f$. There is a particularly nice way to construct
a lift in the case $f\colon\mathcal{J}\to\mathbb{P}^{1}$ has degree $0$.
Assume that $f$ is given by a unimodular row $(A,B)$. Let $U,V\in R$ be such
that $\begin{pmatrix}A&-V\\\ B&U\end{pmatrix}$ has determinant $1$. Then $F$
is given by the matrix $\begin{pmatrix}AU&BU\\\ AV&BV\end{pmatrix}$ which has
trace $1$ and determinant $0$. Composing the map with $\pi$ yields the
$\mathcal{J}\to\mathbb{P}^{1}$ map given by either $[AU:BU]$ or $[AV:BV]$,
whenever they are defined, which coincides with the map corresponding to the
unimodular row $(A,B)$. If $f$ has non-zero degree, there is also concrete
procedure to find a lift of $f$, which we leave to the reader.

Since morphisms $\mathcal{J}\to\mathcal{J}$ can be represented by matrices, it
may seem plausible that one can find a suitable operation on
$[\mathcal{J},\mathcal{J}]^{\mathrm{N}}$ which may help to describe the group
$([\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathbb{A}^{1}},\oplus^{\mathbb{A}^{1}})$.
However, neither addition nor multiplication of matrices equip the set
$[\mathcal{J},\mathcal{J}]^{\mathrm{N}}$ with an operation which is compatible
with the conventional group structure on
$[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathbb{A}^{1}}$. We have verified in
examples that composition of maps in $[\mathcal{J},\mathcal{J}]^{\mathrm{N}}$
descends to the operation $\circ$ on
$[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathrm{N}}$ of [13, Definition 4.5]. As
pointed out in [13, Remark 4.7] the latter does not distribute over the
conventional group structure on
$[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathbb{A}^{1}}$.

We were not able to make a reasonable guess which other operation on
$[\mathcal{J},\mathcal{J}]^{\mathrm{N}}$ might work. We have therefore not
pursued this path further.

##  5 Compatibility with Cazanave’s monoid structure

The goal of this section is to show that the map

$\pi_{\mathrm{N}}^{*}\colon\left([\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathrm{N}},\oplus^{\mathrm{N}}\right)\to\left([\mathcal{J},\mathbb{P}^{1}]^{\mathrm{N}},\oplus\right)$

is a morphism of monoids, where $\oplus^{\mathrm{N}}$ denotes the monoid
operation defined by Cazanave in [13]. We will achieve this goal in theorem
103.

###  5.1 Compatibility with certain degree $0$ maps

We first study an important family of degree $0$ morphisms and their
compatibility with $\oplus$, $\oplus^{\mathrm{N}}$, and
$\pi_{\mathrm{N}}^{*}$.

######  Definition 93.

For $u,v\in k^{\times}$, we write $g_{u,v}$ for the pointed morphism
$\mathcal{J}\to\mathbb{A}^{2}\setminus\\{0\\}$ given by the unimodular row
$\left(x+\frac{v}{u}w,(u-v)y\right)$ in $R$. This unimodular row can be
completed to the $\mathrm{SL}_{2}(R)$-matrix

$m_{u,v}=\begin{pmatrix}x+\frac{v}{u}w&\frac{u-v}{uv}z\\\
(u-v)y&x+\frac{u}{v}w\end{pmatrix}.$

We now develop some basic properties of the maps $g_{u,v}$ which will be
necessary to prove that $\pi_{\mathrm{N}}^{*}$ is a monoid morphism.

######  Lemma 94.

For all $u,v,s\in k^{\times}$, we have the identity $g_{u,v}\oplus
g_{v,s}=g_{u,s}$. In particular, we have $g_{u,v}\oplus g_{v,u}=(1,0)$ and
$g_{u,v}\oplus g_{v,1}=g_{u,1}$.

###### Proof.

A direct computation, using $xw=yz$, shows

$m_{u,v}\cdot
m_{v,s}=\begin{pmatrix}x^{2}+\frac{uv+vs}{uv}xw+\frac{s}{u}w^{2}&\frac{uv-
sv}{usv}xz+\frac{-vs+uv}{usv}zw\\\
(u-s)xy+(u-s)wy&x^{2}+\frac{uv+sv}{sv}xw+\frac{u}{s}w^{2}\end{pmatrix},$

and since $x+w=1$, this simplifies to the matrix $m_{u,s}$. Then
$g_{u,v}\oplus g_{v,u}=g_{u,u}=(1,0)$ and $g_{u,v}\oplus g_{v,1}=g_{u,1}$ are
special cases for respectively $s=u$ and $s=1$. ∎

######  Lemma 95.

Let $u,v,c\in k^{\times}$. Then $[g_{u,v}]=[g_{c^{2}u,c^{2}v}]$.

###### Proof.

By lemma 67, we have

$\displaystyle
g_{u,v}=\left(x+\frac{v}{u}w,(u-v)y\right)\simeq\left(x+\frac{v}{u}w,c^{2}(u-v)y\right)=g_{c^{2}u,c^{2}v}.$

∎

######  Lemma 96.

Let $u,v\in k^{\times}$ and let $v$ be a square. Then we have
$[g_{u,1}]\oplus[g_{v,1}]=[g_{uv,1}]$.

###### Proof.

Using lemma 95 and lemma 94 shows

$g_{u,1}\oplus g_{v,1}\simeq g_{u,1}\oplus g_{1,1/v}=g_{u,1/v}\simeq
g_{uv,1},$

and hence the claim. ∎

Next we study the relationship of the maps $g_{u,v}$ with $\oplus$ and
$\pi_{\mathrm{N}}^{*}$.

######  Lemma 97.

For every $u\in k^{\times}$, we have
$g_{u,1}\oplus\pi=\pi_{\mathrm{N}}^{*}\left(\frac{X}{u}\right)$.

###### Proof.

A direct computation using the facts that $x\begin{bmatrix}y\\\
w\end{bmatrix}=y\begin{bmatrix}x\\\ z\end{bmatrix}$ and $z\begin{bmatrix}y\\\
w\end{bmatrix}=w\begin{bmatrix}x\\\ z\end{bmatrix}$ shows

$\begin{pmatrix}x+\frac{1}{u}w&\frac{u-1}{u}z\\\
(u-1)y&x+uw\end{pmatrix}\begin{pmatrix}\begin{bmatrix}x\\\ z\end{bmatrix}\\\
\begin{bmatrix}y\\\
w\end{bmatrix}\end{pmatrix}=\begin{pmatrix}\begin{bmatrix}x\\\
z\end{bmatrix}\\\ u\begin{bmatrix}y\\\ w\end{bmatrix}\end{pmatrix}$

and hence the result by definition of the maps involved. ∎

######  Lemma 98.

For all $u,v\in k^{\times}$, we have the identity

$\displaystyle
g_{u,v}\oplus\pi_{\mathrm{N}}^{*}\left(\frac{X}{v}\right)=\pi_{\mathrm{N}}^{*}\left(\frac{X}{u}\right).$

###### Proof.

Using lemmas 94 and 97 and definition 81 we get

$g_{u,v}\oplus\pi_{\mathrm{N}}^{*}\left(\frac{X}{v}\right)=(g_{u,v}\oplus
g_{v,1})\oplus\pi=g_{u,1}\oplus\pi=\pi_{\mathrm{N}}^{*}\left(\frac{X}{u}\right)$

and hence the result. ∎

We are now ready to prove a key result for the compatibility of
$\pi_{\mathrm{N}}^{*}$ with the monoid operations.

######  Proposition 99.

Let $u\in k^{\times}$ and $f\colon\mathbb{P}^{1}\to\mathbb{P}^{1}$ be a
pointed morphism. Then we have

$\displaystyle\pi_{\mathrm{N}}^{*}\left(\frac{X}{u}\oplus^{\mathrm{N}}f\right)\simeq
g_{u,1}\oplus\left(\pi_{\mathrm{N}}^{*}\left(\frac{X}{1}\oplus^{\mathrm{N}}f\right)\right).$

###### Proof.

Let $\pi_{\mathrm{N}}^{*}(f)=[f_{1},f_{2}]$, and let $n$ denote its degree.
The pair of generating sections of $\mathcal{P}_{n+1}$ which determines the
morphism $\pi_{\mathrm{N}}^{*}\left(\frac{X}{u}\oplus^{\mathrm{N}}f\right)$ is
given by the matrix product

$\pi_{\mathrm{N}}^{*}\left(\frac{X}{u}\oplus^{\mathrm{N}}f\right)=\begin{pmatrix}\begin{bmatrix}x\\\
z\end{bmatrix}&-\frac{1}{u}\begin{bmatrix}y\\\ w\end{bmatrix}\\\
u\begin{bmatrix}y\\\ w\end{bmatrix}&0\end{pmatrix}\cdot\begin{pmatrix}f_{1}\\\
f_{2}\end{pmatrix}.$

Note that here we use the isomorphism given in Proposition 28 to identify the
product of a pair of column vectors with its image in $\mathcal{P}_{n+1}$. The
pair of generating sections which determines the morphism
$g_{u,1}\oplus\left(\pi^{*}\left(\frac{X}{1}\oplus^{\mathrm{N}}f\right)\right)$
is given by the matrix product

$g_{u,1}\oplus\left(\pi_{\mathrm{N}}^{*}\left(\frac{X}{1}\oplus^{\mathrm{N}}f\right)\right)=\begin{pmatrix}x+\frac{1}{u}w&\frac{u-1}{u}z\\\
(u-1)y&x+uw\end{pmatrix}\cdot\begin{pmatrix}\begin{bmatrix}x\\\
z\end{bmatrix}&-\begin{bmatrix}y\\\ w\end{bmatrix}\\\ \begin{bmatrix}y\\\
w\end{bmatrix}&0\end{pmatrix}\cdot\begin{pmatrix}f_{1}\\\ f_{2}\end{pmatrix}.$

Note that we have the following equality of matrix products

$\displaystyle\begin{pmatrix}x+\frac{1}{u}w&\frac{u-1}{u}z\\\
(u-1)y&x+uw\end{pmatrix}\cdot\begin{pmatrix}\begin{bmatrix}x\\\
z\end{bmatrix}&-\begin{bmatrix}y\\\ w\end{bmatrix}\\\ \begin{bmatrix}y\\\
w\end{bmatrix}&0\end{pmatrix}$
$\displaystyle=\begin{pmatrix}\begin{bmatrix}x\\\
z\end{bmatrix}&-(x+\frac{1}{u})\begin{bmatrix}y\\\ w\end{bmatrix}\\\
u\begin{bmatrix}y\\\ w\end{bmatrix}&(1-u)y\begin{bmatrix}y\\\
w\end{bmatrix}\end{pmatrix}$ $\displaystyle=\begin{pmatrix}\begin{bmatrix}x\\\
z\end{bmatrix}&-\frac{1}{u}\begin{bmatrix}y\\\ w\end{bmatrix}\\\
u\begin{bmatrix}y\\\
w\end{bmatrix}&0\end{pmatrix}\cdot\begin{pmatrix}1&\frac{u-1}{u}y\\\
0&1\end{pmatrix}.$

We claim that the rows of the matrix $h(T)$ defined by the following product

$\displaystyle h(T)$ $\displaystyle=\begin{pmatrix}\begin{bmatrix}x\\\
z\end{bmatrix}&-\frac{1}{u}\begin{bmatrix}y\\\ w\end{bmatrix}\\\
u\begin{bmatrix}y\\\
w\end{bmatrix}&0\end{pmatrix}\cdot\begin{pmatrix}1&\frac{u-1}{u}yT\\\
0&1\end{pmatrix}\cdot\begin{pmatrix}f_{1}\\\ f_{2}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}f_{1}\begin{bmatrix}x\\\
z\end{bmatrix}+\frac{u-1}{u}yTf_{2}\begin{bmatrix}x\\\
z\end{bmatrix}-\frac{1}{u}f_{2}\begin{bmatrix}y\\\ w\end{bmatrix}\\\
uf_{1}\begin{bmatrix}y\\\ w\end{bmatrix}+(u-1)yTf_{2}\begin{bmatrix}y\\\
w\end{bmatrix}\end{pmatrix}$

provide generating sections of $\mathcal{P}_{n+1}$ and define the desired
pointed homotopy from
$h(0)=\pi_{\mathrm{N}}^{*}\left(\frac{X}{u}\oplus^{\mathrm{N}}f\right)$ to
$h(1)=g_{u,1}\oplus\left(\pi_{\mathrm{N}}^{*}\left(\frac{X}{1}\oplus^{\mathrm{N}}f\right)\right)$.

Now we prove the remaining claim. Since $f$ is a morphism
$\mathbb{P}^{1}\to\mathbb{P}^{1}$, we can write it as a rational function
$\frac{X^{n}+a_{n-1}X^{n-1}+\ldots a_{0}}{b_{n-1}X^{n-1}+\ldots+b_{0}}$. We
then define

$F=(F_{0},F_{1})=(\alpha^{n}+\ldots+a_{0}\beta^{n},b_{n-1}\alpha^{n-1}\beta+\ldots+b_{0}\beta^{n})\in(R[\alpha,\beta]_{n})^{2}.$

Using the construction of definition 34, we observe that
$\sigma((F_{0},F_{1}))=[f_{0},f_{1}]=\pi^{*}(f)$. Let $H$ be the following
product of $R[\alpha,\beta,T]$-matrices

$\displaystyle H(T)$ $\displaystyle=\begin{pmatrix}\alpha&-\frac{1}{u}\beta\\\
u\beta&0\end{pmatrix}\cdot\begin{pmatrix}1&\frac{u-1}{u}yT\\\
0&1\end{pmatrix}\cdot\begin{pmatrix}F_{1}\\\ F_{2}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}F_{1}\alpha+\frac{u-1}{u}yTF_{2}\alpha-\frac{1}{u}F_{2}\alpha\\\
uF_{1}\beta+(u-1)yTF_{2}\beta\end{pmatrix}.$

We have $\sigma(H)=h$. Since $F$ has unit resultant, lemma 149 implies that
the pair $(F_{1}+\frac{u-1}{u}yTF_{2},F_{2})$ also has unit resultant. By
lemma 150 the resultant of $H(T)$ is a unit as well. By lemma 35 this implies
that $h$ consists of a pair of generating sections and finishes the proof. ∎

To give a concrete example of the homotopy constructed in the proof of
proposition 99, we look at the special case $f=X/1$:

######  Example 100.

For every $u\in k^{\times}$, the morphism $H$ defined by

$H=\left[\begin{bmatrix}x^{2}\\\
z^{2}\end{bmatrix}+T\frac{u-1}{u}y\begin{bmatrix}xy\\\
zw\end{bmatrix}-\left(x+\frac{1}{u}w\right)\begin{bmatrix}y^{2}\\\
w^{2}\end{bmatrix},u\begin{bmatrix}xy\\\
zw\end{bmatrix}+(T(u-1)-(u-1)y)\begin{bmatrix}y^{2}\\\
w^{2}\end{bmatrix}\right]$

is a homotopy between $H(0)=g_{u,1}\oplus 2\pi$ and
$H(1)=\pi_{\mathrm{N}}^{*}\left(\frac{X}{u}\oplus^{\mathrm{N}}\frac{X}{1}\right)$.

###  5.2 The map $\pi_{\mathrm{N}}^{*}$ is a monoid morphism

We will now prove that $\pi_{\mathrm{N}}^{*}$ is a morphism of monoids.

######  Lemma 101.

We have

$\displaystyle\pi_{\mathrm{N}}^{*}\left(\frac{X}{1}\oplus^{\mathrm{N}}\cdots\oplus^{\mathrm{N}}\frac{X}{1}\right)\simeq\pi\oplus\cdots\oplus\pi,$

where there are $n$ summands on both sides.

###### Proof.

Since
$\nu_{\mathbb{P}^{1}}\left(\frac{X}{1}\right)=\operatorname{id}_{\mathbb{P}^{1}}$
and $\nu_{\mathbb{P}^{1}}$ is a morphism of monoids by [13, Proposition 3.23],
we have the equality
$n[\operatorname{id}]=\left[\frac{X}{1}\oplus^{\mathrm{N}}\cdots\oplus^{\mathrm{N}}\frac{X}{1}\right]$
in $[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathbb{A}^{1}}$. Thus
$\pi_{\mathrm{N}}^{*}\left(\frac{X}{1}\oplus^{\mathrm{N}}\cdots\oplus^{\mathrm{N}}\frac{X}{1}\right)$
is naively homotopic to $n\pi$. By definition,
$[\pi]\oplus\cdots\oplus[\pi]=[n\pi]$, hence the result follows. ∎

######  Proposition 102.

For $u_{1},\ldots,u_{n}\in k^{\times}$ we have

$\displaystyle\pi_{\mathrm{N}}^{*}\left(\frac{X}{u_{1}}\oplus^{\mathrm{N}}\frac{X}{u_{2}}\oplus^{\mathrm{N}}\cdots\oplus^{\mathrm{N}}\frac{X}{u_{n}}\right)\simeq\pi_{\mathrm{N}}^{*}\left(\frac{X}{u_{1}}\right)\oplus\pi_{\mathrm{N}}^{*}\left(\frac{X}{u_{2}}\right)\oplus\cdots\oplus\pi_{\mathrm{N}}^{*}\left(\frac{X}{u_{n}}\right).$

###### Proof.

We use that $\oplus^{\mathrm{N}}$ and $\oplus$ are commutative and that
$\oplus$ is associative and apply proposition 99 and lemma 101 to get

$\displaystyle\pi_{\mathrm{N}}^{*}\left(\frac{X}{u_{1}}\oplus^{\mathrm{N}}\cdots\oplus^{\mathrm{N}}\frac{X}{u_{n}}\right)$
$\displaystyle\simeq
g_{u_{1},1}\oplus\pi_{\mathrm{N}}^{*}\left(\frac{X}{1}\oplus^{\mathrm{N}}\frac{X}{u_{2}}\oplus^{\mathrm{N}}\cdots\oplus^{\mathrm{N}}\frac{X}{u_{n}}\right)$
$\displaystyle\simeq g_{u_{1},1}\oplus
g_{u_{2},1}\oplus\pi_{\mathrm{N}}^{*}\left(\frac{X}{1}\oplus^{\mathrm{N}}\frac{X}{1}\oplus^{\mathrm{N}}\frac{X}{u_{3}}\oplus^{\mathrm{N}}\cdots\oplus^{\mathrm{N}}\frac{X}{u_{n}}\right)$
$\displaystyle\simeq g_{u_{1},1}\oplus\cdots\oplus g_{u_{n},1}\oplus n\pi$
$\displaystyle\simeq(g_{u_{1},1}\oplus\pi)\oplus(g_{u_{2},1}\oplus\pi)\oplus\cdots\oplus(g_{u_{n},1}\oplus\pi)$
$\displaystyle\simeq\pi_{\mathrm{N}}^{*}\left(\frac{X}{u_{1}}\right)\oplus\cdots\oplus\pi_{\mathrm{N}}^{*}\left(\frac{X}{u_{n}}\right).$

For the final step we used lemma 97. ∎

######  Theorem 103.

The map
$\pi_{\mathrm{N}}^{*}\colon\left([\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathrm{N}},\oplus^{\mathrm{N}}\right)\to\left([\mathcal{J},\mathbb{P}^{1}]^{\mathrm{N}},\oplus\right)$
induced by $\pi$ is a morphism of monoids.

###### Proof.

Let $f,g:\mathbb{P}^{1}\to\mathbb{P}^{1}$ be two pointed morphisms. By [13,
Lemma 3.13], $[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathrm{N}}$ is generated by
elements in degree $1$. Hence we can assume
$f\simeq\frac{X}{u_{1}}\oplus^{\mathrm{N}}\frac{X}{u_{2}}\oplus^{\mathrm{N}}\cdots\oplus^{\mathrm{N}}\frac{X}{u_{n}}$
and
$g\simeq\frac{X}{v_{1}}\oplus^{\mathrm{N}}\frac{X}{v_{2}}\oplus^{\mathrm{N}}\cdots\oplus^{\mathrm{N}}\frac{X}{v_{m}}$
for some $u_{1},\ldots,u_{n},v_{1},\ldots,v_{m}\in k^{\times}$. Then
proposition 102 implies the identity

$\pi_{\mathrm{N}}^{*}\left([f]\oplus^{\mathrm{N}}[g]\right)=\left[\pi_{\mathrm{N}}^{*}(f)\right]\oplus\left[\pi_{\mathrm{N}}^{*}(g)\right]$

and hence the result. ∎

##  6 Group completion

The morphism $\pi\colon\mathcal{J}\to\mathbb{P}^{1}$ induces the following
commutative diagram of solid arrows.

(108)

######  Lemma 109.

We have the identity of morphisms
$\xi\circ\pi_{\mathrm{N}}^{*}=\nu_{\mathbb{P}^{1}}$.

###### Proof.

Since the outer diagram in (108) commutes, we have
$(\pi^{*}_{\mathbb{A}^{1}})^{-1}\circ\nu_{\mathcal{J}}\circ\pi_{\mathrm{N}}^{*}=\nu_{\mathbb{P}^{1}}$.
Since $\xi=(\pi^{*}_{\mathbb{A}^{1}})^{-1}\circ\nu_{\mathcal{J}}$ by
definition, this shows

$\displaystyle\xi\circ\pi_{\mathrm{N}}^{*}=\nu_{\mathbb{P}^{1}}$

as desired. ∎

In [13, Theorem 3.22] Cazanave proves that the canonical map
$\nu_{\mathbb{P}^{1}}\colon\left([\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathrm{N}},\oplus^{\mathrm{N}}\right)\to\left([\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathbb{A}^{1}},\oplus^{\mathbb{A}^{1}}\right)$
is a group completion. Hence there exists a unique group homomorphism

$\displaystyle\psi\colon\left([\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathbb{A}^{1}},\oplus^{\mathbb{A}^{1}}\right)\to\left([\mathcal{J},\mathbb{P}^{1}]^{\mathrm{N}},\oplus\right)$

making the lower triangle in diagram (108) commute.

We will show in this section that $\pi_{\mathrm{N}}^{*}$ has image in a
certain subgroup and induces a group completion. Together with Cazanave’s
result this implies that we have a canonical isomorphism between the two group
completions induced by $\pi_{\mathrm{N}}^{*}$ and $\nu_{\mathbb{P}^{1}}$,
respectively. The main result is proven in theorem 129.

###  6.1 Motivic Brouwer degree

In [21] Morel describes the analog of the topological Brouwer degree map in
$\mathbb{A}^{1}$-homotopy theory. For pointed endomorphisms of
$\mathbb{P}^{1}$ it defines a homomorphism

$\deg^{\mathbb{A}^{1}}\colon[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathbb{A}^{1}}\to\mathrm{GW}(k).$

We recall that by the work of Cazanave [13, Corollary 3.10] and Morel [22,
Theorem 7.36] the map given by

$\displaystyle f\mapsto\left(\deg^{\mathbb{A}^{1}}(f),\mathrm{res}(f)\right),$

where $\mathrm{res}(f)$ denotes Cazanave’s resultant of [13] described in
proposition 30, induces an isomorphism of groups

(110)
$\rho\colon[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathbb{A}^{1}}\xrightarrow{\cong}\mathrm{GW}(k)\times_{k^{\times}/k^{\times
2}}k^{\times}.$

Since our definition of $\deg$ is compatible with the notion of degree of a
rational function used by Cazanave in [13], the work of Cazanave and Morel
implies that, for every pointed morphism
$f\colon\mathbb{P}^{1}\to\mathbb{P}^{1}$ we have

$\displaystyle\deg([f])=\mathrm{rank}\left(\deg^{\mathbb{A}^{1}}([f])\right),$

where $\mathrm{rank}$ denotes the homomorphism $\mathrm{GW}(k)\to\Z$ induced
by the rank of a quadratic form. For a pointed morphism
$g\colon\mathcal{J}\to\mathbb{P}^{1}$ with $\xi([g])=[f]$, we have

$\displaystyle\deg([g])=\mathrm{rank}\left(\deg^{\mathbb{A}^{1}}(\xi[g])\right).$

Hence we have the commutative diagram

(117)

where $\mathrm{GW}(k)_{0}\times_{k^{\times}/k^{\times 2}}k^{\times}$ denotes
the kernel of the rank homomorphism.

######  Proposition 118.

The map $\pi_{\mathrm{N}}^{*}$ is injective.

###### Proof.

By lemma 109 we know $\xi\circ\pi_{\mathrm{N}}^{*}=\nu_{\mathbb{P}^{1}}$.
Since $\xi$ is a bijection, it suffices to show that $\nu_{\mathbb{P}^{1}}$ is
injective. The isomorphism $\rho$ fits into the commutative diagram

where $\mathrm{MW}^{s}(k)$ denotes the stable monoid of symmetric bilinear
forms as in [13, Definition 3.8] and $\hat{w}$ is the group completion induced
by the group completion $w\colon\mathrm{MW}^{s}(k)\to\mathrm{GW}(k)$. Since
the vertical maps are isomorphisms by [13, Corollary 3.10] and [22, §7.3],
$\nu_{\mathbb{P}^{1}}$ is injective if and only if $\hat{w}$ is injective. To
show that $\hat{w}$ is injective, it suffices to show that the group
completion $w\colon\mathrm{MW}^{s}(k)\to\mathrm{GW}(k)$ is injective. Since
$\mathrm{MW}^{s}(k)$ satisfies the cancellation property by the definition of
$\mathrm{MW}^{s}(k)$ in [13, Definition 3.8], respectively by Witt’s
cancellation theorem, the map $w$ is indeed injective. This proves the
assertion. ∎

######  Proposition 119.

Let $u,v\in k^{\times}$. If $u\neq v$, then $[g_{u,1}]\neq[g_{v,1}]$.

###### Proof.