# Cancellation and splitting of Symplectic modules in the critical range and
Euler class group

Rakesh Pawar  UMPA, École normale supérieure de Lyon (ENS de Lyon), France
<EMAIL_ADDRESS>and  Husney Parvez Sarwar  Department of Mathematics,
Indian Institute of Technology Kharagpur, Kharagpur 721302, West Bengal, India
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
###### Abstract.

In this paper, we shall discuss the cancellation of symplectic modules and the
splitting of symplectic modules. The symplectic cancellation can be thought of
as a projective module cancellation which is a recent result of Fasel. The
symplectic splitting is similar to Murthy’s splitting theorem. To prove the
cancellation and splitting, we shall carefully analyze the Postnikov towers in
the $\mathbb{A}^{1}$-homotopy category. Then we shall prove the vanishing of
top cohomology with coefficients in some homotopy sheaf. As another
application of the vanishing results, we answer partially a question of Mrinal
Das about the isomorphism of $(d-1)$-th Euler class group and $(d-1)$-th Chow
group, where $d$ is the dimension of the underlying smooth affine variety.

###### Key words and phrases:

projective modules; motivic homotopy theory; Euler class groups

###### 2020 Mathematics Subject Classification:

13C10, 14F20, 14F35

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Ingredients from motivic homotopy theory
    2. 2.2 Symplectic groups and symplectic bundles
    3. 2.3 Affine representability results for symplectic groups
  3. 3 Some vanishing results
  4. 4 Cancellation and splitting of symplectic modules
    1. 4.1 Cancellation of symplectic module
    2. 4.2 Splitting of symplectic module
  5. 5 Comparison of Euler class group and Chow group

##  1\. Introduction

Let us begin with two classical results. Let $R$ be a commutative Noetherian
ring of dimension $d$ and let $P$ be a finitely generated projective
$R$-module of rank $r$. For $r>d$, Serre’s theorem ([23]) says that $P\cong
Q\oplus R$ (Serre’s splitting theorem) and Bass’ theorem ([10]) says that
$P\oplus Q\cong P^{\prime}\oplus Q$ implies that $P\cong P^{\prime}$ (Bass’s
cancellation theorem). For $rank(P)=\dim R$, the non-trivial tangent bundle
over the real algebraic sphere which is stably free, asserts that both the
above theorems are false in the case $r=d$. However, on an algebraically
closed field, Murthy [22] and Suslin [24] proved the following two results on
splitting and cancellation respectively. Let $R$ be a smooth affine algebra
over an algebraically closed field of dimension $d$ and $P$ a finitely
generated projective $R$-module with $rank(P)=d$. Murthy [22, Theorem 3.8]
proved that $c_{d}(P)=0$ if and only if $P\cong Q\oplus R$ where $c_{d}(P)$ is
the top Chern class of $P$ in the Chow group $CH^{d}({\rm Spec\,}(R))$. Suslin
[24] proved that $P\oplus Q\cong P^{\prime}\oplus Q$ implies that $P\cong
P^{\prime}$ even if $R$ is not a smooth algebra. Recently, Fasel [17] proved
that if $rank(P)=d-1$, then also the cancellation holds, i.e., $P\oplus Q\cong
P^{\prime}\oplus Q$ implies that $P\cong P^{\prime}$. Fasel’s result is also
optimal in the view of Mohan Kumar’s example [19]. Similar to Suslin’s
cancellation conjecture, there is a conjecture for the splitting called
Murthy’s splitting conjecture [5].

######  Conjecture 1.1 (Murthy’s splitting conjecture).

Let $R$ be a smooth affine algebra over an algebraically closed field of Krull
dimension $d\geq 3$. Let $P$ be a projective $R$-module of rank $d-1$. Then
the top Chern class $c_{d-1}(P)=0$ in $CH^{d-1}({\rm Spec\,}(R))$ if and only
if $P\cong Q\oplus R$ for some $R$-module $Q$.

Murthy’s splitting conjecture remains open in general. However when $d=3,4$,
the conjecture has been solved affirmatively by Asok–Fasel (see [3], [5]).
Recently Murthy’s conjecture is resolved in the positive when the field $k$ is
of characteristic 0 by combining the works [3], [5] and [1].

In this paper the objective is two-fold. In the first, we study the symplectic
modules and offer the following two results which are analogous to Suslin’s
cancellation (or Murthy’s splitting) conjecture and Fasel’s theorems in the
context of symplectic modules.

Let us denote by $H(R)$ the standard Hyperbolic rank 2 symplectic module on a
ring $R$.

######  Theorem 1.2.

(co-rank 2 symplectic cancellation) Let $X={\rm Spec\,}R$ be a smooth affine
variety of dimension $2n$ ($n\geq 1$) over an algebraically closed field $k$.
Let $(2n)!$ be invertible in $k.$ Then the map

$V_{2n-2}^{Sp}(X)\to V_{2n}^{Sp}(X)$

sending $[P]\mapsto[P\perp H(R)]$ is injective. More explicitly, if two
symplectic rank $(2n-2)$-projective modules $E$ and $E^{\prime}$ are stably
isomorphic i.e., $E\oplus H(R)\simeq E^{\prime}\oplus H(R)$, then $E\simeq
E^{\prime}$.

######  Theorem 1.3.

(Lower co-rank symplectic splitting) Let $R$ be a smooth affine variety over a
field $k$ and $X={\rm Spec\,}(R)$. Let $P$ be a symplectic $R$-module of rank
$2n$ ($n\geq 1$) and $H(R)$ the free hyperbolic space of rank $2$.

$(1)$ Assume that $dim(R)=2n$. Then $P\cong Q\oplus H(R)$ if and only if the
Euler class $e_{2n}(P)=0$ in $\widetilde{CH}^{2n}(X)$.

$(2)$ Assume that $dim(R)=2n+1$, $k$ is algebraically closed field and
$char(k)=0$. Then $P\cong Q\oplus H(R)$ if and only if the Euler class
$e_{2n}(P)=0$ in $\widetilde{CH}^{2n}(X)$.

$(3)$ Assume $dim(R)=3$, $rank(P)=2$ and $k=\overline{\mathbb{F}}_{p}$. Then
$P\cong H(R)$ if and only if $e_{2}(P)=0$ in $\widetilde{CH}^{2}(X)$.

The symplectic modules have been studied by Vaserstein [26], Basu–Rao [11] and
Du [14]. The approach we take throughout this paper is to exploit the
obstruction theoretic machinery developed by Morel-Voevodsky [20] and Morel
[21, Appendix B] which reduces the lifting problems to understanding Nisnevich
cohomology groups of homotopy sheaves. This strategy is successfully deployed
by Asok-Fasel [3], [5] and also Du [14] and Asok-Bachmann-Hopkins [1] amongst
others in a series of works to study the classification/splitting/enumeration
problems for projective modules on smooth affine algebras over a field. In
this paper, we use the vanishing theorem of Fasel [17] at crucial places and
in addition to that use the following result which lists the conditions under
which the top cohomology groups vanish of certain homotopy sheaves of Quadrics
on smooth affine variety. We believe that the following result is of
independent interest and would serve a purpose in other obstruction theoretic
arguments.

######  Theorem 1.4.

Let $X$ be a smooth affine variety over an algebraically closed field $k$ of
Krull dimension $d$. Then

  1. $(1)$

for $d=2$, $H^{2}(X,\pi_{2}^{\mathbb{A}^{1}}(Q_{2}))=0.$

  2. $(2)$

for $d=3$, $H^{3}(X,\pi_{3}^{\mathbb{A}^{1}}(Q_{4}))=0$ if
$k=\overline{\mathbb{F}}_{p}$.

  3. $(3)$

for $d=4$, $H^{4}(X,\pi_{4}^{\mathbb{A}^{1}}(Q_{6}))=0.$

  4. $(4)$

for $d\geq 5$, $H^{d}(X,\pi_{d}^{\mathbb{A}^{1}}(Q_{2d-2}))=0,$ provided
$char(k)=0$.

  5. $(5)$

if $d\geq 5$, then $H^{d}(X,{\bm{\pi}}^{\mathbb{A}^{1}}_{d}(Q_{2d-2}))$ is
$(d-1)!(d-2)!$-torsion.

  6. $(6)$

for $d\geq 5$, $H^{d}(X,\pi_{d}^{\mathbb{A}^{1}}(Q_{2d-2}))=0,$ provided
Conjecture 3.8 (due to Asok–Fasel) holds.

Some comments about the vanishing of the cohomology group
$H^{d}(X,\pi_{d}^{\mathbb{A}^{1}}(Q_{2d-2}))$ are in order.

######  Remark 1.5.

  1. (1)

The case $d=2$ complements the result proved in [17, Theorem 3] as one notices
that $Q_{2}$ is $\mathbb{A}^{1}$-equivalent to $\mathbb{A}^{2}-0$.

  2. (2)

Conjecture 3.8 roughly says that the Nisnevich sheaf
$\pi_{d-1}^{\mathbb{A}^{1}}(\mathbb{A}^{d-1}-0)$ is sandwiched between two
known sheaves ${\mathbf{K}}^{\rm M}_{d+1}/24$ and $\mathbf{GW}^{d-1}_{d}$.

Perhaps unsurprisingly, the above vanishing result, Theorem 1.4 has
application in addressing the following question posed by Mrinal Das [13].

######  Question 1.6.

([13, Question 7.1]) Let $p\neq 2$ be a prime and let
$k=\overline{\mathbb{F}}_{p}$. Let $X={\rm Spec\,}R$ be a smooth affine
variety of dimension $d\geq 4$. Is the natural map

$\psi:E^{d-1}(R)\to CH^{d-1}(X)$

an isomorphism?

We note that the question was answered in the case $d=3$ in [13, Theorem 7.2].
We have the following result which associates the positive answer to Question
1.6 with the vanishing of certain vanishing of cohomology.

######  Theorem 1.7.

Let $X={\rm Spec\,}(R)$ be a smooth affine variety of Krull dimension $d\geq
4$ over an algebraically closed field $k$. Consider the map

$\psi:E^{d-1}(R)\to CH^{d-1}({\rm Spec\,}R).$

Then

  1. (1)

the map $\psi$ is always surjective.

  2. (2)

for $d=4$, $\psi$ is an isomorphism.

  3. (3)

for $d\geq 5$, $\psi$ is an isomorphism provided $char(k)=0$.

  4. (4)

the kernel is $(d-1)!(d-2)!$-torsion if $d\geq 5.$

  5. (5)

further for $d\geq 5$, if Conjecture 3.8 holds, then the map $\psi$ is
injective.

##  2\. Preliminaries

In this section, we recollect some basic notions and results that will form
the framework for the results proved in this paper.

###  2.1. Ingredients from motivic homotopy theory

First, we recall some basic notions in the homotopy theory.

Let $S$ be a finite dimension Noetherian scheme. $Sm/S$ denotes the category
of finite type smooth $S$-schemes. $(Sm/S)_{Nis}$ denotes the site with the
Nisnevich topology on the category $Sm/S$. The category of simplicial
Nisnevich sheaves $Spc_{S}:=\rm{\Delta^{op}Shv}(Sm/S)_{Nis}$ is a simplicial
model category with the model structure as in [20, Theorem 1.4]. By [20,
section 3.2, page 105], the above simplicial model category
$Spc^{\mathbb{A}^{!}}_{S}:=\rm{\Delta^{op}Shv}(Sm/S)_{Nis}$ is endowed with
the $\mathbb{A}^{1}$-model category structure. An object of
$Spc^{\mathbb{A}^{!}}_{S}$ is simply referred to as a Space. Inverting the
$\mathbb{A}^{1}$-weak equivalences on $Spc^{\mathbb{A}^{!}}_{S}$ the resulting
homotopy category is the homotopy category of smooth schemes over $S$ denoted
by $\mathcal{H}^{\mathbb{A}^{1}}(S).$ For the current paper, we take $S={\rm
Spec\,}k$ for a field $k$ and denote
$\mathcal{H}^{\mathbb{A}^{1}}(k):=\mathcal{H}^{\mathbb{A}^{1}}({\rm Spec\,}k)$
for the $\mathbb{A}^{1}$-homotopy category of the field $k.$ For any object
$\mathcal{X},\mathcal{Y}\in Spc^{\mathbb{A}^{!}}_{k}$, we denote
$[\mathcal{X},\mathcal{Y}]_{\mathbb{A}^{1}}:={\rm
Hom}_{\mathcal{H}^{\mathbb{A}^{1}}(k)}(\mathcal{X},\mathcal{Y})$. For a smooth
$k$-scheme $X$, we denote by $X$ itself the corresponding object ${\rm
Hom}_{Spc^{\mathbb{A}^{1}}_{k}}(-,X)$ in $Spc^{\mathbb{A}^{!}}_{k}$.
Similarly, we can define the category of pointed spaces
$Spc^{\mathbb{A}^{1}}_{\bullet,k}$, and the resulting pointed
$\mathbb{A}^{1}$-homotopy category
$\mathcal{H}^{\mathbb{A}^{1}}_{\bullet}(k)$.

######  Definition 2.1.

Let $\mathcal{X}$ be a pointed space. For $n\geq 1$,
${\bm{\pi}}^{\mathbb{A}^{1}}_{n}(\mathcal{X})$ denotes the $n^{th}$
$\mathbb{A}^{1}$-homotopy sheaf defined as the Nisnevich sheafification of the
presheaf

$U\mapsto[S_{s}^{n}\wedge U_{+},\mathcal{X}]_{\mathbb{A}^{1}}$

where $U\in Sm/k$.

######  Remark 2.2.

For a pointed space $\mathcal{X}$,
${\bm{\pi}}^{\mathbb{A}^{1}}_{1}(\mathcal{X})$ is a sheaf of groups which is
strongly $\mathbb{A}^{1}$-invariant and for $n\geq 2$,
${\bm{\pi}}^{\mathbb{A}^{1}}_{n}(\mathcal{X})$ is strictly
$\mathbb{A}^{1}$-invariant sheaf of abelian groups. [21, Theorem 5.1, Theorem
4.47]

The main tool we will be using in the current paper is the existence of the
Moore-Postnikov tower for a given fiber sequence in the homotopy category.

######  Proposition 2.3.

Let $F\to E\xrightarrow{q}B$ be an $\mathbb{A}^{1}$-fiber sequence where
$F,E,B$ are pointed $\mathbb{A}^{1}$-fibrant connected spaces. Let
$G:={\bm{\pi}}^{\mathbb{A}^{1}}_{1}(B)$. Then for each $n$ there are maps
$q_{n}:E_{n}\to E_{n-1}$, $i_{n}:E\to E_{n}$ such that there is a commutative
diagram

$\textstyle{E_{n+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q_{n+1}}$$\scriptstyle{p_{n+1}}$$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{n+1}}$$\scriptstyle{i_{n}}$$\scriptstyle{i_{n-1}}$$\textstyle{E_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{n}}$$\scriptstyle{q_{n}}$$\textstyle{B}$$\textstyle{E_{n-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q_{n-1}}$$\scriptstyle{p_{n-1}}$

such that

  1. (1)

$p_{0}:E_{0}\to B$ is an $\mathbb{A}^{1}$-weak equivalence.

  2. (2)

for each $n$, $p_{n}\circ i_{n}=q$.

  3. (3)

The morphism $q_{n}$ is an $\mathbb{A}^{1}$-fibration with homotopy fiber
$K({\bm{\pi}}^{\mathbb{A}^{1}}_{n}(F),n)$. Further there is a unique map
$E_{n-1}\to K^{G}({\bm{\pi}}^{\mathbb{A}^{1}}_{n}(F),n+1)$ such that the
diagram

---
$\textstyle{E_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q_{n}}$$\textstyle{BG\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{n-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K^{G}({\bm{\pi}}^{\mathbb{A}^{1}}_{n}(F),n+1)}$

is hopotopy pullback.

  4. (4)

The morphism $E\to\operatorname{{\rm{hlim}}}\limits_{n}E_{n}$ induced by the
maps $\\{i_{n}:E\to E_{n}\\}_{n\geq 1}$ is an $\mathbb{A}^{1}$-weak
equivalence.

######  Remark 2.4.

In this paper, we will apply Proposition 2.3 to the $\mathbb{A}^{1}$-fiber
sequence

$F^{\prime}_{n}\to BSp_{2n}\to BSp$

for various natural numbers $n$. Here $F^{\prime}_{n}$ denotes the homotopy
fiber of the natural map

$BSp_{2n}\to BSp.$

###  2.2. Symplectic groups and symplectic bundles

Let us quickly recall symplectic groups. Let $R$ be a commutative ring with
the unity and ${\rm GL}\,_{n}(R)$ the group of all invertible matrices over
$R$. Let $\psi_{2}=\left(\begin{smallmatrix}0&1\\\
-1&0\end{smallmatrix}\right)$ and let $\psi_{2n}=\perp_{1}^{n}\psi_{2}$ be the
standard symplectic form of rank $2n$. Now we recall the definition of the
even-order symplectic groups.

${Sp}_{2n}(R)=\\{M\in GL_{2n}(R):{{}^{t}M}\psi M=\psi\\}.$

######  Definition 2.5.

A symplectic $R$-module is a pair $(P,\phi)$ where $P$ is a finitely generated
projective $R$-module and $\phi:P\times P\xrightarrow{}R$ is a bi-linear map
with $\phi(x,x)=0$ for all $x\in P$. We have an adjoint map
$adj:P\xrightarrow{}P^{*}:=Hom(P,R)$ given as $x\xrightarrow{}\phi(x,-)$.

  1. (1)

For an even rank, symplectic $R$-module $P$, we say that $P$ is non-degenerate
if the adjoint map $adj$ is an isomorphism.

  2. (2)

For an odd rank symplectic $R$-module $P$, we say that $P$ is non-degenerate
if the adjoint map $adj$ splits with the kernel of rank one.

In the paper symplectic modules are always non-degenerate.

###  2.3. Affine representability results for symplectic groups

Let $k$ be an infinite field. Let $X={\rm Spec\,}(R)$ be a smooth affine
variety over $k$ of dimension $n$. For every integer $r\geq 1$, let
$V_{r}^{Sp}(X)$ be the set of isomorphism classes of rank $r$ symplectic
modules over $X.$ From ([9, Theorem 4.1.3, page 1025]), we have

$V_{r}^{Sp}(X)\simeq[X,BSp_{r}]_{\mathbb{A}^{1}}$

for even integer $r\geq 2$.

##  3\. Some vanishing results

In this section, we organize some cohomology vanishing results for certain
homotopy sheaves of a particular class of spheres in the Motivic homotopy
category.

For a field $k$, let us recall the affine quadrics

$Q_{2n-1}={\rm Spec\,}\dfrac{k[x_{1},y_{1},\cdots,x_{n},y_{n}]}{\sum
x_{i}y_{i}-1}$

and

$Q_{2n}={\rm Spec\,}\dfrac{k[x_{1},y_{1},\cdots,x_{n},y_{n},z]}{\sum
x_{i}y_{i}-z(1-z)}.$

We consider $Q_{2n}$ and $Q_{2n-1}$ as pointed spaces in the motivic homotopy
category and their $\mathbb{A}^{1}$-homotopy sheaves. In this section we
mainly consider the following question.

######  Question 3.1.

(Vanishing of $H^{d}(X,{\bm{\pi}}^{\mathbb{A}^{1}}_{d}(Q_{2d-2}))$) Let $X$ be
an affine variety over an algebraically closed field $k$ of dimension $d$.
Under what conditions on $k$, $X$, $d$, is

$H^{d}(X,\pi_{d}^{\mathbb{A}^{1}}(Q_{2d-2}))=0?$

The need to establish this will be clearer in the section 5. In the vain of
answering the above question, we prove the following proposition.

######  Proposition 3.2.

Let $k$ be a field that is not formally real. Then the map of Nisnevich
sheaves

${\bm{\pi}}^{\mathbb{A}^{1}}_{d-1}(\mathbb{A}^{d-1}-0))\to{\bm{\pi}}^{\mathbb{A}^{1}}_{d}(Q_{2d-2})$

  1. (1)

is an epimorphism for $d\geq 4$ and

  2. (2)

an isomorphism for $d\geq 5.$

###### Proof.

By [8, Proposition 5.3.10 (1)] for $d\geq 3$ we have a fiber sequence

$Q_{2d-3}\to\Omega Q_{2d-2}\to\Omega Q_{4d-5}$

hence the exact sequence of sheaves of homotopy groups

$\textstyle{{\bm{\pi}}^{\mathbb{A}^{1}}_{d+1}(Q_{4d-5})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\bm{\pi}}^{\mathbb{A}^{1}}_{d-1}(Q_{2d-3})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\bm{\pi}}^{\mathbb{A}^{1}}_{d}(Q_{2d-2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\bm{\pi}}^{\mathbb{A}^{1}}_{d}(Q_{4d-5})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{{\bm{\pi}}^{\mathbb{A}^{1}}_{d-2}(Q_{2d-3})}$$\textstyle{{\bm{\pi}}^{\mathbb{A}^{1}}_{d}(\mathbb{A}^{2d-2}-0)}$

The $\mathbb{A}^{1}$-equivalence $Q_{4d-5}\simeq
S^{2d-3}\wedge\mathbb{G}_{m}^{2d-2}\simeq\mathbb{A}^{2d-2}-0$ induces
isomorphism

${\bm{\pi}}^{\mathbb{A}^{1}}_{d}(Q_{4d-5})\simeq{\bm{\pi}}^{\mathbb{A}^{1}}_{d}(\mathbb{A}^{2d-2}-0)\simeq
0\ \text{for}\ d\geq 4$

and

${\bm{\pi}}^{\mathbb{A}^{1}}_{d+1}(Q_{4d-5})\simeq{\bm{\pi}}^{\mathbb{A}^{1}}_{d+1}(\mathbb{A}^{2d-2}-0)\simeq
0\ \text{for}\ d\geq 5.$

Hence we have epimorphism

${\bm{\pi}}^{\mathbb{A}^{1}}_{d-1}(Q_{2d-3})\to{\bm{\pi}}^{\mathbb{A}^{1}}_{d}(Q_{2d-2})$

of Nisnevich sheaves if $d\geq 4$ and isomorphism for $d\geq 5$. Since
$Q_{2d-3}$ is $\mathbb{A}^{1}$-equivalent to $\mathbb{A}^{d-1}-0,$ we have for
$d\geq 4$ the epimorphism of Nisnevich sheaves

${\bm{\pi}}^{\mathbb{A}^{1}}_{d-1}(\mathbb{A}^{d-1}-0))\to{\bm{\pi}}^{\mathbb{A}^{1}}_{d}(Q_{2d-2})$

and it is an isomorphism for $d\geq 5.$

∎

We recall the following result from [1].

######  Proposition 3.3.

(Asok–Bachmann–Hopkins [1]) Let $R$ be a smooth affine algebra over an
algebraically closed field $k$ with $char(k)=0$ of dimension $d$ ($d\geq 3$).
Set $X={\rm Spec\,}(R)$. Then

$H^{d}(X,{\bm{\pi}}^{\mathbb{A}^{1}}_{d-1}(\mathbb{A}^{d-1}-0))=0.$

###### Proof.

This is implicit in the proof of [1, Theorem 7.1.1] ∎

######  Proposition 3.4.

Let $R$ be a smooth affine algebra over an algebraically closed field $k$ of
dimension $d$ ($d\geq 5$). Set $X={\rm Spec\,}(R)$. Then
$H^{d}(X,{\bm{\pi}}^{\mathbb{A}^{1}}_{d-1}(\mathbb{A}^{d-1}-0))$ is
$(d-1)!(d-2)!$-torsion.

###### Proof.

The proof follows along the lines as in [17, Proposition 5.0.3]. We provide
the details for the convenience of the reader.

Consider the Suslin map $u_{d-1}:\mathbb{A}^{d-1}-0\to GL_{d-1}\to GL$ from
[17, section 2.4] which factors through

$\mathbb{A}^{d-1}-0\to\Omega_{\mathbb{P}^{1}}^{-d+1}\to GL$

Now by [7, §4.4]
${\bm{\pi}}^{\mathbb{A}^{1}}_{d-1}(\Omega_{\mathbb{P}^{1}}^{-d+1})\simeq
GW_{d}^{d-1}$ and by [3, Theorem 3.7.1] $H^{d}(X,GW_{d}^{d-1})=0$. In
particular this implies that the induced map on cohomology

$H^{d}(X,{\bm{\pi}}^{\mathbb{A}^{1}}_{d-1}(\mathbb{A}^{d-1}-0))\to
H^{d}(X,{\bm{\pi}}^{\mathbb{A}^{1}}_{d-1}(GL_{d-1}))\to
H^{d}(X,{\bm{\pi}}^{\mathbb{A}^{1}}_{d-1}(GL))$

is 0. We claim that the kernel of the 2nd map

${\rm Ker\ }(H^{d}(X,{\bm{\pi}}^{\mathbb{A}^{1}}_{d-1}(GL_{d-1}))\to
H^{d}(X,{\bm{\pi}}^{\mathbb{A}^{1}}_{d-1}(GL))$

is $(d-1)!$-torsion. Hence, the image of the group
$H^{d}(X,{\bm{\pi}}^{\mathbb{A}^{1}}_{d-1}(\mathbb{A}^{d-1}-0))$ is
$(d-1)!$-torsion.

Proof of the claim:

First we observe that the kernel

${\rm Ker\ }(H^{d}(X,{\bm{\pi}}^{\mathbb{A}^{1}}_{d-1}(GL_{d-1}))\to
H^{d}(X,{\bm{\pi}}^{\mathbb{A}^{1}}_{d-1}(GL_{d}))\to
H^{d}(X,{\bm{\pi}}^{\mathbb{A}^{1}}_{d-1}(GL_{d+1}))\to
H^{d}(X,{\bm{\pi}}^{\mathbb{A}^{1}}_{d-1}(GL)))$

is $(d-1)!$-torsion. (Actually the 2nd and 3rd maps are isomorphism). The
third map is isomorphism as it follows from the isomorphism of sheaves
${\bm{\pi}}^{\mathbb{A}^{1}}_{d-1}(GL_{d+1})\simeq{\bm{\pi}}^{\mathbb{A}^{1}}_{d-1}(GL)$.
To see the 2nd map is isomorphism, consider the fiber sequence

$GL_{d}\to GL_{d+1}\to\mathbb{A}^{d+1}-0$

hence the long exact sequence

${\bm{\pi}}^{\mathbb{A}^{1}}_{d}(GL_{d+1})\to{\bm{\pi}}^{\mathbb{A}^{1}}_{d}(\mathbb{A}^{d+1}-0))\to{\bm{\pi}}^{\mathbb{A}^{1}}_{d-1}(GL_{d})\to{\bm{\pi}}^{\mathbb{A}^{1}}_{d-1}(GL_{d+1})\to{\bm{\pi}}^{\mathbb{A}^{1}}_{d-1}(\mathbb{A}^{d+1}-0)=0.$

Breaking the long exact sequence as

${\bm{\pi}}^{\mathbb{A}^{1}}_{d}(GL_{d+1})\to{\bm{\pi}}^{\mathbb{A}^{1}}_{d}(\mathbb{A}^{d+1}-0))\to
A\to 0$

and

$0\to
A\to{\bm{\pi}}^{\mathbb{A}^{1}}_{d-1}(GL_{d})\to{\bm{\pi}}^{\mathbb{A}^{1}}_{d-1}(GL_{d+1}).$

By [17, Lemma 2.4.1] $H^{d}(X,A)$ is $d!$-torsion. By [16] the cohomology
group

$H^{d}(X,{\bm{\pi}}^{\mathbb{A}^{1}}_{d}(\mathbb{A}^{d+1}-0))\simeq
H^{d}(X,{\mathbf{K}}^{\rm MW}_{d+1})\simeq H^{d}(X,{\mathbf{K}}^{\rm
M}_{d+1})$

is uniquely divisible. Hence $H^{d}(X,A)=0$ and thus,

$H^{d}(X,{\bm{\pi}}^{\mathbb{A}^{1}}_{d-1}(GL_{d}))\to
H^{d}(X,{\bm{\pi}}^{\mathbb{A}^{1}}_{d-1}(GL_{d+1}))$

is an ismorphism.

For the 1st map to see that

${\rm Ker\ }(H^{d}(X,{\bm{\pi}}^{\mathbb{A}^{1}}_{d-1}(GL_{d-1}))\to
H^{d}(X,{\bm{\pi}}^{\mathbb{A}^{1}}_{d-1}(GL_{d})))$

is $(d-1)!$-torsion, consider the fiber sequence

$GL_{d-1}\to GL_{d}\to\mathbb{A}^{d}-0$

hence the long exact sequence

${\bm{\pi}}^{\mathbb{A}^{1}}_{d}(GL_{d})\to{\bm{\pi}}^{\mathbb{A}^{1}}_{d}(\mathbb{A}^{d}-0)\to{\bm{\pi}}^{\mathbb{A}^{1}}_{d-1}(GL_{d-1})\to{\bm{\pi}}^{\mathbb{A}^{1}}_{d-1}(GL_{d}).$

By [17, Lemma 2.3.3], cokernel of the map

${\rm Coker\ }(H^{d}(X,{\bm{\pi}}^{\mathbb{A}^{1}}_{d}(GL_{d}))\to
H^{d}(X,{\bm{\pi}}^{\mathbb{A}^{1}}_{d}(\mathbb{A}^{d}-0)))$

is $(d-1)!$-torsion. Hence the kernel of the map

${\rm Ker\ }(H^{d}(X,{\bm{\pi}}^{\mathbb{A}^{1}}_{d-1}(GL_{d-1}))\to
H^{d}(X,{\bm{\pi}}^{\mathbb{A}^{1}}_{d-1}(GL_{d})))$

is $(d-1)!$. Thus, the claim is proved.

Further consider the maps from [17, section 2.4]

$\mathbb{A}^{d-1}-0\xrightarrow{u_{d-1}^{\prime}}GL_{d-1}\xrightarrow{r_{d-1}}\mathbb{A}^{d-1}-0$
$(x_{1},\cdots,x_{d-1})\mapsto(x_{1}^{(d-2)!},\cdots,x_{d-1})$

which induce the map

$H^{d}(X,{\bm{\pi}}^{\mathbb{A}^{1}}_{d-1}(\mathbb{A}^{d-1}-0))\xrightarrow{}H^{d}(X,{\bm{\pi}}^{\mathbb{A}^{1}}_{d-1}(GL_{d-1}))\to
H^{d}(X,{\bm{\pi}}^{\mathbb{A}^{1}}_{d-1}(\mathbb{A}^{d-1}-0))$

given by multiplication by $(d-2)!$, since $d\geq 5$ and by the previous
discussion the image of the first map $(d-1)!$-torsion.

Thus combining the above observations, we get that the group
$H^{d}(X,{\bm{\pi}}^{\mathbb{A}^{1}}_{d-1}(\mathbb{A}^{d-1}-0))$ is
$(d-1)!(d-2)!$-torsion. ∎

######  Remark 3.5.

By [17, Theroem 3] if in addition we assume that $d!\in k^{\times}$, then
$H^{d}(X,{\bm{\pi}}^{\mathbb{A}^{1}}_{d}(\mathbb{A}^{d}-0))=0$. In the above
argument with the additional hypothesis of $d!\in k^{\times}$, we can improve
on the torsion in the sense that, we can conclude that the group
$H^{d}(X,{\bm{\pi}}^{\mathbb{A}^{1}}_{d-1}(\mathbb{A}^{d-1}-0))$ is
$(d-2)!$-torsion.

As a consequence we have the following result.

######  Corollary 3.6.

Let $R$ be a smooth affine algebra over an algebraically closed field $k$. Set
$X={\rm Spec\,}(R)$. Then

  1. (1)

if $char(k)=0$ and $d\geq 4$, then
$H^{d}(X,{\bm{\pi}}^{\mathbb{A}^{1}}_{d}(Q_{2d-2}))=0$ and

  2. (2)

if $d\geq 5$, then $H^{d}(X,{\bm{\pi}}^{\mathbb{A}^{1}}_{d}(Q_{2d-2}))$ is
$(d-1)!(d-2)!$-torsion.

###### Proof.

  1. (1)

This follows from Proposition 3.3 and Proposition 3.2.

  2. (2)

This follows from Proposition 3.4 and Proposition 3.2.

∎

######  Remark 3.7.

In general, though we suspect it to be so, it seems difficult to determine
whether the cohomology groups
$H^{d}(X,{\bm{\pi}}^{\mathbb{A}^{1}}_{d-1}(\mathbb{A}^{d-1}-0))$ or
$H^{d}(X,{\bm{\pi}}^{\mathbb{A}^{1}}_{d}(Q_{2d-2}))$ are trivial or not.

We note that the vanishing of the cohomology groups

$H^{d}(X,{\bm{\pi}}^{\mathbb{A}^{1}}_{d-1}(\mathbb{A}^{d-1}-0))\ \text{or}\
H^{d}(X,{\bm{\pi}}^{\mathbb{A}^{1}}_{d}(Q_{2d-2}))$

follows from the following deep conjecture

######  Conjecture 3.8.

(Asok–Fasel [4, Conjecture 7]) For $d\geq 5,$ the following sequence

$(*)\ \ \ {\mathbf{K}}^{\rm
M}_{d+1}/24\to{\bm{\pi}}^{\mathbb{A}^{1}}_{d-1}(\mathbb{A}^{d-1}-0)\to\mathbf{GW}^{d-1}_{d}$

is exact and after $(d-4)$-fold contractions it is surjective on the right.

######  Proposition 3.9.

Assume Conjecture 3.8 holds. Let $k$ be an algebraically closed field. Let $X$
be a smooth affine variety over $k$ of dimension $d\geq 5$. Then

$H^{d}(X,\pi_{d}^{\mathbb{A}^{1}}(Q_{2d-2}))=0.$

###### Proof.

By Proposition 3.2, we have the epimorphism of sheaves

${\bm{\pi}}^{\mathbb{A}^{1}}_{d-1}(\mathbb{A}^{d-1}-0))\to{\bm{\pi}}^{\mathbb{A}^{1}}_{d}(Q_{2d-2}).$

Hence it suffices to show that
$H^{d}(X,{\bm{\pi}}^{\mathbb{A}^{1}}_{d-1}(\mathbb{A}^{d-1}-0))=0$.

In view of conjecture 3.8, for $d\geq 5,$ the following sequence

${\mathbf{K}}^{\rm
M}_{d+1}/24\to\pi_{d-1}^{\mathbb{A}^{1}}(\mathbb{A}^{d-1}-0)\to\mathbf{GW}^{d-1}_{d}$

is exact and surjective on the right after $(d-4)$-fold contractions. Hence it
is enough to prove the vanishing of $H^{d}(X,\mathbf{GW}^{d-1}_{d})$ and
$H^{d}(X,{\mathbf{K}}^{\rm M}_{d+1}/24)$.

The vanishing of $H^{d}(X,\mathbf{GW}^{d-1}_{d})$ follows from [3, Proposition
3.6.4]. For this we will need to show that $Ch^{d}(X)=CH^{d}(X)/2=0$. Since
the cohomological dimension of $X$ is atmost $d$, we can observe that

$Ch^{d}(X)=CH^{d}(X)/2\simeq H^{d}(X,{\mathbf{K}}^{\rm M}_{d}/2).$

The vanishing of $H^{d}(X,{\mathbf{K}}^{\rm M}_{d}/2)$ follows from the
Gersten complex for the sheaf ${\mathbf{K}}^{\rm M}_{d}/2$. The group
$H^{d}(X,{\mathbf{K}}^{\rm M}_{d}/2)$ is a quotient of $\bigoplus_{x\in
X^{(d)}}k(x)^{*}/2=0$, since $k(x)=k$ is an algebraically closed field.

It remains to show the vanishing of $H^{d}(X,{\mathbf{K}}^{\rm M}_{d+1}/24)$.
This follows from the Gersten complex for the sheaf ${\mathbf{K}}^{\rm
M}_{d+1}/24$. The cohomology group $H^{d}(X,{\mathbf{K}}^{\rm M}_{d+1}/24)$ is
a quotient of $\bigoplus_{x\in X^{(d)}}k(x)^{*}/24=0$ since $k(x)=k$ is an
algebraically closed field. ∎

For the rest of this section we answer the Question 3.1 for $d=2,3$ and 4.

######  Proposition 3.10.

Let $X$ be a smooth affine variety over an algebraically closed field of
dimension $d=2$. Then

$H^{2}(X,{\bm{\pi}}^{\mathbb{A}^{1}}_{2}(Q_{2}))=0.$

###### Proof.

From [2, Theorem 2], we have
$Q_{2}\sim_{\mathbb{A}^{1}}S^{1}\wedge\mathbb{G}_{m}\sim_{\mathbb{A}^{1}}\mathbb{P}^{1}.$

We have an $\mathbb{A}^{1}$-fiber sequence

$\mathbb{A}^{2}-\\{0\\}\to\mathbb{P}^{1}\to B\mathbb{G}_{m}$

which induces the long exact sequence of sheaves

$\cdots\to{\bm{\pi}}^{\mathbb{A}^{1}}_{3}(B\mathbb{G}_{m})\to{\bm{\pi}}^{\mathbb{A}^{1}}_{2}(\mathbb{A}^{2}-0)\to{\bm{\pi}}^{\mathbb{A}^{1}}_{2}(\mathbb{P}^{1})\to{\bm{\pi}}^{\mathbb{A}^{1}}_{2}(B\mathbb{G}_{m})\to\cdots$

One observes that ${\bm{\pi}}^{\mathbb{A}^{1}}_{3}(B\mathbb{G}_{m})=0$ and
${\bm{\pi}}^{\mathbb{A}^{1}}_{2}(B\mathbb{G}_{m})=0,$ hence we have the
isomorphism

${\bm{\pi}}^{\mathbb{A}^{1}}_{2}(\mathbb{A}^{2}-0)\xrightarrow{\sim}{\bm{\pi}}^{\mathbb{A}^{1}}_{2}(\mathbb{P}^{1})$

of sheaves. Thus, it is enough to show that
$H^{2}(X,{\bm{\pi}}^{\mathbb{A}^{1}}_{2}(\mathbb{A}^{2}-0))=0$. Further, since
$\mathbf{SL}_{2}\sim_{\mathbb{A}^{1}}\mathbb{A}^{2}-0$, we have

${\bm{\pi}}^{\mathbb{A}^{1}}_{2}(\mathbb{A}^{2}-0)\simeq{\bm{\pi}}^{\mathbb{A}^{1}}_{2}(\mathbf{SL}_{2}).$

Hence it is enough to show that
$H^{2}(X,{\bm{\pi}}^{\mathbb{A}^{1}}_{2}(\mathbf{SL}_{2}))=0$. From [5, page
15], we have the following short exact sequence of the form

$0\to\mathbf{T}_{4}^{\prime}\to\pi_{2}^{\mathbb{A}^{1}}(\mathbf{SL}_{2})\to\mathbf{GW}^{2}_{3}\to
0.$

Hence we get the long exact sequence of cohomology groups

$H^{2}(X,\mathbf{T}^{\prime}_{4})\to
H^{2}(X,\pi_{2}^{\mathbb{A}^{1}}(\mathbf{SL_{2}}))\to
H^{2}(X,\mathbf{GW}^{2}_{3})\to H^{3}(X,\mathbf{T}_{4}^{\prime})=0.$

Here $H^{3}(X,\mathbf{T}_{4}^{\prime})=0$, since cohomological dimension of
$X$ is 2. This reduces the proposition to show that under the assumptions of
the proposition

$H^{2}(X,\mathbf{T}^{\prime}_{4})=0\ \ \text{and}\ \
H^{2}(X,\mathbf{GW}^{2}_{3})=0.$

Now we have the following sequence by [5, Theorem 3]

$\mathbf{I}^{5}\to\mathbf{T}^{\prime}_{4}\to\mathbf{S}_{4}^{\prime}\to 0$

which is exact (not necessarily on the left). Hence again by the long exact
seqeuence of the cohomologies and also that the Nisnevich cohomological
dimension of $X$ is atmost 2, we get

$H^{2}(X,{\mathbf{K}}^{\rm M}_{4}/12)\twoheadrightarrow
H^{2}(X,\mathbf{S}_{4}^{\prime})$

is surjective.

We calculate $H^{2}(X,{\mathbf{K}}^{\rm M}_{4}/12)$ using the Gersten complex
resolution for ${\mathbf{K}}^{\rm M}_{4}/12.$ Thus, it will a quotient of

$\bigoplus_{x\in X^{(2)}}{\mathbf{K}}^{\rm M}_{2}(k(x))/12.$

We observe that for $x\in X^{(2)}$, ${\mathbf{K}}^{\rm M}_{2}(k(x))/12=0$
follows from [27, Chapter VI, Corollary 1.3.1, Theorem 1.6] since $k(x)=k$ is
an algebraically closed field. Thus, we get $H^{2}(X,{\mathbf{K}}^{\rm
M}_{4}/12)=0$, and hence $H^{2}(X,\mathbf{S}_{4}^{\prime})=0.$

Now it remains to show $H^{2}(X,\mathbf{GW}^{2}_{3})=0.$ We have the following
short exact sequence from [3, section 3.6, page15]

$0\to\mathbf{A}\to\mathbf{GW}_{3}^{2}\to\mathbf{B}\to 0.$

We refer the reader to [3, section 3.6] and also [5, section 4, pages 16-17]
for the definition and properties of $A$ and $B$, which will be used crucially
in what follows.

By the above exact sequence, to prove $H^{2}(X,\mathbf{GW}_{3}^{2})=0$, it is
enough to prove that

$H^{2}(X,\mathbf{A})=0\ \ \text{and}\ \ H^{2}(X,\mathbf{B})=0.$

By [5, Lemma 4.15], we have $H^{2}(X,\mathbf{B})\cong
H^{2}(X,\mathbf{K}_{2}^{Q}/2)$. We have an exact sequence

$0\to 2\cdot\mathbf{K}_{2}^{Q}\to\mathbf{K}_{2}^{Q}\to\mathbf{K}_{2}^{Q}/2\to
0$

Hence

$H^{2}(X,\mathbf{K}_{2}^{Q})\xrightarrow{2\cdot}H^{2}(X,\mathbf{K}_{2}^{Q})\twoheadrightarrow
H^{2}(X,\mathbf{K}_{2}^{Q}/2)$

is exact. We observe that by Bloch’s formula,

$H^{2}(X,\mathbf{K}_{2}^{Q})=CH^{2}(X)\simeq F^{2}K_{0}(X).$

Since $F^{2}K_{0}(X)$ is 2-divisible for a smooth affine variety $X$ over an
algebraically closed field, we conclude that $H^{2}(X,\mathbf{K}_{2}^{Q}/2)$.
Hence $H^{2}(X,\mathbf{B})=0$.

Thus, it remains to prove $H^{2}(X,\mathbf{A})=0$. Note that by [5, Lemma
4.11]

$(\mathbf{A})_{-2}\simeq\mathbf{K}_{1}^{Q}/2.$

From the Gersten resolution for the sheaf $\mathbf{A}$, we have the following
surjection

$\bigoplus_{x\in X^{(2)}}\mathbf{K}_{1}^{Q}/2(k(x))\simeq\bigoplus_{x\in
X^{(2)}}(\mathbf{A})_{-2}(k(x))\twoheadrightarrow H^{2}(X,\mathbf{A})$

Since the field $k(x)=k$ is an algebraically closed field, the left hand side
group vanishes. Hence $H^{2}(X,\mathbf{A})=0$. Combining all the vanishing
results above we get the proposition. ∎

We note the following vanishing result.

######  Proposition 3.11.

Let $A$ be a smooth affine algebra of dimension 3 over an algebraically closed
field $k$ and $X={\rm Spec\,}(A)$. Then
$H^{3}(X,{\bm{\pi}}^{\mathbb{A}^{1}}_{2}(Q_{3}))\simeq
H^{3}(X,{\bm{\pi}}^{\mathbb{A}^{1}}_{2}(\mathbb{A}^{2}-0))=0$.

###### Proof.

We have the following short exact sequence from [5, page 15]

$0\to\mathbf{T}_{4}^{\prime}\to\pi_{2}^{\mathbb{A}^{1}}(\mathbb{A}^{2}-0)\to\mathbf{GW}^{2}_{3}\to
0$

hence we get the long exact sequence of cohomology groups

$H^{3}(X,\mathbf{T}^{\prime}_{4})\to
H^{3}(X,\pi_{2}^{\mathbb{A}^{1}}(\mathbb{A}^{2}-0))\to
H^{3}(X,\mathbf{GW}^{2}_{3})\to H^{4}(X,\mathbf{T}_{4}^{\prime})=0\
\text{~{}~{}(because of dimension)}$

This reduces to show that under the same assumptions, we have

$H^{3}(X,\mathbf{T}^{\prime}_{4})=0\ \ \text{and}\ \
H^{3}(X,\mathbf{GW}^{2}_{3})=0.$

Vanishing of $H^{3}(X,\mathbf{T}^{\prime}_{4})$:

We have the following sequence by [5, Theorem 3]

$\mathbf{I}^{5}\to\mathbf{T}^{\prime}_{4}\to\mathbf{S}_{4}^{\prime}\to 0$

which is exact (not necessarily on the left). Hence again by the long exact
sequence of the cohomologies and also that the Nisnevich cohomological
dimension of $X$ is atmost 3, we get

$H^{3}(X,{\mathbf{K}}^{\rm M}_{4}/12)\twoheadrightarrow
H^{3}(X,\mathbf{S}_{4}^{\prime})$

is surjective.

We calculate $H^{3}(X,{\mathbf{K}}^{\rm M}_{4}/12)$ using the Gersten complex
resolution for ${\mathbf{K}}^{\rm M}_{4}/12.$ Thus, it will be a quotient of

$\bigoplus_{x\in X^{(3)}}{\mathbf{K}}^{\rm M}_{1}(k(x))/12$

which is $0$ (since the field is algebraically closed $k(x)=k$ implies
${\mathbf{K}}^{\rm M}_{1}(k(x))/12=k(x)^{*}/12=0$). So we get
$H^{3}(X,{\mathbf{K}}^{\rm M}_{4}/12)=0$, and hence
$H^{3}(X,\mathbf{S}_{4}^{\prime})=0.$ This together with $H^{3}(X,I^{5})=0$
which follows from [5, Proposition 5.2], since $cd_{2}(k)\leq 1$ (for
algebraically closed field $k$), implies that
$H^{3}(X,\mathbf{T}_{4}^{\prime})=0.$
Vanishing of $H^{3}(X,\mathbf{GW}^{2}_{3})$:

We have the following short exact sequence from [3, section 3.6, page15]

$0\to\mathbf{A}\to\mathbf{GW}_{3}^{2}\to\mathbf{B}\to 0.$

We refer the reader to [3, section 3.6] and also [5, section 4, pages 16-17]
for the definition and properties of $\mathbf{A}$ and $\mathbf{B}$, which will
be used crucially in what follows.

By the above exact sequence, to prove $H^{3}(X,\mathbf{GW}_{3}^{2})=0$, it is
enough to prove that

$H^{3}(X,\mathbf{A})=0\ \ \text{and}\ \ H^{3}(X,\mathbf{B})=0.$

By similar arguments as in [5, Lemma 4.15], we have $H^{3}(X,\mathbf{B})=0$.

Thus, it remains to prove $H^{3}(X,\mathbf{A})=0$. From [5, Lemma 4.11], we
have an epimorphism of sheaves

$\mathbf{K}_{3}^{Q}\twoheadrightarrow\mathbf{A}.$

Since dimension of $X$ is $3$, we have

$H^{3}(X,\mathbf{K}_{3}^{Q})\twoheadrightarrow H^{3}(X,\mathbf{A})$

But $CH^{3}(X)$ is a divisible group, hence $H^{3}(X,\mathbf{A})$ is also in
particular, 2-divisible. Now by Gersten complex for $A$,

$\bigoplus_{x\in X^{(3)}}\mathbb{Z}/2\simeq\bigoplus_{x\in
X^{(3)}}\mathbf{K}_{0}^{Q}/2(k(x))\simeq\bigoplus_{x\in
X^{(3)}}(\mathbf{A})_{-3}(k(x))\twoheadrightarrow H^{3}(X,\mathbf{A})$

The group $\bigoplus_{x\in X^{(3)}}\mathbb{Z}/2$ is 2-torsion, hence the
quotient $H^{3}(X,\mathbf{A})$ is also 2-torsion. Thus, the group
$H^{3}(X,\mathbf{A})$ is both 2-torsion and 2-divisible, hence
$H^{3}(X,\mathbf{A})=0$.

Hence the proposition. ∎

For $d=3$ in the case of the field $k=\overline{\mathbb{F}}_{p}$ we have the
following vanishing result.

######  Proposition 3.12.

Let $A$ be a smooth affine algebra of dimension $3$ over an algebraically
closed field $k$. Set $X={\rm Spec\,}(A)$. If $k=\overline{\mathbb{F}}_{p}$,
then $H^{3}(X,{\bm{\pi}}^{\mathbb{A}^{1}}_{3}(Q_{4}))=0$.

###### Proof.

We recall from [2, Example 4.3.1] that up to change of co-ordinates

$Q_{7}={\rm
Spec\,}\bigg{(}\dfrac{k[x_{11},x_{12},x_{21},x_{22},y_{11},y_{22},y_{12},y_{21}]}{x_{11}x_{22}-x_{12}x_{21}-y_{11}y_{22}+y_{12}y_{21}-1}\bigg{)}$

and

$Q_{4}={\rm
Spec\,}\bigg{(}\dfrac{k[z_{1},z_{2},z_{3},z_{4},z_{5}]}{z_{1}z_{4}-z_{2}z_{3}-z_{5}(1+z_{5})}\bigg{)}.$

The motivic Hopf map $\nu:Q_{7}\to Q_{4}$ is given by

$(M_{1},M_{2})\mapsto(M_{1}M_{2},\det M_{2}).$

where $M_{1},M_{2}$ are $2\times 2$ matrices
$M_{1}=\begin{pmatrix}x_{11}&x_{12}\\\ x_{21}&x_{22}\end{pmatrix}$,
$M_{2}=\begin{pmatrix}y_{11}&y_{12}\\\ y_{21}&y_{22}\end{pmatrix}$ such that
$\det M_{1}-\det M_{2}=1$ and setting $\begin{pmatrix}z_{1}&z_{2}\\\
z_{3}&z_{4}\end{pmatrix}=M_{1}M_{2}$ and $z_{5}=\det M_{2}$. We note that
$\nu:Q_{7}\to Q_{4}$ is an $SL_{2}$-torsor. Recall that the scheme $Q_{4}$ is
pointed at $(0,0,0,0,0)$. Consider the associated $\mathbb{A}^{1}$-fiber
sequence

$SL_{2}\to Q_{7}\to Q_{4}$

hence the long exact sequence of homotopy sheaves

${\bm{\pi}}^{\mathbb{A}^{1}}_{3}(SL_{2})\to{\bm{\pi}}^{\mathbb{A}^{1}}_{3}(Q_{7})\to{\bm{\pi}}^{\mathbb{A}^{1}}_{3}(Q_{4})\to{\bm{\pi}}^{\mathbb{A}^{1}}_{2}(SL_{2})\to
0.$

Note $SL_{2}\simeq\mathbb{A}^{2}-0$ hence by Proposition 3.11,
$H^{3}(X,{\bm{\pi}}^{\mathbb{A}^{1}}_{2}(\mathbb{A}^{2}-0))=0$. Also note that
${\mathbf{K}}^{\rm MW}_{4}\simeq{\bm{\pi}}^{\mathbb{A}^{1}}_{3}(Q_{7})$ hence
we can deduce that the map

$H^{3}(X,{\mathbf{K}}^{\rm MW}_{4})\to
H^{3}(X,{\bm{\pi}}^{\mathbb{A}^{1}}_{3}(Q_{4}))$

is surjective. In the case $k=\overline{\mathbb{F}}_{p}$, the cohomology group
$H^{3}(X,{\mathbf{K}}^{\rm MW}_{4})=0$ (see Proposition 3.13), hence
$H^{3}(X,{\bm{\pi}}^{\mathbb{A}^{1}}_{3}(Q_{4}))=0$. ∎

######  Proposition 3.13.

Let $\mathbb{F}_{p}$ ($p$ a prime) be the field with $p$ elements and let
$\mathbb{F}_{p}\subset k\subset\overline{\mathbb{F}}_{p}$ be a field. Let $A$
be a smooth affine algebra of dimension $d\geq 3$ over $k$ and $X={\rm
Spec\,}(A)$. Then $H^{d}(X,K^{MW}_{d+1})=(0)$.

###### Proof.

By [15, Theorem 4.9], we have $H^{d}(X,K^{MW}_{d+1})\simeq
Um_{d+1}(A)/E_{d+1}(A)$. If $k$ is any finite field, then $A$ is a finitely
generated ring. Hence by [25, Theorem 18.2], we have
$Um_{d+1}(A)/E_{d+1}(A)=(0)$.

Now let $k=\overline{\mathbb{F}}_{p}$ and $\bar{a}=(a_{1},\ldots,a_{d+1})$ be
a unimodular vector in $Um_{d+1}(A)$. Then $a$ is defined over $A^{\prime}$
where $A^{\prime}$ is a finitely generated $L$-algebra such that $L$ is a
finite field and $A\cong A^{\prime}\otimes_{L}\overline{\mathbb{F}}_{p}$.
Again by [25, Theorem 18.2], there exists $\sigma\in
E_{d+1}(A^{\prime})\subset E_{d+1}(A)$ such that
$\sigma(\bar{a})=(1,0,\ldots,0)$. Hence in this case also, we have
$Um_{d+1}(A)/E_{d+1}(A)=(0)$.∎

######  Remark 3.14.

We should note that in the setting of Proposition 3.12, for general
algebraically closed field $k$, the argument of the proof of Proposition 3.12
actually shows that the cohomology group
$H^{3}(X,{\bm{\pi}}^{\mathbb{A}^{1}}_{3}(Q_{4}))$ is a quotient of the
cokernel

${\rm Coker\ }(H^{3}(X,{\bm{\pi}}^{\mathbb{A}^{1}}_{3}(SL_{2}))\to
H^{3}(X,{\mathbf{K}}^{\rm MW}_{4})).$

We already know that $H^{3}(X,{\mathbf{K}}^{\rm MW}_{4}))$ is uniquely
divisible from [16]. If we know some torsion properties of this cokernel, we
can conclude the desired vanishing of
$H^{3}(X,{\bm{\pi}}^{\mathbb{A}^{1}}_{3}(Q_{4}))$ for any algebraically closed
field.

In the case $d=4$ we have the following vanishing result.

######  Proposition 3.15.

Let $X$ be a smooth affine variety over algebraically closed field $k$ of
dimension $4$. Then $H^{4}(X,\pi_{4}^{\mathbb{A}^{1}}(Q_{6}))=0.$

###### Proof.

By [4, Theorem 4] there is a surjection

$H^{4}(X,\pi_{3}^{\mathbb{A}^{1}}(\mathbb{A}^{3}-0))\to
H^{4}(X,\pi_{4}^{\mathbb{A}^{1}}(Q_{6})).$

Since $k$ is an algebraically closed field, from [3, proof of Theorem 2, page
30], we have

$H^{4}(X,\pi_{3}^{\mathbb{A}^{1}}(\mathbb{A}^{3}-0))=0.$

∎

We list the above results in the section for the convenience.

######  Theorem 3.16.

Let $X$ be a smooth affine variety over an algebraically closed field $k$ of
Krull dimension $d$. Then

  1. $(1)$

for $d=2$, $H^{2}(X,\pi_{2}^{\mathbb{A}^{1}}(Q_{2}))=0.$

  2. $(2)$

for $d=3$, $H^{3}(X,\pi_{3}^{\mathbb{A}^{1}}(Q_{4}))=0$ if
$k=\overline{\mathbb{F}}_{p}$.

  3. $(3)$

for $d=4$, $H^{4}(X,\pi_{4}^{\mathbb{A}^{1}}(Q_{6}))=0.$

  4. $(4)$

for $d\geq 5$, $H^{d}(X,\pi_{d}^{\mathbb{A}^{1}}(Q_{2d-2}))=0,$ provided
$char(k)=0$.

  5. $(5)$

if $d\geq 5$, then $H^{d}(X,{\bm{\pi}}^{\mathbb{A}^{1}}_{d}(Q_{2d-2}))$ is
$(d-1)!(d-2)!$-torsion.

  6. $(6)$

for $d\geq 5$, $H^{d}(X,\pi_{d}^{\mathbb{A}^{1}}(Q_{2d-2}))=0,$ provided
Conjecture 3.8 holds.

###### Proof.

(1) follows from Proposition 3.10. (2) follows from Proposition 3.12. (3)
follows from Proposition 3.15. (5) follows from Proposition 3.6(2). (6)
follows from Corollary 3.6 and Proposition 3.9(1). ∎

##  4\. Cancellation and splitting of symplectic modules

In this section, we prove the main results of the paper about the symplectic
modules.

Let $k$ be a field. Let $X={\rm Spec\,}R$ be a smooth affine variety of
dimension $2n$. For $r$ even, let $V_{r}^{Sp}(X)$ be the set of isomorphism
classes of rank $r$ symplectic bundles on $X.$ By the representability
results, we have a bijection

$V_{r}^{Sp}(X)\simeq[X,BSp_{r}]_{\mathbb{A}^{1}}.$

###  4.1. Cancellation of symplectic module

We prove the following result about the cancellation of symplectic modules.

######  Theorem 4.1.

Let $k$ be an algebraically closed field. Let $X={\rm Spec\,}R$ be a smooth
affine variety of dimension $2n$. Let $(2n)!$ be invertible in $k.$ Then the
map

$V_{2n-2}^{Sp}(X)\to V_{2n}^{Sp}(X)$

is injective. In other words, if two symplectic rank $2n-2$-vector bundles
$\mathcal{E}$ and $\mathcal{E}^{\prime}$ are stably isomorphic i.e.,
$\mathcal{E}\oplus H(R)\simeq\mathcal{E}^{\prime}\oplus H(R)$, then
$\mathcal{E}\simeq\mathcal{E}^{\prime}$.

Digression: To aid in the proof of the above theorem we need to establish some
vanishing result for a cohomology group of a homotopy sheaf. In order to do
that let us first recall some notation from [14, Section 8]. We have the
following commutative diagram

(4.1)
$\textstyle{\mathbb{A}^{2n}-0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{A}^{2n}-0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\star\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F^{\prime}_{n-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{BSp_{2n-2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{BSp\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F^{\prime}_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{BSp_{2n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{BSp}$

where all the rows and columns are $\mathbb{A}^{1}$-fiber sequences, where
$F^{\prime}_{n-1}$ and $F^{\prime}_{n}$ are the homotopy fibers of the
naturals maps $BSp_{2n-2}\rightarrow BSp$ and $BSp_{2n}\rightarrow BSp$
respectively. From [14, Section 8], we have the following computations of the
$\mathbb{A}^{1}$-homotopy sheaves of $F^{\prime}_{n-1}$ and $F^{\prime}_{n}$.

$\displaystyle{\bm{\pi}}^{\mathbb{A}^{1}}_{i}(F^{\prime}_{n-1})=\begin{cases}0&\text{for}\
\ i\leq 2n-2\\\ {\mathbf{K}}^{\rm MW}_{2n}&\text{for}\ \ i=2n-1\end{cases}$

and

$\displaystyle{\bm{\pi}}^{\mathbb{A}^{1}}_{i}(F^{\prime}_{n})=\begin{cases}0&\text{for}\
\ i\leq 2n\\\ {\mathbf{K}}^{\rm MW}_{2n+2}&\text{for}\ \ i=2n+1.\end{cases}$

Moreover we have the following long exact sequence of
$\mathbb{A}^{1}$-homotopy sheaves associated to the leftmost column fiber
sequence in the diagram (4.1).

$\cdots\to{\bm{\pi}}^{\mathbb{A}^{1}}_{2n+1}(F^{\prime}_{n})\to{\bm{\pi}}^{\mathbb{A}^{1}}_{2n}(\mathbb{A}^{2n}-0)\to{\bm{\pi}}^{\mathbb{A}^{1}}_{2n}(F^{\prime}_{n-1})\to{\bm{\pi}}^{\mathbb{A}^{1}}_{2n}(F^{\prime}_{n})=0$

As a consequence, we have the following epimorphism of Nisnevich sheaves

(4.2)
${\bm{\pi}}^{\mathbb{A}^{1}}_{2n}(\mathbb{A}^{2n}-0)\to{\bm{\pi}}^{\mathbb{A}^{1}}_{2n}(F^{\prime}_{n-1}).$

In particular, $H^{2n}(X,{\bm{\pi}}^{\mathbb{A}^{1}}_{2n}(\mathbb{A}^{2n}-0))$
maps onto $H^{2n}(X,{\bm{\pi}}^{\mathbb{A}^{1}}_{2n}(F^{\prime}_{n-1}))$ since
the Nisnevich cohomological dimension of $X$ is at most $2n$. From Fasel [17,
Theorem 3], $H^{2n}(X,{\bm{\pi}}^{\mathbb{A}^{1}}_{2n}(\mathbb{A}^{2n}-0))=0$.
(This is one place where the assumptions on the characteristic of the field
$k$ come into play.) Hence it follows that

(4.3) $H^{2n}(X,{\bm{\pi}}^{\mathbb{A}^{1}}_{2n}(F^{\prime}_{n-1}))=0.$

We now proceed with the proof of the Theorem 4.1.

###### Proof of Theorem 4.1.

By the representability result in subsection 2.3, we have the commutative
diagram

$\textstyle{V_{2n-2}^{Sp}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{V_{2n}^{Sp}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{[X,BSp_{2n-2}]_{\mathbb{A}^{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{[X,BSp_{2n}]_{\mathbb{A}^{1}}}$

Thus to show the injectivity of the top horizontal arrow it suffices to show
the injectivity of the bottom horizontal map. We show this by analyzing the
Moore-Postnikov tower (see Proposition 2.3 for the notation) for the
$\mathbb{A}^{1}$-fiber sequence

$F^{\prime}_{n-1}\to BSp_{2n-2}\to BSp.$

Note that ${\bm{\pi}}^{\mathbb{A}^{1}}_{1}(BSp)=0$. The non-trivial part of
the Moore-Postnikov tower is

$\textstyle{K({\bm{\pi}}^{\mathbb{A}^{1}}_{2n}(F^{\prime}_{n-1}),2n)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{2n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K({\mathbf{K}}^{\rm
MW}_{2n},2n-1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{2n-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\textstyle{E_{2n-2}=BSp\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K({\mathbf{K}}^{\rm
MW}_{2n},2n)}$

We have the commutative diagram

$\textstyle{\lim_{r}[X,E_{r}]_{\mathbb{A}^{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{[X,BSp]_{\mathbb{A}^{1}}}$$\textstyle{[X,BSp_{2n-2}]_{\mathbb{A}^{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{[X,BSp_{2n}]_{\mathbb{A}^{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$

where the leftmost vertical map is bijective since
$BSp_{2n-2}\simeq\mathop{\varprojlim}\limits_{r}E_{r}$ and we can deduce taht
the righmost vertical map is bijective by considering the Moore-Postnikov
tower for the map $BSp_{2n}\to BSp.$ From this it is sufficient to prove the
injectivity of the top horizontal map in the above commutative diagram. Hence
it is enough to prove that

  1. (1)

the map $[X,E_{2n}]_{\mathbb{A}^{1}}\to[X,E_{2n-1}]_{\mathbb{A}^{1}}$ is
injective and

  2. (2)

the map $[X,E_{2n-1}]_{\mathbb{A}^{1}}\to[X,BSp]_{\mathbb{A}^{1}}$ is
injective.

For part (1): we have the following $\mathbb{A}^{1}$-fiber sequence

$K({\bm{\pi}}^{\mathbb{A}^{1}}_{2n}(F^{\prime}_{n-1}),2n)\to E_{2n}\to
E_{2n-1}$

hence the following exact sequence

$H^{2n}(X,{\bm{\pi}}^{\mathbb{A}^{1}}_{2n}(F^{\prime}_{n-1}))\to[X,E_{2n}]_{\mathbb{A}^{1}}\to[X,E_{2n-1}]_{\mathbb{A}^{1}}.$

By (4.3), $H^{2n}(X,{\bm{\pi}}^{\mathbb{A}^{1}}_{2n}(F^{\prime}_{n-1}))=0$,
hence the map

$[X,E_{2n}]_{\mathbb{A}^{1}}\to[X,E_{2n-1}]_{\mathbb{A}^{1}}$

is injective.

For part (2): consider the following $\mathbb{A}^{1}$-fiber sequence

$K({\mathbf{K}}^{\rm MW}_{2n},2n-1)\to E_{2n-1}\to BSp$

which yields the following long exact sequence

$[X,Sp]\simeq[X,\Omega BSp]\xrightarrow{\Delta}H^{2n-1}(X,{\mathbf{K}}^{\rm
MW}_{2n})\to[X,E_{2n-1}]_{\mathbb{A}^{1}}\xrightarrow{q_{*}}[X,BSp]_{\mathbb{A}^{1}}.$

Let $\xi$ and $\xi^{\prime}$ be two elements in
$[X,E_{2n-1}]_{\mathbb{A}^{1}}$ such that $q_{*}(\xi)=q_{*}(\xi^{\prime})$.
Then by the exactness of homotopy groups (and the pointed sets), there is
$\alpha\in H^{2n-1}(X,{\mathbf{K}}^{\rm MW}_{2n})$ such that
$\alpha\cdot\xi=\xi^{\prime}.$ We point the sets at $\xi$ so that we have the
long exact sequence

$[X,Sp]\simeq[X,\Omega BSp]\xrightarrow{\Delta}H^{2n-1}(X,{\mathbf{K}}^{\rm
MW}_{2n})\to[X,E_{2n-1}]_{\mathbb{A}^{1}}\to[X,BSp]_{\mathbb{A}^{1}}$

where the image of $\Delta$ is the stabilizer of $\xi$. We have the
commutative diagram

---
$\textstyle{SL_{2n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{H}$$\textstyle{Sp_{4n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Sp\simeq\Omega
BSp\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Omega
b_{2n}}$$\textstyle{K({\mathbf{K}}^{\rm
MW}_{2n},2n-1)}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\textstyle{Q_{4n-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s_{2n}}$$\textstyle{SL_{2^{2n-1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{SL\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

By the arguments as in [14, pages 14860-14861] we have for any $\mu:X\to
Q_{4n-1}$

(4.4) $\Delta(H[(s_{2n})_{*}(\mu)])=\pm 2(2n-1)![(\kappa_{2n})_{*}(\mu)].$

Recall from previous paragraph $g\in H^{2n-1}(X,{\mathbf{K}}^{\rm M}_{2n})$
such that $g\cdot\xi=\xi^{\prime}.$ Since $H^{2n-1}(X,{\mathbf{K}}^{\rm
M}_{2n})$ is divisible coprime to characteristic of $k$, there is $\alpha\in
H^{2n-1}(X,{\mathbf{K}}^{\rm MW}_{2n})$ such that $g=\pm 2(2n-1)!\alpha$.

Consider the fiber sequence

$\mathbb{A}^{2n}-0<2n>\to K({\mathbf{K}}^{\rm MW}_{2n},2n-1)\to
K({\bm{\pi}}^{\mathbb{A}^{1}}_{2n}(\mathbb{A}^{2n}-0),2n+1)$

where $\mathbb{A}^{2n}-0<2n>$ is the $2n^{th}$ term of the Moore-Postnikov
tower asscoiated to $\mathbb{A}^{2n}-0$. Thus we have the exact sequence

$[X,\mathbb{A}^{2n}-0]_{\mathbb{A}^{1}}\simeq[X,\mathbb{A}^{2n}-0<2n>]_{\mathbb{A}^{1}}\to
H^{2n-1}(X,{\mathbf{K}}^{\rm M}_{2n})\to 0.$

Furthermore
$[X,Q_{4n-1}]_{\mathbb{A}^{1}}\simeq[X,\mathbb{A}^{2n}-0]_{\mathbb{A}^{1}}.$
Thus we observe that $\alpha=(\kappa_{2n})_{*}(\mu)$ for some $\mu:X\to
Q_{4n-1}$. Combining (4.4) and the above observation implies that

$g=\pm 2(2n-1)!\alpha=\pm
2(2n-1)!(\kappa_{2n})_{*}(\mu)=\Delta(H[(s_{2n})_{*}(\mu)]).$

Hence $g$ is in the image of $\Delta$, thus $g$ is a stabilizer of $\xi$. Thus
$\xi=g\cdot\xi=\xi^{\prime}.$ Thus,

$[X,E_{2n-1}]_{\mathbb{A}^{1}}\to[X,BSp]_{\mathbb{A}^{1}}$

is injective. ∎

######  Remark 4.2.

Let $p$ be odd prime number and $X={\rm Spec\,}(A)$ a smooth affine variety
over an algebraically closed field $k$ of dimension $p+2$. Recall that for
every integer $r\geq 0$, $V_{r}^{Sp}(X)$ denotes the set of isomorphism
classes of rank $r$ symplectic bundles on $X.$ The map
$\phi_{r}:V_{r}^{Sp}(X)\xrightarrow{}V_{r+2}^{Sp}(X)$ send
$(V,\psi_{r})\mapsto(V\oplus H(A),\psi_{r}\perp\psi_{2})$. Now we would like
to investigate what happens to the following sequence of maps

(4.5)
$V_{p}^{Sp}(X)\xrightarrow{\phi_{p}}V_{p+2}^{Sp}(X)\xrightarrow{\phi_{p+2}}\cdots$

First, we note that $\phi_{n}$ are injective for $n\geq p+2$ if ${\rm
Pic}(A)=(A)$. For this, observe the following commutative diagram

$\textstyle{V_{p+1}^{Sp}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{p+1}}$$\textstyle{V_{p+3}^{Sp}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V_{p+2}^{Sp}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{p+2}}$$\textstyle{V_{p+4}^{Sp}(X)}$

where vertical maps send $(V,\psi)\mapsto(V\oplus A,\psi\perp 0)$. Note that
there is a retraction to the verical maps which send $(V\oplus A,\psi\perp
0)\mapsto(V,\psi)$. By a result of Peng Du [14], $\phi_{p+n}$ are injective
for odd $n\geq 1$. Let $(V\oplus L),(V^{\prime}\oplus L^{\prime})\in
V_{p+2}^{Sp}(X)$ such that $(V\oplus L\oplus H(A))\cong(V^{\prime}\oplus
L^{\prime}\oplus H(A))$. Using retraction and Peng Du’s result, we have
$V\cong V^{\prime}$. Since ${\rm Pic}(X)=0$, we have $L\cong L^{\prime}$. This
implies $(V\oplus L)\cong(V^{\prime}\oplus L^{\prime})$. Same arguments prove
that $\phi_{n}$ are injective for all $n\geq p+2$.

We claim that the map $\phi_{p}$ is not injective in (4.5) even if ${\rm
Pic}(A)$ is a trivial group. For this, we look at the following commutative
diagram

$\textstyle{[(V,\beta)]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V_{p-1}^{Sp}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{p-1}}$$\textstyle{V_{p+1}^{Sp}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{[(V\perp
A,\beta\perp
0)]}$$\textstyle{V_{p}^{Sp}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{p}}$$\textstyle{V_{p+2}^{Sp}(X)}$

where the maps are described as before. In view of the commutative diagram and
the retraction maps of the vertical maps, it is enough to prove that
$\phi_{p-1}$ is not injective. It is indeed true, in view of the result of
Matthias Wendt [28, Corollary 6.2].

###  4.2. Splitting of symplectic module

We prove the following result about the splitting of a symplectic module.

######  Theorem 4.3.

Let $A$ be a smooth affine variety over a perfect field $k$ and $X={\rm
Spec\,}(A)$. Let $P$ be a symplectic $A$-module of rank $2n$ ($n\geq 1$) and
$H(A)$ the free hyperbolic space of rank $2$.

  1. (1)

Assume that $dim(A)=2n$. Then $P\cong Q\oplus H(A)$ if and only if the Euler
class $e_{2n}(P)=0$ in $\widetilde{CH}^{2n}(X)$.

  2. (2)

Assume that $dim(A)=2n+1$, $k$ is algebraically closed and $char(k)=0$. Then
$P\cong Q\oplus H(A)$ if and only if the Euler class $e_{2n}(P)=0$ in
$\widetilde{CH}^{2n}(X)$.

  3. (3)

Assume $dim(A)=3$, $rank(P)=2$. Then $P\cong H(A)$ if and only if $e_{2}(P)=0$
in $\widetilde{CH}^{2}(X)$.

###### Proof.

  1. (1)

It is clear that if $P\cong Q\oplus H(A)$ , then the Euler class $e_{2n}(P)=0$
in $\widetilde{CH}^{2n}(X)$. Conversely let us assume that $P$ is rank $2n$
symplectic module such that $e_{2n}(P)=0$ in $\widetilde{CH}^{2n}(X)$. Let
$V_{2n}^{Sp}$ denote the isomorphism class of rank $2n$ symplectic bundle over
$X$. From [9], we have $V_{2n}^{Sp}(X)\cong[X,BSp_{2n}]_{\mathbb{A}^{1}}$.
Hence the projective module $P$ corresponds to a map $\xi:X\to BSp_{2n}$ in
the homotopy category. We need to show that if $e_{2n}(P)=0$ in
$\widetilde{CH}^{2n}(X)$ then the associated map $\xi:X\to BSp_{2n}$ lifts to
a map $\xi^{\prime}:X\to BSp_{2n-2}$ upto homotopy.

$\textstyle{BSp_{2n-2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\xi^{\prime}}$$\scriptstyle{\xi}$$\textstyle{BSp_{2n}}$

Consider the Moore-Postnikov tower for the fiber sequence

$F^{\prime}_{n-1}\to BSp_{2n-2}\to BSp.$
$\textstyle{K({\bm{\pi}}^{\mathbb{A}^{1}}_{2n}(F^{\prime}_{n-1}),2n)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{2n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q_{2n}}$$\textstyle{K({\mathbf{K}}^{\rm
MW}_{2n},2n-1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{2n-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q_{2n-1}}$$\textstyle{K(\pi_{2n}(F^{\prime}_{n-1}),2n+1)}$$\textstyle{BSp\simeq
E_{2n-2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K({\mathbf{K}}^{\rm
MW}_{2n},2n)}$

The above tower yields the following two fiber sequences

(4.6)
$E_{2n}\xrightarrow{q_{2n}}E_{2n-1}\xrightarrow{}K(\pi_{2n}(F^{\prime}_{n-1}),2n+1)$
(4.7) $E_{2n-1}\xrightarrow{q_{2n-1}}E_{2n-2}\xrightarrow{}K({\mathbf{K}}^{\rm
MW}_{2n},2n)$

From (4.6), we have the following exact sequence

$[X,E_{2n}]\xrightarrow{(q_{2n})_{*}}[X,E_{2n-1}]\xrightarrow{}[X,K(\pi_{2n}(F^{\prime}_{n-1}),2n+1)]$

Note that $[X,K(\pi_{2n}(F^{\prime}_{n-1}),2n+1)]\cong
H^{2n+1}(X,\pi_{2n}(F^{\prime}_{n-1}))$ which is a trivial group, since
$2n+1>2n=\dim X$. Hence $(q_{2n})_{*}$ is surjective.

From (4.7), we have the following exact sequence

$[X,E_{2n-1}]\xrightarrow{(q_{2n-1})_{*}}[X,E_{2n-2}]\xrightarrow{}[X,K({\mathbf{K}}^{\rm
MW}_{2n},2n)]$

The diagram

$\textstyle{[X,E_{2n-2}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{[X,K({\mathbf{K}}^{\rm
MW}_{2n},2n)]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{[X,BSp_{2n}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b_{n}}$$\textstyle{[X,K({\mathbf{K}}^{\rm
MW}_{2n},2n)]}$

commutes such that the map

$[X,BSp_{2n}]\xrightarrow{b_{n}}[X,K({\mathbf{K}}^{\rm MW}_{2n},2n)]$

is taking the $n^{th}$ Borel class $b_{n}(\xi)$ which agrees with the Euler
class $e_{2n}(P)$ (see [14, page 14858 (40)]). Note that we have
$[X,K({\mathbf{K}}^{\rm MW}_{2n},2n)]\cong H^{2n}(X,{\mathbf{K}}^{\rm
MW}_{2n})$ which is isomorphic to $\widetilde{CH}^{2n}(X)$.

The Euler class $e_{2n}(P)=0$ in $\widetilde{CH}^{2n}(X)$ i.e., $b_{n}(\xi)=*$
implies that the sequence of pointed sets

$[X,E_{2n-1}]\xrightarrow{(q_{2n-1})_{*}}[X,E_{2n-2}]\xrightarrow{b_{n}}[X,K({\mathbf{K}}^{\rm
MW}_{2n},2n)]$

is exact. Hence there is $a\in[X,E_{2n-1}]$ such that $(q_{2n-1})_{*}(a)=\xi.$
As observed above the map

$[X,E_{2n}]\xrightarrow{(q_{2n})_{*}}[X,E_{2n-1}]$

is surjective hence there is $a^{\prime}\in[X,E_{2n}]$ such that
$(q_{2n})_{*}(a^{\prime})=a$. Furthermore, since the $\dim X=2n$ the maps

$[X,BSp_{2n-2}]\to[X,E_{r}]\to[X,E_{2n}]$

are surjective for $r\geq 2n$. Hence the preimage
$\xi^{\prime}\in[X,BSp_{2n-2}]$ of $a^{\prime}$ is the desired lift of the map
$\xi\in[X,BSp_{2n}].$

  2. (2)

We follow the same arguments as in part $(1)$. By (4.2), there is an
epimorphism of Nisnevich sheaves

$\pi_{2n}(\mathbb{A}^{2n}-\\{0\\})\to\pi_{2n}(F^{\prime}_{n-1}).$

This induces a surjective map of abelian groups

$H^{2n+1}(X;\pi_{2n}(\mathbb{A}^{2n}-\\{0\\}))\xrightarrow{}H^{2n+1}(X,\pi_{2n}(F^{\prime}_{n-1}))$

since $\dim X=2n+1$.

In view of Theorem 3.3, we observe that
$H^{2n+1}(X,\pi_{2n}(F^{\prime}_{n-1}))=0$. Rest of the arguments are same as
in part $(1)$.

$(3)$ This part follows using similar arguments as in part $(2)$ since for
$n=1$, $F^{\prime}_{n-1}=F^{\prime}_{0}$ is a point.

∎

##  5\. Comparison of Euler class group and Chow group

Let $R$ be a smooth affine algebra over a field $k$ of Krull dimension $d$ and
$X={\rm Spec\,}(R)$. In this section, we study the map between $(d-1)$-th
Euler class group $E^{d-1}(R)$ and the $(d-1)$-the Chow group $CH^{d-1}(X)$
using the Chow-Witt groups and certain homotopy classes of maps in
$\mathbb{A}^{1}$-homotopy category.

For the definition and properties of the Chow-group, we refer to [18]. We
begin by recalling the definition of Euler class group $E^{d-1}(R)$ from [13].
Let $R$ be an affine algebra of dimension $d\geq 3$ over
$\overline{\mathbb{F}}_{p}$ and $I$ an ideal of $R$ of height $d-1$ such that
$I/I^{2}$ is generated by $d-1$ elements. Let
$\alpha,\beta:(R/I)^{d-1}\twoheadrightarrow I/I^{2}$ be two surjections. We
say $\alpha$ and $\beta$ are related if there exists $\sigma\in SL_{d-1}(R/I)$
such that $\alpha\sigma=\beta$. One checks that this relation is an
equivalence relation on the set of all surjections from $(R/I)^{d-1}$ to
$I/I^{2}$. An equivalence class of surjections from
$(R/I)^{d-1}\twoheadrightarrow I/I^{2}$ is called a local orientaion. A local
orientation $\omega_{I}$ of $I$ is called a global orientation if a surjection
(hence all the surjections in the equivalence class by [13, Lemma 4.1]) in the
class of $\omega_{I}$ can be lifted to a surjection from
$R^{d-1}\twoheadrightarrow I$.

Let $G$ be the free abelian group on the set $B$ of pairs
$(\mathcal{I},\omega_{\mathcal{I}})$, where $\mathcal{I}\subset R$ is an ideal
of height $d-1$ with the property that ${\rm Spec\,}(R/\mathcal{I})$ is
connected, $\mu(\mathcal{I}/\mathcal{I}^{2})=d-1$ and
$\omega_{\mathcal{I}}:(R/\mathcal{I})^{d-1}\twoheadrightarrow\mathcal{I}/\mathcal{I}^{2}$
is a local orientation. Now let $I$ be any ideal of height $d-1$ such that
$I/I^{2}$ is generated by $d-1$ elements. Then $I$ has a unique decomposition,
$I=\mathcal{I}_{1}\cap\ldots\cap\mathcal{I}_{k}$, where each of ${\rm
Spec\,}(R/\mathcal{I}_{i})$ is connected, $\mathcal{I}_{i}$ are pairwise
maximal, and $ht(\mathcal{I}_{i})=d-1$. If $\omega_{I}$ is a local
orientation, then it naturally induces local orientation
$\omega_{\mathcal{I}_{i}}:(R/\mathcal{I}_{i})^{d-1}\twoheadrightarrow\mathcal{I}_{i}/\mathcal{I}_{i}^{2}$
for each $i,$ $1\leq i\leq k$. Thus by $(I,\omega_{I})$, we mean
$\Sigma(\mathcal{I}_{i},\omega_{i})\in G$.

Let $H$ be the subgroup of $G$ generated by the set $S(\subset B)$ of pairs
$(I,\omega_{I})$ in $G$ such that $\omega_{I}$ is a global orientation. We
define the $(d-1)$-th Euler class group of $R$ as $E^{d-1}(R):=G/H$.

######  Remark 5.1.

Let $A$ be a commutative noetherian ring of dimension $d$. Bhatwadekar and
Raja Sridharan [12, §4] defined the Euler class group $E^{n}(A)$ for any $n$
such that $2n\geq d+3$. However the group is well defined for any $n$ though
the desired properties are proved in the above mentioned range. For our
convenience, we denote it by $E^{n}_{BS}$. It is this Euler class group which
is used by Asok and Fasel [6, Remark 3.16].

If $A$ is an affine algebra over $\overline{\mathbb{F}}_{p}$ where
${\mathbb{F}_{p}}$ is a finite field with $p$ elements, then Das [13] defined
$E^{d-1}(A)$. We denote this by $E^{d-1}_{MD}(A)$. It is observed by Das [13]
that for $d\geq 5$ the natural map $E^{d-1}_{BS}(A)\rightarrow
E^{d-1}_{MD}(A)$ is an isomorphism.

In fact, for $d=4$, the same arguments also proves that the natural map
$E^{3}_{BS}(A)\rightarrow E^{3}_{MD}(A)$ is an isomorphism as
$E_{3}(A/I)=SL_{3}(A/I)$ where $A$ is an affine algebra over
$\overline{\mathbb{F}}_{p}$ and $I$ is an ideal in $A$ of height $3$.

Therefore, for $d\geq 4$, we shall write $E^{d-1}(A)$ for
$E^{d-1}_{BS}(A)\cong E^{d-1}_{MD}(A)$.

We recall the map

$\phi:E^{d-1}(A)\to CH^{d-1}(X):(I,w_{I})\mapsto(I)$

where $(I)$ denotes the fundamental cycle associated to the the closed
subscheme ${\rm Spec\,}A/I$ in ${\rm Spec\,}A$.

If $A$ is an affine algebra over $\overline{\mathbb{F}}_{p}$, then in [13,
Definition 4.1], a well-defined Euler class $e_{d-1}(P)$ is defined in
$E^{d-1}(A)$ for a projective $A$-module $P$ of rank $d-1$ with
$\wedge^{d-1}P\cong A$. We note that the above map $\phi$ take Euler class
$e_{d-1}(P)$ to Chern class $c_{d-1}(P)$ in $CH^{d-1}(A)$.

######  Proposition 5.2.

Let $X$ be a smooth affine variety of dimension $d$ over an infinite field
$k$. For $d\geq 4$, we have the Hurewicz map

$H:[X,Q_{2d-2}]_{\mathbb{A}^{1}}\to\widetilde{CH}^{d-1}(X)=H^{d-1}(X,{\mathbf{K}}^{\rm
MW}_{d-1})=[X,K(\pi_{d-1}^{\mathbb{A}^{1}}(Q_{2d-2}),d-1)]_{\mathbb{A}^{1}}$

Then

  1. (1)

$H$ is surjective.

  2. (2)

the cohomology group $H^{d}(X,\pi_{d}^{\mathbb{A}^{1}}(Q_{2d-2}))$ surjects
onto the kernel of the Hurewicz map $H$.
In particular if $H^{d}(X,\pi_{d}^{\mathbb{A}^{1}}(Q_{2d-2}))=0$ then $H$ is
injective.

###### Proof.

The proof follows from [5, Proposition 6.2] ∎

######  Remark 5.3.

In the above proposition, under the hypothesis $d\geq 4$ the set
$[X,Q_{2d-2}]_{\mathbb{A}^{1}}$ has a group structure (see [6, Theorem
1.3.4]).

At this point, we do not know whether the injectivity of $H$ implies
$H^{d}(X,\pi_{d}^{\mathbb{A}^{1}}(Q_{2d-2}))=0$.

######  Proposition 5.4.

Let $k$ be an infinite field with $cd_{2}(k)=0$ and $X$ a smooth affine
variety of dimension $d\geq 2$ over $k$. Then the natural map

$\widetilde{CH}^{d-1}(X)\to CH^{d-1}(X)$

is an isomorphism.

###### Proof.

We consider the map

$\widetilde{CH}^{d-1}(X)=H^{d-1}(X,{\mathbf{K}}^{\rm MW}_{d-1})\to
CH^{d-1}(X)=H^{d-1}(X,{\mathbf{K}}^{\rm M}_{d-1})$

We have a short exact sequence of Zariski/Nisnevich sheaves

$0\to\mathbf{I}^{d}\to{\mathbf{K}}^{\rm MW}_{d-1}\to{\mathbf{K}}^{\rm
M}_{d-1}\to 0$

which gives a long exact sequence of cohomlogies

$\cdots\to H^{d-1}(X,\mathbf{I}^{d})\to H^{d-1}(X,{\mathbf{K}}^{\rm
MW}_{d-1})\to H^{d-1}(X,{\mathbf{K}}^{\rm M}_{d-1})\to
H^{d}(X,\mathbf{I}^{d})\to\cdots$

Now by [5, Proposition 5.2], (for $cd_{2}(k)=r=0$),
$H^{d-1}(X,\mathbf{I}^{d})=0$ and $H^{d}(X,\mathbf{I}^{d})=0$.

Thus, the map

$H^{d-1}(X,{\mathbf{K}}^{\rm MW}_{d-1})\to H^{d-1}(X,{\mathbf{K}}^{\rm
M}_{d-1})$

is an isomorphism, hence the result. ∎

######  Theorem 5.5.

Let $k$ be an algebraically closed field and $A$ be a smooth affine algebra of
dimension $d\geq 4$ over $k$. Let $X={\rm Spec\,}(R)$. Consider the map

$\psi:E^{d-1}(A)\to CH^{d-1}(X).$

Then

  1. (1)

the map $\psi$ is surjective.

  2. (2)

further under the assumption $H^{d}(X,\pi_{d}^{\mathbb{A}^{1}}(Q_{2d-2}))=0$,
the map $\psi$ is isomorphism.

###### Proof.

(1) We first claim that the following diagram

$\textstyle{E^{d-1}(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\scriptstyle{s}$$\scriptstyle{\psi}$$\textstyle{[X,Q_{2d-2}]_{\mathbb{A}^{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{H}$$\textstyle{\widetilde{CH}^{d-1}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq~{}by~{}Prop.\ref{prop2}}$$\textstyle{CH^{d-1}(X)}$

commutes. To see this let $[(I,\omega)]\in E^{d-1}(A)$. Then
$\psi(I,\omega)=(I)$ the fundamental class associated to ${\rm Spec\,}R/I$ in
$X$. Under the Segre class map

$s:E^{d-1}(A)\to\pi_{0}(Sing_{*}(Q_{2d-2})(R))\simeq[X,Q_{2d-2}]_{\mathbb{A}^{1}}$

$s(I,\omega)=([\underline{a},\underline{b},s])=f$ where $I=<\underline{a},s>$
and

$f:X={\rm Spec\,}A\to Q_{2d-2}={\rm
Spec\,}\dfrac{k[\underline{x},\underline{y},z]}{\sum x_{i}y_{i}-z(1-z)}$

is the morphism corresponding to

$x_{i}\mapsto a_{i},\ y_{i}\mapsto b_{i},\ z\mapsto s.$

Under the Hurewicz map

$H:[X,Q_{2d-2}]_{\mathbb{A}^{1}}\to\widetilde{CH}^{d-1}(X)=H^{d-1}(X,{\mathbf{K}}^{\rm
MW}_{d-1})$ $f\mapsto f^{*}(\alpha)$

where $\alpha\in H^{d-1}(X,{\mathbf{K}}^{\rm MW}_{d-1})$ corresponding to
$<1>\otimes\overline{x}_{1}\wedge\cdots\overline{x}_{d-1}\in{\mathbf{K}}^{\rm
MW}_{0}(\kappa(w))\otimes\det\mathfrak{m}_{w}/\mathfrak{m}_{w}^{2}$ where $w$
is the generic point of the codimension $d-1$ irreducible closed subscheme $W$
of $Q_{2d-2}$ given by

$x_{1}=\cdots=x_{d-1}=z=0.$

We have the pullback map

$f^{*}:H^{d-1}(Q_{2d-2},{\mathbf{K}}^{\rm MW}_{d-1})\to
H^{d-1}(X,{\mathbf{K}}^{\rm MW}_{d-1})$

so that

$f^{*}(\alpha)=f^{*}(<1>\otimes\overline{x}_{1}\wedge\cdots\wedge\overline{x}_{d-1})=\sum_{u}l_{u}<1>\otimes\overline{a}_{1}\wedge\cdots\wedge\overline{a}_{d-1}$

where the sum varies over the generic points $u$ of $f^{-1}(W)$ and $l_{u}$ is
the length of the Artinian local ring $\mathcal{O}_{f^{-1}(W),u}$.

Under the map

$H^{d-1}(X,{\mathbf{K}}^{\rm MW}_{d-1})\to H^{d-1}(X,{\mathbf{K}}^{\rm
M}_{d-1})$

the element $f^{*}(\alpha)$ maps to $\sum_{u}l_{u}[u]$ which is precisely the
fundamental class associated to the closed subscheme $f^{-1}(W)$, but we note
that $f^{-1}(W)={\rm Spec\,}A/I$.

By [6, Theorem 1(4)], we have $E^{d-1}(A)\to[X,Q_{2d-2}]_{\mathbb{A}^{1}}$ is
an isomorphism. Hence the composition map $\psi$ is surjective.

(2) This follows directly from Proposition 5.2(2), as the assumption implies
that $H$ is an isomorphism. Hence the composition $\psi$ is an isomorphism. ∎

As a consequence of combining the above theorem with the vanishing results
from section 3, we prove the following result.

######  Corollary 5.6.

Let $k$ be an algebraically closed field. Let $X={\rm Spec\,}A$ be a smooth
affine $k$-variety of dimension $d\geq 4$.

Then the map

$\psi:E^{d-1}(A)\to CH^{d-1}(X)$

  1. (1)

is always surjective.

  2. (2)

is injective if $d=4$.

  3. (3)

the kernel is $(d-1)!(d-2)!$-torsion if $d\geq 5.$

  4. (4)

further for $d\geq 5$, if Conjecture 3.8 holds, then the map $\psi$ is
injective .

###### Proof.

It follows by combining Remark 5.1 and Theorem 5.5. Case (2) follows from
Theorem 5.5 and Theorem 3.16 (3). Case (3) follows from Theorem 3.16 (5). Case
(4) follows from Theorem 3.16(6). ∎

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