Relative $\mathbb{A}^1$-homology and its applications

TL;DR

This paper develops a relative $ ext{A}^1$-homology theory, proving key theorems like Whitehead and excision, and applies these to compute homology in projective space embeddings.

Contribution

It introduces a general theory of relative $ ext{A}^1$-homology and homotopy sheaves, extending foundational results in $ ext{A}^1$-homotopy theory.

Findings

Proved an $ ext{A}^1$-homology version of the Whitehead theorem.

Established an excision theorem for $ ext{A}^1$-homology and related sheaves.

Computed the relative $ ext{A}^1$-homology of a hyperplane embedding in projective space.

Abstract

In this paper, we prove an $\mathbb{A}^1$-homology version of the Whitehead theorem with dimension bound. We also prove an excision theorem for $\mathbb{A}^1$-homology, Suslin homology and $\mathbb{A}^1$-homotopy sheaves. In order to prove these results, we develop a general theory of relative $\mathbb{A}^1$-homology and $\mathbb{A}^1$-homotopy sheaves. As an application, we compute the relative $\mathbb{A}^1$-homology of a hyperplane embedding $\mathbb{P}^{n-1} \hookrightarrow \mathbb{P}^{n}$.

Equations

$${\mathbf{H}}_i^{{\mathbb{A}}^1}(U)\to {\mathbf{H}}_i^{{\mathbb{A}}^1}(X)$$

$${\pi }_i^{{\mathbb{A}}^1}(U,\ast )\to {\pi }_i^{{\mathbb{A}}^1}(X,\ast )$$

$${\mathbf{H}}_i^{{\mathbb{A}}^1}({\mathbb{P}}^n,{\mathbb{P}}^{n-1})\cong \{\begin{array}{ll}{\underline{\mathbf{K}}}_n^{MW}& (i=n)\\ 0& (i<n).\end{array}$$

$${\pi }_i^{{\mathbb{A}}^1}(f)=\{\begin{array}{ll}{\pi }_{i-1}^{{\mathbb{A}}^1}(F_{{\mathbb{A}}^1}(f))& (i>0)\\ \mathrm{Spec}k& (i=0),\end{array}$$

$$\begin{gathered} \cdots \to {\pi }_i^{{\mathbb{A}}^1}(\mathcal{X},\ast )\to {\pi }_i^{{\mathbb{A}}^1}(\mathcal{Y},\ast )\to {\pi }_i^{{\mathbb{A}}^1}(f)\to {\pi }_{i-1}^{{\mathbb{A}}^1}(\mathcal{X},\ast )\to \cdots \\ \cdots \to {\pi }_0^{{\mathbb{A}}^1}(\mathcal{X},\ast )\to {\pi }_0^{{\mathbb{A}}^1}(\mathcal{Y},\ast )\to 0. \end{gathered}$$

$$h:{\pi }_n^{{\mathbb{A}}^1}(\mathcal{X},\ast )\to {\mathbf{H}}_n^{{\mathbb{A}}^1}(\mathcal{X};\mathbb{Z})$$

$$h:{\pi }_n^{{\mathbb{A}}^1}(f)\to {\mathbf{H}}_n^{{\mathbb{A}}^1}(f;\mathbb{Z})$$

$${\mathrm{Hom}}_{\mathcal{A}{b}_k}({\pi }_n^{{\mathbb{A}}^1}(f),A)\cong {\mathrm{Hom}}_{\mathcal{A}{b}_k}({\mathbf{H}}_n({\mathrm{Ex}}_{{\mathbb{A}}^1}(f)),A)$$

$$\begin{array}{rl}{\mathrm{Hom}}_{\mathcal{A}{b}_k}({\mathbf{H}}_n({\mathrm{Ex}}_{{\mathbb{A}}^1}(f)),A)& \cong {\mathrm{Hom}}_{\mathbf{D}(k,\mathbb{Z})}(C({\mathrm{Ex}}_{{\mathbb{A}}^1}(f);\mathbb{Z}),A[n])\\ & \cong {\mathrm{Hom}}_{\mathbf{D}(k,\mathbb{Z})}(C^{{\mathbb{A}}^1}({\mathrm{Ex}}_{{\mathbb{A}}^1}(f);\mathbb{Z}),A[n]).\end{array}$$

$${\mathrm{Hom}}_{\mathbf{D}(k,\mathbb{Z})}(C^{{\mathbb{A}}^1}({\mathrm{Ex}}_{{\mathbb{A}}^1}(f);\mathbb{Z}),A[n])\cong {\mathrm{Hom}}_{\mathcal{A}{b}_k}({\mathbf{H}}_n^{{\mathbb{A}}^1}({\mathrm{Ex}}_{{\mathbb{A}}^1}(f);\mathbb{Z}),A).$$

$$\begin{array}{ccccccc}{C}^{{\mathbb{A}}^1}(\mathcal{X};\mathbb{Z})& \to & C^{{\mathbb{A}}^1}(\mathcal{Y};\mathbb{Z})& \to & C^{{\mathbb{A}}^1}(f;\mathbb{Z})& \to & \\ \cong \downarrow & & \cong \downarrow & & \downarrow & & \\ C^{{\mathbb{A}}^1}({\mathrm{Ex}}_{{\mathbb{A}}^1}(\mathcal{X});\mathbb{Z})& \to & C^{{\mathbb{A}}^1}({\mathrm{Ex}}_{{\mathbb{A}}^1}(\mathcal{Y});\mathbb{Z})& \to & C^{{\mathbb{A}}^1}({\mathrm{Ex}}_{{\mathbb{A}}^1}(f);\mathbb{Z})& \to & \end{array}$$

$${\mathrm{Hom}}_{\mathcal{A}{b}_k}({\mathbf{H}}_n^{{\mathbb{A}}^1}({\mathrm{Ex}}_{{\mathbb{A}}^1}(f);\mathbb{Z}),A)\cong {\mathrm{Hom}}_{\mathcal{A}{b}_k}({\mathbf{H}}_n^{{\mathbb{A}}^1}(f;\mathbb{Z}),A).$$

$${\mathrm{Hom}}_{\mathcal{A}{b}_k}({\pi }_n^{{\mathbb{A}}^1}(f),A)\cong {\mathrm{Hom}}_{\mathcal{A}{b}_k}({\mathbf{H}}_n^{{\mathbb{A}}^1}(f;\mathbb{Z}),A)={\mathrm{Hom}}_{\mathcal{A}{b}_k^{{\mathbb{A}}^1}}({\mathbf{H}}_n^{{\mathbb{A}}^1}(f;\mathbb{Z}),A)$$

$$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 21.8792pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&\crcr}}}\ignorespaces{\hbox{\kern-21.8792pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\mathop{\mathcal{M}od_{k}}\nolimits(R)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 35.25351pt\raise 10.63832pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.45056pt\hbox{$\scriptstyle{H_{0}\circ L_{\mathop{\mathbb{A}^{1}}\nolimits}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 75.8792pt\raise 4.30554pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 45.8792pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 75.8792pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\mathop{\mathcal{M}od_{k}^{\mathbb{A}^{1}}}\nolimits(R)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 21.8792pt\raise-4.30554pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces,$$

$$h^{\prime }:{\pi }_n^{{\mathbb{A}}^1}(f)\to H_0(L_{{\mathbb{A}}^1}({\pi }_n^{{\mathbb{A}}^1}(f)))$$

$${\mathrm{Hom}}_{\mathcal{A}{b}_k^{{\mathbb{A}}^1}}({\mathbf{H}}_n^{{\mathbb{A}}^1}(f;\mathbb{Z}),A)\cong {\mathrm{Hom}}_{\mathcal{A}{b}_k}({\pi }_n^{{\mathbb{A}}^1}(f),A)\cong {\mathrm{Hom}}_{\mathcal{A}{b}_k^{{\mathbb{A}}^1}}(H_0(L_{{\mathbb{A}}^1}({\pi }_n^{{\mathbb{A}}^1}(f))),A).$$

$$H_0(L_{{\mathbb{A}}^1}({\pi }_n^{{\mathbb{A}}^1}(f)))\cong {\mathbf{H}}_n^{{\mathbb{A}}^1}(f;\mathbb{Z}).$$

$$h:{\pi }_n^{{\mathbb{A}}^1}(f)\stackrel{h^{\prime }}{\to }{H}_0(L_{{\mathbb{A}}^1}({\pi }_n^{{\mathbb{A}}^1}(f)))\cong {\mathbf{H}}_n^{{\mathbb{A}}^1}(f;\mathbb{Z})$$

$${\pi }_1^{{\mathbb{A}}^1}(\mathcal{Y},\ast )\to {\pi }_1^{{\mathbb{A}}^1}(f)\to {\pi }_0^{{\mathbb{A}}^1}(\mathcal{X},\ast ).$$

$${\mathrm{Hom}}_{\mathcal{M}o{d}_k^{{\mathbb{A}}^1}(R)}({\mathbf{H}}_n^{{\mathbb{A}}^1}(f;R),A)$$

$${\mathrm{Hom}}_{{\mathbf{NST}}_k^{{\mathbb{A}}^1}(R)}({\mathbf{H}}_i^S(f;R),A)$$

$${\mathrm{Hom}}_{{\mathbf{D}}_{tr}(k,R)}(R_{tr}(f),A[n])\stackrel{\cong }{\to }{\mathrm{Hom}}_{\mathbf{D}(k,R)}(R(f),A[n])$$

$$\begin{array}{ccc}{\mathrm{Hom}}_{{\mathbf{NST}}_k^{{\mathbb{A}}^1}(R)}({\mathbf{H}}_n^S(f;R),A)& \stackrel{\cong }{\to }& {\mathrm{Hom}}_{{\mathbf{D}}_{tr}(k,R)}(R_{tr}(f),A[n])\\ \downarrow & & \cong \downarrow \\ {\mathrm{Hom}}_{\mathcal{M}o{d}_k^{{\mathbb{A}}^1}(R)}({\mathbf{H}}_n^{{\mathbb{A}}^1}(f;R),A)& \stackrel{\cong }{\to }& {\mathrm{Hom}}_{\mathbf{D}(k,R)}(R(f),A[n])\end{array}$$

$$H_{Nis}^n(f;A)={\mathrm{Hom}}_{\mathbf{D}(k,\mathbb{Z})}(C(f;\mathbb{Z}),A[n]).$$

$$H_{Nis}^{m+1}(f;A)\cong {\mathrm{Hom}}_{\mathcal{M}o{d}_k^{{\mathbb{A}}^1}(R)}({\mathbf{H}}_{m+1}^{{\mathbb{A}}^1}(f;R),A)$$

$$\cdots \to H_{Nis}^m(X;A)\to H_{Nis}^{m+1}(f;A)\to H_{Nis}^{m+1}(Y;A)\to \cdots$$

$${\mathrm{Hom}}_{{\mathbf{D}}_{tr}(k,R)}(R_{tr}(f),A[m+1])\cong {\mathrm{Hom}}_{{\mathbf{NST}}_k^{{\mathbb{A}}^1}(R)}({\mathbf{H}}_{m+1}^S(f;R),A)$$

$$0={\pi }_1^{{\mathbb{A}}^1}(X,\ast )\to {\pi }_1^{{\mathbb{A}}^1}(Y,\ast )\to {\pi }_1^{{\mathbb{A}}^1}(f)=0$$

$$H_{Nis}^i(X,U;A)\cong {\mathrm{Hom}}_{\mathcal{M}o{d}_k^{{\mathbb{A}}^1}(R)}({\mathbf{H}}_i^{{\mathbb{A}}^1}(X,U;R),A)$$

$${\mathbf{H}}_i^{{\mathbb{A}}^1}({\mathbb{P}}^n,{\mathbb{P}}^{n-1};R)\cong \{\begin{array}{ll}{\underline{\mathbf{K}}}_n^{MW}{\otimes }^{{\mathbb{A}}^1}R& (i=n)\\ 0& (i<n).\end{array}$$

$${\mathbf{H}}_n^{{\mathbb{A}}^1}({\mathbb{P}}^{n+1};R)\cong \mathrm{Coker}({\underline{\mathbf{K}}}_{n+1}^{MW}{\otimes }^{{\mathbb{A}}^1}R\to {\mathbf{H}}_n^{{\mathbb{A}}^1}({\mathbb{P}}^n;R)).$$

Full text

1 Introduction

In this paper, we study a relative version of the $\mathop{\mathbb{A}^{1}}\nolimits$-homology sheaves of smooth schemes over a field and give applications to motives and $\mathop{\mathbb{A}^{1}}\nolimits$-homotopy theory. Let $k$ be a field. In [Vo], Voevodsky constructed a triangulated category of motives $\mathbf{DM}_{-}^{eff}(k)$ over $k$ as a triangulated subcategory of the derived category of Nisnevich sheaves with transfers, with a functor $M$ from the category of smooth $k$-schemes $\mathop{\mathcal{S}m_{k}}\nolimits$ to $\mathbf{DM}_{-}^{eff}(k)$. For $X\in\mathop{\mathcal{S}m_{k}}\nolimits$, $M(X)$ is called the motive of $X$. Its homology sheaves with transfers $\mathbf{H}^{S}_{*}(X)=H_{*}(M(X))$ are called the Suslin homology sheaves (cf. [As, Section 2]), whose sections over $\mathop{\mathrm{Spec}}\nolimits k$ give the Suslin homology group introduced by Suslin-Voevodsky [SV] when $k$ is perfect.

In [MV], Morel-Voevodsky established the $\mathop{\mathbb{A}^{1}}\nolimits$-homotopy theory and defined an $\mathop{\mathbb{A}^{1}}\nolimits$-version of homotopy groups, called $\mathop{\mathbb{A}^{1}}\nolimits$-homotopy sheaves, as Nisnevich sheaves on $\mathop{\mathcal{S}m_{k}}\nolimits$. Morel [Mo2] introduced an $\mathop{\mathbb{A}^{1}}\nolimits$-version of the singular homology, called $\mathop{\mathbb{A}^{1}}\nolimits$-homology sheaves, as an analogue of Suslin homology by instead using Nisnevich sheaves without transfers. As with motives, there is a functor $C^{\mathop{\mathbb{A}^{1}}\nolimits}$ from $\mathop{\mathcal{S}m_{k}}\nolimits$ to a triangulated subcategory of the derived category of Nisnevich sheaves on $\mathop{\mathcal{S}m_{k}}\nolimits$. The $\mathop{\mathbb{A}^{1}}\nolimits$-homology sheaves $\mathbf{H}^{\mathop{\mathbb{A}^{1}}\nolimits}_{*}(X)$ of $X\in\mathop{\mathcal{S}m_{k}}\nolimits$ are defined as the homology sheaves $H_{*}(C^{\mathop{\mathbb{A}^{1}}\nolimits}(X))$.

The purpose of this paper is threefold. Firstly, we prove an $\mathop{\mathbb{A}^{1}}\nolimits$-homological Whitehead theorem with dimension bound and the excision theorem. Secondly, as a tool for proving them, we develop a general theory of relative $\mathop{\mathbb{A}^{1}}\nolimits$-homology, namely $\mathop{\mathbb{A}^{1}}\nolimits$-homology sheaves for morphisms $f:X\to Y$. Thirdly, as an example, we compute the relative $\mathop{\mathbb{A}^{1}}\nolimits$-homology of a hyperplane embedding $\mathbb{P}^{n-1}\hookrightarrow\mathbb{P}^{n}$.

Our $\mathop{\mathbb{A}^{1}}\nolimits$-Whitehead theorem detects whether a morphism $X\to Y$ is an $\mathop{\mathbb{A}^{1}}\nolimits$-weak equivalence in terms of the $\mathop{\mathbb{A}^{1}}\nolimits$-fundamental groups and the $\mathop{\mathbb{A}^{1}}\nolimits$-homology up to degree ${\max\{\dim X+1,\dim Y\}}$. See [Ar, Thm. 6.4.15] for the classical homological Whitehead theorem in topology.

Theorem 1.1** (see Theorem 3.2).**

Assume $k$ perfect. Let $f:(X,*)\to(Y,*)$ be a morphism of $\mathop{\mathbb{A}^{1}}\nolimits$-simply connected pointed smooth $k$-schemes and let $d={\max\{\dim X+1,\dim Y\}}$. If $f$ induces an isomorphism $\mathbf{H}^{\mathop{\mathbb{A}^{1}}\nolimits}_{i}(X)\xrightarrow{\cong}\mathbf{H}^{\mathop{\mathbb{A}^{1}}\nolimits}_{i}(Y)$ for all $2\leq i<d$ and an epimorphism $\mathbf{H}^{\mathop{\mathbb{A}^{1}}\nolimits}_{d}(X)\twoheadrightarrow\mathbf{H}^{\mathop{\mathbb{A}^{1}}\nolimits}_{d}(Y)$, then $f$ is an $\mathop{\mathbb{A}^{1}}\nolimits$-weak equivalence.

The Whitehead theorem for $\mathop{\mathbb{A}^{1}}\nolimits$-homotopy sheaves is established by Morel-Voevodsky [MV], and the novelty here is the detection by $\mathop{\mathbb{A}^{1}}\nolimits$-homology sheaves and the degree bound $d={\max\{\dim X+1,\dim Y\}}$. Next, our excision theorem for $\mathop{\mathbb{A}^{1}}\nolimits$-homology sheaves is stated as follows.

Theorem 1.2** (see Theorem 3.5).**

Let $X$ be a smooth $k$-scheme and $U$ a Zariski open set of $X$ whose complement has codimension $r$. Then the morphisms

[TABLE]

are isomorphisms for every $i<r-1$ and epimorphisms for $i=r-1$.

Asok [As, Prop. 3.8] proved a similar result in degree [math]. We also obtain the $\mathop{\mathbb{A}^{1}}\nolimits$-homotopy version of the excision theorem.

Corollary 1.3** (see Corollary 3.6).**

Assume $k$ perfect. Let $(X,*)$ be a pointed smooth $k$-scheme and $(U,*)$ a pointed Zariski open set of $(X,*)$ whose complement has codimension $r$. If $(X,*)$ and $(U,*)$ are $\mathop{\mathbb{A}^{1}}\nolimits$-simply connected, then the morphism

[TABLE]

is an isomorphism for every $i<r-1$ and an epimorphism for $i=r-1$.

This is a variation of the $\mathop{\mathbb{A}^{1}}\nolimits$-excision theorem of Asok-Doran [AD] which assumes that $\pi^{\mathop{\mathbb{A}^{1}}\nolimits}_{i}(X,*)=0$ for all $i<r-1$ and $k$ infinite. In order to prove these results, we develop a general theory of relative $\mathop{\mathbb{A}^{1}}\nolimits$-homotopy and $\mathop{\mathbb{A}^{1}}\nolimits$-homology sheaves. If $f:X\to Y$ is a morphism of smooth $k$-schemes, we define its $\mathop{\mathbb{A}^{1}}\nolimits$-homotopy $\pi^{\mathop{\mathbb{A}^{1}}\nolimits}_{i}(f)$, $\mathop{\mathbb{A}^{1}}\nolimits$-homology $\mathbf{H}^{\mathop{\mathbb{A}^{1}}\nolimits}_{i}(f)$ and Suslin homology $\mathbf{H}^{S}_{i}(f)$.

Finally, as an application of the results above, we compute the relative $\mathop{\mathbb{A}^{1}}\nolimits$-homology sheaves of the pair $(\mathbb{P}^{n},\mathbb{P}^{n-1})$. Let $\underline{\mathbf{K}}^{MW}_{n}$ be the unramified Milnor-Witt $K$-theory defined by Morel [Mo2].

Theorem 1.4** (see Theorem 4.1).**

Assume $k$ perfect. For $0\leq i\leq n$, $n>0$, we have

[TABLE]

In particular, when $i<n$, we have $\mathbf{H}^{\mathop{\mathbb{A}^{1}}\nolimits}_{i}(\mathbb{P}^{n})\cong\mathbf{H}^{\mathop{\mathbb{A}^{1}}\nolimits}_{i}(\mathbb{P}^{i+1})$.

Similar stabilization $\mathbf{H}^{S}_{i}(\mathbb{P}^{n})\cong\mathbf{H}^{S}_{i}(\mathbb{P}^{i+1})$ in $i<n$ also holds for the Suslin homology (Corollary 4.3). The $\mathop{\mathbb{A}^{1}}\nolimits$-homology $\mathbf{H}^{\mathop{\mathbb{A}^{1}}\nolimits}_{i}(\mathbb{P}^{i+1})$ can be described in terms of $\underline{\mathbf{K}}^{MW}_{i}$ (Corollary 4.2).

This paper is organized as follows. In Section 2, we prove a weak version of the relative $\mathop{\mathbb{A}^{1}}\nolimits$-Hurewicz theorem. In Section 3, we prove Theorems 1.1-1.2. In Section 4, we prove Theorem 1.4.

Notation**.**

In this paper, we fix a field $k$. We denote by $\mathop{\mathcal{S}m_{k}}\nolimits$ the category of smooth $k$-schemes. Every sheaf is considered on the Nisnevich topology on $\mathop{\mathcal{S}m_{k}}\nolimits$. Objects of an abelian category are regarded as complexes concentrated in degree zero.

Acknowledgments**.**

I would like to thank my adviser Shohei Ma for many useful advices. I would also like to thank Tom Bachmann for pointing out a mistake in the previous version. I would like to thank the referee for many detailed comments which helped us to improve the presentation.

2 Relative $\mathop{\mathbb{A}^{1}}\nolimits$-homotopy and $\mathop{\mathbb{A}^{1}}\nolimits$-homology

In this section, we give basic definitions of relative $\mathop{\mathbb{A}^{1}}\nolimits$-homotopy and $\mathop{\mathbb{A}^{1}}\nolimits$-homology, and establish a weak relative $\mathop{\mathbb{A}^{1}}\nolimits$-Hurewicz theorem. We also compare $\mathop{\mathbb{A}^{1}}\nolimits$-homology and Suslin homology. We refer to [MV], [MVW], [SV], [Mo2] and [As] for the basic theory of $\mathop{\mathbb{A}^{1}}\nolimits$-homology and $\mathop{\mathbb{A}^{1}}\nolimits$-homotopy.

2.1 Basic definitions

Let $\mathop{\mathcal{S}pc_{k}}\nolimits$ be the category of simplicial Nisnevich sheaves on $\mathop{\mathcal{S}m_{k}}\nolimits$ (called $k$-spaces) equipped with the $\mathop{\mathbb{A}^{1}}\nolimits$-model structure of [MV]. We denote by $\pi_{0}^{\mathop{\mathbb{A}^{1}}\nolimits}(\mathcal{X})$ the sheaf of $\mathop{\mathbb{A}^{1}}\nolimits$-connected components of a $k$-space $\mathcal{X}$ and denote by $\pi_{n}^{\mathop{\mathbb{A}^{1}}\nolimits}(\mathcal{X},*)$ the $n$-th $\mathop{\mathbb{A}^{1}}\nolimits$-homotopy sheaf of a pointed $k$-space $(\mathcal{X},*)$ for $n\geq 0$. A $k$-space $\mathcal{X}$ is called $\mathop{\mathbb{A}^{1}}\nolimits$-connected if $\pi_{0}^{\mathop{\mathbb{A}^{1}}\nolimits}(\mathcal{X})\cong\mathop{\mathrm{Spec}}\nolimits k$, and a pointed $k$-space $(\mathcal{X},*)$ is called $\mathop{\mathbb{A}^{1}}\nolimits$-$n$-connected if $\mathcal{X}$ is $\mathop{\mathbb{A}^{1}}\nolimits$-connected and if $\pi_{i}^{\mathop{\mathbb{A}^{1}}\nolimits}(\mathcal{X},*)=0$ for all $1\leq i\leq n$. Especially, $(\mathcal{X},*)$ is called $\mathop{\mathbb{A}^{1}}\nolimits$-simply connected if it is $\mathop{\mathbb{A}^{1}}\nolimits$-$1$-connected. We consider a relative version of these definitions.

Definition 2.1**.**

For a morphism of pointed $k$-spaces $f:(\mathcal{X},*)\to(\mathcal{Y},*)$, the $i$-th $\mathop{\mathbb{A}^{1}}\nolimits$-homotopy sheaf of $f$ is defined by

[TABLE]

where $F_{\mathop{\mathbb{A}^{1}}\nolimits}(f)$ is the homotopy fiber with respect to the $\mathop{\mathbb{A}^{1}}\nolimits$-model structure of $\mathop{\mathcal{S}pc_{k}}\nolimits$. When $f$ is an inclusion, we write $\pi^{\mathop{\mathbb{A}^{1}}\nolimits}_{i}(\mathcal{Y},\mathcal{X})=\pi^{\mathop{\mathbb{A}^{1}}\nolimits}_{i}(f)$. A morphism (or a pair) is called $\mathop{\mathbb{A}^{1}}\nolimits$-$n$-connected if its $\mathop{\mathbb{A}^{1}}\nolimits$-homotopy sheaves in degree $\leq n$ are isomorphic to $\mathop{\mathrm{Spec}}\nolimits k$.

Since $F_{\mathop{\mathbb{A}^{1}}\nolimits}(f)\to\mathcal{X}\xrightarrow{f}\mathcal{Y}$ is a homotopy fiber sequence under the $\mathop{\mathbb{A}^{1}}\nolimits$-model structure, we obtain by [AE, Prop. 4.21] the long exact sequence

[TABLE]

We fix a commutative unital ring $R$. Let $\mathop{\mathcal{M}od_{k}}\nolimits(R)$ be the category of Nisnevich sheaves of $R$-modules on $\mathop{\mathcal{S}m_{k}}\nolimits$ and $\mathbf{D}(k,R)$ be its unbounded derived category. We denote by $\mathbf{D}_{\mathop{\mathbb{A}^{1}}\nolimits}(k,R)$ the full subcategory of $\mathbf{D}(k,R)$ consisting of $\mathop{\mathbb{A}^{1}}\nolimits$-local complexes and $\mathop{\mathcal{M}od_{k}^{\mathbb{A}^{1}}}\nolimits(R)$ for the full subcategory of $\mathop{\mathcal{M}od_{k}}\nolimits(R)$ consisting of strictly $\mathop{\mathbb{A}^{1}}\nolimits$-invariant sheaves. We write $\mathop{\mathcal{A}b_{k}}\nolimits=\mathop{\mathcal{M}od_{k}}\nolimits(\mathbb{Z})$ and $\mathop{\mathcal{A}b_{k}^{\mathbb{A}^{1}}}\nolimits=\mathop{\mathcal{M}od_{k}^{\mathbb{A}^{1}}}\nolimits(\mathbb{Z})$. For $\mathcal{X}\in\mathop{\mathcal{S}pc_{k}}\nolimits$, we denote by $R(\mathcal{X})$ the simplicial Nisnevich sheaf of $R$-modules freely generated by $\mathcal{X}$ and $C(\mathcal{X};R)$ for its normalized chain complex. Let $L_{\mathop{\mathbb{A}^{1}}\nolimits}$ be a left adjoint of the inclusion $\mathbf{D}_{\mathop{\mathbb{A}^{1}}\nolimits}(k,R)\hookrightarrow\mathbf{D}(k,R)$ (see [CD, Thm. 2.5]). We write $C^{\mathop{\mathbb{A}^{1}}\nolimits}(\mathcal{X};R)=L_{\mathop{\mathbb{A}^{1}}\nolimits}(C(\mathcal{X};R))$ and $\mathbf{H}_{*}^{\mathop{\mathbb{A}^{1}}\nolimits}(\mathcal{X};R)=H_{*}(C(\mathcal{X};R))$. We consider a relative version of these definitions.

Definition 2.2**.**

Let $f:\mathcal{X}\to\mathcal{Y}$ be a morphism of $k$-spaces. We denote by $C(f;R)$ the mapping cone of $C(\mathcal{X};R)\to C(\mathcal{Y};R)$. We write $C^{\mathop{\mathbb{A}^{1}}\nolimits}(f;R)=L_{\mathop{\mathbb{A}^{1}}\nolimits}(C(f;R))$. We define the $i$-th $\mathop{\mathbb{A}^{1}}\nolimits$-homology sheaf $\mathbf{H}^{\mathop{\mathbb{A}^{1}}\nolimits}_{i}(f;R)$ as the homology of $C^{\mathop{\mathbb{A}^{1}}\nolimits}(\mathcal{X};R)$ in degree $i$. When $f$ is an inclusion, we write $C^{\mathop{\mathbb{A}^{1}}\nolimits}(\mathcal{Y},\mathcal{X};R)=C^{\mathop{\mathbb{A}^{1}}\nolimits}(f;R)$ and $\mathbf{H}^{\mathop{\mathbb{A}^{1}}\nolimits}_{i}(\mathcal{Y},\mathcal{X};R)=\mathbf{H}^{\mathop{\mathbb{A}^{1}}\nolimits}_{i}(f;R)$.

2.2 Relative $\mathop{\mathbb{A}^{1}}\nolimits$-Hurewicz theorem

We denote by $\mathcal{G}r_{k}^{\mathop{\mathbb{A}^{1}}\nolimits}$ the category of strongly $\mathop{\mathbb{A}^{1}}\nolimits$-invariant sheaves of groups on $\mathop{\mathcal{S}m_{k}}\nolimits$ (see [Mo2, Def. 1.7]). For a functor $F:\mathscr{C}\to\mathscr{D}$ between categories, a morphism $f:A\to A^{\prime}$ in $\mathscr{D}$ is called universal with respect to $F$ if it induces a bijection $\mathop{\mathrm{Hom}}\nolimits_{\mathscr{D}}(A^{\prime},F(B))\cong\mathop{\mathrm{Hom}}\nolimits_{\mathscr{D}}(A,F(B))$ for every $B\in\mathscr{C}$. Morel [Mo2] proved the following $\mathop{\mathbb{A}^{1}}\nolimits$-Hurewicz theorem which relates $\mathop{\mathbb{A}^{1}}\nolimits$-homotopy and $\mathop{\mathbb{A}^{1}}\nolimits$-homology.

Theorem 2.3** (see [Mo2, Thm. 6.35 and 6.37]).**

Let $(\mathcal{X},*)$ be a pointed $k$-space. Then there exists a natural morphism

[TABLE]

such that if $(\mathcal{X},*)$ is $\mathop{\mathbb{A}^{1}}\nolimits$-$(n-1)$-connected for $n\geq 1$, then $h$ is universal with respect to the inclusion $\mathop{\mathcal{A}b_{k}^{\mathbb{A}^{1}}}\nolimits\hookrightarrow\mathcal{G}r^{\mathop{\mathbb{A}^{1}}\nolimits}_{k}$.

Morel proves that $h$ is an isomorphism assuming $k$ perfect; for the above assertion his argument works for general $k$. The following is a relative version of Theorem 2.3.

Proposition 2.4**.**

Let $f:(\mathcal{X},*)\to(\mathcal{Y},*)$ be a morphism from an $\mathop{\mathbb{A}^{1}}\nolimits$-simply connected pointed $k$-space to an $\mathop{\mathbb{A}^{1}}\nolimits$-connected $k$-space. Suppose that $f$ is $\mathop{\mathbb{A}^{1}}\nolimits$-$(n-1)$-connected for $n\geq 2$. Then there exists a universal morphism

[TABLE]

with respect to the inclusion $\mathop{\mathcal{A}b_{k}^{\mathbb{A}^{1}}}\nolimits\hookrightarrow\mathcal{G}r^{\mathop{\mathbb{A}^{1}}\nolimits}_{k}$.

Proof.

We write $\mathbf{H}_{n}(-)=H_{n}(C(-;\mathbb{Z}))$. Let $\mathrm{Ex}_{\mathop{\mathbb{A}^{1}}\nolimits}$ be the resolution functor as in [MV, §3.2]. By applying the relative Hurewicz theorem of simplicial sets [GJ, Thm. 3.11] to all stalks, we have a natural isomorphism $\pi^{\mathop{\mathbb{A}^{1}}\nolimits}_{i}(f)\cong\mathbf{H}_{i}(\mathrm{Ex}_{\mathop{\mathbb{A}^{1}}\nolimits}(f))$ for all $1\leq i\leq n$ and $\mathbf{H}_{0}(\mathrm{Ex}_{\mathop{\mathbb{A}^{1}}\nolimits}(f))=0$. Thus we obtain $\mathbf{H}_{i}(\mathrm{Ex}_{\mathop{\mathbb{A}^{1}}\nolimits}(f))=0$ for every $i\leq n-1$. By the isomorphism $\pi^{\mathop{\mathbb{A}^{1}}\nolimits}_{n}(f)\cong\mathbf{H}_{n}(\mathrm{Ex}_{\mathop{\mathbb{A}^{1}}\nolimits}(f))$, there exists an isomorphism

[TABLE]

for every $A\in\mathop{\mathcal{A}b_{k}^{\mathbb{A}^{1}}}\nolimits$. Since $\mathbf{H}_{i}(\mathrm{Ex}_{\mathop{\mathbb{A}^{1}}\nolimits}(f))=0$ for all $i\leq n-1$, the adjunction on $L_{\mathop{\mathbb{A}^{1}}\nolimits}$ shows that

[TABLE]

Then $H_{i}(C^{\mathop{\mathbb{A}^{1}}\nolimits}(\mathrm{Ex}_{\mathop{\mathbb{A}^{1}}\nolimits}(f);\mathbb{Z}))=0$ for all $i\leq n-1$ by [Mo2, Thm. 6.22]. Thus

[TABLE]

The morphism of distinguished triangles

[TABLE]

in $\mathbf{D}(k,\mathbb{Z})$ induced by the natural transformation $\mathrm{Id}\to\mathrm{Ex}_{\mathop{\mathbb{A}^{1}}\nolimits}$ gives a quasi-isomorphism $C^{\mathop{\mathbb{A}^{1}}\nolimits}(f;\mathbb{Z})\to C^{\mathop{\mathbb{A}^{1}}\nolimits}(\mathrm{Ex}_{\mathop{\mathbb{A}^{1}}\nolimits}(f);\mathbb{Z})$. Therefore, we obtain

[TABLE]

By the isomorphisms (2.1)-(2.4), we have

[TABLE]

for all $A\in\mathop{\mathcal{A}b_{k}^{\mathbb{A}^{1}}}\nolimits$. On the other hand, [Mo2, Thm. 6.22] leads to the adjunction

[TABLE]

and this induces a universal morphism

[TABLE]

with respect to $\mathop{\mathcal{A}b_{k}^{\mathbb{A}^{1}}}\nolimits\hookrightarrow\mathop{\mathcal{A}b_{k}}\nolimits$. By the isomorphism (2.5), we have

[TABLE]

Thus Yoneda’s lemma in $\mathop{\mathcal{A}b_{k}^{\mathbb{A}^{1}}}\nolimits$ shows that

[TABLE]

Therefore, the composite morphism

[TABLE]

is universal with respect to $\mathop{\mathcal{A}b_{k}^{\mathbb{A}^{1}}}\nolimits\hookrightarrow\mathop{\mathcal{A}b_{k}}\nolimits$. Since $\pi^{\mathop{\mathbb{A}^{1}}\nolimits}_{n}(f)$ and $\mathbf{H}^{\mathop{\mathbb{A}^{1}}\nolimits}_{n}(f;\mathbb{Z})$ are strongly $\mathop{\mathbb{A}^{1}}\nolimits$-invariant, the morphism $h$ is universal with respect to the inclusion $\mathop{\mathcal{A}b_{k}^{\mathbb{A}^{1}}}\nolimits\hookrightarrow\mathcal{G}r^{\mathop{\mathbb{A}^{1}}\nolimits}_{k}$. ∎

When $k$ is perfect, Proposition 2.4 gives an isomorphism between the relative $\mathop{\mathbb{A}^{1}}\nolimits$-homotopy and the $\mathop{\mathbb{A}^{1}}\nolimits$-homology sheaves.

Corollary 2.5**.**

Let $f$ be as in Proposition 2.4. If $k$ is perfect, then there exists a natural isomorphism $\pi^{\mathop{\mathbb{A}^{1}}\nolimits}_{n}(f)\cong\mathbf{H}^{\mathop{\mathbb{A}^{1}}\nolimits}_{n}(f;\mathbb{Z})$.

Proof.

Since $\pi^{\mathop{\mathbb{A}^{1}}\nolimits}_{n}(f)\in\mathop{\mathcal{A}b_{k}^{\mathbb{A}^{1}}}\nolimits$ by [Mo2, Cor. 6.2], Yoneda’s lemma in $\mathop{\mathcal{A}b_{k}^{\mathbb{A}^{1}}}\nolimits$ gives a natural isomorphism $\pi^{\mathop{\mathbb{A}^{1}}\nolimits}_{n}(f)\cong\mathbf{H}^{\mathop{\mathbb{A}^{1}}\nolimits}_{n}(f;\mathbb{Z})$. ∎

By Corollary 2.5, we obtain the following.

Corollary 2.6**.**

Assume $k$ perfect. Let $f:(\mathcal{X},*)\to(\mathcal{Y},*)$ be a morphism of $\mathop{\mathbb{A}^{1}}\nolimits$-simply connected pointed $k$-spaces and $n\geq 2$ an integer. If $\mathbf{H}^{\mathop{\mathbb{A}^{1}}\nolimits}_{i}(f;\mathbb{Z})=0$ for all $2\leq i\leq n$, then $f$ is $\mathop{\mathbb{A}^{1}}\nolimits$-$n$-connected.

Proof.

We use induction on $i$. The assertion is clear for $i=0$. We next consider the case $i=1$. There exists an exact sequence

[TABLE]

Since $\pi^{\mathop{\mathbb{A}^{1}}\nolimits}_{1}(\mathcal{Y},*)=\pi^{\mathop{\mathbb{A}^{1}}\nolimits}_{0}(\mathcal{X},*)=0$, we have $\pi^{\mathop{\mathbb{A}^{1}}\nolimits}_{1}(f)=0$. Finally, let $i\geq 2$ and $\pi^{\mathop{\mathbb{A}^{1}}\nolimits}_{i-1}(f)=0$. Then Corollary 2.5 shows that $\pi^{\mathop{\mathbb{A}^{1}}\nolimits}_{i}(f)\cong\mathbf{H}^{\mathop{\mathbb{A}^{1}}\nolimits}_{i}(f;\mathbb{Z})=0$. ∎

2.3 $\mathop{\mathbb{A}^{1}}\nolimits$-homology and Suslin homology

Next, we compare $\mathop{\mathbb{A}^{1}}\nolimits$-homology and Suslin homology. Let $\mathbf{NST}_{k}(R)$ be the category of Nisnevich sheaves with transfers with coefficients in $R$, $\mathbf{D}_{tr}(k,R)$ be the unbounded derived category of $\mathbf{NST}_{k}(R)$, and $R_{tr}:\mathop{\mathcal{S}m_{k}}\nolimits\to\mathbf{NST}_{k}(R)$ be the functor as in [MVW, Def. 2.8] (with $R$-coefficients). Following [Vo], we denote by $\mathbf{DM}^{eff}(k,R)$ the full subcategory of $\mathbf{D}_{tr}(k,R)$ consisting of $\mathop{\mathbb{A}^{1}}\nolimits$-local complexes. Let $L^{tr}_{\mathop{\mathbb{A}^{1}}\nolimits}$ be a left adjoint of the inclusion $\mathbf{DM}^{eff}(k,R)\hookrightarrow\mathbf{D}_{tr}(k,R)$ (see [CD, Thm. 2.5]). We write $M(X;R)=L^{tr}_{\mathop{\mathbb{A}^{1}}\nolimits}(R_{tr}(X))$ for each $X\in\mathop{\mathcal{S}m_{k}}\nolimits$. The homology sheaves $\mathbf{H}_{*}^{S}(X;R)=H_{*}(M(X;R))$ are called the Suslin homology sheaves of $X$ (cf. [SV]). We introduce a relative version.

Definition 2.7**.**

Let $f:X\to Y$ be a morphism in $\mathop{\mathcal{S}m_{k}}\nolimits$. Then we denote by $R_{tr}(f)$ the mapping cone of the morphism $R_{tr}(X)\to R_{tr}(Y)$. We write $M(f;R)=L_{\mathop{\mathbb{A}^{1}}\nolimits}^{tr}(R_{tr}(f))$. We define the $i$-th Suslin homology sheaf $\mathbf{H}^{S}_{i}(f;R)$ as the homology of $M(f;R)$ in degree $i$. When $f$ is an embedding, we write $R_{tr}(Y,X)=R_{tr}(f)$ and $\mathbf{H}^{S}_{i}(Y,X;R)=\mathbf{H}^{S}_{i}(f;R)$.

Let $\mathbf{NST}_{k}^{\mathop{\mathbb{A}^{1}}\nolimits}(R)$ be the full subcategory of $\mathbf{NST}_{k}(R)$ consisting of strictly $\mathop{\mathbb{A}^{1}}\nolimits$-invariant sheaves. If $f$ is a morphism in $\mathop{\mathcal{S}m_{k}}\nolimits$, we have a morphism $\mathbf{H}^{\mathop{\mathbb{A}^{1}}\nolimits}_{*}(f;R)\to\mathbf{H}_{*}^{S}(f;R)$ in $\mathop{\mathcal{M}od_{k}^{\mathbb{A}^{1}}}\nolimits(R)$. The following is an analogue of the result of Asok [As, Cor. 3.4] in higher degree.

Proposition 2.8**.**

Let $f:X\to Y$ be a morphism in $\mathop{\mathcal{S}m_{k}}\nolimits$ and let $n\geq 0$. If $\mathbf{H}^{\mathop{\mathbb{A}^{1}}\nolimits}_{i}(f;R)=0$ for all $i<n$, then the natural morphism $\mathbf{H}^{\mathop{\mathbb{A}^{1}}\nolimits}_{n}(f;R)\to\mathbf{H}^{S}_{n}(f;R)$ is universal with respect to the canonical functor $\mathbf{NST}^{\mathop{\mathbb{A}^{1}}\nolimits}_{k}(R)\to\mathop{\mathcal{M}od_{k}^{\mathbb{A}^{1}}}\nolimits(R)$.

Proof.

By induction on $n$ and Yoneda’s lemma in $\mathbf{NST}^{\mathop{\mathbb{A}^{1}}\nolimits}_{k}(R)$, we may assume that $\mathbf{H}^{S}_{i}(f;R)=0$ for all $i<n$. For $A\in\mathbf{NST}_{k}^{\mathop{\mathbb{A}^{1}}\nolimits}(R)$, we have

[TABLE]

On the other hand, the adjunction $\mathbf{D}(k,R)\rightleftarrows\mathbf{D}_{tr}(k,R)$ gives

[TABLE]

such that the diagram

[TABLE]

commutes. ∎

By Proposition 2.8 and Yoneda’s lemma in $\mathbf{NST}^{\mathop{\mathbb{A}^{1}}\nolimits}_{k}(R)$, if $\mathbf{H}^{\mathop{\mathbb{A}^{1}}\nolimits}_{i}(f;R)=0$ for all $i<n$, then $\mathbf{H}^{S}_{i}(f;R)=0$ for all $i<n$. By using the $\mathop{\mathbb{A}^{1}}\nolimits$-Hurewicz theorem, we obtain an analogue of the Hurewicz theorem relating $\mathop{\mathbb{A}^{1}}\nolimits$-homotopy and Suslin homology.

Corollary 2.9**.**

(1) Let $(X,*)$ be a pointed smooth $k$-scheme which is $\mathop{\mathbb{A}^{1}}\nolimits$-$(n-1)$-connected for $n\geq 1$. Then there exists a universal morphism $\pi^{\mathop{\mathbb{A}^{1}}\nolimits}_{n}(X,*)\to\mathbf{H}^{S}_{n}(X;\mathbb{Z})$ with respect to the functor $\mathbf{NST}^{\mathop{\mathbb{A}^{1}}\nolimits}_{k}(\mathbb{Z})\to\mathcal{G}r_{k}^{\mathop{\mathbb{A}^{1}}\nolimits}$.

(2) Let $f$ be an $\mathop{\mathbb{A}^{1}}\nolimits$-$(n-1)$-connected morphism of $\mathop{\mathbb{A}^{1}}\nolimits$-simply connected pointed smooth $k$-schemes for $n\geq 2$. Then there exists a universal morphism $\pi^{\mathop{\mathbb{A}^{1}}\nolimits}_{n}(f)\to\mathbf{H}^{S}_{n}(f;\mathbb{Z})$ with respect to the functor $\mathbf{NST}^{\mathop{\mathbb{A}^{1}}\nolimits}_{k}(\mathbb{Z})\to\mathcal{G}r^{\mathop{\mathbb{A}^{1}}\nolimits}_{k}$.

Proof.

(1) follows from Proposition 2.8 and Theorem 2.3, and (2) follows from Propositions 2.8 and 2.4. ∎

3 Proof of the main theorems

In this section, we prove Theorems 1.1 and 1.2.

3.1 $\mathop{\mathbb{A}^{1}}\nolimits$-Whitehead theorem with dimension bound

For the proof of Theorems 1.1 and 1.2, we consider the Nisnevich cohomology of morphisms. For a morphism $f:X\to Y$ in $\mathop{\mathcal{S}m_{k}}\nolimits$, $A\in\mathop{\mathcal{A}b_{k}}\nolimits$ and $n\geq 0$, we define

[TABLE]

We write $H_{Nis}^{n}(X,U;A)=H_{Nis}^{n}(i;A)$ for an embedding $i:U\hookrightarrow X$.

Proposition 3.1**.**

Let $f:X\to Y$ be a morphism in $\mathop{\mathcal{S}m_{k}}\nolimits$. We write $d={\max\{\dim X+1,\dim Y\}}$.

(1) If $\mathbf{H}^{\mathop{\mathbb{A}^{1}}\nolimits}_{i}(f;R)=0$ for all $i\leq d$, then $f$ induces an isomorphism $C^{\mathop{\mathbb{A}^{1}}\nolimits}(X;R)\cong C^{\mathop{\mathbb{A}^{1}}\nolimits}(Y;R)$ in $\mathbf{D}(k,R)$.

(2) If $\mathbf{H}^{S}_{i}(f;R)=0$ for all $i\leq d$, then $f$ induces an isomorphism of motives $M(X;R)\cong M(Y;R)$ in $\mathbf{DM}^{eff}(k,R)$.

Proof.

(1) For each $m>d$, we only need to show that if $\mathbf{H}^{\mathop{\mathbb{A}^{1}}\nolimits}_{i}(f;R)=0$ for all $i\leq m$, then $\mathbf{H}^{\mathop{\mathbb{A}^{1}}\nolimits}_{m+1}(f;R)=0$. By (2.7), there exists a natural isomorphism

[TABLE]

for every $A\in\mathop{\mathcal{M}od_{k}^{\mathbb{A}^{1}}}\nolimits(R)$. The left hand side vanishes by the exact sequence

[TABLE]

and [Ni, Thm. 1.32]. Therefore, Yoneda’s lemma in $\mathop{\mathcal{M}od_{k}^{\mathbb{A}^{1}}}\nolimits(R)$ shows that $\mathbf{H}^{\mathop{\mathbb{A}^{1}}\nolimits}_{m+1}(f;R)=0$.

(2) For each $m>d$, we only need to show that if $\mathbf{H}^{S}_{i}(f;R)=0$ for all $i\leq m$, then $\mathbf{H}^{S}_{m+1}(f;R)=0$. By (2.8), there exists a natural isomorphism

[TABLE]

for every $A\in\mathbf{NST}_{k}^{\mathop{\mathbb{A}^{1}}\nolimits}(R)$. By (2.9), the left hand side coincides with $H_{Nis}^{m+1}(f;A)$, and thus vanishes by the proof of (1). Since Yoneda’s lemma in $\mathbf{NST}^{\mathop{\mathbb{A}^{1}}\nolimits}_{k}(R)$ shows that $\mathbf{H}^{\mathop{\mathbb{A}^{1}}\nolimits}_{m+1}(f;R)=0$, we have $M(f;R)=0$. ∎

We can now prove the $\mathop{\mathbb{A}^{1}}\nolimits$-Whitehead theorem with dimension bound.

Theorem 3.2**.**

Assume $k$ perfect. Let $f:(X,*)\to(Y,*)$ be a morphism of $\mathop{\mathbb{A}^{1}}\nolimits$-simply connected pointed smooth $k$-schemes. If $\mathbf{H}^{\mathop{\mathbb{A}^{1}}\nolimits}_{i}(f;R)=0$ for all $2\leq i\leq{\max\{\dim X+1,\dim Y\}}$, then $f$ is an $\mathop{\mathbb{A}^{1}}\nolimits$-weak equivalence.

Proof.

The morphism $\mathbf{H}^{\mathop{\mathbb{A}^{1}}\nolimits}_{0}(X;\mathbb{Z})\to\mathbf{H}^{\mathop{\mathbb{A}^{1}}\nolimits}_{0}(Y;\mathbb{Z})$ is an isomorphism by [As, Prop. 3.5]. Moreover, $\mathbf{H}^{\mathop{\mathbb{A}^{1}}\nolimits}_{1}(X;\mathbb{Z})=\mathbf{H}^{\mathop{\mathbb{A}^{1}}\nolimits}_{1}(Y;\mathbb{Z})=0$ by [Mo2, Thm. 6.35]. Thus we have $\mathbf{H}^{\mathop{\mathbb{A}^{1}}\nolimits}_{0}(f;\mathbb{Z})=\mathbf{H}^{\mathop{\mathbb{A}^{1}}\nolimits}_{1}(f;\mathbb{Z})=0$. Therefore, our assumption and Proposition 3.1 show that $\mathbf{H}^{\mathop{\mathbb{A}^{1}}\nolimits}_{i}(f;\mathbb{Z})=0$ for all $i\in\mathbb{Z}$. Thus we have $\pi^{\mathop{\mathbb{A}^{1}}\nolimits}_{i}(f)=0$ for every $i\geq 0$ by Corollary 2.6. Since $f$ induces $\pi^{\mathop{\mathbb{A}^{1}}\nolimits}_{i}(X,*)\cong\pi^{\mathop{\mathbb{A}^{1}}\nolimits}_{i}(Y,*)$ for all $i\geq 0$, the morphism $f$ is an $\mathop{\mathbb{A}^{1}}\nolimits$-weak equivalence by [MV, Prop. 2.14]. ∎

Theorem 3.2 implies the following.

Corollary 3.3**.**

Assume $k$ perfect. Let $f:(X,*)\to(Y,*)$ be a morphism of pointed smooth $k$-schemes and let $d={\max\{\dim X+1,\dim Y\}}$. Suppose that $(X,*)$ is $\mathop{\mathbb{A}^{1}}\nolimits$-simply connected and $(Y,*)$ is $\mathop{\mathbb{A}^{1}}\nolimits$-connected. If $f$ is $\mathop{\mathbb{A}^{1}}\nolimits$-$d$-connected, then $f$ is an $\mathop{\mathbb{A}^{1}}\nolimits$-weak equivalence.

Proof.

By $d\geq 1$, the exact sequence

[TABLE]

shows that $(Y,*)$ is $\mathop{\mathbb{A}^{1}}\nolimits$-simply connected. On the other hand, by Proposition 2.4, $\mathbf{H}^{\mathop{\mathbb{A}^{1}}\nolimits}_{i}(f;R)=0$ for all $i\leq d$. Thus $f$ is an $\mathop{\mathbb{A}^{1}}\nolimits$-weak equivalence by Theorem 3.2. ∎

Proposition 3.1 also has the following application.

Corollary 3.4**.**

Assume $k$ perfect. Let $f:(X,*)\to(Y,*)$ be a morphism of pointed smooth $k$-schemes. We assume that $\mathbf{H}^{\mathop{\mathbb{A}^{1}}\nolimits}_{i}(f;R)=0$ for all $i\leq{\max\{\dim X+1,\dim Y\}}$. Then the morphism $S^{2}\wedge f:S^{2}\wedge X\to S^{2}\wedge Y$ is an $\mathop{\mathbb{A}^{1}}\nolimits$-weak equivalence. Moreover, if $X$ and $Y$ are $\mathop{\mathbb{A}^{1}}\nolimits$-connected, then the morphism $S^{1}\wedge f:S^{1}\wedge X\to S^{1}\wedge Y$ is an $\mathop{\mathbb{A}^{1}}\nolimits$-weak equivalence.

Proof.

For a $k$-space $\mathcal{X}$, the suspension $S^{1}\wedge\mathcal{X}$ is $\mathop{\mathbb{A}^{1}}\nolimits$-connected by [Mo2, Thm. 6.38]. Similarly, if $\mathcal{X}$ is $\mathop{\mathbb{A}^{1}}\nolimits$-connected, then $S^{1}\wedge\mathcal{X}$ is $\mathop{\mathbb{A}^{1}}\nolimits$-simply connected. Since $f$ induces isomorphisms for all $\mathop{\mathbb{A}^{1}}\nolimits$-homology sheaves by Proposition 3.1, so does $S^{1}\wedge f$. Therefore, Corollary 2.6 shows that $S^{2}\wedge f$ is an $\mathop{\mathbb{A}^{1}}\nolimits$-weak equivalence. Similarly, $S^{1}\wedge f$ is an $\mathop{\mathbb{A}^{1}}\nolimits$-weak equivalence when $X$ and $Y$ are $\mathop{\mathbb{A}^{1}}\nolimits$-connected. ∎

3.2 $\mathop{\mathbb{A}^{1}}\nolimits$-excision theorem

We next prove an excision theorem for $\mathop{\mathbb{A}^{1}}\nolimits$- and Suslin homology. This is an analogue of [As, Prop. 3.8] in higher degree.

Theorem 3.5**.**

Let $X$ be a smooth $k$-scheme and $U$ a Zariski open set of $X$ whose complement has codimension $r$. Then the morphisms $\mathbf{H}^{\mathop{\mathbb{A}^{1}}\nolimits}_{i}(U;R)\to\mathbf{H}^{\mathop{\mathbb{A}^{1}}\nolimits}_{i}(X;R)$ and $\mathbf{H}^{S}_{i}(U;R)\to\mathbf{H}^{S}_{i}(X;R)$ are isomorphisms for every $i<r-1$ and epimorphisms for $i=r-1$.

Proof.

It suffices to prove that $\mathbf{H}^{\mathop{\mathbb{A}^{1}}\nolimits}_{i}(X,U;R)=\mathbf{H}^{S}_{i}(X,U;R)=0$ for all $i<r$. By Proposition 2.8, we only need to prove this for the $\mathop{\mathbb{A}^{1}}\nolimits$-homology sheaves. We use induction on $i$. The case $i<0$ follows from [Mo2, Thm. 6.22]. We assume $i\geq 0$ and $\mathbf{H}^{\mathop{\mathbb{A}^{1}}\nolimits}_{j}(X,U;R)=0$ for all $j<i$. By (2.7), we have

[TABLE]

for every $A\in\mathop{\mathcal{M}od_{k}^{\mathbb{A}^{1}}}\nolimits(R)$. Then the left hand side vanishes by [Mo1, Lem. 6.4.4]. Therefore, we have $\mathbf{H}^{\mathop{\mathbb{A}^{1}}\nolimits}_{i}(X,U;R)=0$ by Yoneda’s lemma in $\mathop{\mathcal{M}od_{k}^{\mathbb{A}^{1}}}\nolimits(R)$. ∎

Theorem 3.5 gives the excision theorem for $\mathop{\mathbb{A}^{1}}\nolimits$-homotopy.

Corollary 3.6**.**

Assume $k$ perfect. Let $(X,*)$ be a pointed smooth $k$-scheme and $(U,*)$ a pointed Zariski open set of $(X,*)$ whose complement has codimension $r$. If $(X,*)$ and $(U,*)$ are $\mathop{\mathbb{A}^{1}}\nolimits$-simply connected, then the morphism $\pi^{\mathop{\mathbb{A}^{1}}\nolimits}_{i}(U)\to\pi^{\mathop{\mathbb{A}^{1}}\nolimits}_{i}(X)$ is an isomorphism for every $i<r-1$ and an epimorphism for $i=r-1$. In other words, the pair $(X,U)$ is $\mathop{\mathbb{A}^{1}}\nolimits$-$(r-1)$-connected.

Proof.

Since $\mathbf{H}^{\mathop{\mathbb{A}^{1}}\nolimits}_{i}(X,U;\mathbb{Z})=0$ for all $i<r$ by Theorem 3.5, the pair $(X,U)$ is $\mathop{\mathbb{A}^{1}}\nolimits$-$(r-1)$-connected by Corollary 2.6. ∎

4 $\mathop{\mathbb{A}^{1}}\nolimits$-homology of a hyperplane embedding

In this section, as an example of relative $\mathop{\mathbb{A}^{1}}\nolimits$-homology, we compute the $\mathop{\mathbb{A}^{1}}\nolimits$-homology of a hyperplane embedding $\mathbb{P}^{n-1}\hookrightarrow\mathbb{P}^{n}$ in degree $\leq n$. By a suitable linear change of coordinates, we may regard $\mathbb{P}^{n-1}$ as the hyperplane in $\mathbb{P}^{n}$ defined by $x_{n}=0$, where $x_{0},\ldots,x_{n}$ denote homogeneous coordinates on $\mathbb{P}^{n}$. Let $\underline{\mathbf{K}}^{MW}_{n}$ be the unramified Milnor-Witt $K$-theory defined by Morel [Mo2]. For $A\in\mathop{\mathcal{A}b_{k}}\nolimits$, we write $A\otimes^{\mathop{\mathbb{A}^{1}}\nolimits}R=H_{0}(L_{\mathop{\mathbb{A}^{1}}\nolimits}(A\otimes R))$ which is called $\mathop{\mathbb{A}^{1}}\nolimits$-tensor product by Morel [Mo1]. Our main result is the following.

Theorem 4.1**.**

For $0\leq i\leq n$, $n>0$, we have

[TABLE]

In particular, when $i<n$, we have $\mathbf{H}^{\mathop{\mathbb{A}^{1}}\nolimits}_{i}(\mathbb{P}^{n};R)\cong\mathbf{H}^{\mathop{\mathbb{A}^{1}}\nolimits}_{i}(\mathbb{P}^{i+1};R)$.

When $i=n$, we have the following description.

Corollary 4.2**.**

There exists a morphism $\underline{\mathbf{K}}^{MW}_{n+1}\otimes^{\mathop{\mathbb{A}^{1}}\nolimits}R\to\mathbf{H}^{\mathop{\mathbb{A}^{1}}\nolimits}_{n}(\mathbb{P}^{n};R)$ such that

[TABLE]

Proof.

By Theorem 4.1, the homology exact sequence

[TABLE]

gives an isomorphism

[TABLE]

∎

Theorem 4.1 and Proposition 2.8 imply the following.

Corollary 4.3**.**

For all $i<n$, we have $\mathbf{H}^{S}_{i}(\mathbb{P}^{n},\mathbb{P}^{n-1};R)=0$. Moreover, there exists a universal morphism $\underline{\mathbf{K}}^{MW}_{n}\otimes^{\mathop{\mathbb{A}^{1}}\nolimits}R\to\mathbf{H}^{S}_{n}(\mathbb{P}^{n},\mathbb{P}^{n-1};R)$ with respect to the canonical functor $\mathbf{NST}^{\mathop{\mathbb{A}^{1}}\nolimits}_{k}(R)\to\mathop{\mathcal{M}od_{k}^{\mathbb{A}^{1}}}\nolimits(R)$. In particular, when $i<n$ we have

[TABLE]

Remark 4.4**.**

The vanishing $\mathbf{H}^{\mathop{\mathbb{A}^{1}}\nolimits}_{i}(\mathbb{P}^{n},\mathbb{P}^{n-1};R)=0$ for $i<n$ is an example where the Lefschetz hyperplane theorem holds. However, it is not true in general that $\mathbf{H}^{\mathop{\mathbb{A}^{1}}\nolimits}_{i}(X,H;R)=0$ for $i<\dim X$ and $H\subseteq X$ a very ample divisor. Indeed, let $C\subseteq\mathbb{P}^{2}$ be a smooth plane curve of degree $\geq 3$. Since $C$ is not rational, it is not $\mathop{\mathbb{A}^{1}}\nolimits$-connected by [AM, Prop. 2.1.12]. Therefore, the canonical morphism $\mathbf{H}^{\mathop{\mathbb{A}^{1}}\nolimits}_{0}(C;R)\to R$ is not an isomorphism by [As, Thm. 4.14]. Thus the morphism $\mathbf{H}^{\mathop{\mathbb{A}^{1}}\nolimits}_{0}(C;R)\to\mathbf{H}^{\mathop{\mathbb{A}^{1}}\nolimits}_{0}(\mathbb{P}^{2};R)\cong R$ is not an isomorphism.

4.1 A basic distinguished triangle

For the proof of Theorem 4.1, we compute the mapping cone of $C^{\mathop{\mathbb{A}^{1}}\nolimits}(\mathbb{A}^{n}-\{0\};R)\to C^{\mathop{\mathbb{A}^{1}}\nolimits}(\mathbb{P}^{n-1};R)$. We first give a Zariski excision result for $\mathop{\mathbb{A}^{1}}\nolimits$-homology.

Lemma 4.5**.**

Let $\{U,V\}$ be a Zariski covering of a smooth $k$-scheme $X$. Then the morphism $(U,U\cap V)\to(X,V)$ induces a quasi-isomorphism

[TABLE]

Proof.

Since the functor $L_{\mathop{\mathbb{A}^{1}}\nolimits}$ is exact (see, e.g., [CD, Thm. 2.5]), we only need to show that the morphism ${(U,U\cap V)\to(X,V)}$ induces a quasi-isomorphism

[TABLE]

For each open set $W\subseteq X$, we regard $R(W)$ as a subsheaf of $R(X)$. Then by the short exact sequence in [AD, proof of Prop.3.32], we see that $R(U\cap V)=R(U)\cap R(V)$ and $R(X)=R(U)+R(V)$. Thus we have an isomorphism

[TABLE]

Finally, $R(U)/R(U\cap V)$ and $R(X)/R(V)$ are canonically quasi-isomorphic to $C(U,U\cap V;R)$ and $C(X,V;R)$, respectively. Therefore, we obtain (4.1). ∎

We obtain the following distinguished triangle in $\mathbf{D}(k,R)$.

Proposition 4.6**.**

For $n\geq 1$ and $*\in\mathbb{P}^{n}(k)$, we have a distinguished triangle

[TABLE]

in $\mathbf{D}(k,R)$, where the first morphism is induced by the $\mathbb{G}_{m}$-quotient and the second morphism is induced by $(\mathbb{P}^{n-1},\emptyset)\to(\mathbb{P}^{n},*)$.

Proof.

We write $U={\mathbb{P}^{n}-\{(0:\ldots:0:1)\}}$. Since the projection $\rho:U\to\mathbb{P}^{n-1}$ is a vector bundle, it is an $\mathop{\mathbb{A}^{1}}\nolimits$-weak equivalence by [MV, Example 2.2]. We denote by $V$ the Zariski open set of $\mathbb{P}^{n}$ defined by $x_{n}\neq 0$. Since the diagram

[TABLE]

commutes, we obtain the commutative diagram

[TABLE]

On the other hand, Lemma 4.5 gives the commutative diagram

[TABLE]

By the diagrams (4.2) and (4.3), we obtain an isomorphism of triangles

[TABLE]

Since the lower triangle is distinguished, so is the upper triangle. Therefore, we only need to show that for a $k$-rational point $*\in V(k)$, the morphism $(\mathbb{P}^{n-1},*)\to(\mathbb{P}^{n-1},V)$ induces a quasi-isomorphism $C^{\mathop{\mathbb{A}^{1}}\nolimits}(\mathbb{P}^{n},*;R)\cong C^{\mathop{\mathbb{A}^{1}}\nolimits}(\mathbb{P}^{n-1},V;R)$. Since $V\cong\mathbb{A}^{n}$, the exact sequence

[TABLE]

shows that $\mathbf{H}^{\mathop{\mathbb{A}^{1}}\nolimits}_{i}(\mathbb{P}^{n};R)\cong\mathbf{H}^{\mathop{\mathbb{A}^{1}}\nolimits}_{i}(\mathbb{P}^{n},V;R)$ for all $i\geq 2$. For $i=0,1$, there exists an exact sequence

[TABLE]

Then the morphism $\mathbf{H}^{\mathop{\mathbb{A}^{1}}\nolimits}_{0}(V;R)\to\mathbf{H}^{\mathop{\mathbb{A}^{1}}\nolimits}_{0}(\mathbb{P}^{n};R)$ is an isomorphism by [As, Prop. 3.5]. Therefore, we have $\mathbf{H}^{\mathop{\mathbb{A}^{1}}\nolimits}_{0}(\mathbb{P}^{n},*;R)=\mathbf{H}^{\mathop{\mathbb{A}^{1}}\nolimits}_{0}(\mathbb{P}^{n},V;R)=0$ and $\mathbf{H}^{\mathop{\mathbb{A}^{1}}\nolimits}_{1}(\mathbb{P}^{n};R)\cong\mathbf{H}^{\mathop{\mathbb{A}^{1}}\nolimits}_{1}(\mathbb{P}^{n},V;R)$. Thus $C^{\mathop{\mathbb{A}^{1}}\nolimits}(\mathbb{P}^{n},*;R)\to C^{\mathop{\mathbb{A}^{1}}\nolimits}(\mathbb{P}^{n-1},V;R)$ is a quasi-isomorphism. ∎

4.2 Proof of Theorem 4.1

We can now prove Theorem 4.1.

(Proof of Theorem 4.1).

We first prove $\mathbf{H}^{\mathop{\mathbb{A}^{1}}\nolimits}_{i}(\mathbb{P}^{n},\mathbb{P}^{n-1};R)=0$ for all $i<n$. The $\mathop{\mathbb{A}^{1}}\nolimits$-weak equivalence $\rho$ as in the proof of Proposition 4.6 gives an isomorphism $\mathbf{H}^{\mathop{\mathbb{A}^{1}}\nolimits}_{i}(\mathbb{P}^{n-1};R)\xrightarrow{\cong}\mathbf{H}^{\mathop{\mathbb{A}^{1}}\nolimits}_{i}(\mathbb{P}^{n}-\{0\};R)$. On the other hand, $\mathbf{H}^{\mathop{\mathbb{A}^{1}}\nolimits}_{i}(\mathbb{P}^{n}-\{0\};R)\to\mathbf{H}^{\mathop{\mathbb{A}^{1}}\nolimits}_{i}(\mathbb{P}^{n};R)$ is an isomorphism for all $i<n-1$ and an epimorphism for $i=n-1$ by Theorem 3.5. Thus we have $\mathbf{H}^{\mathop{\mathbb{A}^{1}}\nolimits}_{i}(\mathbb{P}^{n},\mathbb{P}^{n-1};R)=0$.

Next, we prove $\mathbf{H}^{\mathop{\mathbb{A}^{1}}\nolimits}_{n}(\mathbb{P}^{n},\mathbb{P}^{n-1};R)\cong\underline{\mathbf{K}}^{MW}_{n}\otimes^{\mathop{\mathbb{A}^{1}}\nolimits}R$ for all $n\geq 2$. By Proposition 4.6, there exists a morphism of distinguished triangles

[TABLE]

where $\alpha$ is induced by $(\mathbb{P}^{n},\emptyset)\to(\mathbb{P}^{n},*)$. Taking the homology exact sequence, we obtain $\mathbf{H}^{\mathop{\mathbb{A}^{1}}\nolimits}_{n}(\mathbb{P}^{n},\mathbb{P}^{n-1};R)\cong\mathbf{H}^{\mathop{\mathbb{A}^{1}}\nolimits}_{n-1}(\mathbb{A}^{n}-\{0\};R)$ for all $n\geq 2$ by the five lemma. Note that the adjunctions $\mathop{\mathcal{A}b_{k}}\nolimits\rightleftarrows\mathop{\mathcal{M}od_{k}}\nolimits(R)$ and (2.6) show that the functor $-\otimes^{\mathop{\mathbb{A}^{1}}\nolimits}R:\mathop{\mathcal{A}b_{k}}\nolimits\to\mathop{\mathcal{M}od_{k}^{\mathbb{A}^{1}}}\nolimits(R)$ is left adjoint to the canonical functor $\mathop{\mathcal{M}od_{k}^{\mathbb{A}^{1}}}\nolimits(R)\to\mathop{\mathcal{A}b_{k}}\nolimits$. Moreover, for every $X\in\mathop{\mathcal{S}m_{k}}\nolimits$ which is $\mathop{\mathbb{A}^{1}}\nolimits$-$(n-1)$-connected, the adjunction $\mathop{\mathcal{A}b_{k}}\nolimits\rightleftarrows\mathop{\mathcal{M}od_{k}^{\mathbb{A}^{1}}}\nolimits(R)$ leads to a natural isomorphism

[TABLE]

Hence, we have

[TABLE]

where (4.5) and (4.8) follow from the adjunction ${\mathop{\mathcal{A}b_{k}}\nolimits\rightleftarrows\mathop{\mathcal{M}od_{k}^{\mathbb{A}^{1}}}\nolimits}$, (4.6) from Theorem 2.3 and (4.7) from [Mo2, Thm. 6.40]. Thus Yoneda’s lemma in $\mathop{\mathcal{M}od_{k}^{\mathbb{A}^{1}}}\nolimits(R)$ shows that

[TABLE]

Finally, we prove $\mathbf{H}^{\mathop{\mathbb{A}^{1}}\nolimits}_{1}(\mathbb{P}^{1};R)\cong\underline{\mathbf{K}}^{MW}_{1}\otimes^{\mathop{\mathbb{A}^{1}}\nolimits}R$. For $R=\mathbb{Z}$, this is a direct consequence of [MV, Lem. 2.15 and Cor. 2.18] and [Mo2, Thm. 3.37]. Since $\mathbb{P}^{1}$ is $\mathop{\mathbb{A}^{1}}\nolimits$-connected, the general case follows from (4.4). ∎