Naive vs. genuine A^1-connectedness

TL;DR

This paper investigates the differences between naive and genuine A^1-connectedness in algebraic geometry, revealing that trivial sections do not imply trivial sheaves and providing counterexamples related to A^1-locality.

Contribution

It demonstrates that trivial sections of naive A^1-chain connected components do not guarantee triviality of genuine A^1-connected components, highlighting a key distinction.

Findings

Trivial sections of naive A^1-chain components do not imply trivial genuine A^1-connectedness.

Existence of an A^1-connected scheme where the Morel-Voevodsky singular construction is not A^1-local.

Counterexamples to assumptions about A^1-connectedness properties.

Abstract

We show that the triviality of sections of the sheaf of A^1-chain connected components of a space over finitely generated separable field extensions of the base field is not sufficient to ensure the triviality of the sheaf of its A^1-chain connected components, contrary to the situation with genuine A^1-connected components. As a consequence, we show that there exists an A^1-connected scheme for which the Morel-Voevodsky singular construction is not A^1-local.

Equations

$${\mathrm{Hom}}_{{\mathcal{H}}_s(k)}(U,\mathcal{X})\to {\mathrm{Hom}}_{{\mathcal{H}}_s(k)}(U\times {\mathbb{A}}^1,\mathcal{X}).$$

$$(\mathrm{Sin}{\mathrm{g}}_{\ast }^{{\mathbb{A}}^1}\mathcal{X}{)}_n=\underline{\mathrm{Hom}}({\Delta }_n,{\mathcal{X}}_n),$$

$${\Delta }_n=\mathrm{Spec}\left(\frac{k[x_0,...,x_n]}{({\sum }_i{x}_i=1)}\right)$$

$$\mathcal{S}(\mathcal{X}):={\pi }_0^s(\mathrm{Sin}{\mathrm{g}}_{\ast }^{{\mathbb{A}}^1}\mathcal{X}).$$

$${\pi }_0^{{\mathbb{A}}^1}(\mathcal{X}):={\pi }_0^s(L_{{\mathbb{A}}^1}\mathcal{X}),$$

$$L_{{\mathbb{A}}^1}=Ex\circ (Ex\circ {\mathrm{Sing}}_{\ast }^{{\mathbb{A}}^1}{)}^{\mathbb{N}}\circ Ex,$$

$$\mathcal{F}\to \mathcal{S}(\mathcal{F})\to {\mathcal{S}}^2(\mathcal{F})\to \cdots \to {\mathcal{S}}^n(\mathcal{F})\to \cdots ,$$

$$L(\mathcal{F}):=\underset{n}{\underrightarrow{\lim }}{\mathcal{S}}^n(\mathcal{F}).$$

$$X\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$X\simeq \mathcal{S}(X)\simeq {\pi }_0^{{\mathbb{A}}^1}(X).$$

$$\mathcal{S}(G)\simeq {\pi }_0^{{\mathbb{A}}^1}(G)\simeq L(G).$$

$${\pi }_0^{{\mathbb{A}}^1}(G)(k)\simeq {\mathcal{S}}^2(G)(k)\simeq G(k)/R.$$

$$U/(U-Z\cap U)\stackrel{\sim }{\to }{\mathbb{A}}_{V^{\prime }}^1/({\mathbb{A}}_{V^{\prime }}^1-\pi (Z\cap U)).$$

$${\mathbb{A}}_{V^{\prime }}^1/({\mathbb{A}}_{V^{\prime }}^1-\pi (Z\cap U))\stackrel{\sim }{\to }{\mathbb{P}}_{V^{\prime }}^1/({\mathbb{P}}_{V^{\prime }}^1-\pi (Z\cap U)).$$

$$V^{\prime }\to {\mathbb{A}}_{V^{\prime }}^1\to {\pi }_0^s({\mathrm{Sing}}_{\ast }^{{\mathbb{A}}^1}({\mathbb{A}}_{V^{\prime }}^1/({\mathbb{A}}_{V^{\prime }}^1-\pi (Z\cap U))))\to {\pi }_0^s({\mathrm{Sing}}_{\ast }^{{\mathbb{A}}^1}({\mathbb{P}}_{V^{\prime }}^1/({\mathbb{P}}_{V^{\prime }}^1-\pi (Z\cap U))))$$

$$V\to {\pi }_0^s({\mathrm{Sing}}_{\ast }^{{\mathbb{A}}^1}({\mathbb{P}}_{V^{\prime }}^1/({\mathbb{P}}_{V^{\prime }}^1-\pi (Z\cap U))))=\mathcal{S}({\mathbb{P}}_{V^{\prime }}^1/({\mathbb{P}}_{V^{\prime }}^1-\pi (Z\cap U)))$$

$$\mathcal{S}(\mathcal{X})(\mathrm{Spec}k(V))=\underset{W\hookrightarrow V\text{nonempty open}}{\underrightarrow{\lim }}\mathcal{S}(\mathcal{X})(W)$$

$$V\times \{1\}\cup W\times {\Delta }^1\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$V\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\mathcal{X}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

Full text

1 Introduction

Let $k$ be a field and let $Sm/k$ denote the category of smooth, finite-type schemes over $k$. In the 1990’s, Morel and Voevodsky [14] constructed the $\mathbb{A}^{1}$-homotopy category $\mathcal{H}(k)$ by taking a suitable localization of the category of simplicial sheaves of sets on $Sm/k$ for the Nisnevich topology. Objects of $\mathcal{H}(k)$ are often called spaces. Analgous to algebraic topology, one then studies the $\mathbb{A}^{1}$-homotopy sheaves of a (pointed) space $(\mathcal{X},x)$ - the sheaf of $\mathbb{A}^{1}$-connected components $\pi_{0}^{\mathbb{A}^{1}}(\mathcal{X})$, which is a sheaf of sets and the higher homotopy sheaves $\pi_{n}^{\mathbb{A}^{1}}(\mathcal{X},x)$, for $n\geq 1$, which are sheaves of groups. We will use the notation and terminology of [14]. Any (not necessarily smooth) scheme $X$ over $k$ can be viewed as an object of the $\mathbb{A}^{1}$-homotopy category $\mathcal{H}(k)$ (see the conventions stated at the beginning of Section 2). Recent works in $\mathbb{A}^{1}$-homotopy theory have indicated that the $\mathbb{A}^{1}$-homotopy sheaves of schemes are often related to some of their interesting classical invariants.

The simplest of objects in classical topology are the discrete topological spaces. The analogous notion in $\mathbb{A}^{1}$-homotopy theory is that of $\mathbb{A}^{1}$-invariant sheaves (see Section 2 for precise definitions). In topology, the set of connected components of a topological space and the homotopy groups of a (pointed) topological space are discrete as topological spaces. Analogously, one can ask if the $\mathbb{A}^{1}$-homotopy sheaves of a (pointed) space $\mathcal{X}$ are $\mathbb{A}^{1}$-invariant. It has been shown by Morel [13, Theorem 6.1, Corollary 6.2] that the higher homotopy sheaves $\pi_{n}^{\mathbb{A}^{1}}(\mathcal{X},x)$, for $n\geq 1$, are $\mathbb{A}^{1}$-invariant. In fact, Morel shows much more - these higher $\mathbb{A}^{1}$-homotopy sheaves are strongly $\mathbb{A}^{1}$-invariant in the sense of [13, Definition 1.7] (this shows that the higher $\mathbb{A}^{1}$-homotopy sheaves are very special; for instance, they are birational invariants of smooth, proper schemes). However, $\mathbb{A}^{1}$-invariance of the sheaf of $\mathbb{A}^{1}$-connected components is not yet known; this has been conjectured by Morel [13, Conjecture 1.12]. It is worthwhile to mention that $\pi_{0}^{\mathbb{A}^{1}}$ fails to be a birational invariant of smooth, proper schemes [3, Example 4.8].

There are two notions of $\mathbb{A}^{1}$-connectedness in unstable $\mathbb{A}^{1}$-homotopy theory. The naive notion is that of $\mathbb{A}^{1}$-chain connected components of a space (see Definition 2.2), which is obtained by taking the Nisnevich sheafification of the presheaf that associates with any smooth scheme $U$ the set of morphisms from $U$ to the space in question modulo the equivalence relation generated by naive $\mathbb{A}^{1}$-homotopies. On the other hand, the genuine notion is that of $\mathbb{A}^{1}$-connected components (see Definition 2.3) introduced by Morel-Voevodsky. These two notions do not coincide in general, not even for smooth projective varieties over $\mathbb{C}$ (see [3, Section 4] for the first examples). Given a scheme $X$ over $k$, one can infinitely iterate the construction of $\mathbb{A}^{1}$-chain connected components to obtain the so-called universal $\mathbb{A}^{1}$-invariant quotient of $X$, which is isomorphic to $\pi_{0}^{\mathbb{A}^{1}}(X)$ provided the latter sheaf is $\mathbb{A}^{1}$-invariant (that is, Morel’s conjecture holds for $X$). We recall these notions and known results about them in Section 2.

A natural question is to characterize genuine $\mathbb{A}^{1}$-connectedness of a scheme $X$ over $k$ in terms of triviality of sections of $\pi_{0}^{\mathbb{A}^{1}}(X)$ over field extensions of $k$. A result of Morel states that $\mathbb{A}^{1}$-connectedness of a scheme $X$ over an infinite field $k$ in the genuine sense (that is, triviality of $\pi_{0}^{\mathbb{A}^{1}}(X)$ as a sheaf) is equivalent to the triviality of $\pi_{0}^{\mathbb{A}^{1}}(X)({\rm Spec}~{}F)$, where $F$ runs over all finitely generated separable field extensions of $k$. In this short note, we examine the analogous property for the sheaf of $\mathbb{A}^{1}$-chain connected components in Section 3 (see Theorem 3.2). As a consequence, we obtain an example of an $\mathbb{A}^{1}$-connected singular proper scheme $X$ for which the Morel-Voevodsky singular construction ${\rm Sing_{*}^{\mathbb{A}^{1}}}X$ is not $\mathbb{A}^{1}$-local (see Example 3.6).

2 Connectedness in unstable $\mathbb{A}^{1}$-homotopy theory

We begin this section by setting up the notation and conventions that will be used throughout the paper.

Fix a base field $k$. We will henceforth denote by $Sm/k$ the big Nisnevich site of smooth, finite-type schemes over $k$. We begin with the category of simplicial sheaves over $Sm/k$. Any scheme $X$ over $k$ can be seen as an object of this category as follows: consider the functor of points $h_{X}$ of $X$, which is the sheaf that associates with every $U\in Sm/k$ the set of morphisms of schemes over $k$ from $U$ to $X$. Any Nisnevich sheaf $\mathcal{F}$ on $Sm/k$ can be viewed as a simplicially constant simplicial sheaf. More precisely, one considers the simplicial sheaf in which the sheaf at every level is $\mathcal{F}$ and all the face and degeneracy maps are given by the identity map. We will always denote the simplicial sheaf corresponding to $h_{X}$ for a scheme $X$ over $k$ by the same letter $X$.

A morphism $\mathcal{X}\to\mathcal{Y}$ of simplicial sheaves of sets on $Sm/k$ is a local weak equivalence if it induces an isomorphism on every stalk. The Nisnevich local injective model structure on this category is the one in which the morphism of simplicial sheaves is a cofibration (resp. a weak equivalence) if and only if it is a monomorphism (resp. a local weak equivalence). The corresponding homotopy category is called the simplicial homotopy category and is denoted by $\mathcal{H}_{s}(k)$. The left Bousfield localization of the Nisnevich local injective model structure with respect to the collection of all projection morphisms $\mathcal{X}\times\mathbb{A}^{1}\to\mathcal{X}$, as $\mathcal{X}$ runs over all simplicial sheaves, is called the $\mathbb{A}^{1}$-model structure. The associated homotopy category is called the $\mathbb{A}^{1}$-homotopy category and is denoted by $\mathcal{H}(k)$. We will denote by $\ast$ the trivial one-point sheaf on $Sm/k$. We will abuse the notation and use $\ast$ to also denote a set with one element, whenever there is no confusion.

Definition 2.1

A space (that is, a simplicial Nisnevich sheaf of sets on $Sm/k$) $\mathcal{X}$ is said to be $\mathbb{A}^{1}$-local if the projection map $U\times\mathbb{A}^{1}\to U$ induces a bijection

[TABLE]

for every $U\in Sm/k$. Note that a Nisnevich sheaf $\mathcal{F}$ on $Sm/k$ is $\mathbb{A}^{1}$-local if and only if it is $\mathbb{A}^{1}$-invariant, that is, if the projection map $U\times\mathbb{A}^{1}\to U$ induces a bijection $\mathcal{F}(U)\to\mathcal{F}(U\times\mathbb{A}^{1})$, for every $U\in Sm/k$. Following standard convention, we say that a scheme is $\mathbb{A}^{1}$-rigid if it is $\mathbb{A}^{1}$-local as a space.

Let $\mathcal{X}$ be a space. We now recall the singular construction on $\mathcal{X}$ defined by Morel-Voevodsky [14, p. 87-88]. Define ${\rm Sing_{*}^{\mathbb{A}^{1}}}\mathcal{X}$ to be the simplicial sheaf given by

[TABLE]

where $\Delta_{\bullet}$ denotes the cosimplicial scheme

[TABLE]

with natural face and degeneracy maps analogous to the ones on topological simplices. There exists a natural transformation $Id\to{\rm Sing_{*}^{\mathbb{A}^{1}}}$ such that for any simplicial sheaf $\mathcal{X}$, the morphism $\mathcal{X}\to{\rm Sing_{*}^{\mathbb{A}^{1}}}(\mathcal{X})$ is an $\mathbb{A}^{1}$-weak equivalence. Observe that the singular construction ${\rm Sing_{*}^{\mathbb{A}^{1}}}$ takes naive $\mathbb{A}^{1}$-homotopies to simplicial homotopies.

Given a simplicial sheaf of sets $\mathcal{X}$ on $Sm/k$, we will denote by $\pi_{0}(\mathcal{X})$ the presheaf on $Sm/k$ that associates with $U\in Sm/k$ the coequalizer of the diagram $\mathcal{X}_{1}(U)\rightrightarrows\mathcal{X}_{0}(U)$, where the maps are the face maps coming from the simplicial data of $\mathcal{X}$. We will denote by $\pi_{0}^{s}(\mathcal{X})$ the Nisnevich sheafification of the presheaf $\pi_{0}(\mathcal{X})$.

Definition 2.2

The sheaf of $\mathbb{A}^{1}$-chain connected components of a space $\mathcal{X}$ is defined to be

[TABLE]

Thus, $\mathcal{S}(\mathcal{X})$ is the Nisnevich sheafification of the presheaf that associates with any smooth scheme $U$ the set $\mathcal{X}(U)/\sim$, where $\sim$ is the equivalence relation generated by the image of $\mathcal{X}_{1}\to\mathcal{X}_{0}\times\mathcal{X}_{0}$, where the maps are the face maps coming from the simplicial data of $\mathcal{X}$ (in other words, $\sim$ is the equivalence relation generated by naive $\mathbb{A}^{1}$-homotopies).

Definition 2.3

The sheaf of $\mathbb{A}^{1}$-connected components of a space $\mathcal{X}$ is defined to be

[TABLE]

where ${L_{\mathbb{A}^{1}}}$ denotes an $\mathbb{A}^{1}$-fibrant replacement functor. A space $\mathcal{X}$ is said to be $\mathbb{A}^{1}$-connected if $\pi_{0}^{\mathbb{A}^{1}}(\mathcal{X})\simeq\ast$.

Morel-Voevodsky explicitly describe an $\mathbb{A}^{1}$-fibrant replacement functor as follows:

[TABLE]

where $Ex$ denotes a simplicial fibrant replacement functor on the model category of simplicial Nisnevich sheaves of sets over $Sm/k$ [14, §2, Lemma 2.6, p. 107]. There exists a natural transformation $Id\to L_{\mathbb{A}^{1}}$ which factors through the natural transformation $Id\to{\rm Sing}_{*}^{\mathbb{A}^{1}}$ mentioned above. For any object $\mathcal{X}$, the morphism $\mathcal{X}\to L_{\mathbb{A}^{1}}(\mathcal{X})$ is an $\mathbb{A}^{1}$-weak equivalence. A result of Morel-Voevodsky [14, §2, Corollary 3.22] describes what happens to the natural map ${\rm Sing}_{*}^{\mathbb{A}^{1}}\mathcal{X}\to L_{\mathbb{A}^{1}}(\mathcal{X})$ after applying $\pi_{0}^{s}$; we record it below for the sake of convenience.

Lemma 2.4

The canonical map $\mathcal{S}(\mathcal{X})\to\pi_{0}^{\mathbb{A}^{1}}(\mathcal{X})$ is an epimorphism, for every space $\mathcal{X}$. If ${\rm Sing}_{*}^{\mathbb{A}^{1}}\mathcal{X}$ is $\mathbb{A}^{1}$-local, then the map $\mathcal{S}(\mathcal{X})\to\pi_{0}^{\mathbb{A}^{1}}(\mathcal{X})$ is an isomorphism.

We will henceforth focus on a specific class of spaces, namely, sheaves of sets on the big Nisnevich site on $Sm/k$. We will eventually specialize to the case of schemes. Let $\mathcal{F}$ be a Nisnevich sheaf on $Sm/k$. By Lemma 2.4, we have a sequence of epimorphisms

[TABLE]

where $\mathcal{S}^{n+1}(\mathcal{F})$ is defined inductively to be $\mathcal{S}(\mathcal{S}^{n}(\mathcal{F}))$, for every $n\in\mathbb{N}$. We define

[TABLE]

The following result was proved in [3] (see [3, Theorem 2.13, Remark 2.15, Corollary 2.18]), which shows that $L(\mathcal{F})$ is the universal $\mathbb{A}^{1}$-invariant quotient of $\mathcal{F}$.

Theorem 2.5

Let $\mathcal{F}$ be a sheaf of sets on $Sm/k$. Then the sheaf $L(\mathcal{F})$ is $\mathbb{A}^{1}$-invariant. Moreover, if $\mathcal{G}$ is an $\mathbb{A}^{1}$-invariant sheaf, then any map $\mathcal{F}\to\mathcal{G}$ factors uniquely through the epimorphism $\mathcal{F}\to L(\mathcal{F})$. Moreover, if $\pi_{0}^{\mathbb{A}^{1}}(\mathcal{F})$ is $\mathbb{A}^{1}$-invariant, then the canonical map $L(\mathcal{F})\to\pi_{0}^{\mathbb{A}^{1}}(\mathcal{F})$ is an isomorphism.

We will henceforth focus on schemes over a field. In view of Theorem 2.5, it is clear that a good understanding of $L(X)$ is tantamount to understanding $\pi_{0}^{\mathbb{A}^{1}}(X)$. It is natural to ask the following question.

Question 2.6

Let $X$ be a smooth scheme over $k$. Does there exist $n\in\mathbb{N}$ such that $L(X)\simeq\mathcal{S}^{n}(X)$?

For every scheme $X$ over a field $k$, we have the following commutative diagram in which every morphism is an epimorphism

[TABLE]

where the existence of the map $\pi_{0}^{\mathbb{A}^{1}}(X)\to L(X)$ making the diagram commute is a consequence of the $\mathbb{A}^{1}$-invariance of $L(X)$ (see [3, Lemma 2.8]). The morphism $\pi_{0}^{\mathbb{A}^{1}}(X)\to L(X)$ is an isomorphism if $\pi_{0}^{\mathbb{A}^{1}}(X)$ is $\mathbb{A}^{1}$-invariant. An affirmative answer to Question 2.6 will give a conjectural but explicit geometric description of $\pi_{0}^{\mathbb{A}^{1}}(X)$. We end this section by enlisting the known examples in which such an explicit description is available.

Examples 2.7** ($\mathbb{A}^{1}$-rigid varieties)**

For an $\mathbb{A}^{1}$-rigid variety $X$, one has isomorphisms of Nisnevich sheaves

[TABLE]

Examples of $\mathbb{A}^{1}$-rigid varieties include $\mathbb{G}_{m}$, algebraic tori, abelian varieties, curves of genus $\geq 1$ etc.

Examples 2.8** (Reductive algebraic groups)**

For any sheaf of groups $G$, it is known that $\pi_{0}^{\mathbb{A}^{1}}(G)$ is $\mathbb{A}^{1}$-invariant [7]. We therefore have $\pi_{0}^{\mathbb{A}^{1}}(G)\simeq L(G)=\varinjlim_{n}\mathcal{S}^{n}(G)$. We will now focus on the case where $G$ is a reductive algebraic group over a field.

1. (a)

Isotropic groups. Suppose that $G$ satisfies the following isotropy condition: every almost $k$-simple component of the derived group $G_{\rm der}$ of $G$ contains a $k$-subgroup scheme isomorphic to $\mathbb{G}_{m}$. Under this hypothesis, Asok, Hoyois and Wendt have shown that ${\rm Sing}_{*}^{\mathbb{A}^{1}}G$ is $\mathbb{A}^{1}$-local [1, Theorem 2.3.2]. Therefore, for $G$ satisfying the above isotropy condition, one has

[TABLE]

The sections of this sheaf over fields can often be described explicitly. If $G$ is a semisimple, simply connected group over an infinite field $k$ satisfying the above isotropy hypothesis, then one has isomorphisms $W(k,G)\simeq\mathcal{S}(G)(k)\simeq G(k)/R$. This is a consequence of [1, Theorem 2.3.2] and a classical result [9, Théorème 7.2]. Here $W(k,G)$ denotes the Whitehead group of $G$ and $G(k)/R$ denotes the group of R-equivalence classes (see [9], for example, for precise definitions). 2. (b)

Anisotropic groups. Let us assume that the base field $k$ is infinite and perfect. Suppose now that $G$ does not satisfy the above isotropy hypothesis; that means, the derived group $G_{\rm der}$ of $G$ has at least one almost $k$-simple factor which is anisotropic. In this case, it is known that ${\rm Sing}_{*}^{\mathbb{A}^{1}}G$ fails to be $\mathbb{A}^{1}$-local [5, Theorem 4.7]. We now assume that $G$ is a semisimple anisotropic group. Note that one has $W(k,G)=G(k)$ in this case. A result of Borel-Tits implies in this case that $\mathcal{S}(G)(k)\simeq G(k)$ (see [4, Lemma 3.7] for details). However, one has the following result (see [4, Theorem 4.2], [5, Theorem 3.6]): if $G$ is a semisimple, simply connected group over an infinite perfect field $k$ which does not satisfy the above isotropy hypothesis, then one has canonical isomorphisms

[TABLE]

It is worthwhile to mention here that we do not yet know whether $\pi_{0}^{\mathbb{A}^{1}}(G)$ agrees with $\mathcal{S}^{2}(G)$ as a sheaf. 3. (c)

$\mathbb{A}^{1}$-connected reductive algebraic groups. Recall that a space $\mathcal{X}$ is said to be $\mathbb{A}^{1}$-connected if $\pi_{0}^{\mathbb{A}^{1}}(\mathcal{X})\simeq\ast$. In [5, Theorem 5.2], $\mathbb{A}^{1}$-connected reductive algebraic groups have been characterized: a reductive algebraic group $G$ over an infinite perfect field $k$ is $\mathbb{A}^{1}$-connected if and only if $G$ is semisimple, simply connected and $R$-trivial (that is $G(F)/R$ is trivial for every finitely generated separable field extension $F$ of $k$).

Examples 2.9** (Proper varieties)**

1. (a)

If $X$ is a proper variety over $k$ and if $F$ is a finitely generated field extension of $k$, then one has $\mathcal{S}(X)(F)\simeq\pi_{0}^{\mathbb{A}^{1}}(X)(F)$ (see [2, Theorem 2.4.3]). One also has $\mathcal{S}(X)(F)\simeq\mathcal{S}^{n}(X)(F)$, for every $n$ (see [3, Theorem 3.9, Corollary 3.10]). 2. (b)

The case of proper schemes of dimension $\leq 1$ is very easy. For reduced, proper (possibly singular) schemes $X$ over $k$ of dimension $\leq 1$, one always has $\mathcal{S}(X)\simeq\mathcal{S}^{2}(X)$ ([3, Proposition 3.13]). Consequently, $\mathcal{S}(X)\simeq\pi_{0}^{\mathbb{A}^{1}}(X)$. 3. (c)

If $X$ is a proper, non-uniruled surface over $k$, then one has $\mathcal{S}(X)\simeq\pi_{0}^{\mathbb{A}^{1}}(X)\simeq\mathcal{S}^{2}(X)$ ([3, Theorem 3.14]). 4. (d)

The case of smooth projective ruled surfaces is surprisingly very complicated. If $X$ is a smooth proper rational surface, then one has $\mathcal{S}^{2}(X)=\ast$ (see Corollary 3.3). However, one has $\pi_{0}^{\mathbb{A}^{1}}(X)=\ast$ as well. We do not yet know if the sheaf $\mathcal{S}(X)$ for a rational surface $X$ is trivial.

Let us now assume that the characteristic of $k$ is [math]. The case of ruled surfaces whose minimal model is of the form $\mathbb{P}^{1}\times C$, where $C$ is a smooth projective curve of genus $\geq 1$ (note that such a $C$ is $\mathbb{A}^{1}$-rigid) is the most complicated one. If $E$ is a $\mathbb{P}^{1}$-bundle over $C$, then one has $\mathcal{S}(E)\simeq\pi_{0}^{\mathbb{A}^{1}}(E)\simeq C$. If $X$ is the surface obtained by blowing up one closed point on $E$ and when $k$ is assumed to be algebraically closed, one has $\mathcal{S}(X)\neq\mathcal{S}^{2}(X)$. However, in this case one has $\mathcal{S}^{2}(X)\simeq\mathcal{S}^{3}(X)$. The details will appear in a forthcoming paper [6].

3 Naive $\mathbb{A}^{1}$-connectedness on field-valued points

Let $X$ be a scheme over a field $k$. It is often much simpler to determine sections of the sheaf $\mathcal{S}(X)$ on smooth schemes which are the spectrum of a finitely generated separable field extension of the base field $k$. A result of Morel (see [12, Lemma 6.1.3]) states that a space $\mathcal{X}$ over an infinite field $k$ is $\mathbb{A}^{1}$-connected (that is, $\pi_{0}^{\mathbb{A}^{1}}(\mathcal{X})$ is trivial) if and only if $\pi_{0}^{\mathbb{A}^{1}}(\mathcal{X})({\rm Spec}~{}F)=\ast$, for every finitely generated separable field extension $F$ of $k$. The argument given by Morel also works when the base field is finite, thanks to Gabber’s presentation lemma over finite fields proved in [10]. We wish to study the analogue of this result in the context of the sheaf of $\mathbb{A}^{1}$-chain connected components. The method used here closely follows the one employed by Morel in [12, Section 6.1] and in [11, Section 3.3].

Lemma 3.1

Let $V$ be an irreducible smooth scheme over $k$ and let $W\hookrightarrow V$ be the inclusion of a dense open subscheme. Then $\mathcal{S}(V/W)\simeq\ast$.

**Proof **

Since we have epimorphisms $V\to V/W\to\mathcal{S}(V/W)$, triviality of $\mathcal{S}(V/W)$ follows from the following statement: any point $x\in V$ has an open neighbourhood $U$ such that $\mathcal{S}(U/(W\cap U))$ is trivial.

Let $Z\hookrightarrow V$ be the closed immersion of the complement of $W$, with the reduced induced subscheme structure. By Gabber’s presentation lemma (see [8, Theorem 3.1.1] for the case where $k$ is infinite and [10, Theorem 1.1] for the case where $k$ is finite), $x$ admits an open neighbourhood $U$ and an étale morphism $\pi:U\to\mathbb{A}^{1}_{V^{\prime}}$, for some open subscheme $V^{\prime}$ of $\mathbb{A}^{d-1}$, where $d$ is the dimension of $V$ at $x$, such that $\pi$ induces a closed immersion $Z\cap U\hookrightarrow\mathbb{A}^{1}_{V^{\prime}}$ satisfying $Z\cap U=\pi^{-1}(\pi(Z\cap U))$ and such that $Z\cap U\to V^{\prime}$ is a finite morphism. Therefore, we have an isomorphism of Nisnevich sheaves

[TABLE]

Hence, it suffices to check that $\mathcal{S}(\mathbb{A}^{1}_{V^{\prime}}/(\mathbb{A}^{1}_{V^{\prime}}-\pi(Z\cap U)))$ is trivial. Now, since $Z\cap U\to V^{\prime}$ is a finite morphism, $Z\cap U\to\mathbb{P}^{1}_{V^{\prime}}$ is proper. This closed immersion does not intersect the section at infinity $s_{\infty}:V^{\prime}\to\mathbb{P}^{1}_{V^{\prime}}$. By Mayer-Vietoris excision (see [14, §3, Lemma 1.6]), we have an isomorphism of Nisnevich sheaves

[TABLE]

Also observe that $\mathbb{A}^{1}_{V^{\prime}}\to\mathbb{P}^{1}_{V^{\prime}}/(\mathbb{P}^{1}_{V^{\prime}}-\pi(Z\cap U))$ is onto and that ${\rm Sing}_{*}^{\mathbb{A}^{1}}(\mathbb{A}^{1}_{V^{\prime}})\simeq{\rm Sing}_{*}^{\mathbb{A}^{1}}(V^{\prime})$ (since ${\rm Sing}_{*}^{\mathbb{A}^{1}}$ preserves $\mathbb{A}^{1}$-weak equivalences). Thus, the composition

[TABLE]

is surjective for any section $V^{\prime}\to\mathbb{A}^{1}_{V^{\prime}}$; in particular, for the zero section. But, in $\mathbb{P}^{1}_{V^{\prime}}$, the zero section is $\mathbb{A}^{1}$-homotopic to the section at infinity $s_{\infty}$. Since $s_{\infty}(V^{\prime})\subseteq\mathbb{P}^{1}_{V^{\prime}}-\pi(Z\cap U)$, it follows that

[TABLE]

is the trivial morphism, as desired.

Theorem 3.2

Let $k$ be a field and let $\mathcal{X}$ be a simplicial sheaf of sets on $Sm/k$. Suppose that $\mathcal{S}(\mathcal{X})({\rm Spec}~{}F)=\ast$, for every finitely generated separable field extension $F$ of $k$. Then $\mathcal{S}(Ex{\rm Sing}_{*}^{\mathbb{A}^{1}}\mathcal{X})\simeq\ast$. Consequently, $\mathcal{S}^{2}(\mathcal{X})\simeq\ast$.

**Proof **

We need to show that for every $U\in Sm/k$, the pointed set $\mathcal{S}({\rm Sing}_{*}^{\mathbb{A}^{1}}\mathcal{X})(U)$ is trivial. It suffices to show that for every morphism $U\to\mathcal{S}({\rm Sing}_{*}^{\mathbb{A}^{1}}\mathcal{X})$, there is a Nisnevich cover $\mathcal{V}=\coprod V_{i}\to U$ such that the composite $V\to U\to\mathcal{S}({\rm Sing}_{*}^{\mathbb{A}^{1}}\mathcal{X})$ is trivial.

Claim: For any irreducible, smooth $k$-scheme $V$ and a morphism $\phi:V\to\mathcal{X}$, the composition $V\stackrel{{\scriptstyle\phi}}{{\to}}\mathcal{X}\to Ex{\rm Sing}_{*}^{\mathbb{A}^{1}}\mathcal{X}\to\mathcal{S}(Ex{\rm Sing}_{*}^{\mathbb{A}^{1}}\mathcal{X})$ is trivial.

Proof of the claim: Let $k(V)$ denote the function field of $V$. Since

[TABLE]

is trivial by hypothesis, there exists a dense open subset $W\hookrightarrow V$ such that the composite $W\to V\stackrel{{\scriptstyle\phi}}{{\to}}\mathcal{X}$ is $\mathbb{A}^{1}$-chain homotopic to the trivial morphism. Therefore the composite of this morphism with the morphism $\mathcal{X}\to{\rm Sing}_{*}^{\mathbb{A}^{1}}\mathcal{X}$ is simplicially homotopic to the trivial morphism. Choose a simplicial fibrant replacement ${\rm Sing}_{*}^{\mathbb{A}^{1}}\mathcal{X}\to Ex{\rm Sing}_{*}^{\mathbb{A}^{1}}\mathcal{X}$. The composite $W\hookrightarrow V\stackrel{{\scriptstyle\phi}}{{\to}}\mathcal{X}\to{\rm Sing}_{*}^{\mathbb{A}^{1}}\mathcal{X}\to Ex{\rm Sing}_{*}^{\mathbb{A}^{1}}\mathcal{X}$ continues to be simplicially homotopic to the trivial map. We denote this simplicial homotopy by $H:W\times\Delta^{1}\to Ex{\rm Sing}_{*}^{\mathbb{A}^{1}}\mathcal{X}$, where $H|_{W\times\{0\}}$ is the trivial map and $H|_{W\times\{1\}}$ is induced by $\phi|_{W}$ (here $\Delta^{1}$ denotes the simplicial $1$-simplex). Consider the acyclic cofibration $V\times\{1\}~{}\cup~{}W\times\Delta^{1}\to V\times\Delta^{1}$. The maps $V\times\{1\}\stackrel{{\scriptstyle\sim}}{{\to}}V\stackrel{{\scriptstyle\phi}}{{\to}}\mathcal{X}\to Ex{\rm Sing}_{*}^{\mathbb{A}^{1}}\mathcal{X}$ and $H:W\times\Delta^{1}\to Ex{\rm Sing}_{*}^{\mathbb{A}^{1}}\mathcal{X}$ clearly glue to give a map $\Phi:V\times\{1\}~{}\cup~{}W\times\Delta^{1}\to Ex{\rm Sing}_{*}^{\mathbb{A}^{1}}\mathcal{X}$, which fits in the following commutative diagram.

[TABLE]

We now use the right lifting property of the projection map $Ex{\rm Sing}_{*}^{\mathbb{A}^{1}}\mathcal{X}\to\ast$ with respect to acyclic cofibrations to see that dotted arrow in the above diagram exists. It follows that $\phi:V\to Ex{\rm Sing}_{*}^{\mathbb{A}^{1}}\mathcal{X}$ is simplicially homotopic to a morphism $\phi^{\prime}:V\to Ex{\rm Sing}_{*}^{\mathbb{A}^{1}}\mathcal{X}$ whose restriction to $W$ is trivial. Thus, we get an induced morphism (of spaces) $\bar{\phi^{\prime}}:V/W\to Ex{\rm Sing}_{*}^{\mathbb{A}^{1}}\mathcal{X}$. Now, applying Lemma 3.1 and commutativity of the diagram

[TABLE]

proves the claim.

We now complete the proof of the theorem using the claim. Since the natural map $Ex{\rm Sing}_{*}^{\mathbb{A}^{1}}\mathcal{X}\to\mathcal{S}(Ex{\rm Sing}_{*}^{\mathbb{A}^{1}}\mathcal{X})$ is an epimorphism, there is a Nisnevich covering $\coprod V_{i}\to U$, where $V_{i}$ are irreducible smooth $k$-schemes such that every composite $V_{i}\to U\to\mathcal{S}(Ex{\rm Sing}_{*}^{\mathbb{A}^{1}}\mathcal{X})$ lifts to a morphism $V_{i}\to Ex{\rm Sing}_{*}^{\mathbb{A}^{1}}\mathcal{X}$. Since $Ex{\rm Sing}_{*}^{\mathbb{A}^{1}}\mathcal{X}$ is a simplicial fibrant replacement of $\mathcal{X}$, each map $V_{i}\to Ex{\rm Sing}_{*}^{\mathbb{A}^{1}}\mathcal{X}$ is represented by a map $V_{i}\to{\rm Sing}_{*}^{\mathbb{A}^{1}}\mathcal{X}$ in the simplical homotopy category $\mathcal{H}_{s}(k)$.

[TABLE]

Since the sheaf at simplicial level [math] of ${\rm Sing}_{*}^{\mathbb{A}^{1}}\mathcal{X}$ is $\mathcal{X}_{0}$, and since any map from a space of simplicial dimension [math] to another space is determined by a map at the [math]th simplicial level, this map factors through the monomorphism $\mathcal{X}\to{\rm Sing}_{*}^{\mathbb{A}^{1}}\mathcal{X}$. Thus, the theorem now follows from the claim, applied to each of the maps $V_{i}\to\mathcal{X}$.

Corollary 3.3

Let $X$ be a scheme over a field $k$ such that $\mathcal{S}(X)({\rm Spec}~{}F)=\ast$, for every finitely generated separable field extension $F$ of $k$. Then $\mathcal{S}^{2}(X)\simeq\ast$.

The condition in Corollary 3.3 can be seen to hold when $X$ is a smooth proper rational variety over a field of characteristic [math]. In general, it holds when $X$ is a smooth proper $R$-trivial variety over a field of characteristic [math] (see [2, Theorem 2.4.3, Corollary 2.4.9]).

Corollary 3.4

Let $G$ be an $\mathbb{A}^{1}$-connected reductive algebraic group over an infinite perfect field $k$. Then $\mathcal{S}^{3}(G)\simeq\ast$.

**Proof **

This is a straightforward consequence of Theorem 3.2 and Examples 2.8 (a), (b).

Remark 3.5

Note that if $G$ in Corollary 3.4 is such that every almost $k$-simple factor of $G$ contains a copy of $\mathbb{G}_{m}$, then one has $\mathcal{S}(G)=\ast$ by [1, Theorem 2.3.2].

We end this note with an example of a singular, projective scheme $X$ for which $\mathcal{S}(X)\neq\ast$, but $\mathcal{S}(X)({\rm Spec}~{}F)=\ast$, for every finitely generated field extension $F$ of $k$. This example was pointed out to the author by Chetan Balwe.

Example 3.6

Let $k$ be a field and let $C\hookrightarrow\mathbb{P}^{2}$ be an elliptic curve over $k$. Let $X_{1}$ denote the blow up of $\mathbb{P}^{1}\times C\hookrightarrow\mathbb{P}^{1}\times\mathbb{P}^{2}$ at the closed point $P=((1:0),Q)$, where $Q$ is a closed point on $C$. Let $E$ denote the exceptional divisor. We have an obvious morphism $\pi:X_{1}\to C$, which is the composition of the blow-up morphism $\phi:X_{1}\to\mathbb{P}^{1}\times C$ with the projection map $\mathbb{P}^{1}\times C\to C$. Let $X_{2}$ denote the plane $\{(0:1)\}\times\mathbb{P}^{2}$; so $X_{1}$ and $X_{2}$ intersect in $\{(0:1)\}\times C$. Let $X:=X_{1}\cup X_{2}$. Since $\mathcal{S}(X_{2})=\ast$, it is easy to see that $\mathcal{S}(X)({\rm Spec}~{}F)=\ast$, for every field extension $F$ of $k$. By [2, Theorem 2.4.3], we have $\pi_{0}^{\mathbb{A}^{1}}(X)({\rm Spec}~{}F)=\ast$, for every finitely generated field extension $F$ of $k$. By [12, Lemma 6.1.3], we have $\pi_{0}^{\mathbb{A}^{1}}(X)\simeq\ast$.

Let $U$ be a smooth henselian local scheme of dimension $1$ over $k$ with closed point $u$. Let $\gamma:U\to X_{1}$ be a morphism that maps $u$ on $E$ and the generic point of $U$ outside $E$. We begin by considering naive $\mathbb{A}^{1}$-homotopies of $U$ inside $X_{1}$, starting at $\gamma$. Let $h$ be a morphism $\mathbb{A}^{1}\times U\to X_{1}$ such that $h|_{\{0\}\times\mathbb{A}^{1}}=\gamma$. Since $C$ is $\mathbb{A}^{1}$-rigid, the composition $\pi\circ h:\mathbb{A}^{1}\times U\to C$ factors through the projection $\mathbb{A}^{1}\times U\to U$. Thus, $h:\mathbb{A}^{1}\times U\to X_{1}$ factors through $X_{1}\times_{C}U$. By assumption, $h$ is such that the point $Q$ on $C$ is in the image of $\pi\circ h$, that is, the image of the closed fiber $\mathbb{A}^{1}\times\{u\}$ intersects the exceptional divisor $E$. Hence, $h^{-1}(P)$ is a closed subscheme of $\mathbb{A}^{1}\times U$ with support contained in $\mathbb{A}^{1}\times\{u\}$. Since $\phi\circ h$ can be lifted to $X_{1}$, we see that $h^{-1}(P)$ is a closed subscheme of $\mathbb{A}^{1}\times U$ of codimension $1$. Therefore, the support of $h^{-1}(P)$ must be exactly $\mathbb{A}^{1}\times\{u\}$. Thus, $h$ maps $\mathbb{A}^{1}\times\{u\}$ into the exceptional divisor $E$.

Now, let $h:\mathbb{A}^{1}\times U\to X$ be a morphism such that $h|_{\{0\}\times U}=\gamma$. Since $\mathbb{A}^{1}\times U$ is irreducible, this implies that $h$ factors through the inclusion of $X_{1}$ into $X$. The discussion in the above paragraph shows that $h$ maps the whole closed fiber of $\mathbb{A}^{1}\times U$ on $E$. Hence, we cannot have $\mathcal{S}(X)(U)=\ast$ and consequently, $\mathcal{S}(X)\neq\ast$. However, Theorem 3.2 implies that $\mathcal{S}^{2}(X)=\ast$. Since $\pi_{0}^{\mathbb{A}^{1}}(X)\simeq\ast$ as observed above, ${\rm Sing}_{*}^{\mathbb{A}^{1}}X$ cannot be $\mathbb{A}^{1}$-local in view of Lemma 2.4.

**Acknowledgement **

The author thanks Chetan Balwe for suggesting Example 3.6 as well as for his comments on this article and Marc Hoyois for a helpful discussion during the International Colloquium on $K$-theory at TIFR. The author also thanks the referee for a careful reading of the note and for comments that helped him improve the exposition.