2 $\mathbb {A}^1$ -localization

Definition 2.1

A presheaf $F \in \mathcal{P}(\mathrm{Sm}_S)$ is $\mathbb {A}^1$ -local if the natural map $F(X \times \mathbb {A}^1) \to F(X \times \{ 0\} )$ is an equivalence for all $X$ . We write $L_{\mathbb {A}^1}\mathcal{P}(\mathrm{Sm}_S)$ for the full subcategory of $\mathbb {A}^1$ -local presheaves, and $L_{\mathbb {A}^1 \wedge \mathrm{Nis}}\mathcal{P}(\mathrm{Sm}_S) = \mathrm{Spc}_S^{\mathbb {A}^1}$ for the presheaves that are both Nisnevich local and $\mathbb {A}^1$ -local.

Therefore we get a square of accessible localizations

$\begin{useimager} \[ \begin{tikzcd} \Pre(\Sm_S) \ar[r, "L_{\Nis}"] \ar[d, "\widetilde{L_{\A^1}}"] & L_{\Nis} \Pre(\Sm_S) \equiv \Spc_S \ar[d, "L_{\A^1}"]\\ L_{\A^1} \Pre(\Sm_S) \ar[r, "\widetilde{L_{\Nis}}"] & L_{A^1 \wedge \Nis} \Pre(\Sm_S) \equiv \Spc_S^{\A^1} \equiv \H(S) \end{tikzcd} \] \end{useimager}$

We also write the composite $\mathcal{P}(\mathrm{Sm}_S) \to \mathrm{Spc}_S^{\mathbb {A}^1}$ as $L_{\mathrm{Mot}}$ .

Remark 2.2

Note that if we think of each of these as subcategories, $L_{\mathbb {A}^1}$ and $\widetilde{L_{\mathbb {A}^1}}$ are not the same functors.

Remark 2.3

A representable functor is usually not $\mathbb {A}^1$ -local. Hence if $X \in \mathrm{Sm}_S$ , the resulting sheaf $L_{\mathrm{Mot}} X \in \mathrm{Spc}_S^{\mathbb {A}^1}$ is usually not discrete. If $X$ is already $\mathbb {A}^1$ -local, then we say $X$ is $\mathbb {A}^1$ -rigid. For example, $\mathbb {G}_m$ is $\mathbb {A}^1$ -rigid.

Unlike

L_{\mathrm{Nis}}

, the

\mathbb {A}^1

-localization functor

L_{\mathbb {A}^1}

is not a sheafification functor. Thus, a priori, the only nice property of it we know is that it is a left adjoint. To remedy for this, we describe an explicit construction of

\widetilde{L_{\mathbb {A}^1}}

, and then observe that

Lemma 2.4

$L_{\mathrm{Mot}} \simeq (L_{\mathrm{Nis}} \widetilde{L_{\mathbb {A}^1}})^{\omega }$ .

The functor $\widetilde{L_{\mathbb {A}^1}}$ is better known as $\mathrm{Sing}^{\mathbb {A}^1}$ .

Definition 2.5

Define the “affine $n$ -simplex” $\Delta ^n$ by

\Delta ^n = \operatorname{Spec}k[x_0, \ldots , x_n]/(x_0 + \cdots + x_n = 1).

This forms a cosimplicial scheme in the usual way.

For $X \in \mathcal{P}(\mathrm{Sm}_S)$ , we define $\mathrm{Sing}^{\mathbb {A}^1}X \in \mathcal{P}(\mathrm{Sm}_S)$ by

(\mathrm{Sing}^{\mathbb {A}^1}X)(U) = |X(U \times \Delta ^\bullet )|.

It is then straightforward to check that

Lemma 2.6

$\widetilde{L_{\mathbb {A}^1}} \simeq \mathrm{Sing}^{\mathbb {A}^1}$ .

Corollary 2.7

$\widetilde{L_{\mathbb {A}^1}}$ preserves finite products. Hence so does $L_{\mathrm{Mot}}$ .

Example 2.8

Let $E \to X$ be a (Nisnevich-)locally trivial $\mathbb {A}^n$ -bundle. Then $E \to X$ is an $\mathbb {A}^1$ -equivalence in $L_{\mathrm{Nis}}\mathcal{P}(\mathrm{Sm}_S)$ .

Example 2.9

We claim that $\mathbb {P}^1 \cong \Sigma \mathbb {G}_m = S^1 \wedge \mathbb {G}_m \in \mathrm{Spc}_S^{\mathbb {A}^1}$ . Here we are working with pointed objects so that suspensions make sense.

Indeed, we have a Zariski open cover of $\mathbb {P}^1$ given by

$\begin{useimager} \[ \begin{tikzcd} \A^1 \setminus \{0\} \ar[r] \ar[d] & \A^1 \ar[d, "x^{-1}"]\\ \A^1 \ar[r, "x"] & \P^1 \end{tikzcd} \] \end{useimager}$

Since this is in particular an elementary distinguished square, it is also a pushout in $(\mathrm{Spc}_S)_*$ . Now apply $L_{\mathbb {A}^1}$ to this diagram, which preserves pushouts since it is a left adjoint. Since $L_{\mathbb {A}^1} \mathbb {A}^1 \simeq *$ the claim follows.

If we “replace” only one of the $\mathbb {A}^1$ 's with $*$ , we see that this also says

\Sigma \mathbb {G}_m = \mathbb {A}^1 / (\mathbb {A}^1 \setminus \{ 0\} ).

In general, we have

Lemma 2.10

\begin{aligned} \mathbb {A}^n \setminus \{ 0\} & \cong S^{n - 1} \wedge \mathbb {G}_m^{\wedge n}\\ \mathbb {P}^n / \mathbb {P}^{n - 1} & \cong S^n \wedge \mathbb {G}_m^{\wedge n}. \end{aligned}

It is convenient to introduce the following notation:

Definition 2.11

We define $S^{p, q} = \mathbb {G}_m^{\wedge q} \wedge (S^1)^{\wedge (p - q)}$ whenever it makes sense.

2 A1\mathbb {A}^1A1-localization

2 $\mathbb {A}^1$ -localization