Motivic Homotopy Theory — $\mathbb {A}^1$ -localization

# 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

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

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^ n\\ \mathbb {P}^ n / \mathbb {P}^{n - 1} & \cong S^ n \wedge \mathbb {G}_ m^ 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.