Motivic Homotopy Theory — A1\mathbb {A}^1 -localization

2 A1\mathbb {A}^1 -localization

Definition 2.1

A presheaf FP(SmS)F \in \mathcal{P}(\mathrm{Sm}_ S) is A1\mathbb {A}^1-local if the natural map F(X×A1)F(X×{0})F(X \times \mathbb {A}^1) \to F(X \times \{ 0\} ) is an equivalence for all XX. We write LA1P(SmS)L_{\mathbb {A}^1}\mathcal{P}(\mathrm{Sm}_ S) for the full subcategory of A1\mathbb {A}^1-local presheaves, and LA1NisP(SmS)=SpcSA1L_{\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 A1\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 P(SmS)SpcSA1\mathcal{P}(\mathrm{Sm}_ S) \to \mathrm{Spc}_ S^{\mathbb {A}^1} as LMotL_{\mathrm{Mot}}.

Remark 2.2

Note that if we think of each of these as subcategories, LA1L_{\mathbb {A}^1} and LA1~\widetilde{L_{\mathbb {A}^1}} are not the same functors.

Remark 2.3

A representable functor is usually not A1\mathbb {A}^1-local. Hence if XSmSX \in \mathrm{Sm}_ S, the resulting sheaf LMotXSpcSA1L_{\mathrm{Mot}} X \in \mathrm{Spc}_ S^{\mathbb {A}^1} is usually not discrete. If XX is already A1\mathbb {A}^1-local, then we say XX is A1\mathbb {A}^1-rigid. For example, Gm\mathbb {G}_ m is A1\mathbb {A}^1-rigid.

Unlike LNisL_{\mathrm{Nis}}, the A1\mathbb {A}^1-localization functor LA1L_{\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 LA1~\widetilde{L_{\mathbb {A}^1}}, and then observe that
Lemma 2.4

LMot(LNisLA1~)ωL_{\mathrm{Mot}} \simeq (L_{\mathrm{Nis}} \widetilde{L_{\mathbb {A}^1}})^{\omega }.

The functor LA1~\widetilde{L_{\mathbb {A}^1}} is better known as SingA1\mathrm{Sing}^{\mathbb {A}^1}.

Definition 2.5

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

Δn=Speck[x0,,xn]/(x0++xn=1). \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 XP(SmS)X \in \mathcal{P}(\mathrm{Sm}_ S), we define SingA1XP(SmS)\mathrm{Sing}^{\mathbb {A}^1}X \in \mathcal{P}(\mathrm{Sm}_ S) by

(SingA1X)(U)=X(U×Δ). (\mathrm{Sing}^{\mathbb {A}^1}X)(U) = |X(U \times \Delta ^\bullet )|.

It is then straightforward to check that

Lemma 2.6

LA1~SingA1\widetilde{L_{\mathbb {A}^1}} \simeq \mathrm{Sing}^{\mathbb {A}^1}.

Corollary 2.7

LA1~\widetilde{L_{\mathbb {A}^1}} preserves finite products. Hence so does LMotL_{\mathrm{Mot}}.

Example 2.8

Let EXE \to X be a (Nisnevich-)locally trivial An\mathbb {A}^ n-bundle. Then EXE \to X is an A1\mathbb {A}^1-equivalence in LNisP(SmS)L_{\mathrm{Nis}}\mathcal{P}(\mathrm{Sm}_ S).

Example 2.9

We claim that P1ΣGm=S1GmSpcSA1\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 P1\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 (SpcS)(\mathrm{Spc}_ S)_*. Now apply LA1L_{\mathbb {A}^1} to this diagram, which preserves pushouts since it is a left adjoint. Since LA1A1L_{\mathbb {A}^1} \mathbb {A}^1 \simeq * the claim follows.

If we “replace” only one of the A1\mathbb {A}^1's with *, we see that this also says

ΣGm=A1/(A1{0}). \Sigma \mathbb {G}_ m = \mathbb {A}^1 / (\mathbb {A}^1 \setminus \{ 0\} ).

In general, we have

Lemma 2.10
An{0}Sn1GmnPn/Pn1SnGmn. \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 Sp,q=Gmq(S1)(pq)S^{p, q} = \mathbb {G}_ m^{\wedge q} \wedge (S^1)^{\wedge (p - q)} whenever it makes sense.