Motivic Homotopy TheoryHomotopy sheaves

# 3 Homotopy sheaves

Naively, one would like to define the motivic homotopy groups of $X \in (\mathrm{Spc}_S^{\mathbb {A}^1})_*$ as $[S^{p, q}, X]_*$. This is an abelian group. We can do better than that, and produce $\pi _{p, q} X$ as a sheaf.

Definition 3.1

Let $X \in \mathrm{Spc}_S$. We define $\pi _0^{\mathrm{Nis}}(X)$ to be the Nisnevich sheafification of the presheaf of sets

$U \mapsto [U, X]_{\mathrm{Spc}_S}.$

If $X \in (\mathrm{Spc}_S)_*$ and $n \geq 1$, define $\pi _n^{\mathrm{Nis}}(X)$ to be the Nisnevich sheafification of the presheaf of groups

$U \mapsto [S^n \wedge U_+, X]_{(\mathrm{Spc}_S)_*}.$

In general, if $X \in \mathcal{P}(\mathrm{Sm}_S)$, we define $\pi _n^{\mathrm{Nis}}(X) = \pi _n^{\mathrm{Nis}}(L_{\mathrm{Nis}} X)$. Finally, we define

$\pi _n^{\mathbb {A}^1}(X) = \pi _n^{\mathrm{Nis}}(L_{\mathrm{Mot}}X).$

Corollary 3.2

If $F \to X \to Y$ is a fiber sequence in $(\mathrm{Spc}_S^{\mathbb {A}^1})_*$, then there is a long exact sequence

$\cdots \pi _{n + 1}^{\mathbb {A}^1} Y \to \pi _n^{\mathbb {A}^1} F \to \pi _n^{\mathbb {A}^1} X \to \pi _n^{\mathbb {A}^1} Y \to \cdots$

of Nisnevich sheaves.

Proof
The forgetful functor $\mathrm{Spc}_S^{\mathbb {A}^1} \to \mathcal{P}(\mathrm{Sm}_S)$ is a right adjoint, hence preserves fiber sequences. So fiber sequences in $\mathrm{Spc}_S^{\mathbb {A}^1}$ are computed objectwise. Then note that sheafification is exact.
Proof
Here it is essential that we did not include $L_{\mathbb {A}^1}$ in the definition of $\pi _n^{\mathbb {A}^1}(X)$, since $L_{\mathbb {A}^1}$ is not exact.

Definition 3.3

If $X \in \mathcal{P}(\mathrm{Sm}_S)$, we say $X$ is $\mathbb {A}^1$-connected if the canonical map $X \to S$ induces an isomorphism of sheaves $\pi _0^{\mathbb {A}^1} X \to \pi _0^{\mathbb {A}^1} S = *$.

Given $X \in \mathrm{Spc}_S$, to check if $X$ is $\mathbb {A}^1$-connected, it turns out it suffices to check that $\pi _0^{\mathrm{Nis}}(X)$ is trivial.

Proposition 3.4 (Unstable $\mathbb {A}^1$-connectivity)

Let $X \in \mathcal{P}(\mathrm{Sm}_S)$. Then the canonical map

$X \to L_{\mathrm{Mot}}X$

induces an epimorphism

$\pi _0^{\mathrm{Nis}}X \to \pi _0^{\mathrm{Nis}} L_{\mathrm{Mot}} X = \pi _0^{\mathbb {A}^1} X.$

Proof
Since $\pi _0^{\mathrm{Nis}} X \to \pi _0^{\mathrm{Nis}} L_{\mathrm{Nis}} X$ is an isomorphism, it suffices to show that $\pi _0^{\mathrm{Nis}} X \to \pi _0^{\mathrm{Nis}} \mathrm{Sing}^{\mathbb {A}^1} X(U)$ is always an epimorphism. This follows by inspection.
Proof

The final property of $\pi _n^{\mathrm{Nis}}$ we note is that over a perfect field, $\pi _n^{\mathrm{Nis}}$ is unramified. Roughly speaking, it says

$\pi _n^{\mathrm{Nis}}(X)(U) \to \pi _n^{\mathrm{Nis}}(X)(\operatorname{Spec}k(U))$

is injective for any $U \in \mathrm{Sm}_S$. We cannot exactly say this because $\operatorname{Spec}k(U)$ is in general not smooth over $S$.