Motivic Homotopy TheoryHomotopy sheaves

3 Homotopy sheaves

Naively, one would like to define the motivic homotopy groups of X(SpcSA1)X \in (\mathrm{Spc}_S^{\mathbb {A}^1})_* as [Sp,q,X][S^{p, q}, X]_*. This is an abelian group. We can do better than that, and produce πp,qX\pi _{p, q} X as a sheaf.

Definition 3.1

Let XSpcSX \in \mathrm{Spc}_S. We define π0Nis(X)\pi _0^{\mathrm{Nis}}(X) to be the Nisnevich sheafification of the presheaf of sets

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

If X(SpcS)X \in (\mathrm{Spc}_S)_* and n1n \geq 1, define πnNis(X)\pi _n^{\mathrm{Nis}}(X) to be the Nisnevich sheafification of the presheaf of groups

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

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

πnA1(X)=πnNis(LMotX). \pi _n^{\mathbb {A}^1}(X) = \pi _n^{\mathrm{Nis}}(L_{\mathrm{Mot}}X).

Corollary 3.2

If FXYF \to X \to Y is a fiber sequence in (SpcSA1)(\mathrm{Spc}_S^{\mathbb {A}^1})_*, then there is a long exact sequence

πn+1A1YπnA1FπnA1XπnA1Y \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.

The forgetful functor SpcSA1P(SmS)\mathrm{Spc}_S^{\mathbb {A}^1} \to \mathcal{P}(\mathrm{Sm}_S) is a right adjoint, hence preserves fiber sequences. So fiber sequences in SpcSA1\mathrm{Spc}_S^{\mathbb {A}^1} are computed objectwise. Then note that sheafification is exact.
Here it is essential that we did not include LA1L_{\mathbb {A}^1} in the definition of πnA1(X)\pi _n^{\mathbb {A}^1}(X), since LA1L_{\mathbb {A}^1} is not exact.

Definition 3.3

If XP(SmS)X \in \mathcal{P}(\mathrm{Sm}_S), we say XX is A1\mathbb {A}^1-connected if the canonical map XSX \to S induces an isomorphism of sheaves π0A1Xπ0A1S=\pi _0^{\mathbb {A}^1} X \to \pi _0^{\mathbb {A}^1} S = *.

Given XSpcSX \in \mathrm{Spc}_S, to check if XX is A1\mathbb {A}^1-connected, it turns out it suffices to check that π0Nis(X)\pi _0^{\mathrm{Nis}}(X) is trivial.

Proposition 3.4 (Unstable A1\mathbb {A}^1-connectivity)

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

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

induces an epimorphism

π0NisXπ0NisLMotX=π0A1X. \pi _0^{\mathrm{Nis}}X \to \pi _0^{\mathrm{Nis}} L_{\mathrm{Mot}} X = \pi _0^{\mathbb {A}^1} X.

Since π0NisXπ0NisLNisX\pi _0^{\mathrm{Nis}} X \to \pi _0^{\mathrm{Nis}} L_{\mathrm{Nis}} X is an isomorphism, it suffices to show that π0NisXπ0NisSingA1X(U)\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.

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

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

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