3 Homotopy sheaves
Naively, one would like to define the motivic homotopy groups of X∈(SpcSA1)∗ as [Sp,q,X]∗. This is an abelian group. We can do better than that, and produce πp,qX as a sheaf.
Definition
3.1
Let X∈SpcS. We define π0Nis(X) to be the Nisnevich sheafification of the presheaf of sets
U↦[U,X]SpcS.
If X∈(SpcS)∗ and n≥1, define πnNis(X) to be the Nisnevich sheafification of the presheaf of groups
U↦[Sn∧U+,X](SpcS)∗.
In general, if X∈P(SmS), we define πnNis(X)=πnNis(LNisX). Finally, we define
πnA1(X)=πnNis(LMotX).
Corollary
3.2
If F→X→Y is a fiber sequence in (SpcSA1)∗, then there is a long exact sequence
⋯πn+1A1Y→πnA1F→πnA1X→πnA1Y→⋯
of Nisnevich sheaves.
The forgetful functor
SpcSA1→P(SmS) is a right adjoint, hence preserves fiber sequences. So fiber sequences in
SpcSA1 are computed objectwise. Then note that sheafification is exact.
Here it is essential that we did not include
LA1 in the definition of
πnA1(X), since
LA1 is not exact.
Definition
3.3
If X∈P(SmS), we say X is A1-connected if the canonical map X→S induces an isomorphism of sheaves π0A1X→π0A1S=∗.
Given X∈SpcS, to check if X is A1-connected, it turns out it suffices to check that π0Nis(X) is trivial.
Proposition
3.4
(Unstable A1-connectivity)
Let X∈P(SmS). Then the canonical map
X→LMotX
induces an epimorphism
π0NisX→π0NisLMotX=π0A1X.
Since
π0NisX→π0NisLNisX is an isomorphism, it suffices to show that
π0NisX→π0NisSingA1X(U) is always an epimorphism. This follows by inspection.
The final property of πnNis we note is that over a perfect field, πnNis is unramified. Roughly speaking, it says
πnNis(X)(U)→πnNis(X)(Speck(U))
is injective for any U∈SmS. We cannot exactly say this because Speck(U) is in general not smooth over S.