A presheaf F∈P(SmS) is A1-local if the natural map F(X×A1)→F(X×{0}) is an equivalence for all X. We write LA1P(SmS) for the full subcategory of A1-local presheaves, and LA1∧NisP(SmS)=SpcSA1 for the presheaves that are both Nisnevich local and A1-local.
Therefore we get a square of accessible localizations
We also write the composite P(SmS)→SpcSA1 as LMot.
Remark2.2
Note that if we think of each of these as subcategories, LA1 and LA1 are not the same functors.
Remark2.3
A representable functor is usually not A1-local. Hence if X∈SmS, the resulting sheaf LMotX∈SpcSA1 is usually not discrete. If X is already A1-local, then we say X is A1-rigid. For example, Gm is A1-rigid.
Unlike LNis, the A1-localization functor LA1 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, and then observe that
Lemma2.4
LMot≃(LNisLA1)ω.
The functor LA1 is better known as SingA1.
Definition2.5
Define the “affine n-simplex” Δn by
Δn=Speck[x0,…,xn]/(x0+⋯+xn=1).
This forms a cosimplicial scheme in the usual way.
For X∈P(SmS), we define SingA1X∈P(SmS) by
(SingA1X)(U)=∣X(U×Δ∙)∣.
It is then straightforward to check that
Lemma2.6
LA1≃SingA1.
Corollary2.7
LA1 preserves finite products. Hence so does LMot.
Example2.8
Let E→X be a (Nisnevich-)locally trivial An-bundle. Then E→X is an A1-equivalence in LNisP(SmS).
Example2.9
We claim that P1≅ΣGm=S1∧Gm∈SpcSA1. Here we are working with pointed objects so that suspensions make sense.
Indeed, we have a Zariski open cover of P1 given by
Since this is in particular an elementary distinguished square, it is also a pushout in (SpcS)∗. Now apply LA1 to this diagram, which preserves pushouts since it is a left adjoint. Since LA1A1≃∗ the claim follows.
If we “replace” only one of the A1's with ∗, we see that this also says
ΣGm=A1/(A1∖{0}).
In general, we have
Lemma2.10
An∖{0}Pn/Pn−1≅Sn−1∧Gm∧n≅Sn∧Gm∧n.
It is convenient to introduce the following notation:
Definition2.11
We define Sp,q=Gm∧q∧(S1)∧(p−q) whenever it makes sense.