◆ A1-HOMOTOPY LOCALIZATION

VOEVODSKY A1-INVARIANCE // NISNEVICH TOPOLOGY // P1-SUSPENSION

I. A1-Homotopy Theory

Voevodsky's fundamental insight: replace the unit interval [0,1] in topology with the affine line A^1 in algebraic geometry. A presheaf F on Sm/k is A1-invariant if:

F(X) ≅ F(X × A^1) for all smooth schemes X

The A1-homotopy category H(k) is obtained by Nisnevich-localizing and then A1-localizing the category of simplicial presheaves on Sm/k.

II. Nisnevich Topology

The Nisnevich topology on Sm/k is finer than Zariski but coarser than etale. A covering {U_i → X} is Nisnevich if for every x ∈ X, some U_i → X induces a residue field isomorphism at some point above x.

Nis(X) = {U → X etale | ∃ u ∈ U with k(u) = k(f(u))}

III. P1-Stabilization

The motivic sphere is S^1 ∧ G_m where S^1 is the simplicial circle and G_m = A1 \ {0} is the multiplicative group. The P1-stable category SH(k) is obtained by inverting P1-suspension:

SH(k) = H(k)[Σ^{-1}_{P^1}] where P^1 ≅ S^1 ∧ G_m

Objects of SH(k) are bigraded: π_{p,q} means topological degree p and weight q.

IV. Motivic Eilenberg-MacLane Spaces

H^{p,q}(X, Z) = [Σ^∞ X_+, Σ^{p,q} HZ] in SH(k)

The bigraded motivic cohomology recovers Chow groups: H^{2n,n}(X, Z) ≅ CH^n(X).

C447 // A1-HOMOTOPY // BBOBOP VOID MATHEMATICS