Proof — 𝔸¹-homotopy theory of schemes
Statement
For a Noetherian scheme S of finite Krull dimension, Morel and Voevodsky construct the 𝔸¹-homotopy category H(S): one starts from simplicial Nisnevich sheaves on the site of smooth schemes of finite type over S, imposes 𝔸¹-local equivalences (maps that become weak equivalences after 𝔸¹-invariant replacement), and inverts those equivalences. The resulting category behaves like an unstable motivic homotopy category: the affine line 𝔸¹ plays the role of the unit interval, algebraic K-theory and motivic cohomology receive geometric representatives, and there are purity and blow-up descent statements in the motivic setting. The stable refinement leads to motivic spectra and is the geometric layer behind motivic cohomology operations and Voevodsky’s Milnor-conjecture program. The theory is developed with full proofs in algebraic geometry; formalizable in ZFC once schemes and sheaf categories are encoded.
Formula
H(S) = (Δ^op Shv_Nis(Sm/S))[𝔸¹⁻¹] (𝔸¹-local homotopy category)
Equation
𝔸¹-weak equivalence; S^{p,q} motivic spheres; connectivity and long exact sequences
Proof sketch
Morel, F.; Voevodsky, V. A¹-homotopy theory of schemes. Publications Mathématiques de l’Institut des Hautes Études Scientifiques 90 (2001), 45–143. Builds on Nisnevich descent, 𝔸¹-invariance of algebraic K-theory for regular schemes, and model structures on simplicial presheaves.
Source
Morel, F.; Voevodsky, V. A¹-homotopy theory of schemes. Publ. Math. Inst. Hautes Études Sci. 90 (2001), 45–143.
Domains
mathematics