# A1-Algebraic Topology over a Field by Fabien Morel

By Fabien Morel

This textual content offers with A^{1}-homotopy conception over a base box, i.e., with the normal homotopy concept linked to the class of delicate types over a box within which the affine line is imposed to be contractible. it's a usual sequel to the foundational paper on A^{1}-homotopy idea written including V. Voevodsky. encouraged by way of classical leads to algebraic topology, we current new ideas, new effects and functions relating to the homes and computations of A^{1}-homotopy sheaves, A^{1}-homology sheaves, and sheaves with generalized transfers, in addition to to algebraic vector bundles over affine delicate varieties.

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Proof. The ﬁrst part of Axiom (A1) (ii) follows from Axiom (B4). For the second part we choose a uniformizing element π in Ow , which is still a uniformizing element for Ov and the square ∂π v M∗ (F ) −→ M(∗−1) (κ(v) ↑ ↑ ∂π w M(∗−1) (κ(w) M∗ (E) −→ is commutative by our deﬁnition (D4) (iii). Moreover the morphism M∗ (E) → M∗ (F ) preserve the product by π by (D4) (i). To prove Axiom (A3) we proceed as follows. By assumption we have E ⊂ Ov ⊂ F . Choose a uniformizing element π of v. 3 Z-Graded Strongly A1 -Invariant Sheaves of Abelian Groups 39 E(T ) ⊂ F induced by T → π.

Recall from the beginning of Sect. 2 that we denote Σy∈X (1) ∂y by Hz2 (X; M ) the cokernel of the sum of the residues M∗ (E) −→ ⊕y∈X (1) Hy1 (X; M∗ ). 5) where the homomorphisms denoted ∂zy are deﬁned by the diagram. This diagram is the complex C ∗ ((A1U )0 ; M∗ ). 40. Assume M∗ satisfies all the previous Axioms. 41. Assume M∗ satisfies all the previous Axioms. For each n, ˜ k ) Mn satisfies Axiom (A2’). the unramified sheaves of abelian groups (on Sm Proof. 19, it suﬃces to check this when k is inﬁnite.

It is a tautology in case d ≤ 2. 26. 29. Let d ≥ 0 be an integer. 1) (H1)(d) ⇒ (H2)(d). 29. 29 implies by induction on d that properties (H1)(d) and (H2)(d) hold for any d. 26 above that for any essentially smooth 1 1 k-scheme X with inﬁnite residue ﬁelds, then HZar (X; G) ∼ = HN is (X; G) and 1 1 1 ∼ H (X; G) = HZar (AX ; G). 27 if k is inﬁnite. Assume now k is ﬁnite. Let G 1 be the sheaf π1A (BG) = π1 (LA1 (BG)). 9 of Sect. 11 it satisﬁes, as G the Axioms (A2’), (A5) and (A6). By general properties of base change through a smooth morphism (see [52]) we see that for any henselian k-smooth local ring A, with inﬁnite residue ﬁeld, the morphism G(A) → G (A) is an isomorphism.