By Fabien Morel
This textual content offers with A1-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 A1-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 A1-homotopy sheaves, A1-homology sheaves, and sheaves with generalized transfers, in addition to to algebraic vector bundles over affine delicate varieties.
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Extra info for A1-Algebraic Topology over a Field
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 ) we see that for any henselian k-smooth local ring A, with inﬁnite residue ﬁeld, the morphism G(A) → G (A) is an isomorphism.