By Fabien Morel
This textual content bargains with A1-homotopy thought over a base box, i.e., with the ordinary homotopy thought linked to the class of soft forms over a box within which the affine line is imposed to be contractible. it's a traditional sequel to the foundational paper on A1-homotopy thought written including V. Voevodsky. encouraged via classical ends up in algebraic topology, we current new options, new effects and functions regarding the houses and computations of A1-homotopy sheaves, A1-homology sheaves, and sheaves with generalized transfers, in addition to to algebraic vector bundles over affine tender varieties.
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This suﬃces for our purpose because, the connected component of the base point in an A1 -local space is A1 -local. This follows formally from the fact (see ) that the A1 -localization functor takes a 0-connected space to a 0-connected space. In fact we may also prove directly that the space RHom• (Gm , B(G)) is 1 0-connected. Its π0 is the associated sheaf to the presheaf X → HN is (X × Gm ; G), and this amounts to checking that for X the henselization of point 1 in a smooth k-scheme, then HN is (Gm × X; G).
For d = 1 there is nothing to prove. Assume d ≥ 2. We may easily reduce to the case Z is irreducible with generic point z. We have to show that the composition s(jd ) s(j2 ) s(j1 ) S(X) → · · · → S(Y2 ) → S(Y1 ) → S(Z) ⊂ S(κ(z)) 20 2 Unramified Sheaves and Strongly A1 -Invariant Sheaves doesn’t depend on the choice of the ﬂag Z→Y1 → · · · → . . →X. We may thus replace X by any open neighborhood Ω of z if we want or even by Spec(A) with A := OX,z , which we do. We ﬁrst observe that the case d = 2 follows directly from the Axiom (A4).
We assume throughout this section that M∗ is endowed with the following extra structures. (D4) (i) For any F ∈ Fk a structure of Z[F × /(F ×2 )]-module on M∗ (F ), which we denote by (u, α) →< u > α ∈ Mn (F ) for u ∈ F × and for α ∈ Mn (F ). This structure should be functorial in the obvious sense in Fk . α, functorial (in the obvious sense) in Fk . (D4) (iii) For any discrete valuation v on F ∈ Fk and uniformizing element π a graded epimorphism of degree −1 ∂vπ : M∗ (F ) → M∗−1 (κ(v)) which is functorial, in the obvious sense, with respect to extensions E → F such that v restricts to a discrete valuation on E, with ramiﬁcation index 1, if we choose as uniformizing element an element π in E.
A1-Algebraic Topology over a Field by Fabien Morel