New PDF release: A1-Algebraic Topology over a Field

By Fabien Morel

ISBN-10: 3642295134

ISBN-13: 9783642295133

This textual content offers with A1-homotopy thought over a base box, i.e., with the traditional homotopy concept linked to the class of tender kinds over a box during which the affine line is imposed to be contractible. it's a average sequel to the foundational paper on A1-homotopy idea written including V. Voevodsky. encouraged by means of classical ends up in algebraic topology, we current new concepts, new effects and functions on the topic of 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 gentle varieties.

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Extra resources for A1-Algebraic Topology over a Field

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Lies in K1 (U ; G). α = ∗ as required. Let us now prove (2). Assume (H2) (d) holds. Let’s prove (H1) (d+1). Let X be an irreducible smooth k-scheme (of finite type) of dimension ≤ d+1 with function field F , let u ∈ X ∈ Smk be a point of codimension d + 1 and denote by U its associated local scheme, F its function field. We have to check the exactness at the middle of G(F ) ⇒ Πy∈U (1) Hy1 (U ; G) → Πz∈U (2) Hz2 (U ; G). Let α ∈ K1 (U ; G) ⊂ Πy∈U (1) Hy1 (U ; G). ∗. Let us denote by yi ∈ U the points of 32 2 Unramified Sheaves and Strongly A1 -Invariant Sheaves codimension one in U where α is non trivial.

Proof. One might prove this using our description of those strongly A1 invariant sheaf of groups given in the previous section. We give here another argument. Let BG be the simplicial classifying space of G (see [59] for instance). The assumption that G is strongly A1 -invariant means that it is an A1 -local space. Choose a fibrant resolution BG of BG. We use the pointed function space RHom• (Gm , BG) := Hom• (Gm , BG) It is fibrant and automatically A1 -local, as BG is. Moreover its π1 sheaf is G−1 and its higher homotopy sheaves vanish.

Assume (H2) (d) holds. Let’s prove (H1) (d+1). Let X be an irreducible smooth k-scheme (of finite type) of dimension ≤ d+1 with function field F , let u ∈ X ∈ Smk be a point of codimension d + 1 and denote by U its associated local scheme, F its function field. We have to check the exactness at the middle of G(F ) ⇒ Πy∈U (1) Hy1 (U ; G) → Πz∈U (2) Hz2 (U ; G). Let α ∈ K1 (U ; G) ⊂ Πy∈U (1) Hy1 (U ; G). ∗. Let us denote by yi ∈ U the points of 32 2 Unramified Sheaves and Strongly A1 -Invariant Sheaves codimension one in U where α is non trivial.

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A1-Algebraic Topology over a Field by Fabien Morel


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