By Fabien Morel

ISBN-10: 3642295134

ISBN-13: 9783642295133

This textual content bargains with A^{1}-homotopy idea over a base box, i.e., with the normal homotopy thought linked to the class of delicate kinds over a box during which the affine line is imposed to be contractible. it's a normal sequel to the foundational paper on A^{1}-homotopy concept written including V. Voevodsky. encouraged by way of classical leads to algebraic topology, we current new recommendations, new effects and functions relating to the houses 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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**Extra info for A1-Algebraic Topology over a Field**

**Sample text**

Let G be a strongly A1 -invariant sheaf. Let Y be a smooth k-scheme. Then there is a canonical bijection G−1 (Y ) ∼ = H 1 (T ∧ (Y+ ); G) which is a group isomorphism if G is abelian. Proof. We use the coﬁbration sequence Gm × Y ⊂ A1 × Y → T ∧ (Y+ ) to get a long exact sequence in the usual sense 0 → H 0 (A1 × Y ; G) → H 0 (Gm × Y ; G) ⇒ H 1 (T ∧ (Y+ ); G) → H 1 (A1 × Y ; G) → H 1 (Gm × Y ; G) → . . The pointed map H 1 (Y ; G) = H 1 (A1 × Y ; G) → H 1 (Gm × Y ; G) being split injective (use the evaluation at 1), we get an exact sequence 0 → G(Y ) ⊂ G(Gm × Y ) ⇒ H 1 (T ∧ (Y+ ); G) → ∗ As G−1 (Y ) is the kernel of ev1 : G(Gm × Y ) → G(Y ), this exact sequence implies that the action of G−1 (Y ) on the base point ∗ of H 1 (T ∧ (Y+ ); G) induces the claimed bijection G−1 (Y ) ∼ = H 1 (T ∧ (Y+ ); G).

X may not be in K1 (X; G), but, by Axiom (A2’), its boundary its trivial except on ﬁnitely many points zj of codimension 2 in X. Clearly these points are not in U (2) , thus we may, up to removing the closure of these zj ’s, ﬁnd an open subscheme Ω in X which contains u and the yi ’s and such that the element αΩ ∈ Πy∈Ω (1) Hy1 (X; G), induced by α, is in K1 (Ω ; G). 15, there exists an ´etale morphism U → A1V , with V the localization of a k-smooth of dimension d, such that if Y ⊂ U denotes the reduced closed subscheme whose generic points are the yi , the composition Y → U → A1V is still a closed immersion and such that the composition Y → U → A1V → V is a ﬁnite morphism.

Xd )) → S(κ(z)) doesn’t depend on the choice of (x1 , . . , xd ). 2)] the conditions on smoothness on the members of the associated ﬂag to the sequence (x1 , . . , xd ) is equivalent to the fact the family (x1 , . . , xd ) reduces to a basis of the κ(z)-vector space M/M2 . (xi ) also satisﬁes this assumption. For instance any permutation on the (x1 , . . , xd ) yields an other such sequence. By the case d = 2 which was observed above, we see that if we permute xi and xi+1 the compositions S(A) → S(κ(v)) are the same before or after permutation.

### A1-Algebraic Topology over a Field by Fabien Morel

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