By Fabien Morel
This textual content bargains with A1-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 A1-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 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
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