A1-Algebraic Topology over a Field by Fabien Morel

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.

Show description

Read or Download A1-Algebraic Topology over a Field PDF

Similar abstract books

Geometric Topology: Localization, Periodicity and Galois Symmetry: The 1970 MIT Notes (K-Monographs in Mathematics)

The seminal `MIT notes' of Dennis Sullivan have been issued in June 1970 and have been extensively circulated on the time, yet in simple terms privately. The notes had a huge effect at the improvement of either algebraic and geometric topology, pioneering the localization and of entirety of areas in homotopy concept, together with P-local, profinite and rational homotopy conception, the Galois motion on gentle manifold buildings in profinite homotopy concept, and the K-theory orientation of PL manifolds and bundles.

Towards the Mathematics of Quantum Field Theory, 1st Edition

This bold and unique e-book units out to introduce to mathematicians (even together with graduate scholars ) the mathematical equipment of theoretical and experimental quantum box idea, with an emphasis on coordinate-free displays of the mathematical items in use. This in flip promotes the interplay among mathematicians and physicists by means of providing a typical and versatile language for the nice of either groups, even though mathematicians are the first aim.

Extra info for A1-Algebraic Topology over a Field

Sample text

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 [52]) we see that for any henselian k-smooth local ring A, with infinite residue field, the morphism G(A) → G (A) is an isomorphism.

Download PDF sample

Rated 4.73 of 5 – based on 28 votes