Download Algebraic Cobordism by Marc Levine PDF

By Marc Levine

Following Quillen's method of complicated cobordism, the authors introduce the proposal of orientated cohomology conception at the classification of delicate types over a set box. They end up the life of a common such concept (in attribute zero) referred to as Algebraic Cobordism. strangely, this idea satisfies the analogues of Quillen's theorems: the cobordism of the bottom box is the Lazard ring and the cobordism of a gentle kind is generated over the Lazard ring through the weather of optimistic levels. this suggests specifically the generalized measure formulation conjectured via Rost. The e-book additionally includes a few examples of computations and functions.

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Lr )]A := F (˜ c1 (L1 ), . . , c˜1 (Lr ))(1X ) ∈ Ad−n (X) Similarly, if f : Y → X is in M(X), with X in V, we write [f : Y → X]A for f∗ (1Y ). Using these notations, the equation in axiom (Sect) can be written as [L]A = [Z → Y ]A and that in axiom (FGL) as [FA (L, M )]A = [L ⊗ M ]A . We sometimes omit the subscript A if the meaning is clear from the context. 1. Let X be a smooth and irreducible k-scheme. We let RDim (X) ⊂ Z∗ (X) ∗ denote the subset consiting of all elements of the form [Y → X, L1 , .

Then the homomorphism Φ0 (k) : L0 → Ω0 (k) is an isomorphism and Ω0 (k) is the free abelian group on 1 = [Spec k]. Moreover, given any smooth variety X = Spec A of dimension zero over k, then [X] = dimk (A) · 1 in Ω0 (k). Proof. 4. 11 which implies that Ω0 (k) is generated by classes [Spec L] where L is a (separable) finite field extension of k. Thus Z = L0 → Ω0 (k) is surjective. 13); the map Ω0 (k) → CH0 (k) = Z is a left inverse to L0 → Ω0 (k), which is thus injective. 11): Ω ∗ → Ω∗ . 8 to choose for each (i, j) with i ≤ j an element aij ∈ M+ i+j−1 (k) lifting [aij ] ∈ Ωi+j−1 (k); for j < i we set aji = aij .

Ln , L and M are line bundles on Y. 12. One could imagine that we could have defined algebraic coborGL dism directly as the quotient of L∗ ⊗ Z ∗ by L∗ RSect . However, for RF ∗ ∗ F GL the elements in R∗ to be defined, we need to have some vanishing of products of the c˜1 , which is guaranteed by axiom (Dim). This forces us to start by first. 14. The external product on L∗ ⊗ Ω ∗ thus descends to Ω∗ , making algebraic cobordism an oriented Borel-Moore L∗ -functor with product on Schk . Let’s denote the composite homomorphism L∗ → L∗ ⊗ Ω ∗ (k) → Ω∗ (k) by Φ(k) : L∗ → Ω∗ (k) a → [a].

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