By Marc Levine
Following Quillen's method of complicated cobordism, the authors introduce the suggestion of orientated cohomology concept at the type of tender forms over a hard and fast box. They turn out the life of a common such concept (in attribute zero) known as Algebraic Cobordism. unusually, this concept satisfies the analogues of Quillen's theorems: the cobordism of the bottom box is the Lazard ring and the cobordism of a delicate type is generated over the Lazard ring by means of the weather of confident levels. this means specifically the generalized measure formulation conjectured by way of Rost. The publication additionally includes a few examples of computations and purposes.
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Extra resources for Algebraic Cobordism
Let R∗ be a commutative graded ring with unit. An oriented Borel-Moore R∗ -functor on V, A∗ , is an oriented Borel-Moore functor on V with product, together with a graded ring homomorphism Φ : R∗ → A∗ (k). For such a functor, one gets the structure of an R∗ -module on A∗ (X) for each X ∈ V, by using Φ and the external product. All the operations of projective push-forward, smooth pull-back, and c˜1 of line bundles are R∗ linear. For instance, given an oriented Borel-Moore R∗ -functor A∗ and a homomorphism of commutative graded rings R∗ → S∗ , one can construct an oriented Borel-Moore S∗ -functor, denoted by A∗ ⊗R∗ S∗ , by the assignment X → A∗ (X) ⊗R∗ S∗ .
Take X ∈ V, let α be a k-point of P1 and take x ∈ A∗ (X). 1 Let iX α : X → X × P be the section with constant value α. Then iX ˜1 (p∗2 O(1))(p∗1 (x)). α∗ (x) = c In particular, if α and β are any two k-points of P1 , then X iX α∗ (x) = iβ∗ (x). Proof. 10(A7), with f = IdX , g : P1 → Spec k the structure morphism, we have p∗1 (x) = x × 1P1 By (A8), we have 32 2 The deﬁnition of algebraic cobordism c˜1 (p∗2 O(1))(p∗1 (x)) = x × c˜1 (O(1))(1P1 ). k (1); by (A6), we have The axiom (Sect) implies c˜1 (O(1))(1P1 ) = iSpec α∗ k x × iSpec (1) = iX α∗ α∗ (x) completing the proof.
Um ) = J, ||J||≤1 J where each monomial hJ,J u , J = (j1 , . . , jm ), occurring in HJ has js = 0 if js = 0. K Proof. Write H(u1 , . . , um ) = K hK · u , and let J = (j1 , . . , jm ) with K−J , where the sum is over all K = (k1 , . . , km ) ||J|| ≤ 1. Let HJ = K hK u such that ki ≥ ji for all i, and ki = 0 if ji = 0. Uniqueness is easy and left to the reader. ,nm (u1 , . . , um ) ∈ Ω∗ (k)[[u1 , . . ,nm (u1 , . . , um ). ,nm (u1 , . . 3. Assume m = 2 and n1 = n2 = 1. Then F 1,1 (u, v) = F (u, v) = ai,j ui v j i,j ai,j ui v j =u+v+ i≥1,j≥1 ai,j ui−1 v j−1 , = u + v + uv i≥1,j≥1 so 1,1 F(0,0) = 0, 1,1 F(1,0) (u, v) = 1, 1,1 F(0,1) (u, v) = 1 and 1,1 F(1,1) (u, v) = ai,j ui−1 v j−1 .
Algebraic Cobordism by Marc Levine