By Levine M., Morel F.

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By induction on m it suffices to prove the case r = m; write E for E1 . We have i∗ [H(i∗ L1 , . . , i∗ Ln )] = i∗ H(˜ c1 (i∗ L1 ), . . , c˜1 (i∗ Ln ))(IdE ) = H(˜ c1 (L1 ), . . , c˜1 (Ln ))([E → W ]) = H(˜ c1 (L1 ), . . , c˜1 (Ln )) ◦ c˜1 (OW (E))(IdW ) = c˜1 (OW (E)) H(˜ c1 (L1 ), . . , c˜1 (Ln ))(IdW ) = c˜1 (OW (E))[H(L1 , . . 9. Let W be in Smk , let E be a strict normal crossing divisor on W . Then [E → W ] = [OW (E)]. In particular, let X be a finite type k-scheme, let f : W → X be a projective morphism, and let E, E be strict normal crossing divisors on W with OW (E) ∼ = OW (E ).

If f (X) = X d + i=0 bi X i , let F (X0 , X1 ) be the d−1 homogenized form of f , F (X0 , X1 ) = X1d + i=0 bi X1i X0d−i . Assume first that k is infinite. Choose distinct elements a1 , . . , ad ∈ k, and d let G = i=1 (X1 − ai X0 ). Since f is irreducible, no ai is a root of f . Let H = Y1 · F + Y0 · G, and let W ⊂ P1 × P1 = Projk (k[X0 , X1 ]) × Projk (k[Y0 , Y1 ]) be the closed subscheme defined by H. Using the Jacobian criterion, one checks that W is smooth over k. Via the projection on the second factor, W is finite over P1k , and defines a geometric cobordism over Spec k, with fibers Spec L over (Y0 : Y1 ) = (1 : 0) and d disjoint copies of Spec k over (Y0 : Y1 ) = (0 : 1).

The endomorphism c˜1 (L) of A∗ (X) associated to a line bundle L on X induces the endomorphism c˜1 (L): A∗ (X) → A∗+1 (X). Finally, an external product on A∗ induces an external product A∗ (X) ⊗ A∗ (Y ) → A∗ (X × Y ). The assignment X → A∗ (X) will be called the oriented cohomological functor on Smk associated to A∗ . One can rewrite all the axioms for an oriented Borel-Moore functor A∗ on Smk and in terms of A∗ . Clearly A∗ and A∗ are thus determined by each other and the category of oriented Borel-Moore functors on Smk is equivalent to that of oriented cohomological functors on Smk .

### Algebraic cobordism by Levine M., Morel F.

by William

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