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By Levine M., Morel F.

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Nr (u1 , . . , ur ) ≡ 0 mod L≥1 if |J| > 1. If J = (j1 , . . ,nr (u1 , . . , ur ) ≡ ni mod L≥1 . This yields the desired formula. 8. Let W be in Smk and equi-dimensional, and let E = i=1 Ei be a strict normal crossing divisor on W . ,m} → W be the inclusion. Let H(u1 , . . , um ) be a power series with Ω∗ (k) coefficients, and let L1 , . . , Ln be line bundles on W . Then i∗ [H(i∗ L1 , . . , i∗ Ln )] = c˜1 (OW (E1 )) ◦ . . ◦ c˜1 (OW (Em ))[H(L1 , . . , Ln )]. in Ω∗ (W ). 44 III. FUNDAMENTAL PROPERTIES OF ALGEBRAIC COBORDISM Proof.

J Let iJ : E J → W be the inclusion. 8, we have iJ∗ [FJ (LJ1 , . . , LJm )] = [FJ (L1 , . . , Lm )]. Thus iJ∗ [FJ (˜ c1 (LJ1 ), . . , c˜1 (LJm )] [E → W ] = J = [FJ (L1 , . . , Lm )] J = [F (L1 , . . , Lm )] = F c˜1 (L1 ), . . , c˜1 (Lm ) (IdW ) 1 m = c˜1 L⊗n ⊗ . . ⊗ L⊗n (IdW ) (cf. 2) m 1 = [OW (E)]. 2. Localization Let X be a finite type k-scheme i : Z → X a closed subscheme, and j : U → X i j∗ ∗ the open complement. 2. LOCALIZATION 45 is a complex. 7), at least under the assumption that the base field k admits resolution of singularities.

Proof. 11(A7), with f = IdX , g : P1 → Spec k the structure morphism, we have p∗1 (x) = x × 1P1 By (A8), we have c˜1 (p∗2 O(1))(p∗1 (x)) = x × c˜1 (O(1))(1P1 ). k The axiom (Sect) implies c˜1 (O(1))(1P1 ) = iSpec (1); by (A6), we have α∗ k x × iSpec (1) = iX α∗ α∗ (x) completing the proof. 8. Suppose that k has characteristic zero. Let f : Y → X be a projective morphism with X finite type over k and Y smooth quasi-projective over k. Choose a line bundle L on X generated by sections s1 , . . , sn .

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