Algebraic cobordism by Levine M., Morel F.

By Levine M., Morel F.

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Ur ) with coefficients in the graded ring A∗ (k). Suppose that F is absolutely homogeneous of degree n. Given line bundles (L1 , . . , Lr ) on X ∈ V, the operations c˜1 (L1 ), . , c˜1 (Lr ) are locally nilpotent on A¯∗ (X) (by axiom (Dim)) and commute with each other. 1) F (˜ c1 (L1 ), . . , c˜1 (Lr )) : A¯∗ (X) → A¯∗−n (X) ⊂ A∗−n (X). If X is a smooth equi-dimensional k-scheme of dimension d, we have the class 1X ∈ A¯d (X) and we set [F (L1 , . . , Lr )] := 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] for f∗ (1Y ).

Lr ), M1 , . . , Ms ], where Z is a smooth quasi-projective irreducible k-scheme, (L1 , . . , Lr ) are line bundles on Z, π : Y → Z is a smooth quasi-projective equi-dimensional morphism and r > dimk (Z). 13 that the external product on Z∗ descends to give Z ∗ an external product which makes Z ∗ an oriented BorelMoore functor with product on Schk . 3. Of course, for any X ∈ Schk one has Z n (X) = 0 if n < 0 by construction. 4. Let X be a k-scheme of finite type and L a line bundle on X.

1. An oriented Borel-Moore L∗ -functor A∗ on V is said to be of geometric type if the following three axioms holds: (Dim). For any smooth k-scheme Y and any family (L1 , . . , Ln ) of line bundles on Y with n > dimk (Y ), one has c˜1 (L1 ) ◦ · · · ◦ c˜1 (Ln )(1Y ) = 0 ∈ A∗ (Y ). (Sect). For any smooth k-scheme Y , any line bundle L on Y , any section s of L which is transverse to the zero section of L, one has c˜1 (L)(1Y ) = i∗ (1Z ), where i : Z → Y is the closed immersion of the zero-subscheme of s.

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