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By Levine M., Morel F.

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Projective spaces and Milnor’s hypersurfaces. Let n > 0 and m > 0 be integers. Recall that γn denotes the line bundle on Pn whose sheaf of sections is O(1). Write γn,m for the line bundle p∗1 (γn ) ⊗ p∗2 (γm ) on Pn × Pm . We let i : Hn,m → Pn × Pm denote the smooth closed subscheme defined by the vanishing of a section of γn,m transverse to the zero-section. 4. The smooth projective k-schemes Hn,m are known as the Milnor hypersurfaces [2]. Taking m ≤ n, it is easy to see that, choosing suitable homogeneous coordinates X0 , .

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. (FGL). Let FA ∈ A∗ (k)[[u, v]] be the image by the homomorphism L∗ → A∗ (k) (giving the L∗ -structure) of the power series FL . Then for any smooth k-scheme Y and any pair (L, M ) of line bundles on Y , one has FA (˜ c1 (L), c˜1 (M ))(1Y ) = c˜1 (L ⊗ M )(1Y ) ∈ A∗ (Y ). 2. The axioms (Dim) and (Sect) make sense for any oriented Borel-Moore functor with product, and we will sometimes use them in this less restrictive context.

For X in V, let A¯∗ (X) be the sub-A∗ (k)-module of A∗ (X) generated by ϑA,1 (Z∗ (X)). It is easy to see that this defines an oriented Borel-Moore R∗ -functor on V with product, A¯∗ . We call the element ϑA,1 ([f : Y → X, L1 , . . , Lr ]) := f∗ (˜ c1 (L1 ) ◦ . . c˜1 (Lr )(1Y )) a standard cycle on X, and write this element as [f : Y → X, L1 , . . , Lr ]A . For Y ∈ V with structure morphism f : Y → Spec k, we set 1Y := f ∗ (1), and write [Y ]A for f∗ (1Y ) = [f : Y → Spec k]A . Note that 1Y = [IdY ]A .

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