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By Levine M., Morel F.

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Example text

Suppose the associativity relation F (F (u, v), w) = F (u, F (v, w)) in Ω is satisfied modulo (un , v m+1 , wp+1 ). Write aijl ui v j wl ; F (F (u, v), w) = aijl ui v j wl . F (u, F (v, w)) = ijl ijl Let (a, b, c) be integers, and let OX (a, b, c) denote the line bundle whose sheaf of section is O(a, b, c) := p∗1 O(a) ⊗ p∗2 O(b) ⊗ p∗3 O(c) on X := Pn × Pm × Pp . Then, as ˜ ∗ (X), we have endomorphisms of Ω F (F (˜ c1 (OX (1, 0, 0)), c˜1 (OX (0, 1, 0)), c˜1 (OX (0, 0, 1)) = F (˜ c1 (OX (1, 1, 0)), c˜1 (OX (0, 0, 1)) = c˜1 (OX (1, 1, 1)) = F (˜ c1 (OX (1, 0, 0)), c˜1 (OX (0, 1, 1)) = F (˜ c1 (OX (1, 0, 0)), F (˜ c1 (OX (0, 1, 0), c˜1 (OX (0, 0, 1))).

This homomorphism is not in general an isomorphism. In fact it is not in general a surjection because there are line bundles which have no sections transverse to the zero section. 3. Definition of algebraic cobordism. 1 the Lazard ring L∗ , and the universal formal group law FL (u, v) = i,j ai,j ui v j . L∗ is graded and the degree of ai,j is i + j − 1. Thus FL (u, v) is absolutely homogeneous of degree 1. We also observe that ai,j = 0 when ij = 0 unless (i, j) = (1, 0) or (i, j) = (0, 1), in which case a1,0 = a0,1 = 1.

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 H∗ (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 BorelMoore S∗ -functor, denoted by A∗ ⊗R∗ S∗ , by the assignment X → A∗ (X) ⊗R∗ S∗ .

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Algebraic cobordism by Levine M., Morel F.


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