Algebraic Cobordism by Marc Levine

By Marc Levine

Following Quillen's method of complicated cobordism, the authors introduce the concept of orientated cohomology conception at the classification of soft types over a hard and fast box. They turn out the life of a common such idea (in attribute zero) known as Algebraic Cobordism. strangely, this thought satisfies the analogues of Quillen's theorems: the cobordism of the bottom box is the Lazard ring and the cobordism of a soft kind is generated over the Lazard ring by means of the weather of optimistic levels. this suggests specifically the generalized measure formulation conjectured via Rost. The ebook additionally includes a few examples of computations and functions.

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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 . 13. For Y ∈ V with structure morphism f : Y → Spec k, we write [Y ]A for [f : Y → Spec k]A . Note that 1Y = [IdY ]A and that [Y ]A = f∗ (1Y ). It follows from the additivity of A∗ that 1Y Y = 1Y + 1Y in A∗ (Y Y ) = A∗ (Y ) ⊕ A∗ (Y ), and [Y Y ]A = [Y ]A + [Y ]A .

Moreover, one easily checks that it is in fact the universal one: given an oriented Borel-Moore functor on V with product A∗ , there exists one and only one morphism of oriented Borel-Moore functors with product ϑA : Z∗ → A∗ . 9 is compatible with the external products. 12. 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 A∗ (X) for each X ∈ V, by using Φ and the external product.

6. Let OX be the trivial line bundle on X ∈ Schk . Then the homomorphism c˜1 (OX ) : A¯∗ (X) → A¯∗−1 (X) is the zero homomorphism. In particular, if X is in Smk , then c˜1 (OX )(1X ) = 0. Proof. For any standard cycle [Y → X, L1 , . . , Lr ]A on X, one has c˜1 (OX )([Y → X, L1 , . . , Lr ]A ) = [Y → X, L1 , . . , Lr , OY ]A = [∅ → X, L1 , . . , Lr ]A = 0, by (Sect), because the constant unit section of OX never vanishes. 7. Take X ∈ V, let α be a k-point of P1 and take x ∈ A∗ (X). 1 Let iX α : X → X × P be the section with constant value α.

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