Algebraic cobordism of bundles on varieties

Let k be a field of characteristic 0. Let Sch k be the category of separated schemes of finite type over k, and let Sm k be the full subcategory of smooth quasi-projective k-schemes. A geometric presentation of algebraic cobordism in characteristic 0 via double point relations is given in [7]. 0.2. Double point degenerations. Let Y ∈ Sm k be of pure dimension. A morphism π : Y → P 1 is a double point degeneration over 0 ∈ P 1 if π -1 (0) can be written as

where A and B are smooth codimension one closed subschemes of Y , intersecting transversely. The intersection

is the double point locus of π over 0 ∈ P 1 . We do not require A, B, or D to be connected. Moreover, A, B, and D are allowed to be empty.

Let 0.3. M(X) + . For X ∈ Sch k , let M(X) denote the set of isomorphism classes over X of projective morphisms

with Y ∈ Sm k . The set M(X) is a monoid under disjoint union of domains and is graded by the dimension of Y over k. Let M * (X) + denote the graded group completion of M(X).

Alternatively, M n (X) + is the free abelian group generated by morphisms (0.2) where Y is irreducible and of dimension n over k. Let [f : Y → X] ∈ M * (X) + denote the element determined by the morphism. 0.4. Double point relations. Let X ∈ Sch k , and let p 1 and p 2 denote the projections to the first and second factors of X × P 1 respectively.

Let Y ∈ Sm k be of pure dimension. Let g : Y → X × P 1 be a projective morphism for which the composition

is a double point degeneration over 0 ∈ P 1 . Let

be obtained from the fiber π -1 (0) and the morphism p 1 • g.

Definition 1. Let ζ ∈ P 1 (k) be a regular value of π. We call the map g a double point cobordism with degenerate fiber over 0 and smooth fiber over ζ. The associated double point relation over X is

where

The relation (0.4) depends not only on the morphism g and the point ζ, but also on the choice of decomposition of the fiber

We view (0.4) as an analog of the classical relation of rational equivalence of algebraic cycles.

Let R * (X) ⊂ M * (X) + be the subgroup generated by all double point relations over X. Since (0.4) is a homogeneous element of M * (X) + , R * (X) is a graded subgroup of M * (X) + .

A central result of [7] is the isomorphism

which provides a geometric presentation of algebraic cobordism. Since resolution of singularities and Bertini’s results are used, the isomorphism is established only when k has characteristic 0. 0.5. Over a point. We write Ω * (k) and ω * (k) for Ω * (Spec (k)) and ω * (Spec (k)) respectively. Let L * be the Lazard ring [4]. The canonical map

classifying the formal group law for Ω * is proven to be an isomorphism in [6,Theorem 4.3.7]. By Quillen’s result for complex cobordism (in topology),

and the well-known generators of MU * (pt) Q [11, Chapter VII], we see Ω * (k) ⊗ Z Q is generated as a Q-algebra by the classes of projective spaces. The following result is then a consequence of (0.6),

where the sum is over all partitions λ. The partition λ = ∅ corresponds to [P 0 ] in grade 0.

with Y ∈ Sm k of dimension n, f projective, and E a rank r vector bundle on Y . The set M n,r (X) is a monoid under disjoint union of domains. Let M n,r (X) + denote the group completion of M n,r (X). Double point relations are easily defined in the setting of pairs following [7, Section 13]. Let Y ∈ Sm k be of pure dimension n + 1. Let g : Y → X × P 1 be a projective morphism for which the composition

be obtained from the fiber π -1 (0) and the morphism p 1 • g. Here, E A and E B denote the restrictions of E to A and B respectively. The restriction E P(π) is defined by pull-back from Y via

where

For X ∈ Sch k , let R n,r (X) ⊂ M n,r (X) + be the subgroup generated by all double point relations. Double point cobordism theory for bundles on varieties is defined by

is always a ω * (k)-module via product (and pull-back). If X ∈ Sm k , then ω * ,r (X) is also a module over the ring ω * (X). 0.7. Basis. The main result of the paper is the construction of a basis of ω n,r (k) analogous to the fundamental presentation (0.7). Our basis is indexed by pairs of partitions. A partition pair of size n and type r is a pair (λ, µ) where (i) λ is a partition of n, (ii) µ is a sub-partition of λ of length ℓ(µ) ≤ r. The sub-partition condition means µ is obtained by deleting parts of λ. The partition µ may be empty and may equal λ if ℓ(λ) ≤ r. Subpartitions µ, µ ′ ⊂ λ are equivalent if they differ by permuting equal parts of λ.

Let P n,r be the set of all partition pairs of size n and type r. For example,

To each (λ, µ) ∈ P n,r , we associate an element

by the following construction. Let P λ = P λ 1 × . . . × P λ ℓ(λ)

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