ALGEBRAIC COBORDISM

MARC LEVINE

0. Introduction

The classical construction of the Thom spectrum MU in the stable homotopy category SH is easy to lift to the ℙ^{1}-spectrum MGL in the Morel-Voevodsky 𝔸^{1} stable homotopy over a field k, SH(k). In this talk, we describe our joint work with Fabien Morel, constructing an algebro-geometric theory Ω^{∗} on smooth quasi-projective schemes over a field k over characteristic zero, which should give a description of the “geometric part” ⊕_{n}MGL^{2n,n} of the cohomology theory MGL^{∗,∗} represented by MGL.

Throughout this note, k will be a field, Sm_{k} denotes the category of smooth quasi-projective k-schemes (as a full subcategory of k-schemes) and GrRing is the category of graded, commutative rings with unit. For details of the results described here, we refer to [1] and [2].

1. Oriented cohomology

Abstracting the basic properties of complex cobordism leads to the following definition:

Definition 1. An oriented cohomology theory A on Sm_{k} consists of

D1. An additive functor A^{∗} : Sm_{k}^{op} → GrRing.

D2. For each projective morphism g : Y → X in Sm_{k} of relative dimension d := dim_{k} Y − dim_{k} X, an A^{∗}(X)-linear map

g_{∗} : A^{∗}(Y) → A^{∗−d}(X)

which satisfy A1. id_{∗} = id, (f ∘ g)_{∗} = f_{∗} ∘ g_{∗}.

The author gratefully acknowledges the support of the Humboldt Foundation through the Wolfgang Paul Program, and support of the NSF via grant DMS 0140445.

\xymatrix{W \ar[r]^{f′} \ar[d]_{g′} & Y \ar[d]^{g} \\ Z \ar[r]_{f} & X}

is a transverse cartesian square in Sm_{k} with g projective of pure relative dimension, then f^{∗} ∘ g_{∗} = g′_{∗} ∘ f^{′∗}.

A2. For L → X a line bundle over X ∈ Sm_{k} with 0-section s : X → L, set c_{1}(L) := s^{∗}(s_{∗}(1_{X})), where 1_{X} ∈ A^{0}(X) is the unit. Let E → X be a rank n+1 vector bundle, ℙ(E) → X the associated projective space bundle, O(1) → ℙ(E) the tautological quotient bundle and ξ := c_{1}(O(1)) ∈ A^{1}(ℙ(E)). Then A^{∗}(ℙ(E)) is a free A^{∗}(X)-module with basis 1, ξ, …, ξ^{n}.

A3. Let p : V → X be an affine space bundle. Then p^{∗} : A^{∗}(X) → A^{∗}(V) is an isomorphism.

Examples of oriented cohomology theories:

X ↦ CH^{∗}(X); the Chow ring,
X ↦ K_{0}^{alg}(X)[β, β^{−1}],
X ↦ MGL^{2∗,∗}(X) := ⊕_{n}MGL^{2n,n}(X).

In the second example, the β is a variable with degree −1, put in to keep track of the relative dimension under push-forward:

g_{∗}([E]β^{i}) := [Rg_{∗}E]β^{i+d}.

1.1. The formal group law. Let 𝕃 denote the Lazard ring, i.e., the coefficient ring of the universal formal group law F_{𝕃} ∈ 𝕃[[u, v]]. Let A be an oriented cohomology theory. Applying A2 twice gives the isomorphism

lim_{\overleftarrow{m,n}} A^{∗}(ℙ^{n} × ℙ^{m}) ≅ A^{∗}(k)[[u, v]],

with u going to c_{1}(O(1, 0)) and v going to c_{1}(O(0, 1)). Thus, there is a unique F_{A} ∈ A^{∗}(k)[[u, v]] with F (u, v) going over to c_{1}(O(1, 1)). It follows that

F (c_{1}(L), c_{1}(M)) = c_{1}(L ⊗ M)

for all pairs of line bundles L, M on some X ∈ Sm_{k}. It easily follows that F_{A} defines a formal group law over A^{∗}(k), hence there is a canonical homomorphism ϕ_{A} : 𝕃 → A^{∗}(k) classifying F_{A}.

Example 2. F_{CH}(u, v) = u + v and F_{K_{0}}(u, v) = u + v − βuv.

Theorem 3. Let k be a field of characteristic zero. Then there is a universal oriented cohomology theory Ω^{∗} on Sm_{k}. The map ϕ_{Ω} : 𝕃 → Ω^{∗}(k) is an isomorphism. If i : Z → X is a codimension d closed embedding in Sm_{k}, the sequence

Ω^{∗−d}(Z) ⟶^{i_{∗}} Ω^{∗}(X) ⟶^{j^{∗}} Ω^{∗}(X \ Z) → 0

is exact.

We now assume k has characteristic zero. Fix an irreducible X ∈ Sm_{k}. Let deg : Ω^{∗}(X) → Ω^{∗}(k) be the composition

Ω^{∗}(X) ⟶^{res} Ω^{∗}(k^{∗}(X)) ⟶^{ϕ_{Ω_{k(X)}}^{−1}} 𝕃 ⟶^{ϕ_{Ω_{k}}} Ω^{∗}(k).

For each proper integral subscheme Z ⊂ X, fix a resolution of singularities Z̃ → X, giving the map f_{Z} : Z̃ → X. We have the generalized degree formula

Theorem 4. Let α be in Ω^{∗}(X). There are proper integral subschemes Z_{i} ⊂ X and elements α_{i} ∈ Ω^{∗}(k), i = 1, …, n, such that

α − (deg α) · 1_{X} = ∑_{i=1}^{n} α_{i} · f_{Z∗}(1Z̃).

This result implies the so-called “degree formulas” of Rost et al. for characteristic classes.

1.2. Comparison theorems. For a formal group law (F, R), we have the classifying homomorphism 𝕃 → R, giving the oriented theory Ω_{F,R}^{∗} := Ω^{∗} ⊗_{𝕃} R. We write

Ω_{+}^{∗} := Ω_{u+v,ℤ}^{∗}; Ω_{×}^{∗} := Ω_{u+v−βuv,ℤ[β,β^{−1}]}^{∗}.

Theorem 5. The maps Ω_{+}^{∗} → CH^{∗}, Ω_{×}^{∗} → K_{0}^{alg}[β, β^{−1}] induced by the classifying maps Ω^{∗} → CH^{∗}, Ω^{∗} → K_{0}^{alg}[β, β^{−1}] are isomorphisms. The classifying map Ω^{∗} → MGL^{2∗,∗} is surjective with torsion kernel.

References

[1] Levine, M. and Morel, F., Algebraic cobordism I, preprint, Feb. 2002, http://www.math.uiuc.edu/K-theory/0547/index.html .

[2] Levine, M., Algebraic cobordism II, preprint, June 2002, http://www.math.uiuc.edu/K-theory/0577/index.html .

Department of Mathematics, Northeastern University, Boston, MA 02115, USA

E-mail address: marc@neu.edu