International Congress of Mathematicians

Algebraic Cobordism 61Definition 4.8. Let k be a field of characteristic zero and let X be an irreduciblefinite type fc-scheme with generic point i : x —¥ X. For an element n of 0*(X), definedegn £ ii*(k) to be the element mapping to i*r\ in Q*(k(xj) under the isomorphismsn*(k) =* L* =ë n*(k(x)) given by theorem 3.6(2).Theorem 4.9(generalized degree formula). Let k be a field of characteristiczero. Let X be an irreducible finite type k-scheme, and let n be in 0»(X). Let/o : B 0 —¥ X be a resolution of singularities of X, with B 0 quasi-projective over k.Then there are a, £ 0»(fc), and projective morphisms fi : B t —t X such that1. Each Bi is in Snij, fi : B t —t /(!?,) is birational and f(Bi) is a proper closedsubset of X (for i > 0).2.ÌÌ- (degn)[/ 0 : B 0 -+ X] = E[ = i ««[/« : B t "• x i «" °* W-Proof. It follows from the definitions of 0* that we haveÜ*(k(x))=limÜ*(U),uwhere the limit is over smooth dense open subschemes U of X, and ii*(k(xj) is thevalue at Specfc(ar) of the functor Q* on finite type fc(a:)-schemes. Thus, there is asmooth open subscheme j : U —¥ X of X such that j*n = (degn)fid^] in Q*(U).Since U x x B 0 = U, it follows that j*(n - (degn)[/ 0 ]) = 0 in Q*(U).Let W = X \U. From the localization sequencen.(w) A o,(x) A 04c/) -• 0,we find an element % £ Q»(W) with »*(%) = n — (degn)[/o], and noetherianinduction completes the proof.DRemark 4.10. Applying theorem 4.9 to the class of a projective morphism / :Y —¥ X, with X, F £ Sni/., we have the formular[f : Y -+ X] - (deg/)[idx] = $>[/« : A ^ X]in Q*(X). Also, if dim^X = dim/. Y, deg/ is the usual degree, i.e., the fieldextension degree [k(Y) : k(X)] if / is dominant, or zero if / is not.4.2. Complex cobordismFor a differentiable manifold M, one has the complex cobordism ring MU* (M).Given an embedding a : k —¥ C and an X e Sni/., we let X er (C) denote the complexmanifold associated to the smooth C-scheme IxjC. Sending X to MU 2 *(X cr (C))defines an oriented cohomology theory on Sni/.; by the universality of 0*, we havea natural homomorphismi=l&„ : 0*(X) -• MU 2 *(X a (