International Congress of Mathematicians

Algebraic Cobordism 65is universal for symmetric monoidal functors C —¥ C such that C admits an actionof R via natural transformations. In case R = Z, Z ®N C is the standard groupcompletion C~ 1 C. In general, if {e a | a £ A} is a Z-basis for R, thenR® N C = ]JC- 1 C,awith the A-action given by expressing xx : R —¥ R in terms of the basis {e a }.For each integer n > 0, let L» be the quotient of L» by the ideal of elementsof degree > n. We thus have the formal group (F L („),li" ).We form the category li") ®nÛ.(X), which we grade by total degree. For eachf :Y —¥ X projective, with F £ Sni/., and line bundles L, M, L\,...,L r on F, weadjoin an isomophism PL,M/»(F L(n) (ci(L), ci(M))(id Y ,Li,..., L r )) ^ /»(id ®ëi(L® M)(id Y ,Li,..., L r )).We impose the condition of naturality with respect to the maps in li") ®N I2„(F),in the evident sense; the Chern class transformations extend in the obvious manner.We impose the following commutativity condition: We have the evident isomorphismìL,M '• F L {n)(ci(L),ci(M)) —¥ F h („) (ci(M), ci(Lj) of natural transformations,as well as TL,M '• c\(L® M) —t ci(M ® L), the isomorphism induced by thesymmetry L®M = M®L. Then we impose the identity TL,M°PL,M = PM,L°tL,M-We impose a similar identity between the associativity of the formal group law andthe associativity of the tensor product of line bundles.We also adjoin a • TL,M for all a £ li"), with similar compatibilities as above,respecting the li") -action and sum. This forms the symmetric monoidal categoryQ(«) (X), which inherits a grading from O(X). We have the inverse system of gradedsymmetric monoidal categories:... -• Q(" +1) (X) -• Q(") (X)^ ... .Definition 5.14. Set 0^r(X) := n^BÜ^(X)) and iì m ,r( x )At present, we can only verify the following:Theorem 5.15. There is a natural isomorphism O TOj0 (X) = Q TO (X).: = u 0 ,^)-Proof. First note that iro(Z m (Xj) is a commutative monoid with group completionZ m (X). Next, the natural map 7ro(0» i (X)) + —t 0*(X) i s surjective with kernelgenerated by the classes generating TZ dtm (X). Given such an element ip := (/ :F —t X,n*Li,... ,n*L r , Mi,..., M s ), with n : Y —¥ Z smooth, and r > dim/. Z,suppose that the L, are very ample. We may then choose sections «j : Z —¥ L t withdivisors Di all intersecting transversely. Iterating the isomorphisms 7L ; ,Sì givesa path from tp to 0 in BÙ r _(X). Passing to Bum (X), the group law allows usto replace an arbitrary line bundle witha difference of very ample ones, so all theclasses of this form go to zero in Q^0(X). This shows that the natural mapü^Q(X)^(h^® h ü4X)) mn