International Congress of Mathematicians

0*(fc). Using remark 3.7, this gives the identityAlgebraic Cobordism 63r[Y]^(degf)[X] = Y,m{AiXB i ]in Q*(X), for smooth, projective fc-schemes Ay By and integers rrij, where eachBi admits a projective morphism fi : B t —t X which is birational to its image andnot dominant. Since Sd vanishes on non-trivial products, the only relevant part ofthe sum involves those Bj of dimension zero; such a Bj is identified with the closedpoint bj := fj(Bj) of X. Applying Sd, we haves d (Y) -deg(f)s d (X) = J2 m 3 s d( A j)àeg k (bj).Since Sd(Aj) = prij for suitable integers rij, we haves d (Y) - deg(f)sd(X) = pdeg(^m i n i 6 i ).Taking n = V. rnyrijbj proves theorem 4.11.The proof of theorem 4.12 is similar: Start with the decomposition of [/ :F —t X] — (deg/)[idx] given by remark 4.10. One then decomposes the mapsBi —¥ X = X 0 further by pushing forward to X\ and using theorem 4.9. Iteratingdown the tower gives the identity in Q»(fc)[F] - (deg f)[X] = J2m i [B t 0x...x B«] ;ithe condition (2) implies that, if d\ dim/. B l - for all j = 0,..., r, then p\rrij. Applyingtd, r and using the property (4.1)(4) yields the formula.Di=ljj5. Comparison resultsSuppose we have a formal group (f,R),Q : L* —t Q*(k). Theuniversal property of 0* gives the analogous universal property for QT. R,.In particular, let 0^j_ be the theory with (f(u,v),R) = (« + w,Z),and let 0^ bethe theory with (/(«, v),R) = (u + v — ßuv, Z[ß, ß^1]).We thus have the canonicalnatural transformations of oriented theories on Sni/.O; -• CH*; Q* x -• K^ßJ- 1 ]. (5.2)Theorem 5.13. Letk be a field of characteristic zero. The natural transformations(5.2) are isomorphisms, i.e., CH* is the universal ordinary oriented cohomologytheory and K^ßjß^1]is the universal multiplicative and periodic theory.