International Congress of Mathematicians

58 M. Levine(PB) Let £ be a rank r + 1 locally free coherent sheaf on X, with projective bundleq : W(£) —t X and tautological quotient invertible sheaf q*£ —¥ 0(1). Let£ = ci(0(l)). Then A*(¥(£)) is a free .4*(X)-module with basis 1,£,.. • ,f •Finally, A* satisfies a homotopy property: if p : V —¥ X is an affine-space bundle(i.e., a torsor for a vector bundle over X), then p* : A*(X) —t A*(V) is anisomorphism.Examples 1.1. (1) The theories CH* and H?^(—, Z/n(*)) on Sm/. (also with Z/(*)or Qi(*) coefficients).(2) The theory K 0 [ß, ß^1]on Smj. Here ß is an indeterminant of degree —1, usedto keep track of the relative dimension when taking projective push-forward.Remarks 1.2. (1) In [8], we consider a more general (dual) notion, that of anoriented Borel-Moore homology theory .4». Roughly, this is a functor from a fullsubcategory of Seh/, to graded abelian groups, covariant for projective maps, andcontravariant (with a shift in the grading) for local complete intersection morphisms.In addition, one has external products, and a degree -1 Chern class endomorphismci(L) : A*(X) —¥ -A»_i(X) for each line bundle L on X, defined by ci(L)(n) =S*(S»(JJ)), s : X —t L the zero-section. As for an oriented cohomology theory,there are various compatibilities of push-forward and pull-back, and .4» satisfies aprojective bundle formula and a homotopy property.This allows for a more general category of definition for .4», e.g., the categorySeh/.. As we shall see, the setting of Borel-Moore homology is often more naturalthan cohomology. On Smj, the two notions are equivalent: to pass from Borel-Moore homology to cohomology, one re-grades by setting A n (X) := .4„_dim fe x(X)and uses the l.c.i. pull-back for .4» to give the contravariant functoriality of A*,noting that every morphism of smooth fc-schemes is an l.c.i. morphism. We willstate most of our results for cohomology theories on Sm/., but they extend to thesetting of Borel-Moore homology on Seh/, (see [8] for details).(2) Our notion of oriented cohomology is related to that of Panin [10], but is notthe same.2. The formal group lawLet .4» be an oriented cohomology theory on Smj. As noticed by Quillen [11],a double application of the projective bundle formula (PB) yields the isomorphismof ringsA*(k)[[u,v]] =* lim.4*(P" x P ro ),the isomorphism sending u to ci(p\0(t)) and v to ci(plö(lj). The class ofci(p*0(t) ®P2Ö(1)) thus gives a power series FA(U,V) £ A*(k)[[u,v]] withci(plO(l) ®p* 2 0(l)) = F A (ci(plO(l)),ci(plO(l))).