Bivariant Intersection Theory

Abstract

Our basic intersection construction has assigned to a regular imbedding (or 1.c.i. morphism) f: X → Y of codimension d a collection of homomorphisms

$$ {{f}^{!}}:{{A}_{k}}Y' \to {{A}_{{k - d}}}X' $$

for all Y’ → Y, X’ = X x _y Y’, all k. In this chapter we formalize the study of such operations. For any morphism f: X → Y, a bivariant class c in ${A^p}\left( {X\xrightarrow{f}Y} \right)$ is a collection of homomorphisms from A _k Y’ to A _k-P X’, for all Y’ → Y, all k, compatible with push-forward, pull-back, and intersection products.

The group A ^-k (X → pt.) is canonically isomorphic to A _k (X). The other extreme ${A^k}\left( {X\xrightarrow{{id}}X} \right)$ is defined to be the cohomology group A _K X. The bivariants groups have products

$$ {A^p}\left( {X\xrightarrow{f}Y} \right) \otimes {A^q}\left( {Y\xrightarrow{g}Z} \right)\xrightarrow{ \cdot }{A^{p + q}}\left( {X\xrightarrow{{gf}}Z} \right) $$

which specialize to give a ring structure on A ^* X, and a cap product action of A ^* X on A ^* X.If X is non-singular ${A^ * }X \cong {A_ * }X$, There are also a proper push-forward and a pull-back operation for bivariant groups, generalizing the push-forward on A _* and defining a pull-back on A _* There are compatibilities among these three operations which allow one to manipulate bivariant classes symbolically with a freedom one is accustomed to with homology and cohomology in topology.

Many constructions of previous chapters actually produce classes in appropriate bivariant groups. For example, Chern classes of vector bundles on X live in A _* XFlat and l.c.i. morphisms f:X→Y determine canonical elements in ${A^ * }\left( {X\xrightarrow{f}Y} \right)$, which are denoted [f]. An element c of A ^P (X→Y) determines generalized Gysin homomorphisms ${A_k}Y\xrightarrow{{{c^ * }}}{A_{k - p}}$ and ${A^k}X\xrightarrow{{{c_ * }}}{A^{k + p}}Y$ (for the latter f is assumed to be proper). Intersection formulas such as the excess and residual intersection formulas achieve their sharpest formulation in the bivariant language.

There is a useful criterion which implies that an operation which produces rational equivalence classes on X’ from subvarieties of Y’ (for all Y’→Y), passes to rational equivalence and defines a bivariant class (Theorem 17.1). This will be used in the next chapter to deduce the important properties of local Chem classes.