An Intersection Formula. Variations and Generalizations

Abstract

The first point of this chapter is to develop a commutative diagram similar to that of the Riemann-Roch theorems, and called the Intersection Formula for the K-functor. In particular, this will show how the product in the ring K(X) relates to the geometric intersection of subschemes of X. From this intersection formula for K we deduce a corresponding formula for Gr K, which is analogous to the “excess intersection formula” of [FM], cf. [F 2], Theorem 6.3. Special cases of the intersection formula are contained in [SGA 6] and [Man], but the general version given here for K-theory seems to be new. Our proof eliminates the use of Tor, and gives another striking illustration of the deformation formalism of Chapter II.