Generalizations

Abstract

Much of the intersection theory developed in this text is valid for more general schemes than algebraic schemes over a field. A convenient category, sufficient for applications envisaged at present, is the category of schemes X of finite type over a regular base scheme S. Using an appropriate definition of relative dimension, one has a notion of k-cycle on X, and a graded group A _* (X) of rational equivalence classes, satisfying the main functorial properties of Chaps. 1-6. The Riemann-Roch theorem also holds; in particular

$$ A\begin{array}{*{20}{c}} {}\\ * \end{array}\left( X \right) \otimes \mathbb{Q} \cong K\begin{array}{*{20}{c}} {} \\ ^\circ \end{array}\left( X \right) \otimes \mathbb{Q}$$

The main missing ingredient in such generality is an exterior product If S is one-dimensional, however, there is such a product. In particular,

$$ A\begin{array}{*{20}{c}} {} \\ k \end{array}\left( X \right) \otimes A\begin{array}{*{20}{c}} {} \\ 1 \end{array}\left( Y \right) \to A\begin{array}{*{20}{c}} {} \\ {k + 1} \end{array}\left( {X \times \begin{array}{*{20}{c}} {} \\ s \end{array}Y} \right)$$

if X is smooth over S, then A _* (X) has a natural ring structure.

When S = Spec (R),R a discrete valuation ring, and X is a scheme over S, with general fibre X° and special fibre, there are specialization maps

$$ \sigma :{A_k}({X^0}) \to {A_k}(\overline X ),$$

which are compatible with all our intersection operations. If X is smooth over S, σ is a homomorphism of rings.

For proper intersections on a regular scheme, Serre has defined intersection numbers using Tor. For smooth schemes over a field, these numbers agree with those in § 8.2. Indeed, even for improper intersections, a Riemann-Roch construction shows how to recover intersection classes from Tor, at least with rational coefficients.

Although higher K-theory is outside the scope of this work, the chapter concludes with a brief discussion of Bloch’s formula

$$ {A^p}X = {H^p}(X,{\mathcal{Y}_p})$$