## 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
The main missing ingredient in such generality is an exterior product If
if

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

*S*is one-dimensional, however, there is such a product. In particular,$$
A\begin{array}{*{20}{c}}
{} \\
k
\end{array}\left( X \right) \otimes A\begin{array}{*{20}{c}}
{} \\
1
\end{array}\left( Y \right) \to A\begin{array}{*{20}{c}}
{} \\
{k + 1}
\end{array}\left( {X \times \begin{array}{*{20}{c}}
{} \\
s
\end{array}Y} \right)$$

*X*is smooth over*S*, then*A*_{ * }*(X)*has a natural ring structure.When
which are compatible with all our intersection operations. If

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

*X*is smooth over*S*,*is a homomorphism of rings.***σ**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

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

### Keywords

Filtration Covariance Neron## Preview

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## Copyright information

© Springer Science+Business Media New York 1998