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Generalizations

  • William Fulton

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

Keywords

Finite Type Discrete Valuation Ring Dedekind Domain Exterior Product Proper Intersection 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 1998

Authors and Affiliations

  • William Fulton
    • 1
  1. 1.Department of MathematicsUniversity of MichiganAnn ArborUSA

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