Generalizations

Summary

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

$$A_* \left( X \right) \otimes \mathbb{Q} \cong K_ \circ \left( X \right) \otimes \mathbb{Q}$$

.