Today I finished the first complete version of a chapter on intersection theory. As usual comments and suggestion are very welcome.
The chapter uses Serre’s Tor formula and moving lemmas to define an intersection product on the Chow groups of nonsingular projective varieties over an algebraically closed ground field and that is all it does. You can read the introduction for a little bit more information.
There are some improvements that can be made to this chapter. The first is that some of the material on Serre’s Tor formula belongs properly in one of the chapters on commutative algebra. Of course, there is a lot more one can say about regular local rings and the Tor formula, leading up to recent work on homological conjectures in commutative algebra. Also, some of the arguments in the moving lemmas use geometric arguments on varieties over algebraically closed fields and we need to write more of the API to easily translate these into scheme theoretic language. Finally, the chapter is missing examples and more references to the literature.
What often happens with new chapters is that a few years down the road, we take a second look and make substantial improvements.
One aspect of the material in the new chapter is that it was not as straightforward to write as the material on constructible sheaves which was like butter. The conclusion must therefore be that intersection theory is not like butter!