# Topological definition of intersection multiplicities of algebraic varieties

I posted this question in Stack Exchange and was recommended the appendix of Fulton's Young Tableaux. While I think it's good, it'd be nice to have some books which explain this subject in more detail.

Let $X$ be an algebraic variety over the field of complex numbers. In other words, $X$ is a reduced separable scheme of finite type over the field of complex numbers. Let $U$ and $V$ be irreducible subvarieities of X. Let $W$ be an irreducible component of $U ∩ V$. Suppose $W$ contains a closed non-singular point of $X$. In other words, the local ring of $X$ at $W$ is regular. Then dim $U$ + dim $V$ $≦$ dim $X$ + dim $W$ If the equality holds, one says that $U$ and $V$ intersect properly at $W$. In this case, a non negative integer called intersection multiplicity $i(U, V, W; X)$ is defined algebraically(see, for example, Serre's Local Algebra). I heard that this number can be defined by methods of algebraic topology. Is there any book which explains this in detail?

Edit The notion of intersection multiplicities is at the heart of algebraic geometry. For example, Weil's book "Foundations of algebraic geometry" is almost entirely devoted on this subject.

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Makoto, in the future, please provide links when you cross-post, so people can see what others have written (without having to search for it), so that people don't write a bunch of stuff only to find out someone else already wrote it. math.stackexchange.com/questions/136574/… – B R May 9 '12 at 2:21
I didn't know how to link. Now I know it and I will do so. – Makoto Kato May 9 '12 at 9:28