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 I crossposted this on MathOverflow.