According to 1.34 of Birational Geometry of Algebraic Varieties(J.Kollár, S.Mori), there are at least 4 ways(classical approach, cohomological approach, general intersection theory and topological approach) of defining intersection numbers.
In the book, the conditions are:
$X$: a proper scheme over a field
$Z$: a closed subscheme of dimension $d$
$L_1,\cdots ,L_d$: Cartier divisors on $X$
$(L_1\cdots L_d\cdot Z)$ denotes the intersection number of the divisors $L_1,\cdots ,L_d$ on $Z$.
Under these conditions, are the intersection numbers well-defined by the 4 ways? For example, the intersection number well-defined by (Classical approach) if $X$ is nonsingular?
If not, what conditions should be add for each?
