Editing Intersection number (section)

== Snapper–Kleiman definition of intersection number ==
There is an approach to intersection number, introduced by Snapper in 1959-60 and developed later by Cartier and Kleiman, that defines an intersection number as an Euler characteristic.

Let ''X'' be a scheme over a scheme ''S'', Pic(''X'') the [[Picard group]] of ''X'' and '''''G''''' the Grothendieck group of the category of [[coherent sheaf|coherent sheaves]] on ''X'' whose support is [[proper morphism|proper]] over an [[Artinian subscheme]] of ''S''.

For each ''L'' in Pic(''X''), define the endomorphism ''c''<sub>1</sub>(''L'') of '''''G''''' (called the [[first Chern class]] of ''L'') by
:<math>c_1(L)F= F - L^{-1} \otimes F.</math>
It is additive on '''''G''''' since tensoring with a line bundle is exact. One also has:
*<math>c_1(L_1)c_1(L_2) = c_1(L_1) + c_1(L_2) - c_1(L_1 \otimes L_2)</math>; in particular, <math>c_1(L_1)</math> and <math>c_1(L_2)</math> commute.
*<math>c_1(L)c_1(L^{-1}) =  c_1(L) + c_1(L^{-1}).</math>
*<math>\dim \operatorname{supp} c_1(L)F \le \dim \operatorname{supp} F - 1</math> (this is nontrivial and follows from a [[dévissage argument]].)
The intersection number
:<math>L_1 \cdot {\dots} \cdot L_r</math>
of line bundles ''L''<sub>''i''</sub>'s is then defined by:
:<math>L_1 \cdot {\dots} \cdot L_r \cdot F = \chi(c_1(L_1) \cdots c_1(L_r) F)</math>
where χ denotes the [[Euler characteristic]]. Alternatively, one has by induction:
:<math>L_1 \cdot {\dots} \cdot L_r \cdot F = \sum_0^r (-1)^i \chi(\wedge^i (\oplus_0^r L_j^{-1}) \otimes F).</math>
Each time ''F'' is fixed, <math>L_1 \cdot {\dots} \cdot L_r \cdot F</math> is a symmetric functional in ''L''<sub>''i''</sub>'s.

If ''L''<sub>''i''</sub> = ''O''<sub>''X''</sub>(''D''<sub>''i''</sub>) for some [[Cartier divisor]]s ''D''<sub>''i''</sub>'s, then we will write <math>D_1 \cdot {\dots } \cdot D_r</math> for the intersection number.

Let <math>f:X \to Y</math> be a morphism of ''S''-schemes, <math>L_i, 1 \le i \le m</math> line bundles on ''X'' and ''F'' in '''G''' with <math>m \ge \dim \operatorname{supp}F</math>. Then
:<math>f^*L_1 \cdots f^* L_m \cdot F = L_1 \cdots L_m \cdot f_* F</math>.<ref>{{harvnb|Kollár|1996|loc=Ch VI. Proposition 2.11}}</ref>