Intersecting with pseudo-divisors

Suppose we have a pseudo-divisor on a scheme . Then for each subvariety of , we can define the intersection by pulling back ( restricting ) the corresponding line bundle with section of on to , which will in turn determine a divisor supported in . In case the support of contains , then we can only define that intersection divisor up to rational equivalence ( at level of homology class ). Otherwise, it is uniquely determined at the level of cycle. Note that we want to define a class in , and it is different from defining a class in . A cycle can in fact represent the 0 class in the latter but not in the former. Plus, later we shall need commutativity of intersection, thus we want to define a class in so that no matter in what order we intersect, the classes always be defined in the same space.

Now we can extend the definition by linearity to all the cycles. There is a special case where we can definite intersections at cycle level : when the line bundle of is trivial in a neighborhood of . In that case, in the only situation where the previous failed to determine a cycle i.e. when the support of a subvariety is contained , we can just define the intersection to be the 0 cycle.

As we should expect, if the line bundle that comes with is trivial, then intersecting with always yield 0 on the groups . In fact, if is any variety, then the restriction of on is trivial, hence it defines the 0 homology class. Later, we will know that intersecting with divisors yield a homomorphism of Chow groups ( i.e., intersecting with 0 homology class yield a homology class, as a corollary of commutativity of intersection ).

There are two properties of intersection with divisors which are useful but somewhat not quite trivial to prove :
1- Projection formula
2- Flat pullback

In some situation, when we are sure that Cartier divisors can be pulled backed ( i.e., the image scheme is not contained in the support of the divisor, or when the map is flat ( pull back of nonzero divisors are nonzero divisors , hence we can pull back the ration functions that define the Cartier divisor under flat map ). In general, since the support of a Cartier divisor is contained in that of the pseudo-divisor it represents, the equality involving two intersection classes (with Cartier divisor ) is better, since the classes are defined in smaller spaces.