Pushforward and flat pull-back of intersections « Dung Hoang Nguyen’s Weblog

Pushforward and flat pull-back of intersections

By dunghoangnguyen

These are two quite useful properties of intersecting with divisors.
1- Projection formula :
Let $\pi : X' \to X$ be proper. Then if $D$ is a divisor on $X$ , we want to pull back $D$ , intersect it with a cycle in $X'$ , and then push it back. The projection formula gives a shortcut to that : we can just pushforward the cycle, and then intersect it with $D$ .
$\pi^0_*(\pi^*D \cap \alpha ) = \pi_* \cap \alpha$ .

( Where $\pi^0$ is the induced map from the relevant subschemes : from $\pi^{-1}(|D|) \cap |\alpha|$ to $\pi(|\alpha|) \cap |D|$ . This is important, because we have to adhere to where we define our intersection classes!)

2- Flat pull back
Let $\pi : X' \to X$ be flat. Then if $latex$ F is a divisor on $X$ , we want to pull back the intersection of a cycle $\alpha$ with $D$ . So this formula tells us that we can just pull back $D$ and $\alpha$ and do the intersection on $X'$ . Again, we need to be a bit careful to restrict maps to relevant subscheme.
$\pi_0^*(D \cap \alpha) = \pi^*(D) \cap \pi^*(\alpha)$

The idea of proof :
1- We can assume $\pi$ is a surjective proper morphism of varieties, and the cycle is just $X'$ itself.

The nice thing about projection formula and flat pull back is that it allows us to define Chern classes of vector bundle : we pushforward the Chern classes (intersection classes) defined by the canonical line bundle on the corresponding projective bundle. The projection map from the projective bundle is both proper and flat.

This entry was posted on May 6, 2008 at 11:12 pm and is filed under Intersection Theory. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

Dung Hoang Nguyen’s Weblog

Pushforward and flat pull-back of intersections

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