The Stacks project

Proof. All of these follow readily from the results above.

For (1) it suffices to show that $\text{pr}_{X,*}( \Gamma _ f \cdot \alpha \cdot \beta ) = \text{pr}_{X,*}(\Gamma _ f \cdot \alpha ) \cdot \text{pr}_{X,*}(\Gamma _ f \cdot \beta )$ for cycles $\alpha $, $\beta $ on $X \times Y$. If $\alpha $ is a cycle on $X \times Y$ which intersects $\Gamma _ f$ properly, then it is easy to see that

as cycles because $\Gamma _ f$ is a graph. Thus we get the first equality in

the last step by the projection formula in the flat case (Lemma 43.22.1).

If $g : Y \to Z$ then property (2) follows formally from the observation that

in $Z_*(X \times Y \times Z)$ where $\Gamma = \{ (x, f(x), g(f(x))\} $ and maps isomorphically to $\Gamma _{g \circ f}$ in $X \times Z$. The equality follows from the scheme theoretic equality and Lemma 43.14.3.

For (3) we use the projection formula for flat maps twice

where in the last equality we use the remark on graphs made above. This proves (3).

Property (4) rests on identifying the intersection product $\Gamma _ f \cdot pr_ Y^*\alpha $ in the case $f$ is flat. Namely, in this case if $V \subset Y$ is a closed subvariety, then every generic point $\xi $ of the scheme $f^{-1}(V) \cong \Gamma _ f \cap pr_ Y^{-1}(V)$ lies over the generic point of $V$. Hence the local ring of $pr_ Y^{-1}(V) = X \times V$ at $\xi $ is Cohen-Macaulay. Since $\Gamma _ f \subset X \times Y$ is a regular immersion (as a morphism of smooth projective varieties) we find that

with $d$ the dimension of $\Gamma _ f \cap pr_ Y^{-1}(V)$, see Lemma 43.16.5. Since $\Gamma _ f \cap pr_ Y^{-1}(V)$ maps isomorphically to $f^{-1}(V)$ we conclude. $\square$