Let $f : X \to Y$ be a morphism of nonsingular projective varieties. We define
by the rule
where $\Gamma _ f \subset X\times Y$ is the graph of $f$. Note that in this generality, it is defined only on cycle classes and not on cycles. With the notation $\mathop{\mathrm{CH}}\nolimits ^*$ introduced in Section 43.26 we may think of pullback as a map
in other words, it is a map of graded abelian groups.
Lemma 43.27.1. Let $f : X \to Y$ be a morphism of nonsingular projective varieties. The pullback map on chow groups satisfies:
$f^* : \mathop{\mathrm{CH}}\nolimits ^*(Y) \to \mathop{\mathrm{CH}}\nolimits ^*(X)$ is a ring map,
$(g \circ f)^* = f^* \circ g^*$ for a composable pair $f, g$,
the projection formula holds: $f_*(\alpha ) \cdot \beta = f_*( \alpha \cdot f^*\beta )$, and
if $f$ is flat then it agrees with the previous definition.
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. $\square$
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