The Stacks project

Lemma 43.27.1. Let $f : X \to Y$ be a morphism of nonsingular projective varieties. The pullback map on chow groups satisfies:

  1. $f^* : \mathop{\mathrm{CH}}\nolimits ^*(Y) \to \mathop{\mathrm{CH}}\nolimits ^*(X)$ is a ring map,

  2. $(g \circ f)^* = f^* \circ g^*$ for a composable pair $f, g$,

  3. the projection formula holds: $f_*(\alpha ) \cdot \beta = f_*( \alpha \cdot f^*\beta )$, and

  4. 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

\[ \Gamma _ f \cdot \alpha = \Gamma _ f \cdot \text{pr}_ X^*(\text{pr}_{X,*}(\Gamma _ f \cdot \alpha )) \]

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

\begin{align*} \text{pr}_{X,*}(\Gamma _ f \cdot \alpha \cdot \beta ) & = \text{pr}_{X,*}( \Gamma _ f \cdot \text{pr}_ X^*(\text{pr}_{X,*}(\Gamma _ f \cdot \alpha )) \cdot \beta ) \\ & = \text{pr}_{X,*}(\text{pr}_ X^*(\text{pr}_{X,*}(\Gamma _ f \cdot \alpha )) \cdot (\Gamma _ f \cdot \beta )) \\ & = \text{pr}_{X,*}(\Gamma _ f \cdot \alpha ) \cdot \text{pr}_{X,*}(\Gamma _ f \cdot \beta ) \end{align*}

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

\[ \Gamma = \text{pr}_{X \times Y}^*\Gamma _ f \cdot \text{pr}_{Y \times Z}^*\Gamma _ g \]

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

\begin{align*} f_*(\alpha \cdot pr_{X, *}(\Gamma _ f \cdot pr_ Y^*(\beta ))) & = f_*(pr_{X, *}(pr_ X^*\alpha \cdot \Gamma _ f \cdot pr_ Y^*(\beta ))) \\ & = pr_{Y, *}(pr_ X^*\alpha \cdot \Gamma _ f \cdot pr_ Y^*(\beta ))) \\ & = pt_{Y, *}(pr_ X^*\alpha \cdot \Gamma _ f) \cdot \beta \\ & = f_*(\alpha ) \cdot \beta \end{align*}

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

\[ \Gamma _ f \cdot pr_ Y^*[V] = [\Gamma _ f \cap pr_ Y^{-1}(V)]_ d \]

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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