Lemma 45.3.6. Let $f : Y \to X$ be a morphism of smooth projective schemes over $k$. Let $[\Gamma _ f] \in \text{Corr}^0(X, Y)$ be as in Example 45.3.2. Then

1. pushforward of cycles by the correspondence $[\Gamma _ f]$ agrees with the gysin map $f^! : \mathop{\mathrm{CH}}\nolimits ^*(Y) \to \mathop{\mathrm{CH}}\nolimits ^*(X)$,

2. pullback of cycles by the correspondence $[\Gamma _ f]$ agrees with the pushforward map $f_* : \mathop{\mathrm{CH}}\nolimits _*(Y) \to \mathop{\mathrm{CH}}\nolimits _*(X)$,

3. if $X$ and $Y$ are equidimensional of dimensions $d$ and $e$, then

1. pushforward of cycles by the correspondence $[\Gamma _ f^ t]$ of Remark 45.3.5 corresponds to pushforward of cycles by $f$, and

2. pullback of cycles by the correspondence $[\Gamma _ f^ t]$ of Remark 45.3.5 corresponds to the gysin map $f^!$.

Proof. Proof of (1). Recall that $[\Gamma _ f]_*(\alpha ) = \text{pr}_{2, *}([\Gamma _ f] \cdot \text{pr}_1^*\alpha )$. We have

$[\Gamma _ f] \cdot \text{pr}_1^*\alpha = (f, 1)_*((f, 1)^! \text{pr}_1^*\alpha ) = (f, 1)_*((f, 1)^! \text{pr}_1^!\alpha ) = (f, 1)_*(f^!\alpha )$

The first equality by Chow Homology, Lemma 42.61.6. The second by Chow Homology, Lemma 42.58.5. The third because $\text{pr}_1 \circ (f, 1) = f$ and Chow Homology, Lemma 42.58.6. Then we coclude because $\text{pr}_{2, *} \circ (f, 1)_* = 1_*$ by Chow Homology, Lemma 42.12.2.

Proof of (2). Recall that $[\Gamma _ f]_*(\beta ) = \text{pr}_{1, *}([\Gamma _ f] \cdot \text{pr}_2^*\beta )$. Arguing exactly as above we have

$[\Gamma _ f] \cdot \text{pr}_2^*\beta = (f, 1)_*\beta$

Thus the result follows as before.

Proof of (3). Proved in exactly the same manner as above. $\square$

Comment #5929 by Bjorn Poonen on

In (1), shouldn't these maps go from $\mathrm{CH}^\ast(X)$ to $\mathrm{CH}^\ast(Y)$?

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