The Stacks project

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 $f^! : \mathop{\mathrm{CH}}\nolimits ^*(X) \to \mathop{\mathrm{CH}}\nolimits ^*(Y)$,

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 $f^!$.