The Stacks project

Lemma 45.3.9. Let $f : Y \to X$ be a morphism of smooth projective schemes over $k$. Assume $X$ and $Y$ equidimensional of dimensions $d$ and $e$. Denote $a = [\Gamma _ f] \in \text{Corr}^0(X, Y)$ and $a^ t = [\Gamma _ f^ t] \in \text{Corr}^{d - e}(Y, X)$. Set $\eta _ X = [\Gamma _{X \to X \times X}] \in \text{Corr}^0(X \times X, X)$, $\eta _ Y = [\Gamma _{Y \to Y \times Y}] \in \text{Corr}^0(Y \times Y, Y)$, $[X] \in \text{Corr}^{-d}(X, \mathop{\mathrm{Spec}}(k))$, and $[Y] \in \text{Corr}^{-e}(Y, \mathop{\mathrm{Spec}}(k))$. The diagram

\[ \xymatrix{ X \otimes Y \ar[r]_{a \otimes \text{id}} \ar[d]_{\text{id} \otimes a^ t} & Y \otimes Y \ar[r]_{\eta _ Y} & Y \ar[d]^{[Y]} \\ X \otimes X \ar[r]^{\eta _ X} & X \ar[r]^{[X]} & \mathop{\mathrm{Spec}}(k) } \]

is commutative in the category of correspondences.

Proof. Recall that $\text{Corr}^ r(W, \mathop{\mathrm{Spec}}(k)) = \mathop{\mathrm{CH}}\nolimits _{-r}(W)$ for any smooth projective scheme $W$ over $k$ and given $c \in \text{Corr}^ s(W', W)$ the composition with $c$ agrees with pullback by $c$ as a map $\mathop{\mathrm{CH}}\nolimits _{-r}(W) \to \mathop{\mathrm{CH}}\nolimits _{-r - s}(W')$ (Lemma 45.3.1). Finally, we have Lemma 45.3.6 which tells us how to convert this into usual pushforward and pullback of cycles. We have

\[ (a \otimes \text{id})^* \eta _ Y^* [Y] = (a \otimes \text{id})^* [\Delta _ Y] = (f \times \text{id})_*\Delta _ Y = [\Gamma _ f] \]

and the other way around we get

\[ (\text{id} \otimes a^ t)^* \eta _ X^* [X] = (\text{id} \otimes a^ t)^* [\Delta _ X] = (\text{id} \times f)^![\Delta _ X] = [\Gamma _ f] \]

The last equality follows from Chow Homology, Lemma 42.59.8. In other words, going either way around the diagram we obtain the element of $\text{Corr}^ d(X \times Y, \mathop{\mathrm{Spec}}(k))$ corresponding to the cycle $\Gamma _ f \subset X \times Y$. $\square$

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