The Stacks project

Lemma 45.3.4. There is a contravariant functor from the category of smooth projective schemes over $k$ to the category of correspondences which is the identity on objects and sends $f : Y \to X$ to the element $[\Gamma _ f] \in \text{Corr}^0(X, Y)$.

Proof. In the proof of Lemma 45.3.3 we have seen that this construction sends identities to identities. To finish the proof we have to show if $g : Z \to Y$ is another morphism of smooth projective schemes over $k$, then we have $[\Gamma _ g] \circ [\Gamma _ f] = [\Gamma _{f \circ g}]$ in $\text{Corr}^0(X, Z)$. Arguing as in the proof of Lemma 45.3.3 we see that it suffices to show

\[ [\Gamma _{f \circ g}] = \text{pr}_{13, *}([\Gamma _ f \times Z] \cdot [X \times \Gamma _ g]) \]

in $\mathop{\mathrm{CH}}\nolimits ^*(X \times Z)$ when $X$, $Y$, $Z$ are integral. Denote $Z' \subset X \times Y \times Z$ the image of the closed immersion $(f \circ g, g, 1) : Z \to X \times Y \times Z$. Then $Z' = \Gamma _ f \times Z \cap X \times \Gamma _ g$ scheme theoretically and we conclude using Chow Homology, Lemma 42.62.5 that

\[ [Z'] = [\Gamma _ f \times Z] \cdot [X \times \Gamma _ g] \]

Since it is clear that $\text{pr}_{13, *}([Z']) = [\Gamma _{f \circ g}]$ the proof is complete. $\square$

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