Example 45.3.2. Let $f : Y \to X$ be a morphism of smooth projective schemes over $k$. Denote $\Gamma _ f \subset X \times Y$ the graph of $f$. More precisely, $\Gamma _ f$ is the image of the closed immersion

$(f, \text{id}_ Y) : Y \longrightarrow X \times Y$

Let $X = \coprod X_ d$ be the decomposition of $X$ into its open and closed parts $X_ d$ which are equidimensional of dimension $d$. Then $\Gamma _ f \cap (X_ d \times Y)$ has pure codimension $d$. Hence $[\Gamma _ f] \in \mathop{\mathrm{CH}}\nolimits ^*(X \times Y) \otimes \mathbf{Q}$ is contained in $\text{Corr}^0(X \times Y)$, i.e., $[\Gamma _ f]$ is a correspondence of degree $0$ from $X$ to $Y$.

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