Lemma 41.49.1. In the situation above we have $\Delta ^! \circ \text{pr}_ i^* = 1$ in $A^0(X)$.

Proof. After decomposing $X$ into connected components we may and do assume that $X \to Y$ is smooth of constant relative dimension $d$. Let $X' \to X$ be locally of finite type with $\dim _\delta (X') = n$. Then $\text{pr}_ i^*[X'] = [X \times _ Y X']_{n + d}$. We have a cartesian diagram

$\xymatrix{ X' \ar[d] \ar[r] & X \ar[d]^\Delta \\ X \times _ Y X' \ar[r] & X \times _ Y X }$

The left vertical arrow is a regular immersion of codimension $d$ since it is a section of the smooth morphism $X \times _ Y X' \to X'$, see Divisors, Lemma 30.22.7. It follows that $\Delta ^! \cap [X \times _ Y X']_{n + d} = [X']$ by Lemma 41.48.4. $\square$

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