Lemma 42.59.1. In the situation above we have $\Delta ^! \circ \text{pr}_ i^! = 1$ in $A^0(X)$.

Proof. Observe that the projections $\text{pr}_ i : X \times _ Y X \to X$ are smooth and hence we have gysin maps for these projections as well. Thus the lemma makes sense and is a special case of Lemma 42.58.6. $\square$

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