The Stacks project

Remark 42.59.9. Let $(S, \delta )$ be as in Situation 42.7.1. Let $f : X \to Y$ be a local complete intersection morphism of schemes locally of finite type over $S$. Assume the gysin map exists for $f$. Then $f^! \circ c_ i(\mathcal{E}) = c_ i(f^*\mathcal{E}) \circ f^!$ and similarly for the Chern character, see Lemma 42.59.7. If $X$ and $Y$ satisfy the equivalent conditions of Lemma 42.42.1 and $Y$ is Cohen-Macaulay (for example), then $f^![Y] = [X]$ by Lemma 42.59.8. In this case we also get $f^!(c_ i(\mathcal{E}) \cap [Y]) = c_ i(f^*\mathcal{E}) \cap [X]$ and similarly for the Chern character.