The Stacks project

Lemma 42.59.5. 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 $S$. If the gysin map exists for $f$ and $f$ is flat, then $f^!$ is equal to the bivariant class of Lemma 42.33.2.

Proof. Choose a factorization $f = g \circ i$ with $i : X \to P$ an immersion and $g : P \to Y$ smooth. Observe that for any morphism $Y' \to Y$ which is locally of finite type, the base changes of $f'$, $g'$, $i'$ satisfy the same assumptions (see Morphisms, Lemmas 29.34.5 and 29.30.4 and More on Morphisms, Lemma 37.62.8). Thus we reduce to proving that $f^*[Y] = i^!(g^*[Y])$ in case $Y$ is integral, see Lemma 42.35.3. Set $n = \dim _\delta (Y)$. After decomposing $X$ and $P$ into connected components we may assume $f$ is flat of relative dimension $r$ and $g$ is smooth of relative dimension $t$. Then $f^*[Y] = [X]_{n + s}$ and $g^*[Y] = [P]_{n + t}$. On the other hand $i$ is a regular immersion of codimension $t - s$. Thus $i^![P]_{n + t} = [X]_{n + s}$ (Lemma 42.54.5) and the proof is complete. $\square$

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