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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