In this section we verify that proper pushforward and flat pullback are compatible when this makes sense. By the work we did above this is a consequence of cohomology and base change.
Lemma 42.15.1. Let (S, \delta ) be as in Situation 42.7.1. Let
be a fibre product diagram of schemes locally of finite type over S. Assume f : X \to Y proper and g : Y' \to Y flat of relative dimension r. Then also f' is proper and g' is flat of relative dimension r. For any k-cycle \alpha on X we have
in Z_{k + r}(Y').
Proof. The assertion that f' is proper follows from Morphisms, Lemma 29.41.5. The assertion that g' is flat of relative dimension r follows from Morphisms, Lemmas 29.29.2 and 29.25.8. It suffices to prove the equality of cycles when \alpha = [W] for some integral closed subscheme W \subset X of \delta -dimension k. Note that in this case we have \alpha = [\mathcal{O}_ W]_ k, see Lemma 42.10.3. By Lemmas 42.12.4 and 42.14.4 it therefore suffices to show that f'_*(g')^*\mathcal{O}_ W is isomorphic to g^*f_*\mathcal{O}_ W. This follows from cohomology and base change, see Cohomology of Schemes, Lemma 30.5.2. \square
Lemma 42.15.2. Let (S, \delta ) be as in Situation 42.7.1. Let X, Y be locally of finite type over S. Let f : X \to Y be a finite locally free morphism of degree d (see Morphisms, Definition 29.48.1). Then f is both proper and flat of relative dimension 0, and
for every \alpha \in Z_ k(Y).
Proof. A finite locally free morphism is flat and finite by Morphisms, Lemma 29.48.2, and a finite morphism is proper by Morphisms, Lemma 29.44.11. We omit showing that a finite morphism has relative dimension 0. Thus the formula makes sense. To prove it, let Z \subset Y be an integral closed subscheme of \delta -dimension k. It suffices to prove the formula for \alpha = [Z]. Since the base change of a finite locally free morphism is finite locally free (Morphisms, Lemma 29.48.4) we see that f_*f^*\mathcal{O}_ Z is a finite locally free sheaf of rank d on Z. Hence
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