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 $Z_{k + r}(Y')$.
\[ \xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^ f \\ Y' \ar[r]^ g & Y } \]
\[ g^*f_*\alpha = f'_*(g')^*\alpha \]
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 $\alpha \in Z_ k(Y)$.
\[ f_*f^*\alpha = d\alpha \]
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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