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

$f_*f^*\alpha = d\alpha$

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

$f_*f^*[Z] = f_*f^*[\mathcal{O}_ Z]_ k = [f_*f^*\mathcal{O}_ Z]_ k = d[Z]$

where we have used Lemmas 42.14.4 and 42.12.4. $\square$

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