The Stacks project

42.15 Push and pull

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

\[ \xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^ f \\ Y' \ar[r]^ g & Y } \]

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

\[ g^*f_*\alpha = f'_*(g')^*\alpha \]

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

\[ f_*f^*\alpha = d\alpha \]

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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