82.11 Push and pull

This section is the analogue of Chow Homology, Section 42.14.

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 82.11.1. In Situation 82.2.1 let

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

be a fibre product diagram of good algebraic spaces over $B$. 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 of Spaces, Lemma 67.40.3. The assertion that $g'$ is flat of relative dimension $r$ follows from Morphisms of Spaces, Lemmas 67.34.3 and 67.30.4. It suffices to prove the equality of cycles when $\alpha = [W]$ for some integral closed subspace $W \subset X$ of $\delta$-dimension $k$. Note that in this case we have $\alpha = [\mathcal{O}_ W]_ k$, see Lemma 82.6.3. By Lemmas 82.8.3 and 82.10.5 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 Spaces, Lemma 69.11.2. $\square$

Lemma 82.11.2. In Situation 82.2.1 let $X, Y/B$ be good. Let $f : X \to Y$ be a finite locally free morphism of degree $d$ (see Morphisms of Spaces, Definition 67.46.2). 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 of Spaces, Lemma 67.46.6, and a finite morphism is proper by Morphisms of Spaces, Lemma 67.45.9. 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 of Spaces, Lemma 67.46.5) we see that $f_*f^*\mathcal{O}_ Z$ is a finite locally free sheaf of rank $d$ on $Z$. Thus clearly $f_*f^*\mathcal{O}_ Z$ has length $d$ at the generic point of $Z$. Hence

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

where we have used Lemmas 82.10.5 and 82.8.3. $\square$

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