Lemma 108.4.4. Let $B$ be an algebraic space. Let $\pi : X \to Y$ be a quasi-finite morphism of algebraic spaces which are separated and of finite presentation over $B$. Then $\pi _*$ induces a morphism $\mathcal{C}\! \mathit{oh}_{X/B} \to \mathcal{C}\! \mathit{oh}_{Y/B}$.
Proof. Let $(T \to B, \mathcal{F})$ be an object of $\mathcal{C}\! \mathit{oh}_{X/B}$. We claim
$(T \to B, \pi _{T, *}\mathcal{F})$ is an object of $\mathcal{C}\! \mathit{oh}_{Y/B}$ and
for $T' \to T$ we have $\pi _{T', *}(X_{T'} \to X_ T)^*\mathcal{F} = (Y_{T'} \to Y_ T)^*\pi _{T, *}\mathcal{F}$.
Part (b) guarantees that this construction defines a functor $\mathcal{C}\! \mathit{oh}_{X/B} \to \mathcal{C}\! \mathit{oh}_{Y/B}$ as desired.
Let $i : Z \to X_ T$ be the closed subspace cut out by the zeroth fitting ideal of $\mathcal{F}$ (Divisors on Spaces, Section 71.5). Then $Z \to B$ is proper by assumption (see Derived Categories of Spaces, Section 75.7). On the other hand $i$ is of finite presentation (Divisors on Spaces, Lemma 71.5.2 and Morphisms of Spaces, Lemma 67.28.12). There exists a quasi-coherent $\mathcal{O}_ Z$-module $\mathcal{G}$ of finite type with $i_*\mathcal{G} = \mathcal{F}$ (Divisors on Spaces, Lemma 71.5.3). In fact $\mathcal{G}$ is of finite presentation as an $\mathcal{O}_ Z$-module by Descent on Spaces, Lemma 74.6.7. Observe that $\mathcal{G}$ is flat over $B$, for example because the stalks of $\mathcal{G}$ and $\mathcal{F}$ agree (Morphisms of Spaces, Lemma 67.13.6). Observe that $\pi _ T \circ i : Z \to Y_ T$ is quasi-finite as a composition of quasi-finite morphisms and that $\pi _{T, *}\mathcal{F} = (\pi _ T \circ i)_*\mathcal{G})$. Since $i$ is affine, formation of $i_*$ commutes with base change (Cohomology of Spaces, Lemma 69.11.1). Therefore we may replace $B$ by $T$, $X$ by $Z$, $\mathcal{F}$ by $\mathcal{G}$, and $Y$ by $Y_ T$ to reduce to the case discussed in the next paragraph.
Assume that $X \to B$ is proper. Then $\pi $ is proper by Morphisms of Spaces, Lemma 67.40.6 and hence finite by More on Morphisms of Spaces, Lemma 76.35.1. Since a finite morphism is affine we see that (b) holds by Cohomology of Spaces, Lemma 69.11.1. On the other hand, $\pi $ is of finite presentation by Morphisms of Spaces, Lemma 67.28.9. Thus $\pi _{T, *}\mathcal{F}$ is of finite presentation by Descent on Spaces, Lemma 74.6.7. Finally, $\pi _{T, *}\mathcal{F} $ is flat over $B$ for example by looking at stalks using Cohomology of Spaces, Lemma 69.4.2. $\square$
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