Lemma 75.22.1. Let $S$ be a scheme. Let $Y$ be a Noetherian algebraic space over $S$. Let $f : X \to Y$ be a morphism of algebraic spaces which is locally of finite type and quasi-separated. Let $E \in D(\mathcal{O}_ X)$ such that

$E \in D^ b_{\textit{Coh}}(\mathcal{O}_ X)$,

the support of $H^ i(E)$ is proper over $Y$ for all $i$,

$E$ has finite tor dimension as an object of $D(f^{-1}\mathcal{O}_ Y)$.

Then $Rf_*E$ is a perfect object of $D(\mathcal{O}_ Y)$.

**Proof.**
By Lemma 75.8.1 we see that $Rf_*E$ is an object of $D^ b_{\textit{Coh}}(\mathcal{O}_ Y)$. Hence $Rf_*E$ is pseudo-coherent (Lemma 75.13.7). Hence it suffices to show that $Rf_*E$ has finite tor dimension, see Cohomology on Sites, Lemma 21.47.4. By Lemma 75.13.8 it suffices to check that $Rf_*(E) \otimes _{\mathcal{O}_ Y}^\mathbf {L} \mathcal{F}$ has universally bounded cohomology for all quasi-coherent sheaves $\mathcal{F}$ on $Y$. Bounded from above is clear as $Rf_*(E)$ is bounded from above. Let $T \subset |X|$ be the union of the supports of $H^ i(E)$ for all $i$. Then $T$ is proper over $Y$ by assumptions (1) and (2) and Lemma 75.7.6. In particular there exists a quasi-compact open subspace $X' \subset X$ containing $T$. Setting $f' = f|_{X'}$ we have $Rf_*(E) = Rf'_*(E|_{X'})$ because $E$ restricts to zero on $X \setminus T$. Thus we may replace $X$ by $X'$ and assume $f$ is quasi-compact. We have assumed $f$ is quasi-separated. Thus

\[ Rf_*(E) \otimes _{\mathcal{O}_ Y}^\mathbf {L} \mathcal{F} = Rf_*\left(E \otimes _{\mathcal{O}_ X}^\mathbf {L} Lf^*\mathcal{F}\right) = Rf_*\left(E \otimes _{f^{-1}\mathcal{O}_ Y}^\mathbf {L} f^{-1}\mathcal{F}\right) \]

by Lemma 75.20.1 and Cohomology on Sites, Lemma 21.18.5. By assumption (3) the complex $E \otimes _{f^{-1}\mathcal{O}_ Y}^\mathbf {L} f^{-1}\mathcal{F}$ has cohomology sheaves in a given finite range, say $[a, b]$. Then $Rf_*$ of it has cohomology in the range $[a, \infty )$ and we win.
$\square$

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