The Stacks project

Lemma 36.27.1. Let $S$ be a Noetherian scheme. Let $f : X \to S$ be a morphism of schemes which is locally of finite type. Let $E \in D(\mathcal{O}_ X)$ such that

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

  2. the support of $H^ i(E)$ is proper over $S$ for all $i$, and

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

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

Proof. By Lemma 36.11.3 we see that $Rf_*E$ is an object of $D^ b_{\textit{Coh}}(\mathcal{O}_ S)$. Hence $Rf_*E$ is pseudo-coherent (Lemma 36.10.3). Hence it suffices to show that $Rf_*E$ has finite tor dimension, see Cohomology, Lemma 20.49.5. By Lemma 36.10.6 it suffices to check that $Rf_*(E) \otimes _{\mathcal{O}_ S}^\mathbf {L} \mathcal{F}$ has universally bounded cohomology for all quasi-coherent sheaves $\mathcal{F}$ on $S$. 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 $S$ by assumptions (1) and (2), see Cohomology of Schemes, Lemma 30.26.6. In particular there exists a quasi-compact open $X' \subset X$ 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. Moreover, $f$ is quasi-separated by Morphisms, Lemma 29.15.7. Now

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

by Lemma 36.22.1 and Cohomology, Lemma 20.27.4. By assumption (3) the complex $E \otimes _{f^{-1}\mathcal{O}_ S}^\mathbf {L} f^{-1}\mathcal{F}$ 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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