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
$E \in D^ b_{\textit{Coh}}(\mathcal{O}_ X)$,
the support of $H^ i(E)$ is proper over $S$ for all $i$, and
$E$ 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 20.47.5. 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
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. $\square$
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)