The Stacks project

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

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

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

  3. $E$ 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. $\square$

Comments (0)

Post a comment

Your email address will not be published.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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 08IS. Beware of the difference between the letter 'O' and the digit '0'.