The Stacks project

Lemma 37.59.16. Let $f : X \to Y$ be a morphism of schemes locally of finite type over a base $S$. Let $m \in \mathbf{Z}$. Let $E$ be an object of $D(\mathcal{O}_ Y)$. Assume

  1. $\mathcal{O}_ X$ is pseudo-coherent relative to $Y$1, and

  2. $E$ is $m$-pseudo-coherent relative to $S$.

Then $Lf^*E$ is $m$-pseudo-coherent relative to $S$.

Proof. The problem is local on $X$. Thus we may assume $X$, $Y$, and $S$ are affine. Arguing as in the proof of More on Algebra, Lemma 15.81.13 we can find a commutative diagram

\[ \xymatrix{ X \ar[r]_ i \ar[d]_ f & \mathbf{A}^ d_ Y \ar[r]_ j \ar[ld]^ p & \mathbf{A}^{n + d}_ S \ar[ld] \\ Y \ar[r] & \mathbf{A}^ n_ S } \]

Observe that

\[ Ri_* Lf^*E = Ri_* Li^* Lp^*E = Lp^*E \otimes _{\mathcal{O}_{\mathbf{A}_ Y^ n}}^\mathbf {L} Ri_*\mathcal{O}_ X \]

by Cohomology, Lemma 20.54.4. By assumption and the fact that $Y$ is affine, we can represent $Ri_*\mathcal{O}_ X = i_*\mathcal{O}_ X$ by a complexes of finite free $\mathcal{O}_{\mathbf{A}_ Y^ n}$-modules $\mathcal{F}^\bullet $, with $\mathcal{F}^ q = 0$ for $q > 0$ (details omitted; use Derived Categories of Schemes, Lemma 36.10.2 and More on Algebra, Lemma 15.81.7). By assumption $E$ is bounded above, say $H^ q(E) = 0$ for $q > a$. Represent $E$ by a complex $\mathcal{E}^\bullet $ of $\mathcal{O}_ Y$-modules with $\mathcal{E}^ q = 0$ for $q > a$. Then the derived tensor product above is represented by $\text{Tot}(p^*\mathcal{E}^\bullet \otimes _{\mathcal{O}_{\mathbf{A}_ Y^ n}} \mathcal{F}^\bullet )$.

Since $j$ is a closed immersion, the functor $j_*$ is exact and $Rj_*$ is computed by applying $j_*$ to any representing complex of sheaves. Thus we have to show that $j_*\text{Tot}(p^*\mathcal{E}^\bullet \otimes _{\mathcal{O}_{\mathbf{A}_ Y^ n}} \mathcal{F}^\bullet )$ is $m$-pseudo-coherent as a complex of $\mathcal{O}_{\mathbf{A}^{n + m}_ S}$-modules. Note that $\text{Tot}(p^*\mathcal{E}^\bullet \otimes _{\mathcal{O}_{\mathbf{A}_ Y^ n}} \mathcal{F}^\bullet )$ has a filtration by subcomplexes with successive quotients the complexes $p^*\mathcal{E}^\bullet \otimes _{\mathcal{O}_{\mathbf{A}_ Y^ n}} \mathcal{F}^ q[-q]$. Note that for $q \ll 0$ the complexes $p^*\mathcal{E}^\bullet \otimes _{\mathcal{O}_{\mathbf{A}_ Y^ n}} \mathcal{F}^ q[-q]$ have zero cohomology in degrees $\leq m$ and hence are $m$-pseudo-coherent. Hence, applying Lemma 37.59.10 and induction, it suffices to show that $p^*\mathcal{E}^\bullet \otimes _{\mathcal{O}_{\mathbf{A}_ Y^ n}} \mathcal{F}^ q[-q]$ is pseudo-coherent relative to $S$ for all $q$. Note that $\mathcal{F}^ q = 0$ for $q > 0$. Since also $\mathcal{F}^ q$ is finite free this reduces to proving that $p^*\mathcal{E}^\bullet $ is $m$-pseudo-coherent relative to $S$ which follows from Lemma 37.59.15 for instance. $\square$

[1] This means $f$ is pseudo-coherent, see Definition 37.60.2.

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 09US. Beware of the difference between the letter 'O' and the digit '0'.