The Stacks project

Lemma 37.56.16. 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 20.51.4. 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 representating 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 37.56.10 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 37.56.15 for instance. $\square$

[1] This means $f$ is pseudo-coherent, see Definition 37.57.2.

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