The Stacks project

Lemma 21.45.3. Let $(f, f^\sharp ) : (\mathcal{C}, \mathcal{O}_\mathcal {C}) \to (\mathcal{D}, \mathcal{O}_\mathcal {D})$ be a morphism of ringed sites. Let $E$ be an object of $D(\mathcal{O}_\mathcal {C})$. If $E$ is $m$-pseudo-coherent, then $Lf^*E$ is $m$-pseudo-coherent.

Proof. Say $f$ is given by the functor $u : \mathcal{D} \to \mathcal{C}$. Let $U$ be an object of $\mathcal{C}$. By Sites, Lemma 7.14.10 we can find a covering $\{ U_ i \to U\} $ and for each $i$ a morphism $U_ i \to u(V_ i)$ for some object $V_ i$ of $\mathcal{D}$. By Lemma 21.45.2 it suffices to show that $Lf^*E|_{U_ i}$ is $m$-pseudo-coherent. To do this it is enough to show that $Lf^*E|_{u(V_ i)}$ is $m$-pseudo-coherent, since $Lf^*E|_{U_ i}$ is the restriction of $Lf^*E|_{u(V_ i)}$ to $\mathcal{C}/U_ i$ (via Modules on Sites, Lemma 18.19.5). By the commutative diagram of Modules on Sites, Lemma 18.20.1 it suffices to prove the lemma for the morphism of ringed sites $(\mathcal{C}/u(V_ i), \mathcal{O}_{u(V_ i)}) \to (\mathcal{D}/V_ i, \mathcal{O}_{V_ i})$. Thus we may assume $\mathcal{D}$ has a final object $Y$ such that $X = u(Y)$ is a final object of $\mathcal{C}$.

Let $\{ V_ i \to Y\} $ be a covering such that for each $i$ there exists a strictly perfect complex $\mathcal{F}_ i^\bullet $ of $\mathcal{O}_{V_ i}$-modules and a morphism $\alpha _ i : \mathcal{F}_ i^\bullet \to E|_{V_ i}$ of $D(\mathcal{O}_{V_ i})$ such that $H^ j(\alpha _ i)$ is an isomorphism for $j > m$ and $H^ m(\alpha _ i)$ is surjective. Arguing as above it suffices to prove the result for $(\mathcal{C}/u(V_ i), \mathcal{O}_{u(V_ i)}) \to (\mathcal{D}/V_ i, \mathcal{O}_{V_ i})$. Hence we may assume that there exists a strictly perfect complex $\mathcal{F}^\bullet $ of $\mathcal{O}_\mathcal {D}$-modules and a morphism $\alpha : \mathcal{F}^\bullet \to E$ of $D(\mathcal{O}_\mathcal {D})$ such that $H^ j(\alpha )$ is an isomorphism for $j > m$ and $H^ m(\alpha )$ is surjective. In this case, choose a distinguished triangle

\[ \mathcal{F}^\bullet \to E \to C \to \mathcal{F}^\bullet [1] \]

The assumption on $\alpha $ means exactly that the cohomology sheaves $H^ j(C)$ are zero for all $j \geq m$. Applying $Lf^*$ we obtain the distinguished triangle

\[ Lf^*\mathcal{F}^\bullet \to Lf^*E \to Lf^*C \to Lf^*\mathcal{F}^\bullet [1] \]

By the construction of $Lf^*$ as a left derived functor we see that $H^ j(Lf^*C) = 0$ for $j \geq m$ (by the dual of Derived Categories, Lemma 13.16.1). Hence $H^ j(Lf^*\alpha )$ is an isomorphism for $j > m$ and $H^ m(Lf^*\alpha )$ is surjective. On the other hand, since $\mathcal{F}^\bullet $ is a bounded above complex of flat $\mathcal{O}_\mathcal {D}$-modules we see that $Lf^*\mathcal{F}^\bullet = f^*\mathcal{F}^\bullet $. Applying Lemma 21.44.4 we conclude. $\square$

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