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37.64 Descent finiteness properties of complexes

This section is the continuation of Derived Categories of Schemes, Section 36.12.

Lemma 37.64.1. Let $X \to S$ be locally of finite type. Let $\{ f_ i : X_ i \to X\} $ be an fppf covering of schemes. Let $E \in D_\mathit{QCoh}(\mathcal{O}_ X)$. Let $m \in \mathbf{Z}$. Then $E$ is $m$-pseudo-coherent relative to $S$ if and only if each $Lf_ i^*E$ is $m$-pseudo-coherent relative to $S$.

Proof. Assume $E$ is $m$-pseudo-coherent relative to $S$. The morphisms $f_ i$ are pseudo-coherent by Lemma 37.54.6. Hence $Lf_ i^*E$ is $m$-pseudo-coherent relative to $S$ by Lemma 37.53.16.

Conversely, assume that $Lf_ i^*E$ is $m$-pseudo-coherent relative to $S$ for each $i$. Pick $S = \bigcup U_ j$, $W_ j \to U_ j$, $W_ j = \bigcup W_{j, k}$, $T_{j, k} \to W_{j, k}$, and morphisms $\alpha _{j, k} : T_{j, k} \to X_{i(j, k)}$ over $S$ as in Lemma 37.44.2. Since the morphism $T_{j, K} \to S$ is flat and of finite presentation, we see that $\alpha _{j, k}$ is pseudo-coherent by Lemma 37.54.7. Hence

\[ L\alpha _{j, k}^*Lf_{i(j, k)}^*E = L(T_{i, k} \to S)^*E \]

is $m$-pseudo-coherent relative to $S$ by Lemma 37.53.16. Now we want to descend this property through the coverings $\{ T_{j, k} \to W_{j, k}\} $, $W_ j = \bigcup W_{j, k}$, $\{ W_ j \to U_ j\} $, and $S = \bigcup U_ j$. Since for Zariski coverings the result is true (by the definition of $m$-pseudo-coherence relative to $S$), this means we may assume we have a single surjective finite locally free morphism $\pi : Y \to X$ such that $L\pi ^*E$ is pseudo-coherent relative to $S$. In this case $R\pi _*L\pi ^*E$ is pseudo-coherent relative to $S$ by Lemma 37.53.9 (this is the first time we use that $E$ has quasi-coherent cohomology sheaves). We have $R\pi _*L\pi ^*E = E \otimes ^\mathbf {L}_{\mathcal{O}_ X} \pi _*\mathcal{O}_ Y$ for example by Derived Categories of Schemes, Lemma 36.22.1 and locally on $X$ the map $\mathcal{O}_ X \to \pi _*\mathcal{O}_ Y$ is the inclusion of a direct summand. Hence we conclude by Lemma 37.53.12. $\square$

Lemma 37.64.2. Let $X \to T \to S$ be morphisms of schemes. Assume $T \to S$ is flat and locally of finite presentation and $X \to T$ locally of finite type. Let $E \in D(\mathcal{O}_ X)$. Let $m \in \mathbf{Z}$. Then $E$ is $m$-pseudo-coherent relative to $S$ if and only if $E$ is $m$-pseudo-coherent relative to $T$.

Proof. Locally on $X$ we can choose a closed immersion $i : X \to \mathbf{A}^ n_ T$. Then $\mathbf{A}^ n_ T \to S$ is flat and locally of finite presentation. Thus we may apply Lemma 37.53.17 to see the equivalence holds. $\square$

Lemma 37.64.3. Let $f : X \to S$ be locally of finite type. Let $\{ S_ i \to S\} $ be an fppf covering of schemes. Denote $f_ i : X_ i \to S_ i$ the base change of $f$ and $g_ i : X_ i \to X$ the projection. Let $E \in D_\mathit{QCoh}(\mathcal{O}_ X)$. Let $m \in \mathbf{Z}$. Then $E$ is $m$-pseudo-coherent relative to $S$ if and only if each $Lg_ i^*E$ is $m$-pseudo-coherent relative to $S_ i$.

Proof. This follows formally from Lemmas 37.64.1 and 37.64.2. Namely, if $E$ is $m$-pseudo-coherent relative to $S$, then $Lg_ i^*E$ is $m$-pseudo-coherent relative to $S$ (by the first lemma), hence $Lg_ i^*E$ is $m$-pseudo-coherent relative to $S_ i$ (by the second). Conversely, if $Lg_ i^*E$ is $m$-pseudo-coherent relative to $S_ i$, then $Lg_ i^*E$ is $m$-pseudo-coherent relative to $S$ (by the second lemma), hence $E$ is $m$-pseudo-coherent relative to $S$ (by the first lemma). $\square$


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