Lemma 37.70.1. 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.
37.70 Descent finiteness properties of complexes
This section is the continuation of Derived Categories of Schemes, Section 36.12.
Proof. Assume E is m-pseudo-coherent relative to S. The morphisms f_ i are pseudo-coherent by Lemma 37.60.6. Hence Lf_ i^*E is m-pseudo-coherent relative to S by Lemma 37.59.16.
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 37.48.2. 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 37.60.7. Hence
is m-pseudo-coherent relative to S by Lemma 37.59.16. 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 37.59.9 (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 37.59.12. \square
Lemma 37.70.2. 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 37.59.17 to see the equivalence holds. \square
Lemma 37.70.3. 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 37.70.1 and 37.70.2. 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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