Lemma 76.54.3. Let $S$ be a scheme. Let $i : X \to X'$ be a finite order thickening of algebraic spaces. Let $K' \in D(\mathcal{O}_{X'})$ be an object such that $K = Li^*K'$ is pseudo-coherent. Then $K'$ is pseudo-coherent.
Proof. We first prove $K'$ has quasi-coherent cohomology sheaves; we urge the reader to skip this part. To do this, we may reduce to the case of a first order thickening, see Section 76.9. Let $\mathcal{I} \subset \mathcal{O}_{X'}$ be the quasi-coherent sheaf of ideals cutting out $X$. Tensoring the short exact sequence
\[ 0 \to \mathcal{I} \to \mathcal{O}_{X'} \to i_*\mathcal{O}_ X \to 0 \]
with $K'$ we obtain a distinguished triangle
\[ K' \otimes _{\mathcal{O}_{X'}}^\mathbf {L} \mathcal{I} \to K' \to K' \otimes _{\mathcal{O}_{X'}}^\mathbf {L} i_*\mathcal{O}_ X \to (K' \otimes _{\mathcal{O}_{X'}}^\mathbf {L} \mathcal{I})[1] \]
Since $i_* = Ri_*$ and since we may view $\mathcal{I}$ as a quasi-coherent $\mathcal{O}_ X$-module (as we have a first order thickening) we may rewrite this as
\[ i_*(K \otimes _{\mathcal{O}_ X}^\mathbf {L} \mathcal{I}) \to K' \to i_*K \to i_*(K \otimes _{\mathcal{O}_ X}^\mathbf {L} \mathcal{I})[1] \]
Please use Cohomology of Spaces, Lemma 69.4.4 to identify the terms. Since $K$ is in $D_\mathit{QCoh}(\mathcal{O}_ X)$ we conclude that $K'$ is in $D_\mathit{QCoh}(\mathcal{O}_{X'})$; this uses Derived Categories of Spaces, Lemmas 75.13.6, 75.5.6, and 75.6.1.
Assume $K'$ is in $D_\mathit{QCoh}(\mathcal{O}_{X'})$. The question is étale local on $X'$ hence we may assume $X'$ is affine. In this case the result follows from the case of schemes (More on Morphisms, Lemma 37.71.1). The translation into the language of schemes uses Derived Categories of Spaces, Lemmas 75.4.2 and 75.13.2 and Remark 75.6.3. $\square$
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