Lemma 76.54.4. Let $S$ be a scheme. Consider a cartesian diagram
of algebraic spaces over $S$. Assume $X' \to Y'$ is flat and locally of finite presentation and $Y \to Y'$ is a finite order thickening. Let $E' \in D(\mathcal{O}_{X'})$. If $E = Li^*(E')$ is $Y$-perfect, then $E'$ is $Y'$-perfect.
Proof. Recall that being $Y$-perfect for $E$ means $E$ is pseudo-coherent and locally has finite tor dimension as a complex of $f^{-1}\mathcal{O}_ Y$-modules (Definition 76.52.1). By Lemma 76.54.3 we find that $E'$ is pseudo-coherent. In particular, $E'$ is in $D_\mathit{QCoh}(\mathcal{O}_{X'})$, see Derived Categories of Spaces, Lemma 75.13.6. By Lemma 76.52.3 this reduces us to the case of schemes. The case of schemes is More on Morphisms, Lemma 37.71.2. $\square$
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