Lemma 38.29.4. In Situation 38.29.1 let $K$ be as in Lemma 38.29.2. Then $K$ is pseudo-coherent on $X$.

**Proof.**
Combinging Lemma 38.29.3 and Derived Categories of Schemes, Lemma 36.34.3 we see that $R\Gamma (X, K \otimes ^\mathbf {L} E)$ is pseudo-coherent in $D(A)$ for all pseudo-coherent $E$ in $D(\mathcal{O}_ X)$. Thus it follows from More on Morphisms, Lemma 37.63.4 that $K$ is pseudo-coherent relative to $A$. Since $X$ is of flat and of finite presentation over $A$, this is the same as being pseudo-coherent on $X$, see More on Morphisms, Lemma 37.53.18.
$\square$

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