Lemma 36.35.13. Let f : X \to S be a morphism of schemes which is flat and locally of finite presentation. Let E be a pseudo-coherent object of D(\mathcal{O}_ X). The following are equivalent
E is S-perfect, and
E is locally bounded below and for every point s \in S the object L(X_ s \to X)^*E of D(\mathcal{O}_{X_ s}) is locally bounded below.
Proof. Since everything is local we immediately reduce to the case that X and S are affine, see Lemma 36.35.3. Say X \to S corresponds to \mathop{\mathrm{Spec}}(A) \to \mathop{\mathrm{Spec}}(R) and E corresponds to K in D(A). If s corresponds to the prime \mathfrak p \subset R, then L(X_ s \to X)^*E corresponds to K \otimes _ R^\mathbf {L} \kappa (\mathfrak p) as R \to A is flat, see for example Lemma 36.22.5. Thus we see that our lemma follows from the corresponding algebra result, see More on Algebra, Lemma 15.83.10. \square
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