The Stacks project

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

  1. $E$ is $S$-perfect, and

  2. $E$ 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 15.82.10. $\square$


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