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 15.83.10. $\square$

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0GEH. Beware of the difference between the letter 'O' and the digit '0'.