The Stacks project

Definition 37.59.2. Let $f : X \to S$ be a morphism of schemes which is locally of finite type. Let $E$ be an object of $D(\mathcal{O}_ X)$. Let $\mathcal{F}$ be an $\mathcal{O}_ X$-module. Fix $m \in \mathbf{Z}$.

  1. We say $E$ is $m$-pseudo-coherent relative to $S$ if there exists an affine open covering $S = \bigcup V_ i$ and for each $i$ an affine open covering $f^{-1}(V_ i) = \bigcup U_{ij}$ such that the equivalent conditions of Lemma 37.59.1 are satisfied for each of the pairs $(U_{ij} \to V_ i, E|_{U_{ij}})$.

  2. We say $E$ is pseudo-coherent relative to $S$ if $E$ is $m$-pseudo-coherent relative to $S$ for all $m \in \mathbf{Z}$.

  3. We say $\mathcal{F}$ is $m$-pseudo-coherent relative to $S$ if $\mathcal{F}$ viewed as an object of $D(\mathcal{O}_ X)$ is $m$-pseudo-coherent relative to $S$.

  4. We say $\mathcal{F}$ is pseudo-coherent relative to $S$ if $\mathcal{F}$ viewed as an object of $D(\mathcal{O}_ X)$ is pseudo-coherent relative to $S$.

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Your email address will not be published.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 09UI. Beware of the difference between the letter 'O' and the digit '0'.



View Definition 37.59.2 as pdf