The Stacks project

Lemma 37.59.6. Let $f : X \to S$ be a morphism of schemes which is locally of finite type. Let $E$ be an object of $D(\mathcal{O}_ X)$. Fix $m \in \mathbf{Z}$. The following are equivalent

  1. $E$ is $m$-pseudo-coherent relative to $S$,

  2. for every affine opens $U \subset X$ and $V \subset S$ with $f(U) \subset V$ the equivalent conditions of Lemma 37.59.1 are satisfied for the pair $(U \to V, E|_ U)$.

Moreover, if this is true, then for every open subschemes $U \subset X$ and $V \subset S$ with $f(U) \subset V$ the restriction $E|_ U$ is $m$-pseudo-coherent relative to $V$.

Proof. The final statement is clear from the equivalence of (1) and (2). It is also clear that (2) implies (1). Assume (1) holds. Let $S = \bigcup V_ i$ and $f^{-1}(V_ i) = \bigcup U_{ij}$ be affine open coverings as in Definition 37.59.2. Let $U \subset X$ and $V \subset S$ be as in (2). By Lemma 37.59.5 it suffices to find a standard open covering $U = \bigcup U_ k$ of $U$ such that the equivalent conditions of Lemma 37.59.1 are satisfied for the pairs $(U_ k \to V, E|_{U_ k})$. In other words, for every $u \in U$ it suffices to find a standard affine open $u \in U' \subset U$ such that the equivalent conditions of Lemma 37.59.1 are satisfied for the pair $(U' \to V, E|_{U'})$. Pick $i$ such that $f(u) \in V_ i$ and then pick $j$ such that $u \in U_{ij}$. By Schemes, Lemma 26.11.5 we can find $v \in V' \subset V \cap V_ i$ which is standard affine open in $V'$ and $V_ i$. Then $f^{-1}V' \cap U$, resp. $f^{-1}V' \cap U_{ij}$ are standard affine opens of $U$, resp. $U_{ij}$. Applying the lemma again we can find $u \in U' \subset f^{-1}V' \cap U \cap U_{ij}$ which is standard affine open in both $f^{-1}V' \cap U$ and $f^{-1}V' \cap U_{ij}$. Thus $U'$ is also a standard affine open of $U$ and $U_{ij}$. By Lemma 37.59.4 the assumption that the equivalent conditions of Lemma 37.59.1 are satisfied for the pair $(U_{ij} \to V_ i, E|_{U_{ij}})$ implies that the equivalent conditions of Lemma 37.59.1 are satisfied for the pair $(U' \to V, E|_{U'})$. $\square$


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Your email address will not be published.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