The Stacks project

Lemma 70.4.6. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Let $\mathcal{G}$ be a quasi-coherent $\mathcal{O}_ Y$-module. Assume

  1. $\mathcal{F}$ is flat over $Y$,

  2. $X$ and $Y$ are locally Noetherian, and

  3. the fibres of $f$ are locally Noetherian.


\[ \text{Ass}_ X(\mathcal{F} \otimes _{\mathcal{O}_ X} f^*\mathcal{G}) = \{ x \in \text{Ass}_{X/Y}(\mathcal{F})\text{ such that } f(x) \in \text{Ass}_ Y(\mathcal{G}) \} \]

Proof. Via ├ętale localization, this is an immediate consequence of the result for schemes, see Divisors, Lemma 31.3.1. The result for schemes is more general only because we haven't defined associated points for non-Noetherian algebraic spaces (hence we need to assume $X$ and the fibres of $X \to Y$ are locally Noetherian to even be able to formulate this result). $\square$

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Your email address will not be published.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 0CV1. Beware of the difference between the letter 'O' and the digit '0'.