The Stacks project

Remark 15.7.8. In Situation 15.7.1. Assume $B' \to D'$ is of finite presentation and suppose we are given a $D'$-module $L'$. We claim there is a bijective correspondence between

  1. surjections of $D'$-modules $L' \to Q'$ with $Q'$ of finite presentation over $D'$ and flat over $B'$, and

  2. pairs of surjections of modules $(L' \otimes _{D'} D \to Q_1, L' \otimes _{D'} C' \to Q_2)$ with

    1. $Q_1$ of finite presentation over $D$ and flat over $B$,

    2. $Q_2$ of finite presentation over $C'$ and flat over $A'$,

    3. $Q_1 \otimes _ D C = Q_2 \otimes _{C'} C$ as quotients of $L' \otimes _{D'} C$.

The correspondence between these is given by $Q \mapsto (Q_1, Q_2)$ with $Q_1 = Q \otimes _{D'} D$ and $Q_2 = Q \otimes _{D'} C'$. And for the converse we use $Q = Q_1 \times _{Q_{12}} Q_2$ where $Q_{12}$ the common quotient $Q_1 \otimes _ D C = Q_2 \otimes _{C'} C$ of $L' \otimes _{D'} C$. As quotient map we use

\[ L' \longrightarrow (L' \otimes _{D'} D) \times _{(L' \otimes _{D'} C)} (L' \otimes _{D'} C') \longrightarrow Q_1 \times _{Q_{12}} Q_2 = Q \]

where the first arrow is surjective by Lemma 15.6.5 and the second by Lemma 15.6.6. The claim follows by Lemmas 15.7.5 and 15.7.6.

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Your email address will not be published.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 08KR. Beware of the difference between the letter 'O' and the digit '0'.