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.

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