The Stacks project

Lemma 15.7.5. With $A, A', B, B', C, C', D, D', I$ as in Situation 15.7.1.

  1. Let $(N, M', \varphi )$ be an object of $\text{Mod}_ D \times _{\text{Mod}_ C} \text{Mod}_{C'}$. If $M'$ is flat over $A'$ and $N$ is flat over $B$, then $N' = N \times _{\varphi , M} M'$ is flat over $B'$.

  2. If $L'$ is a $D'$-module flat over $B'$, then $L' = (L \otimes _{D'} D) \times _{(L \otimes _{D'} C)} (L \otimes _{D'} C')$.

  3. The category of $D'$-modules flat over $B'$ is equivalent to the categories of objects $(N, M', \varphi )$ of $\text{Mod}_ D \times _{\text{Mod}_ C} \text{Mod}_{C'}$ with $N$ flat over $B$ and $M'$ flat over $A'$.

Proof. Part (1) follows from part (1) of Lemma 15.6.8.

Part (2) follows from part (2) of Lemma 15.6.8 using that $L' \otimes _{D'} D = L' \otimes _{B'} B$, $L' \otimes _{D'} C' = L' \otimes _{B'} A'$, and $L' \otimes _{D'} C = L' \otimes _{B'} A$, see discussion in Situation 15.7.1.

Part (3) is an immediate consequence of (1) and (2). $\square$

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Your email address will not be published.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 07RW. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 15.7.5 as pdf