The Stacks project

Remark 35.7.8. Being locally free is a property of quasi-coherent modules which does not descend in the fpqc topology. Namely, suppose that $R$ is a ring and that $M$ is a projective $R$-module which is a countable direct sum $M = \bigoplus L_ n$ of rank 1 locally free modules, but not locally free, see Examples, Lemma 110.34.4. Then $M$ becomes free on making the faithfully flat base change

\[ R \longrightarrow \bigoplus \nolimits _{m \geq 1} \bigoplus \nolimits _{(i_1, \ldots , i_ m) \in \mathbf{Z}^{\oplus m}} L_1^{\otimes i_1} \otimes _ R \ldots \otimes _ R L_ m^{\otimes i_ m} \]

But we don't know what happens for fppf coverings. In other words, we don't know the answer to the following question: Suppose $A \to B$ is a faithfully flat ring map of finite presentation. Let $M$ be an $A$-module such that $M \otimes _ A B$ is free. Is $M$ a locally free $A$-module? It turns out that if $A$ is Noetherian, then the answer is yes. This follows from the results of [Bass]. But in general we don't know the answer. If you know the answer, or have a reference, please email stacks.project@gmail.com.

Comments (0)

There are also:

  • 2 comment(s) on Section 35.7: Descent of finiteness properties of modules

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Your email address will not be published.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 05VF. Beware of the difference between the letter 'O' and the digit '0'.



View Remark 35.7.8 as pdf