The Stacks project

Remark 15.89.20. The equivalences of Proposition 15.89.16, Theorem 15.89.18, and Proposition 15.89.19 preserve properties of modules. For example if $M$ corresponds to $\mathbf{M} = (M', M_ i, \alpha _ i, \alpha _{ij})$ then $M$ is finite, or finitely presented, or flat, or projective over $R$ if and only if $M'$ and $M_ i$ have the corresponding property over $S$ and $R_{f_ i}$. This follows from the fact that $R \to S \times \prod R_{f_ i}$ is faithfully flat and descend and ascent of these properties along faithfully flat maps, see Algebra, Lemma 10.83.2 and Theorem 10.95.6. These functors also preserve the $\otimes $-structures on either side. Thus, it defines equivalences of various categories built out of the pair $(\text{Mod}_ R, \otimes )$, such as the category of algebras.


Comments (0)

There are also:

  • 2 comment(s) on Section 15.89: Formal glueing of module categories

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Your email address will not be published.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