The Stacks project

Remark 38.6.3. Note that the $R$-algebras $B_ i$ for all $i$ and $A_ i$ for $i \geq 2$ are of finite presentation over $R$. If $S$ is of finite presentation over $R$, then it is also the case that $A_1$ is of finite presentation over $R$. In this case all the ring maps in the complete dévissage are of finite presentation. See Algebra, Lemma 10.6.2. Still assuming $S$ of finite presentation over $R$ the following are equivalent

  1. $M$ is of finite presentation over $S$,

  2. $M_1$ is of finite presentation over $A_1$,

  3. $M_1$ is of finite presentation over $B_1$,

  4. each $M_ i$ is of finite presentation both as an $A_ i$-module and as a $B_ i$-module.

The equivalences (1) $\Leftrightarrow $ (2) and (2) $\Leftrightarrow $ (3) follow from Algebra, Lemma 10.36.23. If $M_1$ is finitely presented, so is $\mathop{\mathrm{Coker}}(\alpha _1)$ (see Algebra, Lemma 10.5.3) and hence $M_2$, etc.

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Your email address will not be published.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