The Stacks project

Lemma 10.131.5. Let $I$ be a directed set. Let $(R_ i \to S_ i, \varphi _{ii'})$ be a system of ring maps over $I$, see Categories, Section 4.21. Then we have

\[ \Omega _{S/R} = \mathop{\mathrm{colim}}\nolimits _ i \Omega _{S_ i/R_ i}. \]

where $R \to S = \mathop{\mathrm{colim}}\nolimits (R_ i \to S_ i)$.

Proof. This is clear from the defining presentation of $\Omega _{S/R}$ and the functoriality of this described above. $\square$


Comments (4)

Comment #1838 by Keenan Kidwell on

I think there should be something in the statement about being the colimit of the directed system in question.

Comment #6054 by Jonas Ehrhard on

I think this lemma should be stated after the discussion of diagram 10.131.5.1 / 00RQ, since the diagram explains the colimit .

Also, I just spend half an hour searching for the existence of colimits of rings, until I found exercise 109.2.2 / 078J. Maybe this could be referenced?

Comment #6190 by on

OK, I moved the lemma, see this. Yes, there is a general discussion of colimits in the chapter on categories but you are right that we are missing a section in the algebra chapter discussing colimits of rings. Anybody?

There are also:

  • 14 comment(s) on Section 10.131: Differentials

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Your email address will not be published.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 031G. Beware of the difference between the letter 'O' and the digit '0'.