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. $\square$

Comment #1838 by Keenan Kidwell on

I think there should be something in the statement about $S/R$ 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 $\text{colim}_i \Omega_{S_i / R_i}$.

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?

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• 12 comment(s) on Section 10.131: Differentials

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