Lemma 10.134.9 (07BQ)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.134: The naive cotangent complex
Lemma 10.134.9 (cite)

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Lemma 10.134.9. Let $R_ i \to S_ i$ be a system of ring maps over the directed set $I$ . Set $R = \mathop{\mathrm{colim}}\nolimits R_ i$ and $S = \mathop{\mathrm{colim}}\nolimits S_ i$ . Then $\mathop{N\! L}\nolimits _{S/R} = \mathop{\mathrm{colim}}\nolimits \mathop{N\! L}\nolimits _{S_ i/R_ i}$ .

Proof. Recall that $\mathop{N\! L}\nolimits _{S/R}$ is the complex $I/I^2 \to \bigoplus _{s \in S} S\text{d}[s]$ where $I \subset R[S]$ is the kernel of the canonical presentation $R[S] \to S$ . Now it is clear that $R[S] = \mathop{\mathrm{colim}}\nolimits R_ i[S_ i]$ and similarly that $I = \mathop{\mathrm{colim}}\nolimits I_ i$ where $I_ i = \mathop{\mathrm{Ker}}(R_ i[S_ i] \to S_ i)$ . Hence the lemma is clear. $\square$

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