Lemma 92.3.4 (08S9)—The Stacks project

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Table of contents
Part 6: Deformation Theory
Chapter 92: The Cotangent Complex
Section 92.3: The cotangent complex of a ring map
Lemma 92.3.4 (cite)

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Lemma 92.3.4. Let $A_ i \to B_ i$ be a system of ring maps over a directed index set $I$ . Then $\mathop{\mathrm{colim}}\nolimits L_{B_ i/A_ i} = L_{\mathop{\mathrm{colim}}\nolimits B_ i/\mathop{\mathrm{colim}}\nolimits A_ i}$ .

Proof. This is true because the forgetful functor $V : A\textit{-Alg} \to \textit{Sets}$ and its adjoint $U : \textit{Sets} \to A\textit{-Alg}$ commute with filtered colimits. Moreover, the functor $B/A \mapsto \Omega _{B/A}$ does as well (Algebra, Lemma 10.131.5). $\square$

Comments (4)

Comment #2354 by Anthony on January 19, 2017 at 05:05

In the statement, should these be L_{B_i/A_i} instead of L_{A_i/B_i} (and similarly for the colimit)?

Comment #2421 by Johan on February 17, 2017 at 14:09

Thanks, fixed here.

Comment #4353 by . on August 21, 2019 at 09:35

Is there a similar result for the direct limit of schemes?

Comment #4354 by Johan on August 21, 2019 at 11:21

Yes, if the transition morphisms are affine then it just reduces to this lemma because we have the comparison in Lemma 92.24.2.

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