Lemma 92.3.4 (08S9)—The Stacks project

Loading [MathJax]/extensions/tex2jax.js

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)

previous remark

numberstags

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.

There are also:

3 comment(s) on Section 92.3: The cotangent complex of a ring map

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).

Comments (4)

Post a comment