The Stacks project

Lemma 86.3.3. With assumptions as in Lemma 86.3.2 assume that $B/I^ nB \to C/I^ nC$ is a local complete intersection homomorphism for all $n$. Then $H^{-1}(\mathop{N\! L}\nolimits ^\wedge _{B/A} \otimes _ B C) \to H^{-1}(\mathop{N\! L}\nolimits ^\wedge _{C/A})$ is injective.

Proof. By More on Algebra, Lemma 15.33.6 we see that this holds for the map between naive cotangent complexes of the situation modulo $I^ n$ for all $n$. In other words, we obtain a distinguished triangle in $D(C/I^ nC)$ for every $n$. Using Lemma 86.3.1 this implies the lemma; details omitted. $\square$

Comments (0)

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Your email address will not be published.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