The Stacks project

Proposition 92.7.4. Let $A \to B \to C$ be ring maps. There is a canonical distinguished triangle

\[ L_{B/A} \otimes _ B^\mathbf {L} C \to L_{C/A} \to L_{C/B} \to L_{B/A} \otimes _ B^\mathbf {L} C[1] \]

in $D(C)$.

Proof. Consider the short exact sequence of sheaves of Lemma 92.7.1 and apply the derived functor $L\pi _!$ to obtain a distinguished triangle

\[ L\pi _!(g_1^{-1}\Omega _1 \otimes _{\underline{B}} \underline{C}) \to L\pi _!(g_2^{-1}\Omega _2) \to L\pi _!(g_3^{-1}\Omega _3) \to L\pi _!(g_1^{-1}\Omega _1 \otimes _{\underline{B}} \underline{C})[1] \]

in $D(C)$. Using Lemmas 92.7.3 and 92.4.3 we see that the second and third terms agree with $L_{C/A}$ and $L_{C/B}$ and the first one equals

\[ L\pi _{1, !}(\Omega _1 \otimes _{\underline{B}} \underline{C}) = L\pi _{1, !}(\Omega _1) \otimes _ B^\mathbf {L} C = L_{B/A} \otimes _ B^\mathbf {L} C \]

The first equality by Cohomology on Sites, Lemma 21.39.6 (and flatness of $\Omega _1$ as a sheaf of modules over $\underline{B}$) and the second by Lemma 92.4.3. $\square$

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Your email address will not be published.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 08QX. Beware of the difference between the letter 'O' and the digit '0'.