The Stacks project

Lemma 92.10.2. Let $p$ be a prime number. Let $A \to B$ be a ring homomorphism and assume that $p = 0$ in $A$. The map $L_{B/A} \to L_{B/A}$ of Section 92.6 induced by the Frobenius maps $F_ A$ and $F_ B$ is homotopic to zero.

Proof. Let $P_\bullet $ be the standard resolution of $B$ over $A$. By Lemma 92.10.1 the map $P_\bullet \to P_\bullet $ induced by $F_ A$ and $F_ B$ is homotopic to the map $F_{P_\bullet } : P_\bullet \to P_\bullet $ given by Frobenius on each term. Hence we obtain what we want as clearly $F_{P_\bullet }$ induces the zero map $\Omega _{P_ n/A} \to \Omega _{P_ n/A}$ (since the derivative of a $p$th power is zero). $\square$


Comments (2)

Comment #9026 by James on

"zero" is repeated twice in the 3rd line of the proof.


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Your email address will not be published.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 0G5Z. Beware of the difference between the letter 'O' and the digit '0'.