The Stacks project

Lemma 90.10.2. 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 90.6 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 90.10.1 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 zero map $\Omega _{P_ n/A} \to \Omega _{P_ n/A}$ (since the derivative of a $p$th power is zero). $\square$


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