Lemma 90.10.3. Let $p$ be a prime number. Let $A \to B$ be a ring homomorphism and assume that $p = 0$ in $A$. If $A$ and $B$ are perfect, then $L_{B/A}$ is zero in $D(B)$.

**Proof.**
The map $(F_ A, F_ B) : (A \to B) \to (A \to B)$ is an isomorphism hence induces an isomorphism on $L_{B/A}$ and on the other hand induces zero on $L_{B/A}$ by Lemma 90.10.2.
$\square$

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