The Stacks project

Lemma 33.35.7. Let $p > 0$ be a prime number. Let $S$ be a scheme in characteristic $p$. Let $X$ be a scheme over $S$. Then $\Omega _{X/S} = \Omega _{X/X^{(p)}}$.

Proof. This translates into the following algebra fact. Let $A \to B$ be a homomorphism of rings of characteristic $p$. Set $B' = B \otimes _{A, F_ A} A$ and consider the ring map $F_{B/A} : B' \to B$, $b \otimes a \mapsto b^ pa$. Then our assertion is that $\Omega _{B/A} = \Omega _{B/B'}$. This is true because $\text{d}(b^ pa) = 0$ if $\text{d} : B \to \Omega _{B/A}$ is the universal derivation and hence $\text{d}$ is a $B'$-derivation. $\square$


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