The Stacks project

Remark 54.2.3. Let $\mathbf{F}_ p \subset \Lambda \subset R \subset S$ and $\text{Tr}$ be as in Lemma 54.2.2. By de Rham Cohomology, Proposition 50.19.3 there is a canonical map of complexes

\[ \Theta _{S/R} : \Omega _{S/\Lambda }^\bullet \longrightarrow \Omega _{R/\Lambda }^\bullet \]

The computation in de Rham Cohomology, Example 50.19.4 shows that $\Theta _{S/R}(x^ i \text{d}x) = \text{Tr}_ x(x^ i\text{d}x)$ for all $i$. Since $\text{Trace}_{S/R} = \Theta ^0_{S/R}$ is identically zero and since

\[ \Theta _{S/R}(a \wedge b) = a \wedge \Theta _{S/R}(b) \]

for $a \in \Omega ^ i_{R/\Lambda }$ and $b \in \Omega ^ j_{S/\Lambda }$ it follows that $\text{Tr} = \Theta _{S/R}$. The advantage of using $\text{Tr}$ is that it is a good deal more elementary to construct.

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