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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