Lemma 54.2.2. Let $\mathbf{F}_ p \subset \Lambda \subset R \subset S$ be ring extensions and assume that $S$ is isomorphic to $R[x]/(x^ p - a)$ for some $a \in R$. Then there are canonical $R$-linear maps

\[ \text{Tr} : \Omega ^{t + 1}_{S/\Lambda } \longrightarrow \Omega _{R/\Lambda }^{t + 1} \]

for $t \geq 0$ such that

\[ \eta _1 \wedge \ldots \wedge \eta _ t \wedge x^ i\text{d}x \longmapsto \left\{ \begin{matrix} 0
& \text{if}
& 0 \leq i \leq p - 2,
\\ \eta _1 \wedge \ldots \wedge \eta _ t \wedge \text{d}a
& \text{if}
& i = p - 1
\end{matrix} \right. \]

for $\eta _ i \in \Omega _{R/\Lambda }$ and such that $\text{Tr}$ annihilates the image of $S \otimes _ R \Omega _{R/\Lambda }^{t + 1} \to \Omega _{S/\Lambda }^{t + 1}$.

**Proof.**
For $t = 0$ we use the composition

\[ \Omega _{S/\Lambda } \to \Omega _{S/R} \to \Omega _ R \to \Omega _{R/\Lambda } \]

where the second map is Lemma 54.2.1. There is an exact sequence

\[ H_1(L_{S/R}) \xrightarrow {\delta } \Omega _{R/\Lambda } \otimes _ R S \to \Omega _{S/\Lambda } \to \Omega _{S/R} \to 0 \]

(Algebra, Lemma 10.134.4). The module $\Omega _{S/R}$ is free over $S$ with basis $\text{d}x$ and the module $H_1(L_{S/R})$ is free over $S$ with basis $x^ p - a$ which $\delta $ maps to $-\text{d}a \otimes 1$ in $\Omega _{R/\Lambda } \otimes _ R S$. In particular, if we set

\[ M = \mathop{\mathrm{Coker}}(R \to \Omega _{R/\Lambda }, 1 \mapsto -\text{d}a) \]

then we see that $\mathop{\mathrm{Coker}}(\delta ) = M \otimes _ R S$. We obtain a canonical map

\[ \Omega ^{t + 1}_{S/\Lambda } \to \wedge _ S^ t(\mathop{\mathrm{Coker}}(\delta )) \otimes _ S \Omega _{S/R} = \wedge ^ t_ R(M) \otimes _ R \Omega _{S/R} \]

Now, since the image of the map $\text{Tr} : \Omega _{S/R} \to \Omega _{R/\Lambda }$ of Lemma 54.2.1 is contained in $R\text{d}a$ we see that wedging with an element in the image annihilates $\text{d}a$. Hence there is a canonical map

\[ \wedge ^ t_ R(M) \otimes _ R \Omega _{S/R} \to \Omega _{R/\Lambda }^{t + 1} \]

mapping $\overline{\eta }_1 \wedge \ldots \wedge \overline{\eta }_ t \wedge \omega $ to $\eta _1 \wedge \ldots \wedge \eta _ t \wedge \text{Tr}(\omega )$.
$\square$

## Comments (2)

Comment #4245 by Dario Weißmann on

Comment #4420 by Johan on