&=&\bigoplus_{0\leq i_j<p^e}R\cdot x_1^{i_1}\cdots x_n^{i_n}dx_1\wedge\cdots\wedge dx_n\\
&=&\bigoplus_{0\leq i_j<p^e}R^{p^e}x_1^{i_1}\cdots x_n^{i_n}dx_1\wedge\cdots\wedge dx_n.\end{aligned}$$ The trace map $${{\operatorname{Tr}}}_X^e:\Gamma({{\operatorname{Spec}}}R, F_*^e\omega_X)\to \Gamma({{\operatorname{Spec}}}R, \omega_X)$$ is described as follows:

1.  [${{\operatorname{Tr}}}_X^e(x_1^{p^e-1}\cdots x_n^{p^e-1}dx_1\wedge\cdots\wedge dx_n)=dx_1\wedge\cdots\wedge dx_n.$]{}

2.  [${{\operatorname{Tr}}}_X^e(x_1^{i_1}\cdots x_n^{i_n}dx_1\wedge\cdots\wedge dx_n)=0$ if $0\leq i_j<p^e-1$ for some $1\leq j\leq n$.]{}

The following two lemmas are the fundamental properties.

\[D-linearequ\] Let $X$ be a smooth projective variety and let $E$ be an effective [$\mathbb{Z}$]{}-divisor. If $D_1$ and $D_2$ are linearly equivalent [$\mathbb{Z}$]{}-divisors, then the two trace maps ${{\operatorname{Tr}}}_X^e(D_1)$ and ${{\operatorname{Tr}}}_X^e(D_2)$ are the same for every positive integer $e$, that is, there exists a following commutative diagram: $$\begin{CD}
F^e_*(\omega_X(E+p^eD_2))@>{{\operatorname{Tr}}}_X^e(D_2)>> \omega_X(E+D_2)\\
@VV\simeq V @VV\simeq V\\
F^e_*(\omega_X(E+p^eD_1))@>{{\operatorname{Tr}}}_X^e(D_1)>> \omega_X(E+D_1).
\end{CD}$$

Consider the Frobenius map: $$\mathcal O_X(-E)\to F^e_*(\mathcal O_X(-E)).$$ Tensoring $\mathcal O_X(-D_i)$, we obtain $$\begin{aligned}
\mathcal O_X(-E-D_i)
\to F^e_*(\mathcal O_X(-E-p^eD_i)).\end{aligned}$$ Then, we obtain the following commutative diagram: $$\begin{CD}
F^e_*(\mathcal O_X(-E-p^eD_2))@<<< \mathcal O_X(-E-D_2)\\
@AA\times g^{p^e} A @AA\times g A\\
F^e_*(\mathcal O_X(-E-p^eD_1))@<<< \mathcal O_X(-E-D_1)
\end{CD}$$ where $g\in K(X)^*$ is defined by $\mathcal O_X(-D_1)\overset{\times g}\simeq\mathcal O_X(-D_2).$ In particular, the vertical arrows are isomorphisms. Apply the functor $\mathcal Hom_{\mathcal O_X}(-, \omega_X)$ and we obtain the required commutative diagram.

\[e-vs-e+1\] Let $X$ be a smooth projective variety and let $E$ be an effective [$\mathbb{Z}$]{}-divisor. Let $D$ be a [$\mathbb{Z}$]{}-divisor. Then, for every positive integer 