$e$, $${{\operatorname{Tr}}}_{X, E}^{e+1}(D)={{\operatorname{Tr}}}_{X, E}^e(D)\circ F_*^e({{\operatorname{Tr}}}_{X, E}^1(p^e D)),$$ that is, $$\begin{aligned}
{{\operatorname{Tr}}}_{X, E}^{e+1}(D):
F^{e+1}_*(\omega_X(E+p^{e+1}D))&\xrightarrow{F_*^e({{\operatorname{Tr}}}_{X,E}^1(p^eD))}&F^e_*(\omega_X(E+p^eD))\\
&\xrightarrow{{{\operatorname{Tr}}}_{X,E}^{e}(D)}&\omega_X(E+D).\end{aligned}$$

Consider the Frobenius maps: $$\mathcal O_X(-E)\to F^e_*(\mathcal O_X(-E))\to F^{e+1}_*(\mathcal O_X(-E)).$$ Tensoring $\mathcal O_X(-D)$, we obtain $$\begin{aligned}
\mathcal O_X(-E-D)
\to F^e_*(\mathcal O_X(-E-p^eD))\to 
F^{e+1}_*(\mathcal O_X(-E-p^{e+1}D)).\end{aligned}$$ Apply the functor $\mathcal Hom_X(-, \omega_X)$ and we obtain the assertion.

In this paper, we often use the following two commutative diagrams.

\[trace-diagram\] Let $X$ be a smooth projective variety and let $S$ be a smooth prime divisor. Then, there exist the following commutative diagrams.

1.  [$$\begin{CD}
    0@>>> F_*^e\omega_X@>>> F_*^e(\omega_X(S))@>>> F_*^e\omega_S@>>> 0\\
    @. @VV{{\operatorname{Tr}}}_X^e V @VV{{\operatorname{Tr}}}_{X,S}^e V @VV{{\operatorname{Tr}}}_S^e V\\
    0@>>> \omega_X@>>> \omega_X(S)@>>> \omega_S@>>> 0.
    \end{CD}$$]{}

2.  [$$\begin{CD}
    0@>>> F_*^e\omega_X@>>> F_*^e(\omega_X(p^eS))@>>> F_*^e(\omega_{p^eS})@>>> 0\\
    @. @VV{{\operatorname{Tr}}}_X^e V @VV{{\operatorname{Tr}}}_{X}^e(S) V @VV\Psi V\\
    0@>>> \omega_X@>>> \omega_X(S)@>>> \omega_S@>>> 0.
    \end{CD}$$ Moreover, ${{\operatorname{Tr}}}_S^e$ factors through $\Psi$ $:$ $${{\operatorname{Tr}}}_S^e:F_*^e\omega_S\to F_*^e(\omega_{p^eS})\overset{\Psi}\to \omega_S.$$ ]{}

\(1) Consider the following commutative diagram: $$\begin{CD}
0@>>> F_*^e(\mathcal O_X(-S))@>>> F_*^e\mathcal O_X@>>> F_*^e\mathcal O_S@>>> 0\\
@. @AAA @AA F_X^e A @AAF_S^e A\\
0@>>> \mathcal O_X(-S)@>>> \mathcal O_X@>>> \mathcal O_S@>>> 0.
\end{CD}$$ Apply the functor $\mathcal Hom_X(-, \omega_X)$ and we obtain the assertion.\
(2) Consider the following commutative diagram: $$\begin{CD}
0@>>> F_*^e(\mathcal O_X(-p^eS))@>>> F_*^e\mathcal O_X@>>> F_*^e\mathcal O_{p^eS}@>>> 0\\
@. @AAA @AA F_X^e A @AAA\\
0@>>> \mathcal O_X(-S)@>>> \mathcal O_X@>>> 