for $i=n$, we obtain $Z_X^n\simeq F_*\omega_X$. By (3) for $i=n$, we obtain $$0\to B_X^n\to F_*\omega_X\overset{C_X^n}\to \omega_X\to 0.$$

\[cartier-trace\] Let $X$ be an $n$-dimensional smooth projective variety. By Remark \[trace-local\], Cartier operator and the trace map of Frobenius are the same: $C_X^n={{\operatorname{Tr}}}^1_X.$

\[cartier-etale\] Let $\gamma:X\to Y$ be a finite étale morphism between smooth varieties. Then, $$\gamma^*B_Y^i\simeq B_{X}^i\,\,\,{\rm and}\,\,\,\gamma^*Z_Y^i\simeq Z_{X}^i.$$

We may assume $X={{\operatorname{Spec}}}B$ and $Y={{\operatorname{Spec}}}A$. Let $$\varphi: A\to B$$ be the natural homomorphism induced by $\gamma$. Let $$F_A:A\to A\,\,\,{\rm and}\,\,\,F_B:B\to B$$ be the $p$-th power maps respectively. Since $\varphi$ is flat, we see that the natural $B$-module homomorphism $$\begin{aligned}
\theta^i:((F_A)_*\Omega_A^i)\otimes_A B&\to& (F_B)_*(\Omega_A^i\otimes_A B)\\
(\sum_J a_Jdx_J)\otimes_A b&\mapsto& (\sum_J a_Jdx_J)\otimes_A b^p\end{aligned}$$ is an isomorphism where $a_J\in A$ and $dx_J:=dx_{j_1}\wedge\cdots\wedge dx_{j_i}$ for some $x_{j_l}\in A$. Since $\varphi$ is étale, the natural $B$-module homomorphism $$\begin{aligned}
\rho^i:\Omega_A^i\otimes_A B&\to&\Omega_B^i\\
(\sum_J a_Jdx_J)\otimes_A b&\mapsto& \sum \varphi(a_J)bd(\varphi(x_J))\end{aligned}$$ is an isomorphism. It is sufficient to check the commutativity of the following diagram: $$\begin{CD}
(F_B)_*\Omega_B^i@>d>> (F_B)_*\Omega_B^{i+1}\\
@AAF_*\rho^iA @AAF_*\rho^{i+1}A\\
(F_B)_*(\Omega_A^i\otimes_A B)@>d>> (F_B)_*(\Omega_A^{i+1}\otimes_A B)\\
@AA\theta^iA @AA\theta^{i+1}A\\
((F_A)_*\Omega_A^i)\otimes_A B@>d>> ((F_A)_*\Omega_A^{i+1})\otimes_A B.
\end{CD}$$ This follows from $$\begin{aligned}
d\circ F_*\rho^i\circ \theta^i((\sum_J a_Jdx_J)\otimes_A b)
&=&d\circ F_*\rho^i((\sum_J a_Jdx_J)\otimes_A b^p)\\
&=&d(\sum_J \varphi(a_J)b^pd(\varphi(x_J)))\\
&=&\sum_J b^pd(\varphi(a_J))\wedge d(\varphi(x_J))\end{aligned}$$ and $$\begin{aligned}
&&F_*\rho^{i+1}\circ \theta^{i+1}\circ d((\sum_J a_Jdx_J)\otimes_A b)\\
&=&F_*\rho^{i+1}\circ \theta^{i+1}((\sum_J da_J\wedge dx_J)\otimes_A b)\\
&=&F_*\rho^{i+1}((\sum_J da_J\wedge dx_J)\otimes_A 