operator
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In this section, we summarize the facts on Cartier operator. By Remark \[cartier-remark\], we consider the trace map of the Frobenius as Cartier operator.

\[de-rham\] Let $X$ be a smooth variety. Consider the de Rham complex of $X$ $$\Omega_X^{\bullet}:\mathcal O_X\overset{d_0}\to\Omega^1_X\overset{d_1}\to\Omega^2_X
\overset{d_2}\to\cdots$$ where $\Omega^i_X:=\Omega^i_{X/k}.$ Apply $F_*$ and we obtain a complex $$F_*\Omega_X^{\bullet}:F_*\mathcal O_X\overset{F_*d_0}\to F_*\Omega^1_X\overset{F_*d_1}\to
F_*\Omega^2_X\overset{F_*d_2}\to\cdots.$$ Then, it is easy to see that $F_*d_i$ is an $\mathcal O_X$-module homomorphism. We define $$\begin{aligned}
B_X^i&:=&{{\operatorname{Image}}}(F_*d_{i-1}:F_*\Omega_X^{i-1}\to F_*\Omega_X^{i})\\
Z_X^i&:=&{{\operatorname{Ker}}}(F_*d_i:F_*\Omega_X^i\to F_*\Omega_X^{i+1}).\end{aligned}$$ Note that $B_X^i$ and $Z_X^i$ are coherent sheaves.

\[cartier-def\] Let $X$ be a smooth variety. For every $i\in \mathbb Z$ such that $1\leq i\leq \dim X$, consider the map $$\begin{aligned}
C_X^{-1}:\Omega_X^i&\to& Z_X^i/B_X^i\end{aligned}$$ locally defined by $$\begin{aligned}
C_X^{-1}|_{{{\operatorname{Spec}}}R}:\Gamma({{\operatorname{Spec}}}R, \Omega_X^i)&\to& \Gamma({{\operatorname{Spec}}}R, Z_X^i/B_X^i)\\
da_1\wedge\cdots \wedge da_i&\mapsto&
a_1^{p-1}\cdots a_i^{p-1}da_1\wedge\cdots\wedge da_i\end{aligned}$$ where ${{\operatorname{Spec}}}R$ is an open affine subset of $X$ and $a_1,\cdots, a_i\in R.$ This map $C_X^{-1}$ is a well-defined $\mathcal O_X$-module isomorphism. We call $C_X:=(C_X^{-1})^{-1}$ Cartier operator.

See, for example, [@EV Theorem 9.14].

\[cartier-remark\] Let $X$ be an $n$-dimensional smooth variety. We obtain the following exact sequences.

1.  [$0\to \mathcal O_X\to F_*\mathcal O_X\to B_X^1\to 0$.]{}

2.  [$0\to Z_X^i\to F_*\Omega_X^i\to B_X^{i+1}\to 0$ for $1\leq i\leq n$.]{}

3.  [$0\to B_X^i\to Z_X^i\overset{C_X^i}\to \Omega_X^{i}\to 0$ for $1\leq i \leq n$.]{}

By (2) 