split as an $\mathcal O_X$-module homomorphism (cf. [@KM Proposition 5.7(2)]). We obtain the following commutative diagram $$\begin{CD}
F_*^e(n_X)_*(\mathcal O_X(p^e(n_X)^*A))@<<< (n_X)_*(\mathcal O_X((n_X)^*A))\\
@AAA @AA n_X A\\
F_*^e(\mathcal O_X(p^eA))@<<<\mathcal O_X(A)
\end{CD}$$ Apply the functor $\mathcal Hom_{\mathcal O_X}(-, \omega_X)$ and take the cohomology: $$\begin{CD}
H^0(X, \omega_X(p^e(n_X)^*A))@>{{\operatorname{Tr}}}_{X}^e((n_X)^*A))>> H^0(X, \omega_X((n_X)^*A))\\
@VVV @VV n_X^* V\\
H^0(X, \omega_X(p^eA))@>{{\operatorname{Tr}}}_{X}^e(A)>> H^0(X, \omega_X(A)).
\end{CD}$$ Here, $n_X^*$ is surjective by the splitting of $n_X$. Therefore, it is sufficient to show that ${{\operatorname{Tr}}}_{X}^e((n_X)^*A)$ is surjective. By [@Mumford Corollary 3 in Section 6], we obtain $$n_X^*A= \frac{n^2+n}{2}A+\frac{n^2-n}{2}(-1)_X^*A.$$ Note that, since $(-1)_X$ is an automorphism, $(-1)_X^*A$ is ample. Therefore, by the Fujita vanishing theorem ([@Fujita1 Theorem (1)], [@Fujita2 Section 5]), we can find $n\in\mathbb Z_{>0}$ such that ${{\operatorname{Tr}}}_{X}^e((n_X)^*A)$ is surjective.

Let $X$ be a smooth projective variety. We say $X$ is [*F-split*]{} if the Frobenius map $$\mathcal O_X\to F_*\mathcal O_X$$ is split as an $\mathcal O_X$-module homomorphism.

\[F-split-surje\] Let $X$ be an $F$-split smooth projective variety and let $D$ be a [$\mathbb{Z}$]{}-divisor. Then, the trace map $${{\operatorname{Tr}}}_{X}^e(D):H^0(X, \omega_X(p^eD))\to H^0(X, \omega_X(D)).$$ is surjective.

By the definition of $F$-splitting, we see that the Frobenius map $$\mathcal O_X(-D)\to F_*^e(\mathcal O_X(-p^eD))$$ is split. Apply the functor $\mathcal Hom_{\mathcal O_X}(-, \omega_X)$ and we obtain the assertion.

Facts on Cartier 