\mathcal O_S@>>> 0.
\end{CD}$$ Apply the functor $\mathcal Hom_X(-, \omega_X)$ and we obtain the required commutative diagram. Since $$F_S^e:\mathcal O_S\to F_*^e\mathcal O_S$$ factors through $F_*^e\mathcal O_{p^eS}$, we see the assertion.

We would like to know whether the trace map ${{\operatorname{Tr}}}_X^e(D)$ has a good property. It is natural to consider the following question.

\[Tango-cex\] Let $X$ be a smooth projective variety and let $A$ be an ample [$\mathbb{Z}$]{}-divisor on $X$. Then, for every $e\in\mathbb Z_{>0}$, is the trace map $${{\operatorname{Tr}}}_X^e(A):H^0(X, \omega_X(p^eA))\to H^0(X, \omega_X(A))$$ surjective?

The answer to the above question is NO. Indeed, [@Tango] constructs a smooth projective curve $X$ and an ample [$\mathbb{Z}$]{}-divisor $A$ on $X$ such that the trace map ${{\operatorname{Tr}}}_X^e(A)$ is not surjective.

On the other hand, we obtain the affirmative answer for the following two cases: ablelian varieties and $F$-split varieties.

Let $X$ be an abelian variety and let $A$ be an ample [$\mathbb{Z}$]{}-divisor. Then, the trace map $${{\operatorname{Tr}}}_{X}^e(A):H^0(X, \omega_X(p^eA))\to H^0(X, \omega_X(A)).$$ is surjective.

For $n\in\mathbb Z$, let $n_X$ be the $n$-multiplication map of the abelian variety $X$. Let $n\in\mathbb Z_{>0}$ which is not divisible by $p$. Then, $$n_X:X\to X$$ is a finite morphism whose degree is not divisible by $p$. Thus, $$\mathcal O_X\to (n_X)_*\mathcal O_X$$ is 