Frobenius
==========================

In this section, we define the trace map of the Frobenius and we see some fundamental properties. We only use the smooth case. For the singular case, see [@Schwede2 Section 2].

\[tracedef\] Let $X$ be a smooth projective variety. Let $E$ be an effective [$\mathbb{Z}$]{}-divisor and let $D$ be a [$\mathbb{Z}$]{}-divisor. Then, for every positive integer $e$, there exists a natural $\mathcal O_X$-module homomorphism $${{\operatorname{Tr}}}_{X,E}^e(D):F^e_*(\omega_X(E+p^eD))\to 
\omega_X(E+D).$$ We call this a trace map.

Consider the Frobenius map: $$\mathcal O_X\to F^e_*\mathcal O_X,$$ that is, $p^e$-th power map $a\mapsto a^{p^e}$. Since $E$ is effective, we obtain $$\mathcal O_X(-E)\to F^e_*(\mathcal O_X(-p^eE))\hookrightarrow F^e_*(\mathcal O_X(-E)).$$ Tensoring $\mathcal O_X(-D)$, we obtain $$\begin{aligned}
\mathcal O_X(-E-D)
&\to& F^e_*(\mathcal O_X(-E))\otimes \mathcal O_X(-D)\\
&\simeq& F^e_*(\mathcal O_X(-E-p^eD)).\end{aligned}$$ By the duality theorem for finite morphisms, we obtain $$\mathcal Hom_{\mathcal O_X}(F^e_*(\mathcal O_X(-E-p^eD)), \omega_X)\simeq F^e_*(\omega_X(E+p^eD)).$$ Then, apply the functor $\mathcal Hom_{\mathcal O_X}(-, \omega_X)$ and we obtain $$F^e_*(\omega_X(E+p^eD))\to \omega_X(E+D).$$ This is the trace map ${{\operatorname{Tr}}}_{X,E}^e(D)$.

\[pE-vs-E\] By the above construction, ${{\operatorname{Tr}}}_{X,E}^e(D)$ factors through ${{\operatorname{Tr}}}_X^e(E+D):={{\operatorname{Tr}}}_{X, 0}^e(E+D)$: [$${{\operatorname{Tr}}}_{X,E}^e(D):F^e_*(\omega_X(E+p^eD))\hookrightarrow F^e_*(\omega_X(p^eE+p^eD))
\xrightarrow{{{\operatorname{Tr}}}_X^e(E+D)} \omega_X(E+D).$$]{}

\[trace-local\] Let $X$ be a smooth projective variety. Let ${{\operatorname{Spec}}}R\subset X$ be an affine open subset such that $R$ has a $p$-basis $\{x_1, \cdots, x_n\}.$ Then, we obtain $$\Gamma({{\operatorname{Spec}}}R, \omega_X)=Rdx_1\wedge\cdots\wedge dx_n$$ and $$\begin{aligned}
\Gamma({{\operatorname{Spec}}}R, F_*^e\omega_X)
