Suppose $R$ is a $\mathbb{F}_p$-algebra and $E/R$ is an elliptic curve such that the sheaf of invariant differentials is free on $R$. So choosing a basis $\omega$ of $H^0(E, \Omega^1_E)$ we get a dual basis $D$ of the (left) invariant $R$-derivations of $E$. By Serre duality this is also a basis of $H^1(E, \mathcal{O}_E)$. Consider the absolute Frobenius on $E$. $F_{\textrm{abs}}$ induces an action on $H^1(E, \mathcal{O}_E)$ which takes $D$ to its $p$-th iteration $D^p$. I don't understand why this last statement is true (probably missing something obvious). Any help would be appreciated!


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