# Action of Frobenius on Lie Algebra

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!