Exercise 108.62.5 (Frobenius). Let $p$ be a prime number (you may assume $p = 2$ to simplify the formulas). Let $R$ be a ring such that $p = 0$ in $R$.

Show that the map $F : R \to R$, $x \mapsto x^ p$ is a ring homomorphism.

Show that $\mathop{\mathrm{Spec}}(F) : \mathop{\mathrm{Spec}}(R) \to \mathop{\mathrm{Spec}}(R)$ is the identity map.

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